mm-1015 === Subject: Re: Parameter Estimamtion of a Exponential Dist. > However, I plan to use A¹=sqrt(Sample Variance). Does any one have > an idea about the robustness of the estimation ? Sample variance is a biased estimator of population variance: = A^2 (n-1) / n. I can prove that A¹ is a biased estimator of A, but haven¹t been able to derive a closed expression for /A in terms of n. Unfortunately, setting A¹ = sqrt(Sample Variance * n / (n-1)) does not remove the bias. === Subject: Taylor Series Can anyone Žnd Taylor series for 1/sqrt(x) centered at 9? Sorry to hear you lost it. Will 1/sqrt(x) = (1/3)*(1 + (x-9)/9)^(-1/2) help? that is not a taylor series... let f(x) = x^k f^(n)(x) = product(k - i, i = 0..n-1)*x^(k-n) but product(k - i, i = 0..n-1) = (-1)^n*Gamma(n-k)/Gamma(-k) if k < n and k a positive integer, else 0 so the nth derivative of f(x) is f^(n)(c) = (-1)^n*Gamma(n-k)/Gamma(-k)*c^n since I¹ve found the nth coefŽcient for you, I think you can Žnd the rest. (now, there maybe easier ways using more indirect methods, but this is a direct application of the theorem. (and in its most general form for any power of x) and incase you don¹t know what gamma is, its basically factorial, except it can handle non integral arguments. Gamma(n) = (n-1)! for integer n, and Gamma(z) = z*Gamma(z-1)) === Subject: Re: Taylor Series X-RFC2646: Format=Flowed; Response > Can anyone Žnd Taylor series for 1/sqrt(x) centered at 9? >> Sorry to hear you lost it. >> Will 1/sqrt(x) = (1/3)*(1 + (x-9)/9)^(-1/2) help? > that is not a taylor series... > let f(x) = x^k > f^(n)(x) = product(k - i, i = 0..n-1)*x^(k-n) > but > product(k - i, i = 0..n-1) = (-1)^n*Gamma(n-k)/Gamma(-k) if k < n and k a > positive integer, else 0 > so the nth derivative of f(x) is > f^(n)(c) = (-1)^n*Gamma(n-k)/Gamma(-k)*c^n oops that should be k-n on the c. > since I¹ve found the nth coefŽcient for you, I think you can Žnd the > rest. > (now, there maybe easier ways using more indirect methods, but this is a > direct application of the theorem. (and in its most general form for any > power of x) and incase you don¹t know what gamma is, its basically > factorial, except it can handle non integral arguments. Gamma(n) = (n-1)! > for integer n, and Gamma(z) = z*Gamma(z-1)) === Subject: Re: Taylor Series <10s8c7fcg2pq370@corp.supernews.com> Can anyone Žnd Taylor series for 1/sqrt(x) centered at 9? > Sorry to hear you lost it. > Will 1/sqrt(x) = (1/3)*(1 + (x-9)/9)^(-1/2) help? > that is not a taylor series... I did not claim it is. But it can help Žnd one, and I thought, perhaps naively, that a hint is better than serving the Žnal result on a silver plate. > let f(x) = x^k > f^(n)(x) = product(k - i, i = 0..n-1)*x^(k-n) > but > product(k - i, i = 0..n-1) = (-1)^n*Gamma(n-k)/Gamma(-k) if k < n and k a > positive integer, else 0 > so the nth derivative of f(x) is > f^(n)(c) = (-1)^n*Gamma(n-k)/Gamma(-k)*c^n > since I¹ve found the nth coefŽcient for you, I think you can Žnd the rest. > (now, there maybe easier ways using more indirect methods, but this is a > direct application of the theorem. My question: is direct application of the theorem helpful in expanding, say, exp(-x^2/2) into Taylor Series around 0? The derivatives involve polynomials of ever increasing degrees (Hermite polynomials), and this complication is fully avoidable by an indirect method, that is, changing the variable and using Uniqueness Theorem. A simpler problem: the (formulas for) high derivatives of 1/(1+x^2) are also increasingly messy, and the indirect method goes smoothly, as most readers have experienced. Binomial Formula for (1+z)^k was on my mind; it is on the list of standard expansions, and all you need is to apply it to z = (x-9)/9 and divide the expansion by 3. Too sophisticated? > (and in its most general form for any > power of x) and incase you don¹t know what gamma is, its basically > factorial, except it can handle non integral arguments. Gamma(n) = (n-1)! > for integer n, and Gamma(z) = z*Gamma(z-1)) Using a cannon to kill a mosquito? === Subject: Re: Taylor Series X-RFC2646: Format=Flowed; Original > Can anyone Žnd Taylor series for 1/sqrt(x) centered at 9? >> Sorry to hear you lost it. >> Will 1/sqrt(x) = (1/3)*(1 + (x-9)/9)^(-1/2) help? >> that is not a taylor series... > I did not claim it is. But it can help Žnd one, and I thought, perhaps > naively, that a hint is better than serving the Žnal result on a silver > plate. >> let f(x) = x^k >> f^(n)(x) = product(k - i, i = 0..n-1)*x^(k-n) >> but >> product(k - i, i = 0..n-1) = (-1)^n*Gamma(n-k)/Gamma(-k) if k < n and k a >> positive integer, else 0 >> so the nth derivative of f(x) is >> f^(n)(c) = (-1)^n*Gamma(n-k)/Gamma(-k)*c^n >> since I¹ve found the nth coefŽcient for you, I think you can Žnd the >> rest. >> (now, there maybe easier ways using more indirect methods, but this is a >> direct application of the theorem. > My question: is direct application of the theorem helpful in expanding, > say, exp(-x^2/2) into Taylor Series around 0? The derivatives involve > polynomials of ever increasing degrees (Hermite polynomials), and this > complication is fully avoidable by an indirect method, that is, changing > the variable and using Uniqueness Theorem. > A simpler problem: the (formulas for) high derivatives of 1/(1+x^2) are > also increasingly messy, and the indirect method goes smoothly, as most > readers have experienced. > Binomial Formula for (1+z)^k was on my mind; it is on the list of standard > expansions, and all you need is to apply it to z = (x-9)/9 and divide the > expansion by 3. Too sophisticated? >> (and in its most general form for any >> power of x) and incase you don¹t know what gamma is, its basically >> factorial, except it can handle non integral arguments. Gamma(n) = >> (n-1)! >> for integer n, and Gamma(z) = z*Gamma(z-1)) > Using a cannon to kill a mosquito? Are you scared of the Gamma function? My method is much more intuitive than yours. While one without any ingenuity might be lost, if one can do x^k for integer k, there is no difŽculty to extend this to non-integral k... but the same method applies... it is direct and intuitive and simple... what more could you ask for? Me thinks you are afraid of the gamma... just remember, it doesn¹t bite. === Subject: Re: Taylor Series Jon Slaughter a .8ecrit : >Can anyone Žnd Taylor series for 1/sqrt(x) centered at 9? >>Sorry to hear you lost it. >>Will 1/sqrt(x) = (1/3)*(1 + (x-9)/9)^(-1/2) help? > that is not a taylor series... No, but it helps to Žnd it > let f(x) = x^k > f^(n)(x) = product(k - i, i = 0..n-1)*x^(k-n) > but > product(k - i, i = 0..n-1) = (-1)^n*Gamma(n-k)/Gamma(-k) if k < n and k a > positive integer, else 0 > so the nth derivative of f(x) is > f^(n)(c) = (-1)^n*Gamma(n-k)/Gamma(-k)*c^n > since I¹ve found the nth coefŽcient for you, I think you can Žnd the rest. > (now, there maybe easier ways using more indirect methods, but this is a > direct application of the theorem. (and in its most general form for any > power of x) and incase you don¹t know what gamma is, its basically > factorial, except it can handle non integral arguments. Gamma(n) = (n-1)! > for integer n, and Gamma(z) = z*Gamma(z-1)) === Subject: Complex Fittings by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBI3LPp01529; Hello all, I am trying to do a Žtting of many variables (~ 20 - 100) but I need the returned constants to only be positive. I am currently using a least squares Žtting method written in python and I would like to stick with python if possible. I was hoping someone may be able to help me with some theory behind getting only positive values returned. Matthew === Subject: Re: Lebesgue measurable functions R such > that inverse image of a Lebesgue measurable set is not Lebesgue measurable. > Examples through cardinality argument will be accepted. Can you Žnd a Lebesgue measurable f that maps [0,1] bijectively to the > Cantor set? Every subset of the Cantor set is Lebesgue measurable, but > not every subset of [0,1] is. > -- > G. A. Edgar http://www.math.ohio-state.edu/~edgar/ > There is also a continuous (strictly increasing) bijection of [0,1] onto > [0,2] which maps the Cator set (of Lebesgue measure 0) onto a Cantor-like > set of Lebesgue measure 1. It appeared recently: x+C(x) where C is the > Cantor function. Cator set is of course my private nickname for the Cantor set. (Sitting in my ofŽce on Saturday night, ready to answer questions from students writing a Žnal exam...) === Subject: ethics and mathematics posting-account=JKe5pgwAAABshOT4LtiAVjJ9JJ_HZvuQ John Nash proved you have to think not only about yourself. Is possible to prove that people should be ethical with mathematics? If somebody proved that absolute truth exists and, for example, economists read another proof written in business language that they should choose to follow the path of peace and love instead of war and hate, would economists ignore both proofs? Since the intellectual elite and the rich elite study mathematics, can mathematics be a bridge between them? === Subject: Re: ethics and mathematics > John Nash proved you have to think not only about yourself. Why not say that John von Neumann and Oskar Morgenstern showed this? > Is possible to prove that people should be ethical with mathematics? I don¹t think this is the sort of thing that can be mathematically proven. > If somebody proved that absolute truth exists and, for example, > economists read another proof written in business language that they > should choose to follow the path of peace and love instead of war and > hate, would economists ignore both proofs? Sure. It was mathematically demonstrated about half a century ago that much of what mainstream economics teach and practice is mistaken. Mainstream economists just ignore the demonstration. For one of my expositions of some elements of this demonstration, see: By the way, the empirical evidence is that studying economics tends to make one nasty: -- Mostly economics: r c v s a Whether strength of body or of mind, or wisdom, or i m p virtue, are found in proportion to the power or wealth e a e of a man is a question Žt perhaps to be discussed by n e . slaves in the hearing of their masters, but highly @ r c m unbecoming to reasonable and free men in search of d o the truth. -- Rousseau === >Is possible to prove that people should be ethical with mathematics? No. Mathematics proves things from axioms; it cannot manufacture the axioms. >If somebody proved that absolute truth exists and, for example, >economists read another proof written in business language that they >should choose to follow the path of peace and love instead of war >and hate, would economists ignore both proofs? You¹d have to ask the economists. But why do you believe that the politicians would pay attention to the economists in that scenario? >Since the intellectual elite and the rich elite study mathematics, >can mathematics be a bridge between them? Why do you believe that the rich elite study Mathematics? And if they do, what does it have to do with their ethics? Plenty of highly educated and intelligent people have supported evil, including some Mathematicians. -- Shmuel (Seymour J.) Metz, SysProg and JOAT Unsolicited bulk E-mail subject to legal action. I reserve the right to publicly post or ridicule any abusive E-mail. Reply to domain Patriot dot net user shmuel+news to contact me. Do not reply to spamtrap@library.lspace.org === Subject: Re: ethics and mathematics > Since the intellectual elite and the rich elite study mathematics Do they? sirix. === Subject: Re: ethics and mathematics > If somebody proved that absolute truth exists and, for example, > economists read another proof written in business language that they > should choose to follow the path of peace and love instead of war and > hate, would economists ignore both proofs? This is nonsense. Mathematics has nothing to do with peace and love. As for absolute truth, that is easy. ~(p&~p) for any proposition p. That is as absolute as truth gets. Bob Kolker === Subject: Re: job advice - math student posting-account=2Lbxhg0AAACPgQos9yMpmxCwf0XKMqMJ You could check out FindAJob.com ( http://www.Žndajob.com ) for university math jobs or academic jobs in general. They have about 20,000 jobs up now. === Subject: Re: Polysigned Numbers posting-account=NRsM-wwAAAAi5GV8YXCgqBCSyfE5DQT7 I¹ve been toying with the square root of minus one in four-signed since you posted this. How do you prove that it does not exist? So Far I don¹t have an answer, but my method goes like this: (-a+b*c#d)(-a+b*c#d) = -1. As a result: | - 2ad - 2bc | = 1 | + aa + cc + 2bd | = 0 | * 2ab * 2cd | = 0 | # 2ac # bb # dd | = 0 . Or in real values: 2ad + 2bc = 1 aa + cc + 2bd = 0 2ab + 2cd = 0 2ac + bb + dd = 0 Now, we can arbitrarily choose a value for a,b,c,or d since the system is nonorthogonal. So I chose a = 1.0 . Then I boiled down to a large polynomial in c which I couldn¹t solve. So I graphed it in gnuplot and got some roots. But neither of them actually worked out so I must have made a mistake along the way, or the method is bogus. I think that this method is alright, except for the part where I zoom in on the zero solutions of a complicated polynomial in gnuplot. I¹ll try to spend a little more time with it. -Tim === Subject: Re: Polysigned Numbers posting-account=NRsM-wwAAAAi5GV8YXCgqBCSyfE5DQT7 > I¹ve been toying with the square root of minus one in four-signed since > you posted this. > How do you prove that it does not exist? Well I can claim to have veriŽed that sqrt( - 1 ) in four signs does not exist now. That is, assuming I haven¹t made any more errors on top of my old errors. Corrections to the method below... > So Far I don¹t have an answer, but my method goes like this: > (-a+b*c#d)(-a+b*c#d) = -1. > As a result: > | - 2ad - 2bc | = 1 > | + aa + cc + 2bd | = 0 > | * 2ab * 2cd | = 0 > | # 2ac # bb # dd | = 0 . This should be: | - 2ad - 2bc | = 1 + x | + aa + cc + 2bd | = x | * 2ab * 2cd | = x | # 2ac # bb # dd | = x . where x is a magnitude > Or in real values: > 2ad + 2bc = 1 > aa + cc + 2bd = 0 > 2ab + 2cd = 0 > 2ac + bb + dd = 0 Or in real values: 2ad + 2bc = 1 + x aa + cc + 2bd = x 2ab + 2cd = x 2ac + bb + dd = x Now choose a value for one of a,b,c,d, or x. So choosing x = 0 disallows a free choice of a = 1. Well it¹s apparent that some of these must be zero. OK. it shows a = 0, b = 0, c = 0, d = 0. Therefore there is no sqrt( -1 ) since square(0) is zero. But these equations are the general square on the left. Just substitute in something that works to check them... sqrt( # 1 ) should make 2ac + bb + dd = 1 and the rest equate to zero. and so a = 0, c = 0, b = 1, d = 0 is a solution and a = 0, c = 0, b = 0, d = 1 is a solution Yes, that check is good. > Now, we can arbitrarily choose a value for a,b,c,or d since the system > is nonorthogonal. So I chose a = 1.0 . > Then I boiled down to a large polynomial in c which I couldn¹t solve. This step was bogus. I should not have left the magnitude from when a magnitude is equal to zero. e.g. | + x + y | = 0 does not mean that x = - y. It means x = 0, y = 0. whereas if | + x + y | = z then it is ok to say that x = z - y where all values are magnitudes and leaving the magnitude form steps up to the reals with a guarantee that z > x and z > y. > So I graphed it in gnuplot and got some roots. But neither of them > actually worked out so I must have made a mistake along the way, or the > method is bogus. > I think that this method is alright, except for the part where I zoom > in on the zero solutions of a complicated polynomial in gnuplot. I¹ll > try to spend a little more time with it. > -Tim -Tim === Subject: Re: Polysigned Numbers posting-account=BjC-YAwAAADQ91Zm3XkS3aGs3XlaqZ4X I didn¹t check on Your calculation. > Or in real values: > 2ad + 2bc = 1 > aa + cc + 2bd = 0 > 2ab + 2cd = 0 > 2ac + bb + dd = 0 (4 equations with 4 unknown) i calculated Žrst a=f(d,b,c) then c =g (a,b, d) insert this in the Žrst. Both in the remaining equations and by comparing a=b=c=d=0 I¹ve got a nice rule to memorize the sizes in a tetrahedron. On the way to the center start with one corner. Divide the edge or side line of length s 1:1, or go 1/2 of s. Turn right and facing a corner, divide this line 1:2, or go 1/3 of this line (which is sqrt3 / 2 of s long) . Now turn a right angle and face the last corner, divide this line 1:3, or go 1/4 of it ( which is sqrt2/sqrt3 of s long). Have fun Hero === Subject: Re: Double posts > When I notice duplicates, > I delete within Google, but of course Usenet cancel messages > don¹t propagate everywhere. Do you have any tips for sending a cancel message from Mozilla Thunderbird, which is also a little bit buggy? -- Karl M. Bunday P.O. Box 1456, Minnetonka MN 55345 Learn in Freedom (TM) http://learninfreedom.org/ remove .de to email === Subject: Re: Double posts >> When I notice duplicates, >> I delete within Google, but of course Usenet cancel messages >> don¹t propagate everywhere. > Do you have any tips for sending a cancel message from Mozilla > Thunderbird, which is also a little bit buggy? No idea about cancelling, but: what was your problem with Thunderbird? -- Herman Jurjus === Subject: Re: Double posts >> Do you have any tips for sending a cancel message from Mozilla >> Thunderbird, which is also a little bit buggy? > No idea about cancelling, but: what was your problem with Thunderbird? My problem with Thunderbird, a set of habits I carry over from using a Brand X product, is that I forget the message composer for Thunderbird looks the same whether it is sending email or a Usenet message. So once in a while I see a cool Usenet message and desire to tell an email list about it, and then I click compose (now it¹s write in the current release of Thunderbird) and the address of the recipient is automatically Žlled in as the newsgroup I am reading at the time. Filling in an email recipient does NOT suppress sending a copy of my message to the newsgroup; to prevent the newsgroup posting, I have to actively delete the newsgroup address. The last brand name of email/newsgroup client I used brought up different compose programs depending on whether I was posting to a newsgroup or to email, so it was a lot harder to make that mistake. More cancel messages by the original authors would be good, so I¹d like to learn how to send them. Even better will be keeping vigilant to avoid posting mistaken posts in the Žrst place. Technology should make mistakes less likely, not more. http://www.useit.com -- Karl M. Bunday P.O. Box 1456, Minnetonka MN 55345 Learn in Freedom (TM) http://learninfreedom.org/ remove .de to email Supersedes: X-Last-Updated: 1999/08/06 === Subject: Invariant Galilean Transformations (FAQ) On All Laws Summary: All laws/equations are Galilean invariant when expressed in the generalized cartesian coordinates demanded by basic analytic geometry, vector algebra, and measurement theory. Originator: faqserv@penguin-lust.mit.edu Disclaimer: approval for *.answers is based on form, not content. Opponents of the content should Žrst actually Žnd out what it is, then think, then request/submit-to arbitration by the appropriate neutral mathematics authorities. Flaming the hard- working, selžess, *.answers moderators evidences ignorance and despicable netiquette. Archive-Name: physics-faq/criticism/galilean-invariance Version: 0.04.03 Posting-frequency: 15 days Invariant Galilean Transformations (FAQ) On All Laws (c) Eleaticus/Oren C. Webster Thnktank@concentric.net An obvious typo or two corrected. The Brittanica section revised to less Œpussy-footing¹ and to more directly anticipate the elementary measurement theory and basic analytic geometry that is applied to the transformation concept. ------------------------------ === Subject: 1. Purpose The purpose of this document is to provide the student of Physics, especially Relativity and Electromagnetism, the most basic princ- iples and logic with which to evaluate the historic justiŽcation of Relativity Theory as a necessary alternative to the classical physics of Newton and Galileo. We will prove that all laws are invariant under the Galilean transformation, rather than some being non-invariant, after we show you what that means. We shall also show that another primal requirement that SR exist is nonsense: Michelson-Morley and Kennedy-Thorndike do indeed Žt Galilean (c+v) physics. ------------------------------ === Subject: 2. Table of Contents 1. Foreword and Intent 2. Table of Contents 3. The Principle of Relativity 4. The Encyclopedia Brittanica Incompetency. 5. Transformations on Generalized Coordinate Laws 6. The data scale degradation absurdity. 7. The Crackpots¹ Version of the Transforms. 8. What does sci.math have to say about x0¹=x0-vt? 9. But Doesn¹t x.c¹=x.c? 10. But Isn¹t (x¹-x.c¹)=(x-x.c) Actually Two Transformations? 11. But Doesn¹t (x¹-x.c+vt) Prove The Transformation Time Dependent? 12. But Isn¹t (x¹-x.c¹)=(x-x.c) a Tautology? 13. But Isn¹t (x¹-x.c¹)=(x-x.c) Almost the DeŽnition of a Linear Transform? 14. But The Transform Won¹t Work On Time Dependent Equations? 15. But The Transform Won¹t Work On Wave Equations? 16. But Maxwell¹s Equations Aren¹t Galilean Invariant? 17. First and Second Derivative differential equations. ------------------------------ === Subject: 3. The Principle of Relativity and Transformation If a law is different over there than it is here, it is not one law, but at least two, and leaves us in doubt about any third location. This is the Principle of Relativity: a natural law must be the same relative to any location at which a given event may be perceived or measured, and whether or not the observer is moving. The idea of location translates to a coordinate system, largely because any object in motion could be considered as having a coordinate system origin moving with it. If you perceive me moving relative to you - who have your own coordinate system - will your measurements of my position and velocity Žt the same laws my own, different measurements Žt? If a law has the same form in both cases it is called covariant. If it is identical in form, var- ables, and output values, it is called invariant. What we¹re asking is that if the x-coordinate, x, on one coordinate axis works in an equation, does the coordinate, x¹, on some other, parallel axis work? Speaking in terms of the axis on which x is the coordinate, x¹ is the Œtransformed¹ coordinate. The situation is complicated because we¹re talking about coordinates - locations - but in most mean- ingful laws/equations, it is lengths/distances (and time intervals) the equations are about, and x coord- inates that represent good, ratio scale measures of distances are only interval scale measures on the x¹ axis. [See Table of Contents for discussion of scales.] So, if we have an x-coordinate in one system, then we can call the x¹ value that corresponds to the same point/location the transform of x. In particular, the Principle of Relativity is embodied in the form of the Galilean transformation, which relates the original x, y, z, t to x¹, y¹, z¹, t¹ by the transform equations x¹=x-vt, y¹=y, z¹=z, t¹=t in the simpliŽed case where attention is focused only on transforming the x-axis, and not y and z. In the case of Special Relativity, the x¹ transform is the same except that x¹ is then divided by sqrt(1-(v/c)^2), and t¹=(t-xv/cc)/sqrt(1-(v/c)^2). In either case, v is the relative velocity of the coordinate systems; if there is already a v in the equations being trans- formed use u or some other variable name. ------------------------------ === Subject: 4. The Encyclopedia Brittanica Incompetency. One example of the traditional fallacious idea that an equation is not invariant under the galilean transformation comes from the Encyclopedia Brittanica: Before Einstein¹s special theory of relativity was published in 1905, it was usually assumed that the time coordinates measured in all inertial frames were identical and equal to an Œabsolute time¹. Thus, t = t¹. (97) The position coordinates x and x¹ were then assumed to be related by x¹ = x - vt. (98) The two formulas (97) and (98) are called a Galilean transformation. The laws of nonrelativ- istic mechanics take the same form in all frames related by Galilean transformations. This is the restricted, or Galilean, principle of relativity. The position of a light wave front speeding from the origin at time zero should satisfy x^2 - (ct)^2 = 0 (99) in the frame (t,x) and (x¹)^2 - (ct¹)^2 = 0 (100) in the frame (t¹,x¹). Formula (100) does not transform into formula (99) using the transform- ations (97) and (98), however. ................................................. Besides the trivially correct statement of what the Galilean Œtransform¹ equations are, there is exactly one thing they got right. I. Eq-100 is indeed the correct basis for discussing the question of invariance, given that eq-99 is the correct Œstationary¹ (observer S) equation. [Let observer M be the Œmoving¹system observer.] In particular, eq-100 is of exactly the same form [the square of argument one minus the square of argument two equals zero (argument three).] II. It is nonsense to say eq-99 should be derivable from eq-100; for one thing, the transforms are TO x¹ and t¹ from x and t, not the other way around, and the idea that either observer¹s equation should contain within itself the terms to simplify or rearrange to get to the other is ridiculous. As the transform equations say, the relationship of t¹, x¹ to t, x is based on the relative velocity between the two systems, but neither the original (eq-99) equation nor the M observer equation is about a relationship between coordinate systems or observers. One might as well expect the two equations to contain banana export/import data; there is no relevancy. The Œtransform¹ equations are the relationships between x¹ and x, t¹ and t and have nothing to do with what one equation or the other ought to Œsay¹. The equations¹ content is the rate at which light emitted along the x-axes moves. III. Most remarkable, the True Believer SR crackpots who most despise the consequences of measurement theory (demonstrable fact) contained in this document are those who want to argue against our saying the Britt- anica got eq-100 right; They insist that the correct equation is derived directly from x¹=x-vt and t¹=t. Solve for x=x¹+vt and replace t with t¹, then substitute the result in eq-99: (x¹+vt¹)^2 - (ct¹)^2 = 0. Besides the fact that this results in an equation with arguments exactly equal to eq-99, they will insist the transform is not invariant. IV. A major justiŽcation they have for their idea of the correct M system equation on which to base the the discussion of invariance, is that the variables are M system variables, never mind the fact that the arguments are S system values. That argument of theirs is arrant nonsense. The velocity v that S sees for the M system relative to herself is the negative of what the M system sees for the S system relative to himself. In other words, x¹+vt¹ is a mixed frame expression and it is x¹+(-v)t¹ that would be strictly M frame notation, and that equation is far off base. [Work it out for yourself, but make sure you try out an S frame negative v so as not to mislead yourself.] V. In I. we said: given that eq-99 is the correct Œstationary¹ equation. Let¹s look at it closely: x^2 - (ct)^2 = 0 (99) This whole matter is supposed to be about coordinate transforms. Is that what t is, just a coordinate? No. It isn¹t, in general. Suppose you and I are both modelling the same light event and you are using EST and I¹m using PST. ŒJust a time coordinate¹ is just a clock reading amd your t clock reading says the light has been moving three hours longer than my clock reading says. Well, that¹s what the idea that t is a coordinate means. Eq-99 works if and only if t is a time interval, and in particular the elapsed time since the light was emitted. Thus, that equation works only if we understand just what t is, an elapsed time, with emissioon at t=0. However, we don¹t have to Œunderstand¹ anything if we use a more intelligent and insightful form of the equation: (x)^2 - [ c(t-t.e) ]^2 = 0, where t.e is anyone¹s clock reading at the time of light emission, and t is any subsequent time on the same clock. Similarly, x is not just a coordinate, but a distance since emission. (x-x.e)^2 - [ c(t-t.e) ]^2 = 0 (99a) VI. In the spirit of Œthere is exactly one thing they got right¹, the correct M system version of eq-99a is eq-100a: (x¹-x.e¹)^2 - [ c(t¹-t.e¹) ]^2 = 0 (100a) Every observer in the universe can derive their eq-100a from eq-99a and vice versa, not to mention to and from every other observer¹s eq-99a. Now, THAT¹s invariance. [You do realize that every eq-100a reduces to eq-99a, when you back substitute from the transforms, right? t.e¹=t.e, x.e¹=x.e-vt.] ------------------------------ === Subject: 5. Transformations on Generalized Coordinate Laws The traditional Gallilean transform is correct: t¹ = t x¹ = x - vt. But remember this: a transform of x doesn¹t effect just some values of x, but all of them, whether they are in the formula or not. This is important if you want to do things right. The crackpot position is strongly against this sci.math veriŽed position, and the apparently standard coordinate pseudo-transformation they suggest is perhaps the result. {See Table of Contents.] Let¹s use a simple equation: x^2 + y^2 = r^2, which is the formula for a circle with radius r, centered at a location where x=0. But what if the circle center isn¹t at x=0? Well, we¹d want to use the form analytic geometry, vector algebra, and elementary measurement theory tells us to use, a form where we make explicit just where the circle center is, even if it is at x=x0=0: (x-x0)^2 + (y-y0)^2 = r^2. The circle center coordinate, x0, is an x-axis coordinate, just like all the x-values of points on the circle. So, in proper generalized cartesian coordinate forms of laws/equations we want to transform every occurence of x and x0 - by whatever name we call it: x.c, x_e, whatever. So, what is the transformed version of (x-x0)? Why, (x¹-x0¹); both x and x0 are x-coordinates, and every x-coordinate has a new value on the new axis. So, what is the value of (x¹-x0¹) in terms of the original x data? is also true for x0¹=x0-vt: (x¹-x0¹)=[ (x-vt)-(x0-vt) ]=(x-x0). In other words, when we use the generalized coordinate form speciŽed by analytic geometry, we Žnd that the value of (x¹-x0¹) does not depend on either time or velocity in any way, shape, form, or fashion. Similarly for (y-y0). We can treat time the same way if necessary: (t-t0). The above is a proof that any equation in x,y,z,t is invariant under the galilean transforms. Just use the generalized coordinate form, with (x-x0)/etc, in the transformation process, not the incompetently selected privileged form, with just x/etc. [The form is privileged because it assumes the circle center, point of emission, whatever, is at the origin of the axes instead at some less convenient point. After transform the coordinate(s) of the circle center/origin are also changed but the privileged form doesn¹t make this explicit and screws up the calculations, which should be based on (x¹-x0¹) but are calculated as (x¹-0).] The value of (x¹-x0¹) is the same as (x-x0). That makes sense. Draw a circle on a piece of paper, maybe to the right side of the paper. On a transparent sheet, draw x and y coordinate axes, plus x to the right, plus y at the top. Place this axis sheet so the y-axis is at the left side of the circle sheet. Now answer two questions after noting the x-coordinate of the circle center and then moving the axis sheet to the right: (a) did the circle change in any way because you moved the axis sheet (ie because you transformed the coordin- nate axis)? (b) did the coordinate of the circle center change? The circle didn¹t change [although SR will say it did]; that means that (x¹-x0¹) does indeed equal (x-x0). The coordinate of the circle center did change, and it changed at the same rate (-vt) as did every point on the circle. That means that x0¹<>x0, and the fact the circle center didn¹t change wrt the circle, means that the relationship of x0¹ with x0 is the same as that of any x¹ on the circle with the corresponding x: x¹=x-vt; x0¹=x0-vt. This is to prepare you for the True Believer crackpots that say Œconstant¹ coordinates can¹t be transformed; some even say they aren¹t coordinates. These crackpots include some that brag about how they were childhood geniuses, btw. QED: The galilean transformation for any law on generalized Cartesian coordinates is invariant under the Galilean transform. The use of the privileged form explains HOW the transformed equation can be messed up, the next Subject explains what the screwed up effect of the transform is, and how use of the generalized form corrects the screwup. ------------------------------ === Subject: 6. The data scale degradation absurdity. The SR transforms and the Galilean transforms both convert good, ratio scale data to inferior interval scale data. The effect is corrected, allowed for, when the transforms are conducted on the generalized coordinate forms speciŽed by analytic geometry and vector algebra. Both sets of transforms are Œtranslations¹ - lateral movements of an axis, increasing over time in these cases - but with the SR transform also involving a rescaling. It is the translation term, -vt in the x transform to x¹, and -xv/cc in the t transform to t¹, that degrades the ratio scale data to interval scale data. In general, rescaling does not effect scale quality in the size-of-units sense we have here. SR likes to consider its transforms just rotations, however - in spite of the fact Einstein correctly said they were Œtranslations¹ (movements) - and in the case of Œgood¹ rotations, ratio scale data quality is indeed preserved, but SR violates the conditions of good ro- tations; they are not rigid rotations and they don¹t appropriately rescale all the axes that must be rescaled to preserve compatibility. The proof is in the pudding, and the pudding is the combination of simple tests of the transformations. We can tell if the transformed data are ratio scale or interval. Ratio scale data are like absolute Kelvin. A measure- ment of zero means there is zero quantity of the stuff being measured. Ratio scale data support add- ition, subtraction, multiplication, and division. The test of a ratio scale is that if one measure looks like twice as much as another, the stuff being measured is actually twice as much. With absolute Kelvin, 100 degrees really is twice the heat as 50 degrees. 200 degrees really is twice as much as 100. Interval scale data are like relative Celsius, which is why your science teacher wouldn¹t let you use it in gas law problems. There is only one mathematical operation interval scales support, and that has to be between two measures on the same scale: subtraction. 100 degrees relative (household) Celsius is not twice as much as 50; we have to convert the data to absolute Kelvin to tell us what the real ratio of temperatures is. However, whether we use absolute Kelvin or relative Celsius, the difference in the two temperature readings is the same: 50 degrees. Thus, if we know the real quantities of the Œstuff¹ being measured, we can tell if two measures are on a ratio scale by seeing if the ratio of the two measures is the same as the ratio of the known quant- ities. If a scale passes the ratio test, the interval scale test is automatically a pass. If the scale fails the ratio test, the interval scale test becomes the next in line. It isn¹t just the bare differences on an interval scale that provides the test, however. Differences in two interval scale measures are ratio scale, so it is ratios of two differences that tell the tale. Let¹s do some testing, and remember as we do that our concern is for whether or not the data are messed up, not with Œreasons¹, excuses, or avoidance. ------------------------------------------------------ Are we going to take a transformed length (difference) and see whether that length Žts ratio or interval scale deŽnitions? Of course, not. Interval scale data are ratio after one measure is subtracted from another. That is the major reason the SR transforms can be used in science. Let there be three rods, A, B, C, of length 10, 20, 40, respectively. These lengths are on a known ratio scale, our original x-axis, with one end of each rod at the origin, where x=0, and the other end at the coordinate that tells us the correct lengths. Note that these x-values are ratio scale only because one end of each rod is at x=0. That may remind you of the correct way to use a ruler or yard/meter-stick: put the zero end at one end of the thing you are measuring. Put the 1.00 mark there instead of the zero, and you have interval scale measures. Let A,B,C, be 10, 20, 40. Let a,b,c be x¹ at v=.5, t=10. x¹=x-vt. A B C a b c ---------------- -------------------- 10 20 40 5 15 35 ---------------- -------------------- B/A = 2 b/a = 3 C/A = 4 c/a = 7 C/B = 2 c/b = 2.333 Obviously, the transformed values are no longer ratio scale. The effect is less on the greater values. C-A = 10 b-a = 10 C-A = 30 c-a = 30 C-B = 20 c-b = 20 Obviously, the transformed values are now interval scale. This will hold true for any value of time or velocity. (C-A)/(B-A) = 3 (c-a)/(b-a) = 3 (C-B)/(B-A) = 2 (c-b)/(b-a) = 2 Obviously, the ratios of the differences are ratio scale, being identical to the ratios of the corresponding original - ratio scale - differences. The main difference between these results and the SR results is that the differences do not correspond so neatly to the original, ratio scale, differences. This is due only to the rescaling by 1/sqrt(1-(v/c)^2). The ratios of the differences on the transformed values do correspond neatly and exactly to the ratio scale results. Using the generalized coordinate form, such as (x-x0), the transform produces an interval scale x¹ and an interval scale x0¹. That gives us a ratio scale (x¹-x0¹), just like - and equal to - (x-x0). ------------------------------ === Subject: 7. The Crackpots¹ Version of the Transforms. It has become apparent - whether misleading or not - that the crackpot responses to the obvious derive from a common source, whether it be bandwagoning or their SR instructors. Below, in the sci.math subject, we see that all sci.math respondents agree with the basic controversial position of this faq: every coordinate is transformed, whether a supposed constant or not. Think about it, the generalized coordinate of a circle center, x0, applies to inŽnities upon inŽnities of circle locations (given y and z, too); it is a constant only for a given circle, and even then only on a given coordinate axis. And even variables are often held Œconstant¹ during either integration or differentiation. The utility of a variable is that you can discuss all possible particular values without having to single out just one. That utility does not make particular - singled out - values on the variable¹s axis not values of the variable just because they have become named values. In any case, all that is preamble to the incompetent idea they have proposed for a transform of coordinates. It is based on the idea that the circle center, point of emission, whatever, has coordinates that cannot be transformed. Let there be an equation, say (x)^2 - (ict)^2 = 0. What is the transformed version of that equation? Answer: (x¹)^2 - (ict¹)^2 = 0. That¹s the one thing the Brittanica got right. Note that the leading crackpot just criticized this faq for presuming to correct the Britt- anica, but it then and before poses the incompetent pseudo- transform we analyze here in this section. x to x¹ and t to t¹ are obviously coordinate transforms; the x and t coordinates have been replaced by the coord- inates in the primed system. A tranform of an equation from one coordinate system to another is NOT a substitution of the/a deŽnition of x for itself; that is not a coordinate transformation. The most that can said for such a substitution is that it is a change of variable. But the crackpots are calling this a coordinate trans- form of the original equation: (x¹+vt)^2 - (ict¹)^2 = 0. It is not a coordinate transform, of course, except accidentally. (x¹+vt) is not the primed system coordinate, it is another form/expression of x. They get that substitution by solving x¹=x-vt for x; x=x¹+vt. So, by incompetent misnomer, they accomplish what they have been railing against all along. It has been the generalized coordinate form in question all this time: (x-x0)^2 - (ict)^2 = 0. Here they substitute for x instead of transforming to the primed frame: (x¹+vt-x0)^2 - (ict¹)^2. ----- ^ | ^ | It is still x ^ but see what they have accomplished by their mis/malfeasance: [x¹+vt-x0]=[x¹+(vt-x0)]=[x¹-(x0-vt)]. =[x¹-x0¹] The crackpots have been bragging about how you don¹t have to transform the circle center¹s coordinate to transform the circle center¹s coordinate. Bragging that what they were doing was not what they said they were doing. This does give us insight as to some of the crackpot variations on their x0¹<>x0-vt theme, which in all the variations will be discussed in later sections.. They are used to seeing the mixed coordinate form, (x¹+vt-x0) without realizing what it respresented, so - accompanied with a lack of understanding of the term Œdependent¹ - they are used to seeing just the one vt term, and not the one hidden in the deŽ- nition of x¹ and are used to imagining it makes the whole expression time dependent and thus not invariant. About which, let x=10, let, x0=20, v=10, and t variously 10 and 23: (x-x0)=-10. Using their (x¹+vt-x0): For t=10, we have (x¹+vt-x0) = [ (10-10*10) + (10*10) - (20) ] = -90 + 100 - 20 = -10 = (x-x0) For t=23, we have (x¹+vt-x0) = [ (10-10*23) + (10*23) - (20) ] = -220 + 230 - 20 = -10 = (x-x0) The result depends in no way on the value of time; we showed the obvious for a couple of instances of t just so you can see that the crackpots not only do not understand the obvious logic of the algebra { (x¹-x0¹)=[ (-vt)-(x0-vt) ]=(x-x0) } - which shows that the transform has no possible time term effect - but they don¹t understand even a simple arithmetic demonstration of the facts. Oh. Their (x¹+vt-x0) or (x¹+vt¹-x0) reduces the same way since t¹=t: (x-vt+vt-x0)=(x-x0). Their process, which says (x¹+vt¹) is the transform of x, says that (x¹+vt¹) is the moving system location of x, but it can¹t be because x is moving further in the negative direction from the moving viewpoint. That formula will only work out with v<0 which is indeed the velocity the primed system sees the other moving at. However, that formula cannot be derived from x¹=x-vt, the formula for transformation of the coordinates from the unprimed to the primed, ------------------------------ === Subject: 8. What does sci.math have to say about x0¹=x0-vt? The crackpots¹ positions/arguments were put to sci.math in such a way that at least two or three who posted re- sponses thought it was your faq-er who was on the idiot¹s side of the questions. Their responses: ---------------------------------------------------------- I. x0¹ = x0. In other words: x0¹ <> x0-vt, or constant values on the x-axis are not subject to the transform. No. x0¹ = x0 - vt. Well, if you want, you could deŽne constant values on the x-axis, but in the context of the question that is not relevant. The relevant fact is that if the unprimed observer holds an object at point x0, then the primed observer assigns to that object a coordinate x0¹ which is numerically related to x0 by x0¹= x0 -vt. What does this mean? The line x=x0 will give x¹=x-v*t=x0-vt¹, so if x0¹ is to give the coordinate in the (x¹,t¹,)-system, it will be given by x0¹=x0-v*t¹: ie., it is not given by a constant. Thus, being at rest (constant x-coordinate) is a coordinate-dependent concept. Sounds very false. We can say that the representation of the point X0 is the number x0 in the unprimed system, and x0¹ in the primed system. Clearly x0 and x0¹ are different, if vt is not zero. However one may say that (though it sounds/is stupid) the point X0 itself is the same throughout the transformation. However that expression sounds meaningless, since a transform (ok, maybe we should call it a change of basis) is only a function that takes the point¹s representation in one system into the same point¹s representation in another system. It is preferrable to use three notations: X0 for the point itself and x0 and x0¹ for the points¹ representations in some coordinate systems. ------------------------------ === Subject: 9. But Doesn¹t x.c¹=x.c? That idea is one of the most idiotic to come up, and it does so frequently. And in a number of guises. The idea being that x.c¹ <> x.c-vt, with x.c being what we have called x0 above; the notation makes no difference. Some crackpots have managed to maintain that position even after graphs have illustrated that such an idea means that after a while a circle center represented by x.c¹ could be outside the circle. The leading crackpot just make that explicit, as far as one can tell from his befuddled post in response to a line about active transforms, which are actually moving body situations, not coordinate transformations: ------------------------------------------------------------- ------- e>An active transform is not a coordinate transform, ... Right, it is a transform of the center (in the opposite direction) done to effect the change of coordinates without a coordinate transform. ... E: Transform of the center? Center of a circle? He really is saying a circle center moves in the opposite direction of the circle! Right? ------------------------------------------------------------- ------- If r=10 and x.c was at x.c=0, then the points on the circle (10,0), (-10,0), (0,10) and (0,-10) could at some time become (-10,0), (-30,0), (-20,10), and (-20,-10), but with x.c¹=x.c, the circle center would be at (0,0) still! The circle is here but its center is way, way over there! Indeed, although a change of coordinate systems is not movement of any object described in the coordinates, the x.c¹=x.c crackpottery is tantamount to the circle staying put but the center moving away. Or vice versa. ------------------------------ === Subject: 10. But Isn¹t (x¹-x.c¹)=(x-x.c) Actually Two Transformations? One crackpot puts the (x¹-x.c¹)=(x-vt - x.c+vt) relationship like this: (x-vt+vt - x.c). See, he says, that is transforming x (with x-vt - x.c) and then reversing the transform (x-vt+vt - x.c). That¹s just another crackpot form of the idiocy that x.c¹ <> x.c-vt. You¹ll have noticed the implication is that there is no transform vt term relating to x.c. ------------------------------ === Subject: 11. But Doesn¹t (x¹-x.c+vt) Prove The Transformation Time Dependent? That particular crackpottery is perhaps more corrupt than moronic, since it includes deliberately hiding a vt term from view, and pretending it isn¹t there. [However, we have seen above that it is a familiar incompetency, and not likely an original.] Look, the crackpots say, there is a time term in the transformed (x¹ - x.c+vt). The transform isn¹t invariant! It¹s time dependent! Just put x¹ in its original axis form, also, which reveals the other time term, the one they hide: (x¹-x.c+vt) = (x-vt - x.c+vt) = (x-x.c). So, at any and all times, the transform reduces to the original expression, with no time term on which to be dependent. Then there is the fact that if you leave the equation in any of the various notation forms - with or without reducing them algebraicly - the arithmetic always comes down to the same as (x-x.c). That means nothing to crack- pots, but may mean something to you. ------------------------------ === Subject: 12. But Isn¹t (x¹-x.c¹)=(x-x.c) a Tautology? My dictionary relates Œtautology¹ to needless repetition. That¹s another form of the x.c¹ <> x.c-vt idiocy. The repetition involved is the vt transformation term. Apply the -vt term to the x term, and it is needless repetition to apply it anywhere again? The Œagain¹ is to the x.c term. The x.c¹ = x.c crackpot idiocy. The repetition of the vt terms is required by the presence of two x values to be transformed. Be sure to note the next section. ------------------------------ === Subject: 13. But Isn¹t (x¹-x.c¹)=(x-x.c) Almost the DeŽnition of a Linear Transform? Now, how on earth can we relate a tautology to a basic deŽnition in math? we get this deŽnition: -------------------------------------------------------------- A linear transformation, A, on the space is a method of corr- esponding to each vector of the space another vector of the space such that for any vectors U and V, and any scalars a and b, A(aU+bV) = aAU + bAV. ------------------------------------------------------------- Let points on the sphere satisfy the vector X={x,y,z,1}, and the circle center satisfy C={x.c,y.c,z.c,1}. Let a=1, and b=-1. Let A= ( 1 0 0 -ut ) ( 0 1 0 -vt ) ( 0 0 1 -wt ) ( 0 0 0 1 ) A(aX+bC) = aAX + bAC. aX+bC = (x-x.c, y-y.c, z-z.c, 0 ). The left hand side: A( x - x.c , y - y.c, z - z.c, 0 ) = ( x-x.c , y-y.c, z-z.c, 0 ). The right hand side: aAX= ( x-ut, y-vt, z-wt, 1 ). bAC= (-x.c+ut, -y.c+vt, -z.c+wt, -1 ). and aAX+bAC = ( x-x.c, y-y.c, z-z.c, 0 ). Need it be said? Sure: QED. On the galilean transform the deŽnition of a linear transform, A(aU+bV)=aAU + bAV, is completely satisŽed. The generalized form transforms exactly and non-redundantly - with ONE TRANSFORM, not a transform and reverse transform - and non- tautologically, just as the very deŽnition of a linear transform says it should. And does so with absolute invariance, with this galilean transformation. ------------------------------ === Subject: 14. But The Transform Won¹t Work On Time Dependent Equations? The main crackpot that has asserted such a thing was referring to equations such as in Subject 4, above. The Light Sphere equation; for which we have shown repeatedly elsewhere that the numerical calculations are identical for any primed values as for the unprimed values. The presence - before transformation - of a velocity term seems to confuse the crackpots. It turns out there is ex- treme historical reason for this, as you will see in the subject on Maxwell¹s equations. ------------------------------ === Subject: 15. But The Transform Won¹t Work On Wave Equations? See Subject 17, below, for a discussion of Second Derivative forms and the galilean transforms. ------------------------------ === Subject: 16. But Maxwell¹s Equations Aren¹t Galilean Invariant? Oh? Just what is the magical term in them that prevents (x¹-x.c¹)=(x-vt - x.c+vt)=(x-x.c) from holding true? It turns out not to be magic, but reality, that interferes with the application of the galilean transforms to the gen- eralized coordinate form(s) of Maxwell: there are no coordi- nates to transform! When True Believer crackpots are shown the simple demonstration that the galilean transform on generalized cartesian coordinates is invariant, their Žrst defense is usually an incredibly stupid x0¹=x0, because the coordinate of a circle center, or point of emission, etc, is a constant and can¹t be transformed. The last defense is but Maxwell¹s equations are not invariant under that coordinate transform. When asked just what magic occurs in Maxwell that would prevent the simple algebra (x¹-x0¹)=[ (x-vt)-(x0-vt) ]=(x-x0) from working, and when asked them for a demonstration, they will never do so, however many hundreds of times their defense is asserted. The reason may help you understand part of Einstein¹s 1905 paper in which he gave us his absurd Special Relativity derivation: THERE ARE NO COORDINATES IN THE EQUATIONS TO BE TRANSFORMED. Einstein gave the electric force vector as E=(X,Y,Z) and the magnetic force vector as B=(L,M,N), where the force components in the direction of the x axis are X and L, Y and M are in the y direction, Z and N in the z direction. Those values are not, however, coordinates, but values very much like acceleration values. BTW, the current fad is that E and B are ŒŽelds¹, having been Œforce Želds¹ for a while, after being Œforces¹. So, when Einstein says he is applying his coordinate transforms to the Maxwell form he presented, he is either delusive or lying. (a) there are no coordinates in the transform equations he gives us for the Maxwell transforms, where B=beta=1/sqrt(1-(v/c)^2): X¹=X. L¹=L. Y¹=B(Y-(v/c)N). M¹=B(M+(v/c)Z). Z¹=B(Z+(v/c)M). N¹=B(N-(v/c)Y). X is in the same direction as x, but is not a coordinate. Ditto for L. They are not locations, coordinates on the x-axis, but force magnitudes in that direction. Similarly for Y and M and y, Z and N and z. (b) the v of the coordinate transforms is in Maxwell before any transform is imposed; Einstein¹s transform v is the velocity of a coordinate axis, not the velocity he touched it. (c) if they were honest Einsteinian transforms, they¹d be x, which means it is X and L that are supposed to be transformed, not Y and M, and Z and N. And when SR does transform more than one axis, each axis has its own velocity term; using the v along the x-axis as the v for a y-axis and z-axis transform is thus trebly absurd: the axes perpendicular to the motion are not changed according to SR, the v used is not their v, and the v is not a transform velocity anyway. (d) as everyone knows, the effect of E and B are on the direction. Both the speed and direction are changed by E and B, but v - the speed - is a constant in SR. As absurd as are the previously demonstrated Einsteinian blunders, this one transcends error and is an incredible example of True Believer delusion propagating over decades. The components of E and B do differ from point to point, and in the variations that are not coordinate free, they are subject to the usual invariant galilean trans- formation when put in the generalized coordinate form. ------------------------------------------------------------- The SR crackpots don¹t know what coordinates are. The various things they call coordinates include coordin- nates, but also include a variety of other quantities. ------------------------------------------------------ 1. One may express coordinates in a one-axis-at-a-time manner [like x^2+y^2=r^2] but it is the use of vector notation that shows us what is going on. In vector notation the triplet x,y,z [or x1,x2,x3, whatever] represents the three spatial coordinates, but there are so-called basis vectors that underlie them. Those may be called i,j,k. Thus, what we normally treat as x,y,z is a set of three numbers TIMES a basis vector each. 2. These e*i, f*j, g*k products can have a lot of meanings. If e, f, j are distances from the origin of i,j,k then e*i, f*j, g*k are coordinates: distances in the directions of i,j,k respectively, from their origin. That makes the triplet a coordinate vector that we describe as being an x,y,z triplet; perhaps X=(x,y,z). The e*i, f*j, g*k products could be directions; take any of the other vectors described above or below and divide the e,f,g numbers by the length of the vector [sqrt(e^2+f^2+g^2)]. That gives us a vector of length=1.0, the e,f,g values of which show us the direction of the original vector. That makes the triplet a direction vector that we describe as being an x,y,z triplet; perhaps D=(x,y,z). The e*i, f*j, g*k products could be velocities; take any of the unit direction vectors described above and multiply by a given speed, perhaps v. That gives a vector of length v in the direction speciŽed. That makes the triplet a velocity vector that we describe as being an x,y,z triplet; perhaps V=(x,y,z). Each of the three values, e,f,g, is the velocity in the direction of i,j,k respectively. The e*i, f*j, g*k products could be accelerations; take any of the unit direction vectors described above and multiply by a given acceleration, perhaps a. That gives a vector of length a in the direction speciŽed. That makes the triplet an acceleration vector that we describe as being an x,y,z triplet; perhaps A=(x,y,z). Each of the three values, e,f,g, is the acceleration in the direction of i,j,k respectively. The e*i, f*j, g*k products could be forces (much like accel- erations); take any of the unit direction vectors described above and multiply by a given force, perhaps E or B. That gives a vector of length E or B in the direction speciŽed. That makes the triplet a force vector that we describe as being an x,y,z triplet; perhaps E=(x,y,z) or B=(x,y,z). Each of the three values, e,f,g, is the force in the direction of i,j,k respectively. Einstein¹s - and Maxwell¹s - E and B are not coordinate vectors. There is another variety of intellectual befuddlement that misinforms the idea that Maxwell isn¹t invariant under the galilean transform: confusions about velocities. Velocities With Respect to Coordinate Systems. ----------------------------------------------- Aaron Bergman supplied the background in a post to a sci.physics.* newsgroup: Imagine two wires next to each other with a current I in each. Now, according to simple E&M, each current generates a magnetic Želd and this causes either a repulsion or attraction between the wires due to the interaction of the magnetic Želd and the current. Let¹s just use the case where the currents are parallel. Now, suppose you are running at the speed of the current between the wires. If you simply use a galilean transform, each wire, having an equal number of protons and electrons is neutral. So, in this frame, there is no force between the wires. But this is a contradiction. First of all, the invariance of the galilean transform (x¹-x.c¹) =(x-x.c), insures that it is an error to imagine there is any difference between the data and law in one frame and in another; the usual, convenient rest frame is the best frame and only frame required for universal analysis. [Well, (x¹<>x, x,c¹<>x.c, but (x¹-x.c¹)=(x-x.c).] Second, given that you decide unnecessarily to adapt a law to a moving frame, don¹t confuse coordinate systems with meaningful physical objects, like the velocity relative to a coordinate system instead of relative to a physical body or Želd. In other words, what does current velocity with respect to a coordinate system have to do with physics? Nothing. Certainly not anything in the example Bergman gave. What is relevant is not current velocity with respect to a coordinate system, but current velocity with respect to wires and/or a medium. The velocity of an imaginary coordinate sys- tem has absolutely nothing to do with meaningful physical vel- ocity. You can - if you are insightful enough and don¹t violate item (e) - identify a coordinate system and a relevant physical object, but where some v term in the pre-transformed law is in use, don¹t confuse it with the velocity of the coordinate transform. Velocities With Respect to ... What? ----------------------------------------------- Albert Einstein opened his 1905 paper on Special Relativity with this ancient incompetency: The equations of the day had a velocity term that was taken as meaning that moving a magnet near a conductor would create a current in the conductor, but moving a conductor near a wire would not. This was belied by fact, of course. The important velocity quantity is the velocity of the magnet and conductor with respect to each other, not to some absolute coordinate frame (as far as we know) and not to an arbitrary coordinate system. One possible cause was the idea: but the equation says the magnet must be moving wrt the coordinate system or ... the absolute rest frame. There not being anything in the equation(s) to say either of those, it is amazing that folk will still insist the velocity term has nothing to do with velocity of the two bodies wrt each other. ----------------------------------------------------------- ------------------------------ === Subject: 17. First and Second Derivative differential equations. One of the intellectually corrupt ways of denying the very simple demonstration of galilean invariance of all laws expressed in the generalized coordinate form demanded by analytic geometry, vector analysis, and measurement theory [ (x¹-x.c¹)=[ (x-vt)-(x.c-vt) ]=(x-x.c) ] is the assertion that those equations Œover there¹ (usually Maxwell or wave) are somehow immune to the elementary laws of algebra used to demon- strate the invariance. [Unfortunately, the assertions are never accompanied by reference to the magical math that makes elementary al- gebra invalid. Wonder why that is?] Part of the time it is based on the old lore based on the incompetent transformation of the privileged form of an equation instead of the correct form. [Evidence of this is any reference to an effect due to the velocity of the transform; it falls out algebraicly - as you see above - and cancels out arith- metically - as you can see above.] But usually it is just whistling in the dark, waving the cross (zwastika, I¹d say) at the mean old vampire. The most general equation that could be conjured up is a differential with either First or Second Derivatives. Let¹s examine the plausibility of such magical magical, non-invariance assertions. (a) to get a Second Derivative you must have a First Derivative. (b) to get a First Derivative you must have a function to differentiate. (c) to get a Second Derivative you must have a function in the second degree. So, let us examine the question as to whether any such common Maxwell/wave equation will differ for (a) the common, privileged form, represented as ax^2, with a being an unknown constant function. (b) the generalized cartesian form, represented as a(x-x.c)^2 = ax^2 -2ax(x.c) + ax.c^2, with a being an unknown constant function. (c) the transformed generalized cartesian form, represented as a(x-vt -x.c+vt)^2, same as for (b), = ax^2 -2ax(x.c) + ax.c^2, of course, with a being an unknown constant function. I. for (a), remembering that x.c is a constant, and that this version is only correct because x.c=0, otherwise (b) is the correct form: d/dx ax^2 = 2ax (d/dx)^2 ax^2 = 2a II. for (b), remembering that x.c is a constant. d/dx (ax^2 -2ax(x.c) + ax.c^2) = 2ax - 2ax.c (d/dx)^2 (ax^2 -2ax(x.c) + ax.c^2) = 2a III. for (c); same as for (b). So, what we have seen so far is (1) differential equations in the second degree - the wave equations - must clearly be the same for all forms: the privileged form in x, the generalized cartesian form in x and the centroid, x.c, or the transformed generalized cartesian form. That is, anyone who imagines that correct usage gives different results for galilean transformed frames is at Žrst showing his ignorance, and in the end showing his intellectual corruption. (2) As far as the First Derivatives are concerned, the only cases in which there really is a difference between the two forms is where x.c <> 0, and in that case, the use of the privileged form is obviously incompetent. So, how do you correctly use the differential equations? If you are using rest frame data with the centroid at x=0, etc, you can¹t go wrong without trying to go wrong. If you are using rest frame data with the centroid not at x=0, you must use (x-x.c) anyplace x appears in the equation. If you are using moving frame data, you must use the moving frame centroid as well as the light front (or whatever) moving frame data itself, perhaps Žrst calculating (x¹-x.c¹), which equals (x-x.c) which is obviously correct, and which is obviously the plain old correct x of the privileged form. Unless, of course, there really is some magical term or expression that invalidates the obvious and elemen- tary algebra of the invariance demonstration. Or maybe you just whistle when you don¹t want basic algebra to hold true. Eleaticus !---?---!---?---!---?---!---?---!---?---!---?---!---?---!---? ---!---? ! Eleaticus Oren C. Webster ThnkTank@concentric.net ? ! Anything and everything that requires or encourages systematic ? ! examination of premises, logic, and conclusions ? !---?---!---?---!---?---!---?---!---?---!---?---!---?---!---? ---!---? === Subject: Re: Invariant Galilean Transformations (FAQ) On All Laws > Disclaimer: approval for *.answers is based on form, not content. > Opponents of the content should Žrst actually Žnd out what > it is, then think, then request/submit-to arbitration by the > appropriate neutral mathematics authorities. Flaming the hard- > working, selžess, *.answers moderators evidences ignorance > and despicable netiquette. > Archive-Name: physics-faq/criticism/galilean-invariance > Version: 0.04.03 > Posting-frequency: 15 days > Invariant Galilean Transformations (FAQ) On All Laws > (c) Eleaticus/Oren C. Webster > Thnktank@concentric.net [snip 1300 lines of trolled garbage] eleaticus, Oren Webster, is a despised and stooopid troll, http://users.pandora.be/vdmoortel/dirk/Physics/Fumbles/ Crimes.html Several crimes against logic and science Ha ha ha! Originally trolled across sci.physics sci.physics.relativity alt.physics sci.math sci.answers alt.answers news.answers Psychotic ineducable boring troll Eleaticus, Were there to be internal inconsistencies in SR (meaning inconsistencies of a purely mathematical logical nature) that would automatically lead to contradictions in number theory, itself, and arithmetic, since the mathematics of Minkowski geometry is equiconsistent with the theory of real numbers and with arithmetic. Eleaticus explicitly demonstrates that he is completely ignorant of multivariable calculus. He has no concept of the Chain Rule in multivariable calculus. Consider his Galilean Transformation goo and dribble: t¹ = t, x¹ = x - vt, y¹ = y, z¹ = z. His refusal to accept that t¹ must be introduced as a separate variable springs from a massive emprical stupidity re space and time are described as a four-dimensional manifold, with four coordinates instead of a time evolution of a three-dimensional manifold, and that the change of coordinate system should be a change of four coordinates, and not a time-dependent change of three coordinates. This is particularly vital when it comes to Želds over space and time (electric and magnetic Želds for example). The transformation law for the differential operators under the Galilean transformation is given by: d/dt¹ = d/dt + v d/dx, d/dx¹ = d/dx, d/dy¹ = d/dy, d/dz¹ = d/dz. This shows the necessity of introducing a new variable t¹, since partial differentiation with respect to t¹ (constant x¹, y¹, z¹) is a different operation to partial differentiation with respect to t (constant x, y, z). The above transformation law is determined by the Chain Rule: d/dt¹ = dt/dt¹ d/dt + dx/dt¹ d/dx + dy/dt¹ d/dy + dz/dt¹ d/dz, d/dx¹ = dt/dx¹ d/dt + dx/dx¹ d/dx + dy/dx¹ d/dy + dz/dx¹ d/dz, d/dy¹ = dt/dy¹ d/dt + dx/dy¹ d/dx + dy/dy¹ d/dy + dz/dy¹ d/dz, d/dz¹ = dt/dz¹ d/dt + dx/dz¹ d/dx + dy/dz¹ d/dy + dz/dz¹ d/dz. The presence of the term involving d/dx in the expression for d/dt¹ is indicative of the fact that x depends on t¹ (x¹, y¹, z¹, being held constant), as can be seen from the fact that the coefŽcient of d/dx in the expression for d/dt¹ is dx/dt¹. Because of the now demonstrated fact that Eleaticus has no formal education in multivariable calculus, he has managed, somehow, to get it into his head that the presence of the term involving d/dx in the expression for d/dt¹ is indicative of t¹ depending on x (t, y, z, being held constant). Because of his stupidty Eleaticus cannot get the correct transformation law for the differential operators under the Galilean Transformation, and he cannot determine the invariance or otherwise of Maxwell¹s Equations under the Galilean Transformation. The Žrst advice to Eleaticus is to learn multivariable calculus. Eleaticus should not pretend that he can understand how to determine invariance or otherwise of Maxwell¹s Equations under the Galilean Transformation, or under the Lorentz Transformation, until he understands the multivariable calculus which underlies such considerations. Eleaticus is a loud idiot. The homogeneous Maxwell equations are invariant under the Galilean Transformation, with transformation laws: E_x¹ = E_x, E_y¹ = E_y - v B_z, E_z¹ = E_z + v B_y, B_x¹ = B_x, B_y¹ = B_y, B_z¹ = B_z. The derivation of these transformation laws was determined using the transformation laws for the differential operators given above. These transformation laws have the additional advantage that they determine the correct transformation for the force law, thus providing further evidence in favour of the transformation law for the differential operators, as above. The inhomogeneous Maxwell equations are also invariant under the Galilean transformation, with transformation laws: E_x¹ = E_x, E_y¹ = E_y, E_z¹ = E_z, B_x¹ = B_x, B_y¹ = B_y + v/c^2 E_z, B_z¹ = B_z - v/c^2 E_y, rho¹ = rho, J_x¹ = J_x - v rho, J_y¹ = J_y, J_z¹ = J_z. Note the the transformation laws for the charge density and current density are as they should be under the Galilean transformation. Homogeneous equations are invariant under the Galilean Transformation, and inhomogeneous equations are invariant under the Galilean Transformation, but Maxwell¹s Equations as a whole are NOT invariant under the Galilean Transformation, since the transformation laws required for the EM Želd for the two cases are inconsistent with each other. The transformation law for the EM Želd which makes the homogeneous equations invariant will not also make the inhomogeneous equations invariant. The transformation law for the EM Želd which makes the inhomogeneous equations invariant will not also make the homogeneous equations invariant. On the other hand, all of Maxwell¹s equations are invariant under the Lorentz Transformation, with transformation laws: E_x¹ = E_x, E_y¹ = gamma (E_y - v B_z), E_z¹ = gamma (E_z + v B_y), B_x¹ = B_x, B_y¹ = gamma (B_y + v/c^2 E_z), B_z¹ = gamma (B_z - v/c^2 E_y), rho¹ = gamma (rho - v/c^2 J_x), J_x¹ = gamma (J_x - v rho), J_y¹ = J_y, J_z¹ = J_z, where gamma = 1/sqrt(1 - v^2/c^2). Idiot Oren Webster sees himself this way, http://www.mazepath.com/uncleal/effete6.jpg The entire remainder of the planet sees him this way, http://www.mazepath.com/uncleal/effete3.png http://www.mazepath.com/uncleal/sunshine.jpg http://www.apa.org/journals/psp/psp7761121.html http://insti.physics.sunysb.edu/~siegel/quack.html Hey, stooopid troll Eleaticus - Do you want EVIDENCE? Each of the 24 GPS satellites carries either four cesium atomic clocks or three rubidum atomic clocks in orbit, with full relativistic corrections being applied. Internal inconsistencies in SR (meaning inconsistencies of a purely mathematical logical nature) automatically lead to contradictions in number theory, itself, and arithmetic, since the mathematics of Minkowski geometry is equiconsistent with the theory of real numbers and with arithmetic. Mathematics of gravitation Equivalence Principle testing http://arXiv.org/abs/hep-th/0111236 Geometric structure of reality http://arxiv.org/abs/gr-qc/0103044 http://arXiv.org/abs/hep-th/0307140 GR structure, especially Part 4/p. 7 http://arXiv.org/abs/gr-qc/0311039 Experimental constraints on General Relativity http://www.eftaylor.com/pub/projecta.pdf Relativity in the GPS system http://arXiv.org/abs/gr-qc/9909014 falling light Hafele-Keating Experiment http://www.hawaii.edu/suremath/SRtwinParadox.html Twin Paradox http://arXiv.org/abs/astro-ph/0401086 http://arxiv.org/abs/astro-ph/0312071 Deeply relativistic neutron star binaries http://arxiv.org/abs/hep-th/0405160 Black hole evaporation No aether http://fsweb.berry.edu/academic/mans/clane/ No Lorentz violation http://arXiv.org/abs/gr-qc/0409089 Spin-2 gravitons have problems (so does the proposal) http://arXiv.org/abs/gr-qc/0301024 Nordtvedt Effect http://arxiv.org/abs/astro-ph/0403292 http://arXiv.org/abs/astro-ph/0310723 WMAP + Sloane Digital Sky Survey http://arxiv.org/abs/hep-ph/0404175 Dark matter candidates Carroll on what it all means. Special Relativity is physics on a topologically trivial Lorentzian manifold with a metric whose curvature tensor is zero. This is a perfectly diffeomorphism-invariant condition and does not require any particular coordinate choice. It is invariant under the full group of diffeomorphisms. The Poincare group is the group of *isometries* of the metric in special relativity. The Special Relativity metric is *non-dynamical* (unlike GR). It deŽnes the coupling *constants* of your theory. If you change the metric in any nontrivial way you are changing your theory. An operation can only be called a symmetry of a special-relativistic (non-gravitational) theory if it preserves the metric, and therefore the symmetry of special-relativistic theories is the Poincare group only. General Relativity (gravitation) has a dynamic metric. NIM A 355 537 (1995) Physics Letters B 328 103 (1994) Physical Review Letters 64 1697 (1990) Physical Review Letters 39 1051 (1977) Physical Review 135 B1071 (1964) Physics Letters 12 260 (1964) Europhysics Letters 56(2) 170-174 (2001) General Relativity and Gravitation 34(9) 1371 (2002) http://fourmilab.to/etexts/einstein/specrel/specrel.pdf Longitudinal and transverse mass http://arxiv.org/abs/gr-qc/0306076.pdf http://www.navcen.uscg.gov/pubs/gps/gpsuser/gpsuser.pdf http://www.navcen.uscg.gov/pubs/gps/sigspec/default.htm http://www.navcen.uscg.gov/pubs/gps/icd200/default.htm http://www.trimble.com/gps/index.html http://sirius.chinalake.navy.mil/satpred/ http://www.phys.lsu.edu/mog/mog9/node9.html http://egtphysics.net/GPS/RelGPS.htm http://www.schriever.af.mil/gps/Current/current.oa1 http://edu-observatory.org/gps/gps_books.html -- Uncle Al http://www.mazepath.com/uncleal/ (Toxic URL! Unsafe for children and most mammals) http://www.mazepath.com/uncleal/qz.pdf === Subject: Re: Invariant Galilean Transformations (FAQ) On All Laws X-RFC2646: Format=Flowed; Original > [snip 1300 lines of trolled garbage] > eleaticus, Oren Webster, is a despised and stooopid troll, Schwartz is a despised and stooopid troll too, so who cares? Androcles. === Subject: Re: Invariant Galilean Transformations (FAQ) On All Laws > [snip 1300 lines of trolled garbage] > eleaticus, Oren Webster, is a despised and stooopid troll, > Schwartz is a despised and stooopid troll too, so who cares? > Androcles. http://arXiv.org/abs/gr-qc/0311039 Experimental constraints on General Relativity Your ignorance, incompetence, and psychosis are not of interest to the world at large. Quite the contrary. You are not even an interesting laughingstock. http://www.apa.org/journals/psp/psp7761121.html http://insti.physics.sunysb.edu/~siegel/quack.html -- Uncle Al http://www.mazepath.com/uncleal/ (Toxic URL! Unsafe for children and most mammals) http://www.mazepath.com/uncleal/qz.pdf === Subject: Re: Invariant Galilean Transformations (FAQ) On All Laws X-RFC2646: Format=Flowed; Original >> [snip 1300 lines of trolled garbage] >> eleaticus, Oren Webster, is a despised and stooopid troll, >> Schwartz is a despised and stooopid troll too, so who cares? >> Androcles. [snip crap] Put this past your sphincter where you can read it. http://www.mazepath.com/uncleal/sunshine.jpg http://www.fourmilab.ch/etexts/einstein/specrel/www/ Androcles. Supersedes: X-Last-Updated: 1999/10/17 === Subject: (SR) Lorentz t¹, x¹ = Intervals Summary: The Lorentz transforms themselves are proof t¹ and x¹ cannot possibly be just coordinates. Examination of their derivation veriŽes their identity as intervals. Originator: faqserv@penguin-lust.mit.edu Disclaimer: approval for *.answers is based on form, not content. Opponents should Žrst actually Žnd out what the content is, then think, then request/submit-to arbitration by the appropriate neutral mathematics authorities. Flaming the hard- working, selžess, *.answers moderators evidences ignorance and atrocious netiquette. Version: 0.02.1 Archive-name: physics-faq/criticism/lorentz-intervals Posting-frequency: 15 days (SR) Lorentz t¹, x¹ = Intervals (c) Eleaticus/Oren C. Webster Thnktank@concentric.net ------------------------------ === Subject: 1. Introduction with the obvious debunking of the use of Œjust coordinates¹ in any scientiŽc formula. Defenders of the Special Relativity faith are especially fond of telling opponents of their space-time fairy tales that they do not know the difference between coordinates and magnitudes. That may often be so, but the fault lies ultimately with SR dogma. The Lorentz-Einstein transformations cannot possibly be Œjust coordinates¹, which is the interpre- tation required to support the many sideshow carnival acts with which the SR faithful bedazzle the public, and establish their moral and intellectual superiority. If I get in my car and drive steadily for a few hours at 50 kilometers per hour, is 50t the distance I travel? Of course not. The last time my hours-counting Œjust coord- inates¹ clock was set to zero was when Zeno Žrst reported one of his paradoxes to Parmenides. That was a long time ago, so my t is not useful for such purposes unless you also use my clock to established the starting time, perhaps t0, and use the formula 50(t-t0) to calculate the distance. In any case, my t is even then not Œjust a coordinate¹ because it always represents particular elapsed times that can be used in the (t-t0) form to calculate perfectly good time intervals (elapsed times). Alternatively, I could (re)set my clock to zero at the start of some meaningful time interval, in which case my t shows a scientiŽcally perfect current and/or end time. In which case, the Lorentz-Einstein t¹=(t-vx/cc)/g is a function of an elapsed time interval (not Œjust a coordinate¹) and a time interval (-vx/cc; the interval amount the t¹ clock is being screwed up at time t) and thus cannot be Œjust a coordinate¹ since neither of the independent variables is such a Œjust¹ thing. {Their meaning is shown below, step-by-step.] If it takes me 50 minutes to cross the Interstate highway, was x/50 my velocity crossing it? Of course not. The origin of all my axes is at the very spot where Zeno Žrst presented his Žrst paradox to Parmenides. That makes my x equal a couple of thousands of miles, plus, and is not useful for such purposes unless you establish the starting x value, perhaps x0, and use the formula (x-x0)/50 to calculate my velocity. In any case, even then my x is not Œjust a coordinate¹ because it always repesents particular distance intervals that can always be used in the (x-x0) form for any and every scientiŽc purose. Alternatively, I could move my x-axis origin to the starting (zero) point of some meaningful distance, in which case my x shows a scientiŽcally perfect current and/or end distance. In which case, the Lorentz-Einstein x¹=(x-vt)/g is a function of a current/ending distance interval (not Œjust a coordinate¹) and a distance interval (-vt; the interval amount the x¹ axis is being screwed up at time t) and thus cannot be Œjust a coordinate¹ since neither of the independent variables is such a Œjust¹ thing. {Their meaning is shown below, step-by-step.] ------------------------------ === Subject: 2. Table of Contents 1. Introduction with the obvious debunking of the use of Œjust coordinates¹ in any scientiŽc formula. 2. Table of Contents. 3. The Lorentz-Einstein transforms. 4. The Œjust coordinates¹ argument. 5. Single-system, little-purpose ambiguity. 6. Relating two coordinate measures/systems. 7. Distances and moving coordinate axes. 8. Time intervals. 9. Einstein¹s (1905) derivations. 10. A word about intervals. 11. Intervals versus the Twins Paradox. 12. Summary ------------------------------ === Subject: 3. The Lorentz-Einstein transforms Special Relativity¹s space-time circus is based on the Œtransformation¹ equations by which it is believed one can relate a nominally Œstationary¹ system¹s space and time coordinates to those of an inertially (not accelerating) moving other observer. That moving observer¹s own physical body and coordinate system might have been identical in size to those of the stationary observer before the traveller began moving, but are Œseen¹ as very different by the stationary observer when the relative velocity of the two is great enough, a high percentage of the velocity of light. Concerning ourselves - as is customary - with just the spatial coordinate axis that lies parallel to the direction of motion, and with time, Einstein arrived at these formulas that relate the moving system measures or coordinates (x¹ and t¹) to the stationary system coordinates (x and t): x¹ = (x - vt)/sqrt(1-vv/cc) (Eq 1x) t¹ = (t - vx/cc)/sqrt(1-vv/cc) (Eq 1t) The v is for the two systems¹ relative velocity as seen by the stationary observer, and is positive if the dir- ection is toward higher values of x. By concensus, the moving system x¹-axis higher values also lie in that direction, and all axes parallel the other system¹s corresponding axis. We used vv to mean the square of v but might use v^2 for that purpose below. Similarly for c. Because it is believed that no physical object can reach or exceed c, the square-root term in both denominators is presumed always less than one, which means that the formulas say both x¹ and t¹ will tend to be greater than x and t, respectively. However, SRians call the x¹ result Œcontraction¹ - which means shortening - and the t¹ result Œdilation¹ - which means increasing. ------------------------------ === Subject: 4. The Œjust coordinates¹ argument The Œjust coordinates¹ argument is so patently ridiculous that even opponents have a hard time accepting just how simple and obvious its debunking can be, as shown in this section. However, further sections take a more arithmet- ical approach that you¹ll maybe Žnd more professorial. The Œjust coordinates¹ argument is that t is mot a duration, not a time interval; it¹s just an arbitrary clock reading. But what if the moving system observer comes speeding by while you make your annual Œspring forward¹ or Œfall back¹ change? The formula says that the moving system clock¹s Œjust coordinate¹ reading can be calculated from yours: t¹ = (t - vx/cc)/sqrt(1-vv/cc) (Eq 1t) Imagine the moving system oberver¹s confusion if his clock changes its reading while he¹s looking at it! If his clock doesn¹t change when yours does, the formula is wrong; if it is truly a Œjust coordinates¹ formula. And then what happens if you realize you were a day early and put your clock back to what it had said previously? And suppose you are in NYC and your twin in LA and both are watching the moving observer. You¹ll both be using the same v because you are at rest wrt (with respect to) each other. You¹re on Eastern Standard Time and your twin is on PaciŽc Standard Time maybe. You have three hours more on your clock than does your twin. On which Œjust coordinate¹ clock will the Lorentz transforms base the Œjust coordinate¹ time the moving system clock says? The formula applies to both of your t-times: t¹ = (t - vx/cc)/sqrt(1-vv/cc) (Eq 1t) Sure, the idea that you can change someone else¹s clock with no connection of any kind is really ridiculous, but Eqs 1x and 1t aren¹t MY equations. Are they yours? And we aren¹t the ones to say x, t, x¹, and t¹ are just coordinates. If the t¹ formula is actually either an elapsed time formula, or the basis of a t¹/t ratio, then there is no implication that one clock¹s reading has anything to do with the other¹s. It can only be rates of clock ticking, or how one time INTERVAL compares to the other that the formula is about. ------------------------------ === Subject: 5. Single-system, little-purpose ambiguity. Since we¹re going to be comparing measurements on two coordinate systems in the next section, let¹s go to our supply cabinet and get our yard-stick (which we use to measure things in inches) and our meter-stick (which we use to measure things in centimeters). Here, I¹m getting mine. Oh! Oh! There¹s an ant on mine, and he ... she ... sure is hanging on, right at the 3.5 inch mark of the yard- stick. Let¹s see if I can wave the stick around enough that she¹ll let go. Nope. However, before I gave up I waved the stick and the ant Œall over the place. Always, however, the ant was at the 3.5 mark on the yard-stick, and always 3.5 away from the end of the stick, however far and wide I have transported her. Neither of those 3.5 facts means very much. Of the two, the distance aspect meant almost nothing. So the distance was 3.5 from the end. So what? That length, distance, was not in use. And only maybe the ant might have been concerned with just what location, Œjust coordinate¹, on the stick she was at. Just so with x and t. So, is the 3.5 reading just a coordinate? Or a distance/length? It¹s ambiguous in and of itself, and really makes no difference what you say until you try to make use of the number. Hey, my address is 5047 Newton Street. If you are looking for me and you¹re at 4120 Newton, it is helpful information, because it tells you which direction to go. Is that Œjust coordinate¹? Where it really becomes useful, perhaps, is in telling you how far away I am. That¹s not just a coordinate value, that¹s a distance, length, interval. However, it is subtracting 4120 from 5047 that tells you which direction and how far. It is only because both 5047 and 4120 are distances from the same point - ANY same point - that the result means anything. My x - my yardstick reading - is always a distance or length; it is impossible to be otherwise with an honest, competently designed yardstick. Whether or not its reading is of good use in some particular scientiŽc formula depends on whether I put the zero end of the yardstick at some useful place. As in the introduction, we should either put it at the starting location/end, or use two readings from it: (x-x0). ------------------------------ === Subject: 6. Relating two coordinate measures/systems. Taking care to not damage our brave little ant, I place my yard-stick onto the table, zero end to the left, 36 end to the right. Now I place the Œjust coordinate¹ meter-stick on the table in the same orientation, in a random location, and Žnd that the ant¹s coordinate on the meter-stick is 51. The formula relating centimeters to inches is cm=i*2.54 but we want a formula similar to x¹=(x-vt)/sqrt(1-vv/cc). That would be c=i/.03937 approximately, but let¹s use x¹ for the meter-stick reading, and x for the inch reading: x¹=x/.3937. 3.5/.3937 = 8.89 Wait a minute. It¹s not just science but deŽnition that says c=i/.3937=8.89, so something is wrong. 8.89 is not 51. We already knew that 51 cm was just an arbitrary coordinate. Arbitrary not because that point isn¹t 51 cm from the zero end of the meter-stick, but because the zero point was in an arbitrary position. Let¹s put the meter-stick in a position where it¹s zero point is at the yard-stick zero point. What is the centimeter coordinate now? Hey. 8.89, just like the formula says. The only way for a Œtransform¹ like x¹=x/g to work, whatever g might be, is for both coordinate systems to have their zero points aligned, in which case saying the two measures are not intervals is pure idiocy. Noe that with both zero points at the same position both x¹ and x are great measures for scientiŽc purposes, in any and every case where we were smart enough to put those zero points at a useful location. There is one extension of x¹=x/g that will let us use the meter-stick in arbitrary position. When the cm reading was 51, the zero point of the yard-stick read (51-8.89=) 42.11 cm. If we call that point x.z¹ we get x¹ = x.z¹ + x/.3937. = 42.11 + 3.5/.3937 = 42.11 + 8.89 = 51. Obviously, in this formula x/.3937 is the distance from the x¹ coordinate of the location where x=0. An interval. Just as obviously, the fact that we now have the correct formula for relating an x interval to an arbitrary x¹ coordinate, does not mean that x¹ is anything more than nonsense for use in any scientiŽc formula. Unless we were smart enough to put the x zero point in a useful location, and use (x¹-x.z¹) in the scientiŽc formula. (x¹-x.z¹) equals the useful, Ratio Scale value x/.3937. So, we have discovered a basic fact: a transformation formula like x¹=x/g works only if the two zero points of the coordinate systems coincide. That makes it non- sense to say the two coodinates are only coordinates and not intervals. Both must be values that represent distances from their respective zero points unless you take the proper steps to adjust for the discrepancy. Make sure you understand that although the inclusion of x.z¹ made it possible to correctly calculate x¹, the result is nonsense when it comes to use of x¹ for general length/distance purposes; it is x¹-x.z¹ that is a useful number in such cases. It could be that we¹re measuring a sheet of paper with one end at x=0 and the other at x=3.5; x¹=51 is nonsense as a centimeter measure of the paper. But, you say, the Lorentz transform contain a -vt term. ------------------------------ === Subject: 7. Distances and moving coordinate axes. We discovered x¹=x.z¹ + x/g as the correct formula for relating one coordinate to another system¹s. But the Lorentz transform contains another term, -vt/sqrt(1-vv/cc). What is it? Let¹s start with our x¹=51 cm, x=3.5, x.z¹=42.11 example. Every minute, let¹s move the meter-stick one inch to our right. At minute 0, the cm reading was 51 cm. At minute 1, the cm reading is now 50 cm. At minute 2, the cm reading is now 49 cm. In this instance, v=1 inch/minute. And t was 0, 1, 2. What has happened is that we have made our x.z¹ a lie, and increasingly so. -vt/.3937 is the change in x.z¹. x¹ = (x.z - vt/.3937) + x/.3937. Obviously, vt/.3937 is not a coordinate; even most SRians wouldn¹t imagine it was. It is an interval, the distance over which the moving system has moved since t=0. And, of course, x/.3937 is the distance of our brave little ant from the point where x=0 and the centimeter reading is x.z¹-vt/.3937. Yes, every minute the meter- stick moves to the right and the meter-stick coordinate of the spot where x=0 gets less and less - and eventually negative. Make sure you understand that every minute the x¹ coordinate, because of -vt/g, becomes a better measure of, say, the 3.5 paper we might be measuring with the yard-stick, given that 51 was too big a number and -vt is negative. That is, until the two origins coincide at x¹=x=0, and then it gets worse and worse. With -vt positive (because v<0) the situation is different. With 51 and -vt positive, x¹ just gets worse and worse over time. Quite obviously, the fact that we now have the correct formula for relating an x interval to an arbitrary x¹ coordinate even when the x¹ axis is moving, does not mean that x¹is anything more than nonsense for use in any scientiŽc formula. Unless we were smart enough to put the x zero point in a useful location, and use (x¹-x.z¹+vt/.3937) in the scientiŽc formula. (x¹-x.z¹+vt/.3937) equals the useful, Ratio Scale value x/.3937. ------------------------------ === Subject: 8. Time intervals. Instead of using our sticks, let¹s get out two clocks. Mind you, we¹re not going to deal with different clock rates here, just establish the same basics as for distance. Your clock says 9:00 Eastern Standard Time (EST) and we note that t=540 minutes when we put down the clock. Blindly, let¹s turn the setting knob of your twin¹s PaciŽc Standard Time clock and put it down before us. According to what we see, EST¹s 540 minutes (9:00) corre- sponds to PST¹s 14:30; t¹=870. We know the formula relating PST to EST is t¹ (paciŽc) = t (eastern) - 180 (minutes). Thus, it is not correct that the second clock can have an arbitrary setting, because 870 <> 540-180. We know that the two clocks are related by t¹ = t/1 since both are using the same second, hour, etc units. But 870 (14:30 in minutes) is not 540/1-180, so once again we know something is wrong. However, t¹=t.z¹ + t/1 works. EST midnight equals PST 0.0 (midnite) - 180, so t.z¹ = -180, and t¹ = -180 + 540/1 = 360. Since EST-180=PST, 9:00 EST is 6:00 PST = 360 minutes. We see thus that like distance measures/coordinates, time axis origins (zero points) must either be Œlined up¹ or adjusted for. So, the Lorentz/Einstein t¹=t/sqrt(1-vv/cc) must be the moving system elapsed time interval since the time axes were both at a common zero. There is no t.z¹ adjustment: t¹ = (t - vx/cc)/sqrt(1-vv/cc) (Eq 1t) Make sure you understand that in the clock case, if the EST is showing a good number for elapsed time since the travelling observer passed NYC, then the PST clock is silliness. t.z¹ must be zero or must be taken out of time lapse calculations for the PST clock to be used intelligently, just as was true for x.z¹. What is lacking as yet for Lorentz t¹ is the -vx/cc term that corresponds to the x¹ formula -vt term. Break it up into two parts: v/c and x/c. v/c is a scaling factor that changes velocity from whatever kind of unit you are using over to fractions of c. x/c is distance divided by velocity, which is time. x/c is thus the time interval since the two time axes had a common zero point - which they have to have in the Lorentz transforms which do not have the t.z¹ term we learned to use above. Thus, (-vx/cc)/sqrt(1-vv/cc) is the interval amount the moving system clock has been changed - since the common/ adjusted time - over and beyond the elapsed time interval represented by x/sqrt(1-vv/cc). We have discovered that the only way for t¹ to be t/g is for t¹ and t to have a common zero point, just as for x¹ and x. It would be otherwise if the t¹ formula contained an adjustment t.z¹ under some name or other, but the necessity to include such a term correlates 100% with t¹ numbers that aren¹t directly usable. As for x and x¹, our knowledge of how to setup a proper formula relating t and t¹ is of no use unless we use the knowledge in scientiŽc formulas; (t¹-t.z¹+xv/gcc) gives us the only directly useful value: t/g. ------------------------------ === Subject: 9. Einstein¹s (1905) derivations. When we return to Einstein¹s derivations of the transform formulas with a well-focused eye, we Žnd he was a wee bit confused - or at least self-contradictory. When he set up his (at Žrst unknown) tau=moving system time formulas, he created three particular instances of tau. Tau.0 is the time at which light is emitted at the moving origin toward a mirror to the right that is moving at rest wrt that moving origin and at a constant distance from that origin. He lets the stationary time slot have the value t, a constant, the stationary system starting time. Tau.1 is the time at which the light is režected. He lets the stationary time be t+x¹/(c-v); t is still a constant and x¹/(c-v) is the time interval since t. Tau.2 is the time at which the light gets (back) to the moving origin. The stationary time value is put as t + x¹/(c-v) + x¹/(c+v); t is still a constant and x¹/(c-v) + x¹/(c+v) is the time interval since t. On the thesis that the moving observer sees the time to the mirror as the same as the time back to the origin, he sets .5[ tau.0 + tau.2 ] = tau.1. Tau.0 completely drops out of the analysis and leaves no trace, and has no effect. Further, the t you see in tau.0, tau.1, and tau.2 also completely drops out with no trace and no effect, leaving us with exactly what you¹d get if you had explicilty said t¹ is an interval and so is t. What doesn¹t drop out in the stationary time values is x¹/(c-v) and x¹/(c+v), the time interval it takes for light to get to the žeeing mirror, and the time interval it takes for light to get back to the approaching origin. Thus, his resultant t¹ formula is strictly based on time intervals in the stationary system. Time intervals since some starting time, yes, but time intervals. There is absolutely nothing in the derived formulas that depends on arbitrary coordinates like the constant t in the stationary time arguments. Let¹s look at the x dimension; it is x¹=x-vt [as x increases by vt, the effect over time is x¹=(x+vt)-vt)], which Einstein explicitly sets up as a constant stationary distance. He uses that x¹ not just in the time interval parts of the stationary time arguments, but also in the x (distance) stationary system argument for the tau at the time light is režected. x¹ can¹t be the stationary system coordinate of the mirror at that time. That value is x¹+vt. x¹ is explicitly an interval, distance. Thus, the whole tau derivation of the t¹ formula is fully and explicitly based on x¹ - a spatial length/distance/interval - and the two time interals x¹/(c-v) and x¹/(c+v). While we¹re at it, if the starting t is not zero, his x¹=x-vt formula is complete nonsense also. Given that there was some L that was the mirror x-location and length when the light is emitted, if t was already, say, 500, then x¹=L-vt could have been a very negative length. ------------------------------ === Subject: 10. A word about intervals. There are intervals, and there are intervals. If we put our yard stick zero point at one end of a piece of paper and read off the coordinate at the other end of the paper, we have a good measure of the paper¹s length, a Ratio Scale measure. [Absolute temperature scales are ratio scale.] If instead we put the one end of the paper at the one inch mark (or the zero end of the stick one inch Œinto¹ the length of the paper) we get measures that are one inch off the true, ratio scale length. The two messed up measures are still intervals, but they are Interval Scale measures. [Household temperature scales are interval scale, which is why your physics and chemistry professors won¹t let you use them without Žrst converting to the ratio scale absolute temperatures.) t¹=t/g and x¹=x/g represent ratio scale measures, given that t and x were ratio scalae to start with. t¹=t.z¹+t/g and t¹=t/g-vx/gcc are both interval scale measures, even given a good ratio scale t and a good ratio scale x. x¹=x.z¹+x/g and x¹=x/g-vt/g are both interval scale measures, even given a good ratio scale x and a good ratio scale t. Look for the (SR) Lorentz t¹, x¹ = degraded measures document soon at a newsgroup near you. ------------------------------ === Subject: 11. Intervals versus the Twins Paradox. t¹=(t-vx/cc)/g shows t¹ being greater than t. The reason Special Relativity will not allow the use of its basic time equation in determining what SR has to say about the twins¹ ages, is that t¹ and x¹ are supposedly just coordinates, and they say you have to take the coordinate pairs (t¹,x¹) and (x,t) into consideration in both the time and place the twins¹ separation started and the time and place the twins reunited. Since t¹ and x¹ are actually both intervals, not just coordinates, the Œexcuse¹ is spurious, and is so even without use of the obvious (x_b-x_a) and (t_b-t_a) usages. However, SR is right to be embarrassed by their transformation formulas. Look for the (SR) Lorentz t¹, x¹ = degraded measures document at a newsgroup near you. ------------------------------ === Subject: 12. Summary A. t¹=t/g and x¹=x/g can be almost Œjust coordinates¹ in the sense that the values obtained may not be of much use except in the most primal and useless way: how long and how far since/from the time/ place they were zero. Even here, however, the zero points within each of the two scale pairs (t¹,t) and (x¹.x) must have been lined up. If the zero points have been intelligently selected (such as at the starting point and time of a trip) they can be rationally used Œas is¹ in any valid sci- entiŽc equation. B. Even the interval scale t¹=t.z¹ - xv/gcc + t/g and x¹=x.z¹ - vt/g + x/g are not Œjust coordinates¹. They can be used to good effect by establishing the relevant starting times/points and using (t¹-t.z¹+xv/gcc) and (x¹-x.z¹+vt/g), as the situation may require. C. When you see vx/gcc or vt/g in use in any guise with non-zero values, you know the resultant t¹ or x¹ is a degraded, interval scale value. E-X: Anytime you do not see what amounts to t.z¹ and xv/gcc in the time case, or x.z¹ and vt/g in the distance case, you know that the t¹ and/or x¹ in use are intervals. Period. Y: Either set your clock to zero at the start of the relevant time interval, or use (t-t0), with both being readings on the same clock. Either move your x-axis origin to the starting end or point, or use (x-x0), with both being readings on the same axis. Z: In _(SR) Lorentz t¹, x¹ = Degraded (Interval) Scales_ we see that t¹ and x¹ satisfy the mathematical tests for/of interval scales when -vt and -vx/cc are not zero; thus, they must be intervals. When -vt and -vx/cc are zero, t¹ and x¹ satisfy the much better mathematical deŽnition of ratio scales, and are thus not just mere intervals, but (rescaled) good ones. Eleaticus !---?---!---?---!---?---!---?---!---?---!---?---!---?---!---? ---!---? ! Eleaticus Oren C. Webster ThnkTank@concentric.net ? ! Anything and everything that requires or encourages systematic ? ! examination of premises, logic, and conclusions ? !---?---!---?---!---?---!---?---!---?---!---?---!---?---!---? ---!---? === Subject: Re: (SR) Lorentz t¹, x¹ = Intervals [snip lies] > (SR) Lorentz t¹, x¹ = Intervals > (c) Eleaticus/Oren C. Webster > Thnktank@concentric.net [snip 700 lines of trolled garbage] eleaticus, Oren Webster, is a despised and stooopid troll, http://users.pandora.be/vdmoortel/dirk/Physics/Fumbles/ Crimes.html Several crimes against logic and science Ha ha ha! Originally trolled across sci.physics sci.physics.relativity alt.physics sci.math sci.answers alt.answers news.answers Psychotic ineducable boring troll Eleaticus, Were there to be internal inconsistencies in SR (meaning inconsistencies of a purely mathematical logical nature) that would automatically lead to contradictions in number theory, itself, and arithmetic, since the mathematics of Minkowski geometry is equiconsistent with the theory of real numbers and with arithmetic. Eleaticus explicitly demonstrates that he is completely ignorant of multivariable calculus. He has no concept of the Chain Rule in multivariable calculus. Consider his Galilean Transformation goo and dribble: t¹ = t, x¹ = x - vt, y¹ = y, z¹ = z. His refusal to accept that t¹ must be introduced as a separate variable springs from a massive emprical stupidity re space and time are described as a four-dimensional manifold, with four coordinates instead of a time evolution of a three-dimensional manifold, and that the change of coordinate system should be a change of four coordinates, and not a time-dependent change of three coordinates. This is particularly vital when it comes to Želds over space and time (electric and magnetic Želds for example). The transformation law for the differential operators under the Galilean transformation is given by: d/dt¹ = d/dt + v d/dx, d/dx¹ = d/dx, d/dy¹ = d/dy, d/dz¹ = d/dz. This shows the necessity of introducing a new variable t¹, since partial differentiation with respect to t¹ (constant x¹, y¹, z¹) is a different operation to partial differentiation with respect to t (constant x, y, z). The above transformation law is determined by the Chain Rule: d/dt¹ = dt/dt¹ d/dt + dx/dt¹ d/dx + dy/dt¹ d/dy + dz/dt¹ d/dz, d/dx¹ = dt/dx¹ d/dt + dx/dx¹ d/dx + dy/dx¹ d/dy + dz/dx¹ d/dz, d/dy¹ = dt/dy¹ d/dt + dx/dy¹ d/dx + dy/dy¹ d/dy + dz/dy¹ d/dz, d/dz¹ = dt/dz¹ d/dt + dx/dz¹ d/dx + dy/dz¹ d/dy + dz/dz¹ d/dz. The presence of the term involving d/dx in the expression for d/dt¹ is indicative of the fact that x depends on t¹ (x¹, y¹, z¹, being held constant), as can be seen from the fact that the coefŽcient of d/dx in the expression for d/dt¹ is dx/dt¹. Because of the now demonstrated fact that Eleaticus has no formal education in multivariable calculus, he has managed, somehow, to get it into his head that the presence of the term involving d/dx in the expression for d/dt¹ is indicative of t¹ depending on x (t, y, z, being held constant). Because of his stupidty Eleaticus cannot get the correct transformation law for the differential operators under the Galilean Transformation, and he cannot determine the invariance or otherwise of Maxwell¹s Equations under the Galilean Transformation. The Žrst advice to Eleaticus is to learn multivariable calculus. Eleaticus should not pretend that he can understand how to determine invariance or otherwise of Maxwell¹s Equations under the Galilean Transformation, or under the Lorentz Transformation, until he understands the multivariable calculus which underlies such considerations. Eleaticus is a loud idiot. The homogeneous Maxwell equations are invariant under the Galilean Transformation, with transformation laws: E_x¹ = E_x, E_y¹ = E_y - v B_z, E_z¹ = E_z + v B_y, B_x¹ = B_x, B_y¹ = B_y, B_z¹ = B_z. The derivation of these transformation laws was determined using the transformation laws for the differential operators given above. These transformation laws have the additional advantage that they determine the correct transformation for the force law, thus providing further evidence in favour of the transformation law for the differential operators, as above. The inhomogeneous Maxwell equations are also invariant under the Galilean transformation, with transformation laws: E_x¹ = E_x, E_y¹ = E_y, E_z¹ = E_z, B_x¹ = B_x, B_y¹ = B_y + v/c^2 E_z, B_z¹ = B_z - v/c^2 E_y, rho¹ = rho, J_x¹ = J_x - v rho, J_y¹ = J_y, J_z¹ = J_z. Note the the transformation laws for the charge density and current density are as they should be under the Galilean transformation. Homogeneous equations are invariant under the Galilean Transformation, and inhomogeneous equations are invariant under the Galilean Transformation, but Maxwell¹s Equations as a whole are NOT invariant under the Galilean Transformation, since the transformation laws required for the EM Želd for the two cases are inconsistent with each other. The transformation law for the EM Želd which makes the homogeneous equations invariant will not also make the inhomogeneous equations invariant. The transformation law for the EM Želd which makes the inhomogeneous equations invariant will not also make the homogeneous equations invariant. On the other hand, all of Maxwell¹s equations are invariant under the Lorentz Transformation, with transformation laws: E_x¹ = E_x, E_y¹ = gamma (E_y - v B_z), E_z¹ = gamma (E_z + v B_y), B_x¹ = B_x, B_y¹ = gamma (B_y + v/c^2 E_z), B_z¹ = gamma (B_z - v/c^2 E_y), rho¹ = gamma (rho - v/c^2 J_x), J_x¹ = gamma (J_x - v rho), J_y¹ = J_y, J_z¹ = J_z, where gamma = 1/sqrt(1 - v^2/c^2). Idiot Oren Webster sees himself this way, http://www.mazepath.com/uncleal/effete6.jpg The entire remainder of the planet sees him this way, http://www.mazepath.com/uncleal/effete3.png http://www.mazepath.com/uncleal/sunshine.jpg http://www.apa.org/journals/psp/psp7761121.html http://insti.physics.sunysb.edu/~siegel/quack.html Hey, stooopid troll Eleaticus - Do you want EVIDENCE? Each of the 24 GPS satellites carries either four cesium atomic clocks or three rubidum atomic clocks in orbit, with full relativistic corrections being applied. Internal inconsistencies in SR (meaning inconsistencies of a purely mathematical logical nature) automatically lead to contradictions in number theory, itself, and arithmetic, since the mathematics of Minkowski geometry is equiconsistent with the theory of real numbers and with arithmetic. Mathematics of gravitation Equivalence Principle testing http://arXiv.org/abs/hep-th/0111236 Geometric structure of reality http://arxiv.org/abs/gr-qc/0103044 http://arXiv.org/abs/hep-th/0307140 GR structure, especially Part 4/p. 7 http://arXiv.org/abs/gr-qc/0311039 Experimental constraints on General Relativity http://www.eftaylor.com/pub/projecta.pdf Relativity in the GPS system http://arXiv.org/abs/gr-qc/9909014 falling light Hafele-Keating Experiment http://www.hawaii.edu/suremath/SRtwinParadox.html Twin Paradox http://arXiv.org/abs/astro-ph/0401086 http://arxiv.org/abs/astro-ph/0312071 Deeply relativistic neutron star binaries http://arxiv.org/abs/hep-th/0405160 Black hole evaporation No aether http://fsweb.berry.edu/academic/mans/clane/ No Lorentz violation http://arXiv.org/abs/gr-qc/0409089 Spin-2 gravitons have problems (so does the proposal) http://arXiv.org/abs/gr-qc/0301024 Nordtvedt Effect http://arxiv.org/abs/astro-ph/0403292 http://arXiv.org/abs/astro-ph/0310723 WMAP + Sloane Digital Sky Survey http://arxiv.org/abs/hep-ph/0404175 Dark matter candidates Carroll on what it all means. Special Relativity is physics on a topologically trivial Lorentzian manifold with a metric whose curvature tensor is zero. This is a perfectly diffeomorphism-invariant condition and does not require any particular coordinate choice. It is invariant under the full group of diffeomorphisms. The Poincare group is the group of *isometries* of the metric in special relativity. The Special Relativity metric is *non-dynamical* (unlike GR). It deŽnes the coupling *constants* of your theory. If you change the metric in any nontrivial way you are changing your theory. An operation can only be called a symmetry of a special-relativistic (non-gravitational) theory if it preserves the metric, and therefore the symmetry of special-relativistic theories is the Poincare group only. General Relativity (gravitation) has a dynamic metric. NIM A 355 537 (1995) Physics Letters B 328 103 (1994) Physical Review Letters 64 1697 (1990) Physical Review Letters 39 1051 (1977) Physical Review 135 B1071 (1964) Physics Letters 12 260 (1964) Europhysics Letters 56(2) 170-174 (2001) General Relativity and Gravitation 34(9) 1371 (2002) http://fourmilab.to/etexts/einstein/specrel/specrel.pdf Longitudinal and transverse mass http://arxiv.org/abs/gr-qc/0306076.pdf http://www.navcen.uscg.gov/pubs/gps/gpsuser/gpsuser.pdf http://www.navcen.uscg.gov/pubs/gps/sigspec/default.htm http://www.navcen.uscg.gov/pubs/gps/icd200/default.htm http://www.trimble.com/gps/index.html http://sirius.chinalake.navy.mil/satpred/ http://www.phys.lsu.edu/mog/mog9/node9.html http://egtphysics.net/GPS/RelGPS.htm http://www.schriever.af.mil/gps/Current/current.oa1 http://edu-observatory.org/gps/gps_books.html -- Uncle Al http://www.mazepath.com/uncleal/ (Toxic URL! Unsafe for children and most mammals) http://www.mazepath.com/uncleal/qz.pdf === Subject: About Ancle assAls¹ response to: (SR) Lorentz t¹, x¹ = Intervals > The transformation law for the differential operators under the > Galilean transformation is given by: > d/dt¹ = d/dt + v d/dx, > d/dx¹ = d/dx, > d/dy¹ = d/dy, > d/dz¹ = d/dz. > This shows the necessity of introducing a new variable t¹, since > partial differentiation with respect to t¹ (constant x¹, y¹, z¹) is a > different operation to partial differentiation with respect to t > (constant x, y, z). The above transformation law is determined by the > Chain Rule: Actually, I am soOOO surprized you miss the necessity of introducing a new variable v¹ since partial differentiation wrt v¹ is a different operation to partial differentiation with respect to v. Hey, violating the rules of logic by asserting t¹=t when there is no t¹ in the newtonian theoretic material you have such a problem with is one thing, but how about justifying the idiocy of not imposing the actual, factual, not-so-satisfactual (to SR) v¹=-v transform? Not to mention, justifying not differentiating wrt v in terms that don;t also apply to t. eleaticus === Subject: Re: About Ancle assAls¹ response to: (SR) Lorentz t¹, x¹ = Intervals X-RFC2646: Format=Flowed; Original >> The transformation law for the differential operators under the >> Galilean transformation is given by: >> d/dt¹ = d/dt + v d/dx, >> d/dx¹ = d/dx, >> d/dy¹ = d/dy, >> d/dz¹ = d/dz. >> This shows the necessity of introducing a new variable t¹, since >> partial differentiation with respect to t¹ (constant x¹, y¹, z¹) is a >> different operation to partial differentiation with respect to t >> (constant x, y, z). The above transformation law is determined by >> the >> Chain Rule: > Actually, I am soOOO surprized you miss the necessity of introducing a > new > variable v¹ since partial differentiation wrt v¹ is a different > operation to > partial differentiation with respect to v. > Hey, violating the rules of logic by asserting t¹=t when there is no > t¹ in > the newtonian theoretic material you have such a problem with is one > thing, > but how about justifying the idiocy of not imposing the actual, > factual, > not-so-satisfactual (to SR) v¹=-v transform? > Not to mention, justifying not differentiating wrt v in terms that > don;t > also apply to t. > eleaticus I think you are trying to teach an old dog new tricks. Even a street whore wouldn¹t trick with him, and she was desperate. Androcles. === Subject: Re: (SR) Lorentz t¹, x¹ = Intervals sci.answers,alt.answers and news.answers pruned. Followups set exclusively to sci.physics.relativity. And this has probably been debunked somehwere before. But I¹m in a mood to vent at somebody; it¹s been one of those weeks. :-) In sci.math, Eleaticus : > Disclaimer: approval for *.answers is based on form, not content. > Opponents should Žrst actually Žnd out what the content is, > then think, then request/submit-to arbitration by the > appropriate neutral mathematics authorities. Flaming the hard- > working, selžess, *.answers moderators evidences ignorance > and atrocious netiquette. > Version: 0.02.1 > Archive-name: physics-faq/criticism/lorentz-intervals > Posting-frequency: 15 days > > (SR) Lorentz t¹, x¹ = Intervals > (c) Eleaticus/Oren C. Webster > Thnktank@concentric.net > ------------------------------ === > Subject: 1. Introduction with the obvious debunking > of the use of Œjust coordinates¹ in any > scientiŽc formula. > Defenders of the Special Relativity faith are especially > fond of telling opponents of their space-time fairy tales > that they do not know the difference between coordinates > and magnitudes. > That may often be so, but the fault lies ultimately with > SR dogma. The Lorentz-Einstein transformations cannot > possibly be Œjust coordinates¹, which is the interpre- > tation required to support the many sideshow carnival acts > with which the SR faithful bedazzle the public, and establish > their moral and intellectual superiority. > If I get in my car and drive steadily for a few hours at 50 > kilometers per hour, is 50t the distance I travel? Assuming t is in hours, you will be driving 166.6 * t microseconds¹ distance. And yes, this is a weird way of putting it -- but because of the invariance of x^2 - c^2*t^2, perfectly correct as far as SR is concerned. But don¹t tell the speedometer manufacturers... :-) If you prefer, you can use units such as light-second for distance. The velocity can then be expressed as 1.666 light-microseconds per hour, and of course c = 1 in such units. However, there are some minor inconveniences -- this is, after all, the population which still clings to so-called Imperial Units. :-) One might have better luck in Europe. > Of course not. The last time my hours-counting Œjust coord- > inates¹ clock was set to zero was when Zeno Žrst reported > one of his paradoxes to Parmenides. Well, if you want to move your origin, as opposed to the goalposts, Žne...just be aware you¹re doing it. :-) > That was a long time ago, so my t is not useful for such > purposes unless you also use my clock to established the starting > time, perhaps t0, and use the formula 50(t-t0) to calculate the > distance. > In any case, my t is even then not Œjust a coordinate¹ because > it always represents particular elapsed times that can be > used in the (t-t0) form to calculate perfectly good time > intervals (elapsed times). > Alternatively, I could (re)set my clock to zero at the start > of some meaningful time interval, in which case my t shows a > scientiŽcally perfect current and/or end time. > In which case, the Lorentz-Einstein t¹=(t-vx/cc)/g is a function > of an elapsed time interval (not Œjust a coordinate¹) and a time > interval (-vx/cc; the interval amount the t¹ clock is being > screwed up at time t) and thus cannot be Œjust a coordinate¹ > since neither of the independent variables is such a Œjust¹ thing. > {Their meaning is shown below, step-by-step.] An interesting point, but again, beware the shifting origin. [identical dist case snipped for brevity] > ------------------------------ === > Subject: 2. Table of Contents > 1. Introduction with the obvious debunking > of the use of Œjust coordinates¹ in any > scientiŽc formula. > 2. Table of Contents. > 3. The Lorentz-Einstein transforms. > 4. The Œjust coordinates¹ argument. > 5. Single-system, little-purpose ambiguity. > 6. Relating two coordinate measures/systems. > 7. Distances and moving coordinate axes. > 8. Time intervals. > 9. Einstein¹s (1905) derivations. > 10. A word about intervals. > 11. Intervals versus the Twins Paradox. > 12. Summary > ------------------------------ Silly. Usually the TOC (a) goes at the top of the doc and (b) doesn¹t need an entire subject dedicated thereto. === > Subject: 3. The Lorentz-Einstein transforms > Special Relativity¹s space-time circus is based on > the Œtransformation¹ equations by which it is believed > one can relate a nominally Œstationary¹ system¹s space > and time coordinates to those of an inertially (not > accelerating) moving other observer. > That moving observer¹s own physical body and coordinate > system might have been identical in size to those of the > stationary observer before the traveller began moving, > but are Œseen¹ as very different by the stationary observer > when the relative velocity of the two is great enough, a > high percentage of the velocity of light. Actually, it¹s seen in all velocities -- though it¹s hard to observe at standard highway speeds. A car travelling 30 km/s = 67.1 mph will have a shrinkage factor of about 5 * 10^-15. Since that¹s about the width of an atomic nucleus (per meter), there¹s a vanishingly small chance of it being observed directly -- but that¹s what SR predicts. > Concerning ourselves - as is customary - with just > the spatial coordinate axis that lies parallel to > the direction of motion, and with time, Einstein > arrived at these formulas that relate the moving > system measures or coordinates (x¹ and t¹) to the > stationary system coordinates (x and t): > x¹ = (x - vt)/sqrt(1-vv/cc) (Eq 1x) > t¹ = (t - vx/cc)/sqrt(1-vv/cc) (Eq 1t) This equation assumes that both origins are coincident. Of course, you then go on a long-winded tirade about how the origins need to be coincident. So we¹ll schlog through here for awhile... :-) > The v is for the two systems¹ relative velocity as seen > by the stationary observer, and is positive if the dir- > ection is toward higher values of x. By concensus, > the moving system x¹-axis higher values also lie in > that direction, and all axes parallel the other system¹s > corresponding axis. v = -v¹. To argue otherwise invites much silliness. > We used vv to mean the square of v but might use v^2 > for that purpose below. Similarly for c. > Because it is believed that no physical object can > reach or exceed c, the square-root term in both > denominators is presumed always less than one, which > means that the formulas say both x¹ and t¹ will tend to > be greater than x and t, respectively. However, > SRians call the x¹ result Œcontraction¹ - which means > shortening - and the t¹ result Œdilation¹ - which > means increasing. > ------------------------------ === > Subject: 4. The Œjust coordinates¹ argument > The Œjust coordinates¹ argument is so patently ridiculous > that even opponents have a hard time accepting just how > simple and obvious its debunking can be, as shown in this > section. However, further sections take a more arithmet- > ical approach that you¹ll maybe Žnd more professorial. > The Œjust coordinates¹ argument is that t is mot a > duration, not a time interval; it¹s just an arbitrary > clock reading. But what if the moving system observer > comes speeding by while you make your annual Œspring > forward¹ or Œfall back¹ change? The formula says that > the moving system clock¹s Œjust coordinate¹ reading > can be calculated from yours: > t¹ = (t - vx/cc)/sqrt(1-vv/cc) (Eq 1t) > Imagine the moving system oberver¹s confusion if his > clock changes its reading while he¹s looking at it! Hence such artiŽces as UTC, which doesn¹t have time zones. It does have leap-second adjustments, however, which can cause minor confusion. > If his clock doesn¹t change when yours does, the formula > is wrong; if it is truly a Œjust coordinates¹ formula. > And then what happens if you realize you were a day > early and put your clock back to what it had said > previously? > And suppose you are in NYC and your twin in LA and > both are watching the moving observer. You¹ll both be > using the same v because you are at rest wrt (with > respect to) each other. You¹re on Eastern Standard > Time and your twin is on PaciŽc Standard Time > maybe. You have three hours more on your clock than > does your twin. > On which Œjust coordinate¹ clock will the Lorentz > transforms base the Œjust coordinate¹ time the moving > system clock says? The formula applies to both of > your t-times: > t¹ = (t - vx/cc)/sqrt(1-vv/cc) (Eq 1t) > Sure, the idea that you can change someone else¹s > clock with no connection of any kind is really > ridiculous, but Eqs 1x and 1t aren¹t MY equations. > Are they yours? And we aren¹t the ones to say x, t, > x¹, and t¹ are just coordinates. > If the t¹ formula is actually either an elapsed > time formula, or the basis of a t¹/t ratio, then > there is no implication that one clock¹s reading > has anything to do with the other¹s. > It can only be rates of clock ticking, or how one > time INTERVAL compares to the other that the formula > is about. The origin shifts again -- hopefully predictably. > ------------------------------ === > Subject: 5. Single-system, little-purpose ambiguity. > Since we¹re going to be comparing measurements on two > coordinate systems in the next section, let¹s go to > our supply cabinet and get our yard-stick (which we > use to measure things in inches) and our meter-stick > (which we use to measure things in centimeters). > Here, I¹m getting mine. Oh! Oh! > There¹s an ant on mine, and he ... she ... sure is > hanging on, right at the 3.5 inch mark of the yard- > stick. > Let¹s see if I can wave the stick around enough that > she¹ll let go. Nope. > However, before I gave up I waved the stick and the > ant Œall over the place. > Always, however, the ant was at the 3.5 mark on the > yard-stick, and always 3.5 away from the end of the > stick, however far and wide I have transported her. You are now getting into rotational tensor territory, which is unfamiliar to me. > Neither of those 3.5 facts means very much. Of the > two, the distance aspect meant almost nothing. So > the distance was 3.5 from the end. So what? That > length, distance, was not in use. And only maybe > the ant might have been concerned with just what > location, Œjust coordinate¹, on the stick she was > at. > Just so with x and t. > So, is the 3.5 reading just a coordinate? Or a > distance/length? Neither. It¹s a *vector*. SpeciŽcally, it¹s a direction, within an established coordinate system, from the origin to a desired location. Since you¹ve established *two* coordinate systems (C1: you, standing there waving the stick all over the place; C2: the stick, where 0 = the endpoint and the x-axis is the stick itself [the other two axes can be deŽned in a compatible but consistent fashion; e.g., z might be to the stick¹s left, and y might be through the stick going up]), you now get to relate them. Lucky you! Fortunately, tensors are available, although all over the place isn¹t exactly the easiest one to use, mathematically. > It¹s ambiguous in and of itself, > and really makes no difference what you say until > you try to make use of the number. > Hey, my address is 5047 Newton Street. If you > are looking for me and you¹re at 4120 Newton, it > is helpful information, because it tells you which > direction to go. Is that Œjust coordinate¹? Actually, it does not. For starters, there are