Talk:Lorentz transformation/Archive 1

Error in Section "Details"
Quote:

Then, we can write

s^2 = \begin{bmatrix} -1&0&0&0\\ 0&1&0&0 \\ 0&0&1&0 \\ 0&0&0&1 \end{bmatrix} \begin{bmatrix} c \Delta t \\ \Delta x \\ \Delta y \\ \Delta z \end{bmatrix}\. $$

Endquote.

This is clearly not true. Because if you multiply it out, you'll get another vector (which is easy to see when you know a bit of matrix algebra: A n x n matrix multiplied with a n x 1 matrix gives a n x 1 matrix). Which is probably not what the author meant to say. The square of the norm of a vector $$x^{\mu}$$



x^{\mu} = \begin{bmatrix} c \Delta t \\ \Delta x \\ \Delta y \\ \Delta z \end{bmatrix} $$

with respect to a metric $$\eta_{\mu\nu}$$ is:



s^2 = \eta_{\mu\nu} x^{\mu} x^{\nu} $$

where Einstein's Summation Convention is used. More explicit:



s^2 = \sum_{\mu=1}^4 \sum_{\nu=1}^4 \eta_{\mu\nu} x^{\mu} x^{\nu} $$

So in this case with



\eta_{\mu\nu} = \begin{bmatrix} -1&0&0&0\\ 0&1&0&0 \\ 0&0&1&0 \\ 0&0&0&1 \end{bmatrix} $$

We get:



s^2 = \eta_{\mu\nu} x^{\mu} x^{\nu} = -(c \Delta t)^2 + (\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2 = -c^2 (\Delta t)^2 + (\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2 $$


 * OK, I think its fixed. Please check it. PAR 14:30, 17 September 2006 (UTC)


 * Ok, there's some style details. I find it unfortunate how the spacetime events A and B are notated by upper case letters. The norm with respect to the minkowski metric is most naturally written in tensor co/contravariant notation as with the $$s^2 = \eta_{\mu\nu} x^{\mu} x^{\nu}$$, where $$x^\mu$$ denotes a vector x and where its components are denoted by $$x^1, x^2$$ or generally $$x^\nu$$ . There's another clash with the use of x as in x,y and z of the spatial components. This might confuse readers. I'm not sure how to go about this. FlorianPaulSchmidt 19:16, 17 September 2006 (UTC)

Confusing
The below, added to the History section, is confusing: c is then identified to the propagation speed of electromagnetic radiation in vacuum. (Comment from before 2003)

Merge
We must merge this with Lorentz transformation equations

Brute-force merged: now the equations need thorough checking, because they are subtly different in each sub-version... -- The Anome 00:04, 13 Aug 2003 (UTC)

Proper foundation
"Proper foundation for its application" ... I'd love to watch a debate between the author and this bunch who seem to be staking their careers on Lorentz being right after all: "The Einstein assumptions leading to the Special and General Theory of Relativity are shown to be falsified by the extensive experimental data. Contrary to the Einstein assumptions absolute motion is consistent with relativistic effects, which are caused by actual dynamical effects of absolute motion through the quantum foam, so that it is Lorentzian relativity that is seen to be essentially correct." I suggest "proper foundation" is not NPOV; but WP has been sufficiently abusive to me I'm doing no more than to leave this note. -- fmr Kwantus (March 2004)

Lorentz Transformation is Flawed
The Lorentz transformation is actually nothing but a 'cheated' Galilei Transformation: it uses a vectorial velocity addition for light signals (which contradicts the invariance of the speed of light) and then this error is being 'rectified' by changing the original length and time units (see my page http://www.physicsmyths.org.uk/lightspeed.htm for more).


 * In order to understand the interpretation of the invariance of the speed of light, behold the eigenvectors of the Lorentz transformation.
 * 217.81.157.237 20:29, 10 Jan 2005 (UTC)

In your website page illustrated example your hypothesis and conclusion seem to be one and the same: the invariance of the speed of light. Then any "demonstration" would be redundant (i.e. already in the hypothesis). The whole discussion about the Lorentz transformation is the physical validation or invalidation of his mathematical expression of t' in relation to t. If possible please post relevant material of how he arrived at the proposed equation.--Lucian 20:00, 27 May 2005 (UTC)


 * Yeah, I fail to understand how that website proves anything. Perhaps the original poster should read up on the Lorentz group to learn how the Lorentz transformation isn't a 'cheated' Galilean transformation, but rather an extension of rotations.  The most intuitive way to understand the derivation of Lorentz transformations is to see it as finding a transformation that preserves c (rotations) and extending that in a really simple way, rather than pulling a new transformation out of thin air, which is usually how it's presented to freshmen undergrads). --Laura Scudder | Talk 20:34, 27 May 2005 (UTC)

The error made in the website is the statement:
 * "Since obviously the same amount of rope has reeled off at the car as well, the marking at the latter does at this moment also read x1 and since the rope has reeled off with the same speed, the car-clock does also shows T1'=T1=x1/v."

The same amount has not reeled off. In the GBO frame, the GBO-rope is moving at velocity v while the car-rope is at rest. This means that the GBO-rope is Lorentz-contracted, while the car-rope is not. At any given time in the GBO frame, the GBO-rope will measure a greater distance to the car than the car-rope. This is not a paradox, because the spatial distance between two events is not absolute, but depends on intertial frame in which it is measured. 69.143.43.101 15:47, 28 August 2005 (UTC)


 * Yes, see the ladder paradox for a more in depth explanation. Fresheneesz 19:02, 17 April 2006 (UTC)

Derivation of Lorentz Transformation
I am missing a derivation of the Lorentz transformation on Wikipedia. I think this is so vital for the development of Special Relativity (both historically and didactically) that one should not have to refer to external links for this.


 * done... —Preceding unsigned comment added by 131.114.192.226 (talk) 13:53, 11 September 2007 (UTC)

Merging?
Shouldn't this be merged with Lorentz-Fitzgerald contraction hypothesis? This article is about relativity and that one is about luminiferous aether, but it's basically the same contraction used for two different purposes and it seems like it should be one article. Ken Arromdee 18:05, 5 October 2005 (UTC)


 * Maybe - but the contraction hypothesis should be included in the Lorentz transformation article, not vice versa. PAR 23:49, 5 October 2005 (UTC)

Triangle of Velocities proves that invariant speed is mathematical error
The article at http://www.masstheory.org/triangle_of_velocities.pdf presents detailed analysis of Lorentz transformation, it's derivation procedure and formal mathematical proof that introducing "invariant" speed is indeed nothing but mathematical error.
 * I'd be interesting to know if the person who posted this understands what the author of "triangle of velocities" wrote. I read it, and I think he made a mistake in assumptions about the variables - ie what variable corresponds to what - which i admit is very confusing (which is why I wrote the variable definitions out on this page for the SC transforms). Fresheneesz 21:16, 17 April 2006 (UTC)

The article is funny but not serious, and reflects serious mistakes in physics knowledge. The assert x=v+c is wrong. We can speak about (x,t) and (x',t') but when introducing third velocities, we need to use velocity transforms.The Lorentz transforms are valid for spacetime point transforms, not for third velocity transforms. For these cases, Lorentz transforms helps us to get the accurate 'velocity transform'. The article is not serious nor matematicamente rigorous. On the other hand, the speed of the light is not already an observation, is an international definition: 2.99798458E8 m/s.[mailto:melviso@telefonica.net Mel Viso] —Preceding unsigned comment added by 80.25.164.213 (talk) 09:47, 29 January 2008 (UTC)

Was it necessary to say, in the definitions of $$x_1$$ and $$x_2$$, that the "point" was stationary in one frame and moving relative to the other? Couldn't these be the coordinates of a point moving relative to both frames (measured relative to each frame) in which case $$dx_1/dt_1$$ is the speed of the point in one frame and $$dx_2/dt_2$$ its speed in the other? Is there any reason why one of these speeds must be zero? E4mmacro 06:42, 20 April 2006 (UTC)


 * It depends what you're measuring. If one dx/dy is 0 in one frame, then it will neccessarily not be 0 in another frame (a frame at a different velocity). Fresheneesz 09:29, 20 April 2006 (UTC)


 * I think that the definitions as written were confusing. Making the point stationary in one frame may appear to be a simplification, but gives a (false?) impression that this was necessary, that the transformations do not apply unless the point is stationary in one of the frames. E4mmacro 21:23, 20 April 2006 (UTC)


 * I agree, I did a terrible job on that. I was trying to get it all straight in my head. I've tried to think about it more, but my laziness is getting the better of me. I think it works if you consider (t,x,y,z) and the other one to be just two different coordinate systems, but for me - that doesn't help me think about it. If I could get off my ass (metephorically) and rework it, I think I would go back to saying that they're simply points in a coordinate system, but it needs to be well explained - because otherwise they're just confusing equations. Fresheneesz 11:14, 23 April 2006 (UTC)

Remove prime notation and put transforms at top
I know everyone likes to use prime notation, but noone likes reading it. We should use subscripts here. Perhaps like this with front and back subscripts: $$ {}_0 X_s $$. Also, the transforms should go at the top for reference, not burried under the "details" section. User:fresheneesz 128.111.95.147 22:30, 10 April 2006 (UTC)
 * I did it. PLEASE, someone look over it and tell me if I botched anything - because I don't know if I got it all right. Fresheneesz 21:14, 17 April 2006 (UTC)

Variable definitions
I finnally put all the variable definitions down for the SC lorentz transforms - but I would appreciate someone *checking* them to make sure I did it right. It was a very confusing process. Fresheneesz 21:17, 17 April 2006 (UTC)

time dilation and simultaneity
What exactly is this sentence from the article supposed to mean?
 * Time dilation was also used to prove that simultaneity varies between reference frames.

How was time dilation used to prove relativity of simultaneity? Is it just trying to say the Lorentz transformations incoporate both time dilation and reletivity of simultaneity? There is some related discussion in the relativity of simultaneity page. E4mmacro 06:28, 20 April 2006 (UTC)

Not cartesian but...
The first equation under "details" only holds for certain coordinate systems, but I forgot the name of them. The equivalent of Cartesian coordinate systems in 3-space. PAR 00:52, 21 April 2006 (UTC)
 * I'll bet the word is either "Lorentzian" or "Minkowskian" coordinates. -lethe talk [ +] 01:27, 21 April 2006 (UTC)

Simpler
I have been studying special relativity in college for the past semester, and I've learned these equations. That said, they looked a great deal foreign to me when I saw them on this page. In class, I worked with x, x', t, t', v, v', and B (beta). I never saw the hyperbolic transformation, and it seems to me like--although we should keep the original equations for history's sake--we ought to also provide the easier equations. After all, wouldn't it benefit the Wikicommunity at large to add a small section at the bottom with the simpler algebraic equations? -Jess V


 * They're there, but lower on the page. You would have noticed them if the "easier" equations were higher. I think they *should* be higher on the page - I had the same problem as Jess V when I first came to this page. Fresheneesz 05:28, 8 May 2006 (UTC)

Lorentz transormations vs Length contraction
I have been very confused with how the lorentz transformations translate into length contraction. I know about how the time component goes away since length contraction involves changes in time and distance, and the change in time is 0 since both events are measured at the same time (in one person's reference frame). However, it seems to me that if we used lorentz transformations we could get *both* $$L = \frac{L_0}{\gamma}$$ and $$L = \gamma L_0 \ $$. I don't understand why its one and not the other, can anybody help? Fresheneesz 05:32, 8 May 2006 (UTC)


 * I only see this now. This is a common confusion, and it would be good to explain in the article space that two distant clocks (with (t2 - t1 = 0) are read in the stationary frame for establishing two readings of position x2 and x1;
 * and that the readings of corresponding clocks in the moving system are offset due to relativity of simultaneity. The boundary condition t2 - t1 = 0 is essential. Harald88 15:07, 17 September 2006 (UTC)

Animated lorentz transformation
Image:Animated_Lorentz_Transformation.gif

I added this animation to the Links section. It's a big file so I wasn't sure it should be on the main page. Thanks, Jonathan (JDoolin, 13 May 2006)
 * Nice picture, but it is indeed too big. Maybe you could make a non-animated version with a link to the full version or change the image license to allow someone else to do so? Han-Kwang 20:51, 12 July 2006 (UTC)


 * ✅ File:Animated_Lorentz_Transformation frame 0031.gif added with link to full animation. -84user (talk) 18:58, 22 March 2010 (UTC)

Order changed to make it easier for non-experts to follow
The article started out with the section "Lorentz boost in 2 dimensions". This text would be unintelligible to a reader who didn't already understand the formulation, in that it used undefined terms and undefined mathematics (i.e., if the reader didn't already know what "the form xy" is, the article is incomprehensible.) I moved this to come second, and added "in matrix form" to clarify (possibly I should have said "tensor" or "linear algebraic form," but since tensors aren't brought up yet either...)

The word "form" should be clarified with a link to the wikipedia article bilinear form, I think.

History section
Anyone else think History should be at the top as an easy lead in to the subject?--Light current 07:23, 7 October 2006 (UTC)

Causality implies the Lorentz group
Quote:

''If space is homogeneous, then the Lorentz transformation must be a linear transformation. Also, since relativity postulates that the speed of light is the same for all observers, it must preserve the spacetime interval between any two events in Minkowski space''

However it is very common to require linearity from Lorentz transformation, really we don't have to make such strong assumptions. In an 1964 paper, E.C. Zeeman proved that the causality preserving group is equal to the Lorentz group. That is, instead of requiring linearity and isometry from the coordinate transformation between inertial observers, it is enough to require causality preservation. Linearity and isometry are mathematical consequences!

The paper: E. C. Zeeman, Journal of Mathematical Physics -- April 1964 -- Volume 5, Issue 4, pp. 490-493

This result is also found in ''Naber, Gregory L., The Geometry of Minkowski Spacetime, Springer-Verlag, New York, 1992. ISBN 0-387-97848-8 (hardcover), ISBN 0-486-43235-1''. XCelam 08:11, 25 December 2006 (UTC)

It had been proved previously to Zeeman by Andronov in Canadian J. Math. 1957 JFB80 (talk) 10:05, 9 October 2010 (UTC)

Two errors
There is a serious misinterpretation of a paper by Zeeman, E. C. in the section titled, Derivation. The error of that section is this claim:


 * "[Pal's] derivation invokes the natural, but unnecessarily strong assumption of homogeneity of spacetime. A deeper analysis [3] shows that the Lorentz-transformation is the consequence of the only requirement of keeping causality relations on spacetime."

The abstract of Zeeman's paper simply states:


 * "Causality is represented by a partial ordering on Minkowski space, and the group of all automorphisms that preserve this partial ordering is shown to be generated by the inhomogeneous Lorentz group and dilatations." ©1964 The American Institute of Physics.

In other words, Zeeman began with Minkowski space, which was already known to be a homogeneous spacetime.


 * The result linking the light-cone preserving transformations with the Lorentz transformatins is (a) not something that "just Zeeman came up with", (b) runs much deeper that some "misconception of one person" as you're painting it as, (c) has been around along time (since A. A. Robb 1915) in the literature, and (d) is a more-or-less part of the standard folklore in mathematical physics and is widely accepted, and (e) is the basis of the name &ldquo;causal group&rdquo;. The correct statement is this: in any affine space (of dimension 3 or greater), the transformations preserving a light cone structure comprise the causal group. This consists of the inhomogeneous Lorentz transformations, along with global scale change (dilations). For 4 dimensional affine spaces, this comprises a 11-parameter group. This can be generalized to projective spaces (in which case, one gets the 15-parameter conformal group).


 * The underlying geometric basis for the result was laid out back in 1915 by A. A. Robb, who showed that one can systematically reconstruct the spatial part of the space-time geometry (e.g. definition of congruence, angles, orthogonality, etc.) from the light cone structure, alone. Underlying this are such constructions as the Robb rectangle, which is used to define orthogonality between spacelike and timelike intervals (and, indirectly, spatial orthogonality).


 * Of course you're beginning in a Minkowski space (more precisely: in an affine space possessing an invariant light cone structure). That's the whole setting of the discussion in this article! Calling Zeeman misleading on this count is, itself, misleading. Nobody's talking about &ldquo;deriving&rdquo; Minkowski space (whatever that means) here; just the Lorentz transformation; and all the objections you're raising are therefore irrelevant. -- Mark July 2, 2008 —Preceding unsigned comment added by 64.24.187.116 (talk) 06:28, 2 July 2008 (UTC)

The second error is not as serious. The statement, "If space is homogeneous, then the Lorentz transformation must be a linear transformation" is only misleading. It all depends on whatever definition of the Lorentz transformation you begin with. For instance, we could adopt the opening sentence of the article as a definition. In that case, "the Lorentz transformation is a set of equations that converts back and forth between two different observers' measurements of space and time." That would be an error. The truth is, observers are free to use nonlinear clock synchronization schemes and therefore nonlinear coordinate transformations if they like. That choice wouldn't be practical but there is no law of physics against it. See exercise 1 and 2 of the reference Generalized Lorentz Transformations for a valid counterexample. --e.Shubee 17:10, 29 January 2007 (UTC)


 * Zeeman's theorem isn't a tautology. This is quite clear if we think on the fact that Zeeman's statement is valid only if the dimension of spacetime is greather then 2. In 1 space + 1 time dimension there are automorphisms of spacetime which keep the causal structure but aren't linear, consequently they aren't element of the Lorentz-group. XCelam 16:41, 30 January 2007 (UTC)


 * Can you tell me Zeeman's definition of spacetime for which his theorem is true? --e.Shubee 20:34, 31 January 2007 (UTC)


 * The "homogeneity of spacetime" in the sense of Palash B.Pal's paper means that the translation of a rigid rod does not change its length. Zemman does not make such assumptions, and what is more, in the 1+1 dimensional case (in which case Pal derives the Lorentz-transformation) isn't valid at all. It can be that the rod is translated but it's length changes in a reference frame, in spite that the coordinate transformation between the frames preserves causality relation. This means that Zeeman's condition is definitely weaker than Pal's one (beause Zeeman's condition alone does not results in the Lorentz-transformation, only together with the dimension>2 condition, while Pal's condition alone results in the Lorentz-transformation in any dimensions).


 * But you have asked me about the definition of Zeeman's spacetime. Zeeman defines the Minkowskian spacetime in terms of coordinates. This can be done whatever is the transformation rule between the observers' coordinates. Zeeman does not suppose an absolute Minkowskian spacetime with an observer-independent Minkowski-metric. Every observer defines his affine structure and his metric by means of his coordinates and there isn't an assumption that the Minkowskian or Euclidean distance is preserved during a transformation between the observers or during moving a rod. And there is no assumtion that a straight line transforms to a straight line or a uniform motion transforms to a uniform motion. The only criterium is that the trasformation preserves the causality relation. So in this case the "homogeneity of the Minkowski-space" means here nothing because every observer defines the Minkowski-form on his own coordinates and there is no predefined connection between these coordinates. In other words, this can be done independently what realy the spacetime is and whatever the transformation rules are between the observers. XCelam 07:40, 1 February 2007 (UTC)

I believe you're saying that Zeeman assumed a spacetime structure weaker than Minkowski space. Please define Zeeman spacetime. I asked if you can tell me Zeeman's definition of spacetime for which his theorem is true. That was my question. Please realize that you haven't stated a precise definition yet. --e.Shubee 12:07, 1 February 2007 (UTC)


 * No, I say that Zeeman doesn't define general spacetime structure, only each observer defines his own Minkowski-structure on his coordinates. The definition itself is the standard definition: R^4 and an nondegenerate, symmetric, indefinite bilinear form on it with index 1. XCelam 13:45, 1 February 2007 (UTC)


 * I have no clue or idea what an observer-dependent Minkowski structure means. What does it mean physically? --e.Shubee 13:54, 1 February 2007 (UTC)


 * Perhaps the same as the observer-dependent Euclidean structure :-) XCelam 14:03, 1 February 2007 (UTC)

I can't imagine what an observer-dependent Euclidean space would be if it isn't Euclidean space. If the meaning of a Zeeman-Euclidean space refers to an uncountable number of Euclidean spaces, then that's fine but what does that mean physically? I don't see the point of editors editing Wikipedia articles if they don't understand the subject. --e.Shubee 14:27, 1 February 2007 (UTC)


 * I don't understand yor problem. You can define an Euclidean metric on x,y,z,t coordinates, but this will be observer-dependent, isn't it? Can you imagine this? XCelam 14:37, 1 February 2007 (UTC)


 * Euclidean spaces have nothing to do with observers. Euclidean spaces come predefined and they all have the same metric. I can see the physical meaning of an infinite collection of Euclidean spaces and each copy representing one inertial frame of reference. That is the essential starting point of The Axiomatization of Physics - Step 1. But what defines clock time throughout all the Euclidean spaces in Zeeman's universe? There has to be some mathematical link or physical connection between the definition of time in one inertial frame and some another frame. Disjointed realities have no meaning. --e.Shubee 16:00, 1 February 2007 (UTC)

According to JSTOR, Zeeman's theorem states that the group of self-transformations of Minkowski space, preserving causality, is the orthochronous inhomogeneous Lorentz group with dilations. That's an interesting result but no one is calling it a derivation. I therefore believe that your interpretation of Zeeman's theorem is incorrect. --e.Shubee 18:39, 1 February 2007 (UTC)

Update
The paper by H. J. Borchers and G. C. Hegerfeldt titled The structure of space-time transformations generalizes Zeeman's theorem. The physics is clearly stated to be that the constancy of light alone gives us the Lorentz transformation. If I remember correctly, Vladimir Fock proved that in Appendex A of his book, Theory of Space, Time and Gravitation. The 2nd edition of that book is dated June 1964. According to one review at JSTOR, the second edition of The Theory of Space Time and Gravitation by V. Fock is very little different from the first English edition published in 1959. I'm certain that we're talking about a very old result. --e.Shubee 19:27, 1 February 2007 (UTC)

An example for a observer-dependent Minkowski-structure and for a causal automorphism which isn't linear
Quote:
 * I have no clue or idea what an observer-dependent Minkowski structure means. What does it mean physically? --e.Shubee 13:54, 1 February 2007 (UTC)

Dear Shubee, I show you an example of a possible world having 1 time and 1 space dimension, in which there isn't absolute Minkowskian, nor Euclidean structure, still each observer can define his Minkowski-form and define a "causality relation" based on it. In this fancy world, the coordinate transformation rule preserves the observer's causality relations (in spite that there isn't an invariant Minkowski structure), but isn't linear, hence it isn't a Lorentz-transformation.

This world is the following.

Each observer measures his $$x$$ and $$t$$ coordinates by his resting measuring rods and by his resting clocks. Each observer defines his Minkowski-form as $$Q(x,t) = x^2 - t^2 \,$$ and a causality relation $$(x_1,t_1)<\cdot (x_2,t_2) \,$$ defined by $$t_1 \,<t_2$$ and $$Q(x_2-x_1,t_2-t_1) = 0 \,$$ (we don't know what is the physical meaning of this form and this relation but we can't prohibit them to define anything they want.) The physicists of this world realize that the transformation rule between their $$(x,t) \,$$ coordinates and the $$(x',t') \,$$ coordinates of an observer movig with velocity $$v \,$$ with respect them is
 * $$x' = \frac{1}{2}(1+v^2)(t+x)(1+v^2(t+x)^2) - \frac{1}{2}(1+v^4)(t-x)(1+v^4(t-x)^4) \,$$,
 * $$t' = \frac{1}{2}(1+v^2)(t+x)(1+v^2(t+x)^2) + \frac{1}{2}(1+v^4)(t-x)(1+v^4(t-x)^4) \,$$.

This transformation preserves their causality relation, but has nothing to do with the Lorentz-transformation, and what is more, it isn't linear at all.

Zeeman's theorem states that such a strange world isn't possible in higher dimensions; if dim > 2 then all causality-preserving transformation is composed from a Lorentz-transformation, a translation and a dilatation (multiplication with a scalar).XCelam 12:03, 3 February 2007 (UTC)


 * You have explained that clearly. Each observer uses the quadratic form $$x^2 - t^2 \,$$ to mimic the definition of simultaneous events in Minkowski space. Causality is preserved for all observers, not because $$x^2 - t^2 \,$$ is invariant but because $$x'^2 -  t'^2 = f(x,t,v)(x^2 - t^2)\,$$. The proof then shows that all causality preserving automorphisms must be linear in higher dimensions. I still don't think that such a gimmick should be called a derivation of the Lorentz transformation unless it is called that in the literature. And you have shown why it is incorrect to say "the Lorentz-transformation is the consequence of the only requirement of keeping causality relations on spacetime." The proof doesn't presuppose a general definition of causality as that sentence implies. The proof presupposes a Minkowskian definition of causality. Why is that remarkable?


 * I also believe it's unfair to say that "[Pal's] derivation invokes the natural, but unnecessarily strong assumption of homogeneity of spacetime" but that Zeeman's theorem provides "A deeper analysis." We're talking about two completely different ways of coming up with the Lorentz transformation. Zeeman's theorem might be more difficult than Pal's derivation but this is not an instance where it is mathematically correct to say that one assumption is stronger than the other. --e.Shubee 19:14, 3 February 2007 (UTC)


 * Hmmm... So, it seems unfair. OK, I will try to reformulate this sentence. However, that is an indubitable fact that causality-preserving condition is weaker in mathematical sense that homogeneity, because the former works only together with the dim>2 condition, while the latter works in dim=2 case too. But I will try to refine my sentence. XCelam 21:30, 3 February 2007 (UTC)

Error in the article cited in section "Derivation"
The derivation in the cited Nothing but Relativity article contains a serious error.

On Page 2 equation (9) is wrong.

There stands: Suppose there is a rod placed along the x-axis such that its ends are at points x1 and x2 in the frame S, with x2 > x1. In the frame S′, the ends will be at the points X(x1, t, v) and X(x2, t, v), so that the length would be

l′ = X(x2, t, v) − X(x1, t, v).

This relation would be valid only if the simultaneity relation was independent of the frame. But it isn't in the theory of special relativity.

XCelam 02:33, 10 January 2007 (UTC)


 * I don't think it is an error. In the notation of the NbR article, the uppercase X represents a function of lowercase x, t, and v.  The uppercase function performs the Lorentz transformation from the base frame of reference to the frame of reference that is moving at relative velocity v.  Thus, by definition, this function X, which is what the article attempts to derive, provides the correct spatial location of the event in the moving frame of reference given the event's position and time in the base frame of reference.  There is no violation of simultaneity whatsoever.  First Harmonic 02:42, 10 January 2007 (UTC)


 * Please think over again. Length is defined by the distance of simultaneous events.
 * The whole transformation is:
 * x′ = X(x, t, v)
 * t′ = T(x, t, v)
 * So, the $$(x_1,t) \,$$ coordinates transforms to $$(X(x_1,t,v),T(x_1,t,v)) \,$$, while the $$(x_2,t) \,$$ coordinates to $$(X(x_2,t,v),T(x_2,t,v))\,$$.


 * If $$T(x_1,t,v)\neq T(x_2,t,v)$$ then these two events aren't simultaneous in S'. Therefore you don't get any length if you subtract the x coordinates of them. XCelam 07:52, 10 January 2007 (UTC)

Removed Eugene Shubert's original, unpublished research.
I removed the original research by Eugene Shubert because it is massively inappropriate for Wikipedia. I am making a new section on this because I anticipate much arguing between myself and Eugene over the matter.

Jowr 05:21, 9 June 2007 (UTC)


 * Good spot - I hadn't noticed its presence on this article. It is not only massively inappropriate - in the first place it is massive nonsense. He might have put this reference in other articles as well. DVdm 06:26, 9 June 2007 (UTC)

Untitled image?
The image "Animated Lorentz Transformation Frame.png" at the top of the page should have at least some description shouldn't it? --Beast of traal 21:25, 9 June 2007 (UTC)Beast of traal : I agree; also, I think a better image could be found to grab the readers attention in a more informative way. MP (talk•contribs) 21:25, 6 November 2007 (UTC)

Suggestion for derivation section
I have included a derivation show/hide box for the derivation section for primarily 2 reasons:


 * Better aesthetics.
 * There should clearly be more derivations in the article (such as some of the other standard ones), and the show/hide option is ideal for this as it won't make the article inordinately long.

As an aside, the derivation intro. should be improved (I'll try to do this myself, sometime) as there currently appears to be a discussion that goes off into a tangent. MP (talk•contribs) 21:36, 6 November 2007 (UTC)

Error in caption of GIF animation
The caption of Diagram 1 ("Views of spacetime along the world line of a rapidly accelerating observer") says: ''The lower quarter of the diagram shows the events visible to the observer. Upper quarter shows the light cone- those that will be able to see the observer''. I think this is wrong: one can not see oneself in the past. The light cone in this 1+1 dimensional Minkwoski space are the two diagonal lines. The only events that are visible to an observer in the origin are represented by the dots that pass through the diagonal lines in the lower part of the diagram. A beautiful animation, by the way. JocK (talk) 18:14, 31 December 2007 (UTC)

Comment by 66.152.149.8
In the matrix Beta must equal v/c² and gamma must equal (1-(v/c)²)^½ else the whole transformation is lost —Preceding unsigned comment added by 66.152.149.8 (talk • contribs)


 * Please note the presence of the factor c in $$c t$$ and $$c t'$$. I have reverted your edit. DVdm (talk) 14:51, 19 February 2008 (UTC)


 * Even with ct it makes it impossible to get back the transformation. if t'= gamma(t-Beta*x). in the first matrix t' will then only be =gamma(t-v*x/c) when what we need is =gamma(t-(v*x)/(c)²). And as for ct'=gamma(t-Beta*x) as stated currently on the site give t'=gamma((t/c)-(v*x)/(c)²) witch is also incorrect. Do you not agree?  —Preceding unsigned comment added by 66.152.149.8 (talk) Tjd195 (talk) 22:05, 19 February 2008 (UTC)


 * never mind I have noted my mistake —Preceding unsigned comment added by Tjd195 (talk • contribs) 22:30, 19 February 2008 (UTC)

First equation of Lorentz Transformation
Lorentz modified t’ = t – (vt/c) to get his first equation. He made t as a function of variant x by t(x) = x/c. That is fine, according to mathematical regulation of function he could change the function t’(t) to t’(t(x)) = t(x) – (v t(x)/c) or simply t’(x) = ((c-v)/c^2)x. But then that is not what he wanted. That new function t’(x) simply means to the observer at the position x in the stationed system, what will be the arrival time of the ray which was emitted from the origin of the moving system at the time t = t’ = 0 when two origins met.
 * Since that is not what he wanted, he substituted only one of the variant, the second variant, and leave the first variant as an additional variant in the original function. That means, he actually changed the function t’(t) to a new function t’(t, x) because of that mathematical error. Logically both “If A&B then C” and “IF A then C” could be true if B is not related to the reasoning. For example, both “If it is an apple and it is red then it is a fruit.” and “If it is an apple then it is a fruit.” are valid. But “If t’(t) = ((c-v)/c)t and t(x) = x/c then t’(t, x) = t – (vx/c^2).” is valid while “IF t’(t) = ((c-v)/c)t then t’(t, x) = t – (vx/c^2).” is not true.

Actually, when a man measures the actual time period t of an event happened in an inertial system which is moving toward him at constant speed v, and the measured result is t’, then t’ = ((c-v)/c)t. If the inertial system moves away from him at constant speed v, then t’ = ((c+v)/c)t. Time relationship is just so simple.

John C. Huang (talk) 18:12, 16 March 2008 (UTC)

My mistake. Here is the correction:

Lorentz derived his first equation from his two second equations of his transformation and inverse transformation. However, my reasoning provided another possible time formula.

John C. Huang (talk) 04:00, 4 April 2008 (UTC)

Vandalism in 'Matrix Form' section.
In the first matrix shown in the section in question, a fifth line states "lolololol". I'm pretty sure there shouldn't be any "lol" in a matrix... I'd change it, but I don't know how the math formatting works. mj_sklar (talk) 22:23, 7 May 2008 (UTC)

Transformation in hiperbolic form
In the section of the Lorentz tranformation in hiperbolic form there's only the matrix of transformation in the x axis. There shoud be the tranformation for a movement towards any direction. I now this matrix but I'm not familierized with mathematical scripture in wikipedia so if anyone could write it would be useful and would complete the article. I wanted also to say that the hiperbolic form (using rapidity) should be the first one in the article, beacouse it's the most powerful and the most useful one for those who work on general relativity.It should also say that the Lorentz transformation is a Tensor and put a link to the page explaining them.

--Eudald (talk) 10:13, 17 May 2008 (UTC)

On proper Lorentz transformations
In my opinion, there is a mistake in the second following sentence : '' Lorentz transformations with $$\det ({\Lambda^\mu}_\nu)=+1$$ are called proper Lorentz transformations. They consist of spatial rotations and boosts and form a subgroup of the Lorentz group. ''

Because the PT symmetry, which reverse space's and time's directions, is a element of the proper Lorentz transformations group.

Heartily. LyricV (talk) 12:01, 1 June 2008 (UTC)

Group Postulate Derivation
Firstly, calling this Group Postulate Derivation seems to be just a fancy way of saying that the transform must be linear. Is this right ? If so, then why not say that instead ?
 * Thank you Bakken for adding the Group axioms. Delaszk (talk) 07:45, 4 November 2008 (UTC)

Secondly the following statement seems to be incorrect: "Theoretically it can be either infinitely large, which gives Galilean transformation and Euclidean world with absolute time, or it can be finite, which gives Lorentz transformation and Minkowski world of special relativity."

Given two transforms with finite v and v' then the entries of the transform matrix must be finite and so the entries of the inverse matrix must be finite therefore the expression :$$ \frac{v\gamma(v)}{\delta(v)}=\frac{v'\gamma(v')}{\delta(v')} $$ must be finte therefore $$\frac{v\gamma(v)}{\delta(v)}=-c^2$$ must be finte and hence c is finite.

The Galilean transforms may be a limit of the Lorentz as c tends to infinity, but an actual value of inifinity is inconsistent with this derivation of the Lorentz transformation. Delaszk (talk) 13:20, 2 November 2008 (UTC)


 * $$\begin{pmatrix}t' \\ x'\end{pmatrix} = \begin{pmatrix}1 & 0 \\ -v & 1\end{pmatrix} \begin{pmatrix}t \\ x\end{pmatrix};

\qquad \begin{pmatrix}t \\ x\end{pmatrix} = \begin{pmatrix}1 & 0 \\ v & 1\end{pmatrix} \begin{pmatrix}t' \\ x'\end{pmatrix}.$$
 * Those matrices are the inverses of each other, and have finite entries. I'm tweaking the derivation to make the point more rigorous. (If $$\delta(v)$$ is zero and $$v\gamma(v)$$ isn't, $$\frac{v\gamma(v)}{\delta(v)}$$ isn't finite. OTOH, $$\gamma(v)$$ cannot be zero, as $$\gamma^2 + v \delta \gamma = 1$$, and so $$v\gamma(v)$$ is nonzero when $$v$$ is nonzero.)-- Army1987 (t — c) 14:48, 2 November 2008 (UTC)


 * The changes you have made to the article have inverted the numerator and denominator. v is now on the denominator which would force c to equal infinity. Delaszk (talk) 15:25, 2 November 2008 (UTC)


 * That was exactly what I intended. And they don't "force" anything as &gamma;(v) is nonzero. (In the case v = 0 you have a zero-over-zero situation, but in that case the matrix should obviously be the identity.) -- Army1987 (t — c) 15:52, 2 November 2008 (UTC)


 * Ok, but $$\frac{v\gamma(v)}{\delta(v)}=\frac{v'\gamma(v')}{\delta(v')}

$$ is a constant for any value of v, so in particular it is a constant for any value of v for which delta is nonzero, so if there exist any v for which delta is nonzero then c is finite. So c could only be infinite if delta was zero for every possible linear transform. Delaszk (talk) 16:17, 2 November 2008 (UTC)
 * In Galilean relativity, &delta; (the upper right entry in the matrix) is zero for all v, so there is no inconsistency. -- Army1987 (t — c) 16:24, 2 November 2008 (UTC)


 * So Galilean relativity may be self-consistent but only by disallowing certain transforms. Galileo was unaware that he was implicity ignoring such transforms. If you recognise the existence of transforms with nonzero delta then you get special relativity with a finite max-speed built-in. Delaszk (talk) 16:34, 2 November 2008 (UTC)


 * Indeed. &delta;(v) equals −Kv/$\sqrt{1 − Kv^{2}}$; K is 0 in Galilean relativity and 1/c2 in special relativity. (BTW, I've fixed the derivation so that it never divides by terms which could ever possibly be zero.) -- Army1987 (t — c) 17:16, 2 November 2008 (UTC)


 * I'm going to rewrite the section on Group Postulate Derivation. As the above discussion shows, if you start with a general linear transformation then you force the existence of a finite max-speed. There is no choice in the matter. A finite max-speed is a necessary consequence. The Galilean transformations are something else altogether which you get by restricting the types of motion that you are willing to accept. The fact that putting c=infinity into Lorentz transform gives you the Galilean transform is irrelevant because we've already shown that c cannot equal infinity. Changing the article to avoid talking about division, obscures this fact. So I'm going to rewrite the article based on a previous edit.Delaszk (talk) 18:04, 2 November 2008 (UTC)

← I can't understand that. By the principle of relativity you get the transformations I've shown. If K is 0 you get Galilean relativity and if it is positive you get special relativity with c = 1/$\sqrt{K}$. And there is no a priori reason why K should be nonzero – that's just an experimental fact. -- Army1987 (t — c) 18:08, 2 November 2008 (UTC)


 * If you recognise the existence of nonzero delta in the above discussion (and the derivation does assume this at the beginning by using a general linear transform in which delta can be anything) then c must be finite.
 * —This is part of a comment by Delaszk which was interrupted by the following:
 * Indeed, "anything". Which includes the zero function. -- Army1987 (t — c) 13:19, 4 November 2008 (UTC)
 * It is finite if any nonzero delta exist. You cannot let K=0 unless you also go back to the beginning of the derivation and disallow nonzero delta. The form of the Galilean and Lorentz transforms may be similar but the derivations of each have different starting points. c is a constant and has only one value - it cannot vary between finite and infinite just by taking different values of delta. The only way to change its value is to change the starting point of the whole derivation, but given the starting point that we have, then c is finite. Delaszk (talk) 18:19, 2 November 2008 (UTC)


 * We must however avoid dividing by zero which is an invalid operation, therefore we do indeed put delta in the numerator and gamma in the denominator. Gamma is never zero, so the expression is always finite. Delaszk (talk) 20:50, 2 November 2008 (UTC)


 * In fact all arguments based on the possibility of having zero on the denominator are all complete nonsense. You cannot derive any valid conclusions by having zero on the denominator. I have now rewritten the article in a way which makes it obvious that c has to be finite. Delaszk (talk) 21:24, 2 November 2008 (UTC)


 * But c doesn't have to be finite. c happens to be finite. THAT is the difference, and it is not a trivial one.Headbomb {{{sup|ταλκ}}κοντριβς – WP Physics} 03:06, 3 November 2008 (UTC)


 * If you allow frames of reference to be rotated with respect to one another (and why shouldn't you ?) then you must have nonzero delta in the upper-right element of the transform matrix whenever you have such a rotation. If you let K=0 then delta must always be zero therefore K=0 is inconsistent with allowing rotations, therefore K is not equal to zero. I suppose the consideration of rotated frames is the crux of the matter from which the finite-speed emerges. You only get K=0 and hence absolute-time if you disallow frames to be rotated. Delaszk (talk) 08:23, 3 November 2008 (UTC)


 * This is not correct, Delaszk: rotations do not affect the time coordinate. The rotation matrix has zero components with indexes [0,1],[0,2],[0,3] and [1,0],[2,0],[3,0] and a unit element [0,0]. Rotational part of the transformation is identical for Lorentz and Galilean transformation.


 * If that is not correct then there is still the matter that according to the New Scientist article about Feigenbaum's paper, special relativity is a consequence of allowing certain rotations which arise in a natural way. The references are at Temporarily reverted material pending evaluation. Delaszk (talk) 07:59, 4 November 2008 (UTC)

There is nothing wrong in dividing by zero
Delaszk, writing $$1/f(z)$$ instead of $$f(z)$$ does not change a thing. The physical meaning is completely the same. The singularities, if genuine, reveal certain physical meaning. You can't avoid a genuine singularity in a physical theory by mathematical transformations.

For example: the elements of the Lorentz matrix diverge at $$v\to c$$. This is a genuine singularity which prevents transformations with relative velocity larger, than the speed of light. It makes the Lorentz group non-compact with all the consequences. Bakken (talk) 10:13, 3 November 2008 (UTC)


 * That argument is a red herring. Yes $$v\to c$$ is indeed a genuine singularity but that has nothing to do with dividing by zero. Dividing by zero is wrong, especially since it only occurred here by mistake, and has nothing to do with any genuine singularity. What is really going on here is this: Calculating the product of two transformation matrices, one with $$v$$ the other with $$v'$$ and comparing the diagonal elements and cancelling like terms gives: $${\delta(v)}{v'\gamma(v')}={\delta(v')}{v\gamma(v)}$$. The only question is where do you go after this equation. Dividing through by zero is certainly not the answer as this is invalid and produces contradictions. Instead you divide through by the gamma terms which are nonzero and therefore valid to divide by.Delaszk (talk) 11:36, 3 November 2008 (UTC)


 * you seem to believe that $$a=b$$ and $$1/a=1/b$$ are very different things. They are actually the same. See, if, say, $$a=0$$ then the left-hand term of the second equation is infinitely large. The equation can then only be satisfied when the right-hand side is also infinitely large. That means the $$b=0$$. See, the second equation has the same meaning as the first equation even in the case where $$a=b=0$$. Trust me, there is nothing wrong in dividing by zero. In physics we do it quite often. Bakken (talk) 12:33, 3 November 2008 (UTC).


 * I can't believe I'm reading this. It is a fact that a = b and 1/a = 1/b are very different things. The first equation says something about two numbers. The second equation says something about two non-zero numbers. If, say, a = 0, then the second equation is invalid. You can't even say something about its left-hand term to begin with. See, the second equation has NO meaning in the case where a = b = 0. Trust me, everything is wrong in dividing by zero. You are not talking about physics. At best you are talking about laziness and carelessness. DVdm (talk) 12:47, 3 November 2008 (UTC)


 * :) think about the second equation in this way: $$1/(a+i\epsilon)=1/(b+i\epsilon)$$ where $$\epsilon$$ is a small number. Then it has a perfect meaning for $$a=b=0$$, does it not? See, very often the integral of a function converges, despite the fact that the function itself diverges. Many Green's function diverge, the Coulomb cross-section diverges, the classical potential energy of the electron diverges, the S-matrix diverges at eigenstates of the Hamiltonian. I am sorry, but division by zero simply means that a function has a pole at this point. Many functions have poles. Bakken (talk) 13:03, 3 November 2008 (UTC)


 * Sure, you can think about it as you like, and I'm sure that every mathematician does it, but that does not mean we can write it that way. Yes, many functions have poles. These are cases where a function has no value, since division by zero is simply not allowed. And yes, there's a lot of divergence around, and it's all properly covered with the clean and decent mathematics of limits. I notice that your most recent change says "When v --> 0, ...". That is just about barely acceptable. Had you written "When v = 0, ...", it would have been totally unacceptable. Anyway, I have made another little change to remove the predicate "barely". Cheers, DVdm (talk) 13:24, 3 November 2008 (UTC)


 * That's all right with me, to write $$v\to0$$... :) See, division by zero is simply "not defined". There is no such thing as "not allowed" in physics, I don't think so. But I agree that division by zero has to be treated carefully no matter how obvious it might seem to somebody... Cheers, Bakken (talk) 14:00, 3 November 2008 (UTC)


 * Consider two analytical functions, $$f(z)$$ and $$g(z)$$, of a complex variable $$z$$, possibly with zeros and poles. Can you tell me, what is the difference between $$f(z)=g(z)$$ and $$1/f(z)=1/g(z)$$? Bakken (talk) 14:04, 3 November 2008 (UTC)


 * For z for which f(z)=0 or g(z)=0, the string "1/f(z)=1/g(z)" represents nonsense and is, in proper lingo, "not even wrong" :-) DVdm (talk) 17:55, 3 November 2008 (UTC)


 * :) well, I can just as well say that for $$z$$ for which $$f(z)$$ has a pole, the string $$f(z)=g(z)$$ "represents nonsense". What's the principal difference? :) If $$f(z)$$ is an analytical function, $$f(z)$$ and $$1/f(z)$$ contain the same information. You can't say one is any better than the other. They are equivalent. Bakken (talk) 18:19, 3 November 2008 (UTC)


 * For z for which f has a pole, there is no such thing as f(z). Your phrase "for z for which f(z) has a pole" is nonsense before you can even take breath to finish your sentence. Also note that "The reciprocal of an analytic function that is nowhere zero is analytic". DVdm (talk) 18:34, 3 November 2008 (UTC)


 * I am afraid the discussion descends into tautology. I customarily use phrases like "the analytical function $$1/(1+z^2)$$ has poles at $$z=\pm i$$" and everybody so far understood me without problems. If you insist on a more precise language, be my guest, just correct whatever sentences you deem ambiguous. Bakken (talk) 18:47, 3 November 2008 (UTC)


 * Yes, I agree. You customarily use "lazy and careless" language. I try to avoid it. But as I already said, I can live with the current still somewhat sloppy but sufficiently acceptable wording. Cheers, DVdm (talk) 18:56, 3 November 2008 (UTC)

If v=0 then the equation :$$ \frac{\delta(v)}{v\gamma(v)}=\frac{\delta(v')}{v'\gamma(v')} $$ cannot be derived from the equation $${\delta(v)}{v'\gamma(v')}={\delta(v')}{v\gamma(v)}$$

It makes no sense to say that the first equation is valid for all v. It is only valid for nonzero v. Saying it is valid for $$v\to0$$ still doesn't let you say the equation is valid for v=0. Delaszk (talk) 14:58, 3 November 2008 (UTC)


 * Dear Delaszk, let's say we have different opinions on that: you cannot divide by zero, while I can. I suggest we make both opinions visible, by saying that user:Delaszk believes the equation is not valid in the limit $$v\to0$$. Be my guest, write it down, but don't delete context you disagree with or do not understand -- it's vandalism. Bakken (talk) 16:46, 3 November 2008 (UTC)


 * I did not say it was invalid in the limit. I said it is not valid for v=0 and you cannot argue that it is. You cannot ever in any context divide by zero. You might be using the phrase "divide by zero" to mean something else, but I mean the phrase "divide by zero" to mean divide by zero. You also accuse me of vandalism for deleting content I disagree with when you deleted the content about nonzero v because you disagreed with it. So you are criticising me for doing exactly what you did. Delaszk (talk) 17:40, 3 November 2008 (UTC)


 * I did not delete it, I have rewritten it to make it correct. You can see it on the byte count of the article. If you insist on claiming that the equation is not valid for the identical transformation, please, write it down. Bakken (talk) 18:14, 3 November 2008 (UTC)


 * You are playing with words to avoid being seen as a hypocrite. I wrote stuff about nonzero v. You changed it. I changed it back. You accused me of vandalism for doing this. You need to explain why it is ok to divide by zero, otherwise I will revert it again. Delaszk (talk) 18:25, 3 November 2008 (UTC)


 * I don't understand: your phrase "This is valid for nonzero v since γ(v) is always nonzero, as γ2 + vδγ = 1" is not deleted, it is there in the article. What are you talking about? Bakken (talk) 21:20, 3 November 2008 (UTC).


 * First of all, I admit that there can be contexts in which division by zero is possible, but I'm not sure that this article is such a situation. I don't think that using limits can be described as being the same thing as dividing by zero.


 * Now in answer to your question: The equation :$$\gamma^2+v\delta\gamma=1. \,$$ is derived about half way down the section where it says "Thus, γ( − v) = γ(v) and comparing the two matrices, we get".
 * Putting gamma=0 into :$$\gamma^2+v\delta\gamma=1. \,$$ gives 0=1 which is absurd therefore gamma cannot be zero. Now since I was concerned about not allowing zero on the denominator of :$$

\frac{\delta(v)}{v\gamma(v)}=\frac{\delta(v')}{v'\gamma(v')} $$I don't need to worry about gamma because I've just shown that it is nonzero. The only other factor to worry about that could make the denominator zero is if v=0. So I don't want v=0 because then the equation doesn't exist, but for all values of v for which the equation exists we don't need to worry about gamma becoming zero as it can't, therefore for all these nonzero values of v the expression is a well-defined finite constant and not some undefined thing like 0/0. Delaszk (talk) 20:40, 3 November 2008 (UTC)


 * well, in my language the equation exists all right for $$v=0$$, as in my language the functions $$f(z)$$ and $$1/f(z)$$ carry exactly the same information. But I have realized now that it is rather offensive for you. And you know what, if it really is so important for you, just do it -- consider separately $$v\ne0$$ and $$v=0$$ and I won't object. I am sorry, I didn't realise division by zero can be so a touchy issue. I always divided by zero if needed and never got into problems so far. I am sorry for my insensitive stubbornness. Bakken (talk) 21:18, 3 November 2008 (UTC).


 * It's not a question of offence, if we were to define division by zero to be infinity then we would have K is infinite and therefore c=0, and nobody wants lightspeed=0 in either the Lorentz or the Galilean transformations. Delaszk (talk) 08:43, 4 November 2008 (UTC)


 * that would be correct if the nominator were finite. But our nominator itself vanishes at $$v=0$$ such that we have a well defined limit, $$-1/c^2$$ at $$v\to0$$. And that is because our equation is trivially satisfied (due to the identity element axiom, if you like) for identical transformations even before you have divided it by $$v\gamma$$. My point is -- if an equation is already satisfied, it will still be satisfied after whatever identical mathematical transformations you might apply to it. But all right, folk, I surrender: just go ahead and litter the text with "careful there when dividing by zero!" stuff :)) Bakken (talk) 08:59, 4 November 2008 (UTC)]

Look, this is what happens with division by zero. You can't divide by the number zero. Ever. You can divide by a function which may equal zero at an isolated point, and you can even make a statement abut the quotient being "infinitely large" at that point. The reason is the context: if f is an analytic function with an (isolated) zero at a, then g = 1/f is a meromorphic function with an (isolated) pole at a, and if you do the necessary analysis you find that the values of g(z) uniformly approach infinity (in magnitude) as z approaches a. However, this statement about infinity is a statement about the function g, not the "value" g(a). That value is undefined, but it it is embedded into a context where it makes sense to discuss it being infinite. I'm being very careful to distinguish the functional notation g from the value notation g(z), despite the way it's often presented in books: f is a mathematical object defined relative to a certain open set of complex numbers, and assigns to each of them a value f(z); if we remove from that set all the points, such as a, at which f has value 0, then the resulting object can be inverted to form g, which is defined on a different set than f. It cannot be extended, at least not as a complex-valued function. But it can be discussed as a single object, and that includes discussion of its limits at points outside its domain, if those limits exist.

Every argument that Bakken presents in favor of dividing by zero falls into the category of "it's okay to divide by zero if this is done in the limit as part of a whole-function computation", and every counter-argument given here falls into the category of "the function may not be defined at the limit". You're both right, but unfortunately for Bakken, there is no circumstance under which it is acceptable (mathematically) to omit the phrase "assuming that the denominator is nonzero", even in a discussion of limits. Since in the formula under consideration, there is only one value of v (namely zero) when the denominator vanishes, it is prudent simply to consider that case separately; my mathematical experience shows that this usually results in a more meaningful statement anyway, these extremal cases being of independent interest. Ryan Reich (talk) 00:29, 4 November 2008 (UTC)


 * all right, I agree that there are cases where it is prudent to be careful when the denominator vanishes; and although in my view our case is non one of them, I agree to adding phrases like "assuming denominator is nonzero" and considering the case $$v=0$$ separately. I just thought that since it does not change a thing in the end -- why bother the reader with some unnecessary formalities? Our $$v=0$$ is identical transformation -- clearly, there can be no genuine singularities for identical transformation... :) (Bakken (talk) 08:45, 4 November 2008 (UTC))


 * Surely it is perfectly acceptable to have undefined points in a function, or define division by zero as the limit of a/b as b->0, as is done in most treatments of physics and calculus? –OrangeDog (talk • edits) 02:45, 10 February 2009 (UTC)

Inline math rendering
The current version of the revision is okay IMO. I'd just call the axis of relative motion x as it's customary rather than z, using (x, t) rather than (t, x) so that the entries of the matrix are in alphabetic order (I think people oppose renaming them because now gamma has its common meaning), and avoid TeX formulas rendering to PNG when used inline, replacing them with WikiMarkup/HTML, using math tags only for formulas indented in their own line. (The number I called K now is called −κ, but I have no preference about which is better – both conventions make sense, for different reasons.) -- Army1987 (t — c) 13:15, 4 November 2008 (UTC)


 * Unfortunately, there is no standard notation and different textbooks use different notations. Most of the modern textbooks, in my opinion, use the notation $$x^\mu=(t,x,y,z)$$ where $$\mu=0,1,2,3$$ and the Minkowski metric tensor with the signature $$(+1,-1,-1,-1)$$. That's why $$t$$ goes first, because it has index $$0$$. And actually usually it is the $$z$$ (or third) axis which is chosen as the odd one out, e.g.: the $$\sigma_z$$ is usually chosen diagonal, the spherical angle $$\theta$$ is measured from the $$z$$ axis, the third component of the $$W$$ field combines with the $$B$$ field to make the photon. Bakken (talk) 13:52, 4 November 2008 (UTC)


 * And I believe the math tags are very important for future post-processing as they convey contextual information. If we change it to HTML we loose this information and somebody in the future will have to restore the math tags. Bakken (talk) 13:52, 4 November 2008 (UTC)
 * As for the reason for swapping t with z, that assumption was mistaken, the entries wouldn't be in the alpha-beta-gamma-delta with that ordering, either, so, nevermind. As for x vs z, z is usually chosen for an arbitrary axis of rotation, but x is used for translations—see Lorentz transformation. As for the inline math tags, if you want to keep them, at least remove the \, which force PNG rendering for formulas used inline. (Also, progress in the processing of math tags has been stalled for as long as I can remember—see WT:MOSMATH and elsewhere.) -- Army1987 (t — c) 14:01, 4 November 2008 (UTC)


 * All right, I won't object to change all $$(t,z)$$ to $$(t,x)$$ if you believe it is the right thing to do. Concerning in-line maths -- my browser automatically uses HTML whenever it can, and only on few occasions resorts to png rendering. I use Firefox on Ubuntu. I'm afraid we can't avoid math tags writing mathematics. What is your browser? Bakken (talk) 15:38, 4 November 2008 (UTC)
 * It is not a matter of browsers, they are rendered by the WP parser, which convert them either to PNG pictures or to HTML. You can choose how in Special:Preferences. As for me, I have it set to "Recommended for modern browsers" (the default). In that setting, using the  for the small space forces any formula to be rendered as an image, but that's only useful for formulas in their own lines. Probably you have "HTML if possible or else PNG", in which   doesn't force PNG but   does.
 * (Also, I fail to understand why, for $$\mathbf{F}=m\mathbf{a}$$, the rendering "[r]ecommended for modern browsers" is a picture. It doesn't even share the baseline (typography) with the surrounding text. Wouldn't F = m a be good enough? (Ideally, the styles would be in the stylesheet declarations for the classes, but I am dreaming.) But that's another matter...) -- Army1987 (t — c) 10:29, 5 November 2008 (UTC)


 * all right, I will not use "\," or "\!" or any other formatting code in inline maths. Bakken (talk) 14:20, 5 November 2008 (UTC)

We need more citations and less toying around with derivations
We need to proved proper citations for the derivations in this article. For example when the article says "following a classical derivation..." this should be followed by a cite doing precisely such a derivation. This avoids a lot of bickering since then it is quite clear what derivation should be there in the first place. (TimothyRias (talk) 10:57, 3 November 2008 (UTC))


 * Yes. Have a look at the recent fuss on a mere kinetic energy subsection of special relativity in "Removed unneeded second derivation...", and perhaps "Propose removal of paragraph...". No sources. Just algebra and calculus. DVdm (talk) 11:54, 3 November 2008 (UTC)


 * I have added a citation. :) Bakken (talk) 12:20, 3 November 2008 (UTC)

Do we really need $$\kappa$$ instead of $$-1/c^2$$?
Any good reason to introduce yet another letter, $$\kappa$$, for the universal constant $$-1/c^2$$? It only obscures the physical meaning of the constant. And according to Occam's razor principle you don't introduce extra letters unless absolutely needed. So if nobody objects I will change $$\kappa\to-1/c^2$$, OK?. Bakken (talk) 22:42, 4 November 2008 (UTC)


 * Yes, there is a reason. c denotes the speed of light and since we all know where this derivation is going we use it in the derivation of the Lorentz transformation where c then becomes a maximum speed limit and via Maxwell is then identified with the speed of light, whereas the Galilean transformations say nothing about any speed limit, much less anything about light. For the Galilean, $$\kappa=0$$ and that's where the derivation ends. There's no place for c in the Galilean transformations. Delaszk (talk) 21:56, 4 November 2008 (UTC)


 * well, I'd say that this is precisely the point: Galilean transformation is obtained from the Lorentz transformation in the limit of slow motion, $$v\ll c$$ or, equivalently, in the limit when the speed of light is much larger than the velocities involved, $$c\to\infty$$. From the Galilean point of view the speed of light is infinitely large. So it is very intuitive to have $$1/c^2\to0$$ for Galilean transformation -- it is simply the limit of every-day velocities for which the light is just as good as infinitely fast. Bakken (talk) 22:42, 4 November 2008 (UTC)


 * :) any other opinion? Bakken (talk) 22:42, 4 November 2008 (UTC)


 * True, c in the Galilean is useful to show that it is a limit of the Lorentz, but it also misleadingly gives the impression that the Galilean says the speed of light is also infinite. Of course this is not so. A finite speed of light that behaves like ordinary speeds by changing depending on the observer is perfectly consistent with the Galilean.Delaszk (talk) 23:08, 4 November 2008 (UTC)


 * No, Delaszk, the light, as a massless object, *has* to travel with infinite velocity because otherwise it won't exist at all, as it will have no energy. Indeed in the Galilean world the energy of an object is $$E=mv^2/2$$, so if you have $$m=0$$ you need to travel with $$v=\infty$$ to have finite energy. In Lorentz world massless objects travel with the maximal possible speed. That is why the speed of light, or any other massless object, is identical to the maximal possible speed. Bakken (talk) 13:10, 6 November 2008 (UTC)


 * Newton assumed a finite speed of light made up of particles. He did not accept instantaneous transmission of light. - He did not know about the masslessness of light. The Galilean transformations on their own say nothing about light and without a priori knowledge of the massless nature of light then a finite speed is consistent with the Galilean.Delaszk (talk) 13:36, 6 November 2008 (UTC)


 * that is true, without the knowledge of the nature of light you cannot assume it moves with maximal speed... But why do we have to pretend we don't know the nature of light? :) Bakken (talk) 13:44, 6 November 2008 (UTC)


 * We don't. But the principle of relativity does not logically imply that light is massless. IOW, "The principle of relativity is true but light is not massless" is false, but I don't think it is logically inconsistent. -- Army1987 (t — c) 18:17, 7 November 2008 (UTC)


 * No different opinion here. The point is to keep light speed (and thus the symbol c) out of the picture as long as possible.
 * What you could do, is introducing another variable V (with the property kappa = -1/V^2) and call it "the maximum speed". And then, at the very end, you can postulate that there is indeed something in nature that can attain this maximum speed, namely light, i.o.w. postulate that V = c. DVdm (talk) 23:22, 4 November 2008 (UTC)
 * Well, I think that the use of c for the speed of light in vacuum became widespread after special relativity was accepted. (Einstein used V in his 1905 German article, but the 1923 English translation uses c), so, if you want to keep them distinct, calling them the other way round might make more sense. But I think it's better to use &kappa; because &minus;1/c2 might give the wrong impression (that it cannot possibly be zero), which Delaszk had got. -- Army1987 (t — c) 12:19, 6 November 2008 (UTC)


 * &minus;1/c2 might give the wrong impression (that it cannot possibly be zero), which Delaszk had got -- oh, *that* was the problem! I though it is obvious that 299792458 m/s is just as good as infinitely fast for all pedestrians on earth, and (a typical household velocity)^2/(299792458m/s)^2 is just as good as zero (unless you use atomic clock precision measurements).


 * I think there is something wrong with what mathematicians teach their students about infinities and division by zero. Infinity is simply something much larger then (and zero is something much smaller then) the stuff you are concerned with. If you divide by something very small you get something very large. :) Why is it a taboo for mathematicians? Bakken (talk) 13:01, 6 November 2008 (UTC)


 * I had a rather long reply to this question. When I tried to save the page, I got an edit-conflict due to Delaszk's reply, and I could not retrieve my text - page expired. I'm not going to re-type it. I'll just say that "I think there is something wrong with" people who think that physics is an application of sloppy mathematics and logic. DVdm (talk) 14:06, 6 November 2008 (UTC)


 * well, massless things, like light, gravity, neutrinos (if massless) always travel with maximum speed because otherwise they would have no energy and thus would not exist in the physical world. Therefore the speed of light (or any other massless object) is obviously identical to the maximum speed (be it infinitely large or finite) even before we started our derivation. Therefore if there is maximal speed in your theory the light will travel with that speed. If there is no maximal speed in your theory the light will be infinitely fast. That's why your mangling with $$\kappa$$ and maximum speed and then separately speed of light seems to me strange and distracting from the physical meaning of $$c$$. Bakken (talk) 13:20, 6 November 2008 (UTC)


 * This derivation does not posit the masslessness of light. That would be extra information which is not necessary to derive the transformation matrices. So let's keep both c and k. Delaszk (talk) 13:49, 6 November 2008 (UTC)


 * Indeed. It looks like Bakken still doesn't get the point. This looks like confusing physics with logic. DVdm (talk) 14:08, 6 November 2008 (UTC)


 * Ok, but nothing wrong with a bit of friendly rivalry between different disciplines. :-) And all this discussion helps us to locate what the issues with the article are, and has resulted in improvements to the article. Delaszk (talk) 14:18, 6 November 2008 (UTC)


 * Ah well, perhaps I got carried away by that question. Sorry, Bakken.
 * Anyway, I have created a little extract (pdf) of the section "Relativity without c" of the once freely downloadable version of David Morrin's text book and left it on my site. I think that fair use applies here. If not, I will remove it soon. It illustrates "the point". DVdm (talk) 14:40, 6 November 2008 (UTC)


 * :) I find it amusing, folk: you are redefining $$a=1/b$$ in the middle of derivation which obviously does not change a thing in the theory (except for introducing an additional letter) and then discuss it at length and prove that at the end it indeed did not change a thing and than get proud of this discussion. :)) Bakken (talk) 15:11, 6 November 2008 (UTC)

Derivation edit
I editted a small section of the derivation (transformation matrices consistent with group axioms). Just added in a few steps and reworded the explanation a little as I found it a bit difficult to follow. Hopefully it is a little easier now. It is to me anyway. It does add in more math to the article, but I don't think that it is too much. Others may disagree, if so, go ahead and edit, but I think its better.60.230.206.243 (talk) 10:25, 14 February 2009 (UTC)

"basketball" metaphor
One or two editors seem fond of this language, while others (including myself) have objected:


 * Imagine a four-dimensional basketball spinning on a four-dimensional finger. In physics, the Lorentz transformation can be used to describe such a four-dimensional rotation. It is important because Albert Einstein used it in this way in developing his theory of special relativity, uniting space and time in a single four-dimensional system.

I think this confuses more than it illuminates. Most readers are not going to understand why translations between different observers in spacetime should be thought of as a "rotation", or what the "four-dimensional basketball" is supposed to represent. And they might easily be confused into thinking it is about curvature of spacetime, and/or they could easily confuse the "spinning" of the ball in the analogy with the physical motion or frames of reference of moving observers which has already been mentioned (which is especially misleading because rotational motion is not an inertial frame). Also, Lorentz transformations are not the four-dimensional generalization of rotations in the usual sense, in that they are not unitary (and the norm that the Lorentz group preserves is not a true norm since it is indefinite), so that is another reason to be wary of this metaphor.

And I find it rather ridiculous to drag Einstein into this. Einstein gave a number of well-chosen thought experiments about putative physical observers in different reference frames, but he made no analogies about spinning basketballs to my knowledge (the citation claimed by the anon editor to Relativity: The Special and General Theory  was simply false). Just because Einstein talked about some physical objects does not automatically justify any possible metaphor, and putting words into Einstein's mouth is mere sophistry.

I have no objection to a non-technical lede — I added a first stab along those lines recently — but it should emphasize the underlying ideas and impact of Lorentz transformations on the relativity of space and time for different observers, not invite unsuspecting readers to form evocative but misleading pictures.

— Steven G. Johnson (talk) 16:23, 27 July 2009 (UTC)


 * That's one editor (the anonymous edit was mine, sorry). The reason to drag Einstein into this is because, to the general public, the importance of the Lorentz transformation is due to Einstein's use of it in special relativity. Wikipedia is for a general audience, so I think the connection should be made up front.


 * Regarding Einstein and basketballs: you are reading me suspiciously. The Relativity reference was to the "rotation in four dimensions" language, not (of course) to basketballs. I accept that invoking Einstein "does not automatically justify any possible metaphor." I took your barb to be against metaphor generally, hence my response.


 * So, if we can agree that metaphor in itself is not out of the question, then we reach the question whether this metaphor is helpful or not. That in turn hinges on whether saying "The Lorentz transformation can be thought of as a four-dimensional rotation" is the most helpful starting point for a general audience. In suggesting that it is, I would point to the way that this language emphasizes the unity of space and time in a single system, which is the central insight of special relativity. My assumption is that a majority of readers of this article are looking for a quick handle on the Lorentz transformation and its broader significance; I see myself as advocating for a simple one-liner for those that want it. Such simplicity comes at the cost of nuance, certainly. But that's what the rest of the article is for.


 * If you knew someone was going to take only one short sentence away from this article, what would you want that sentence to be?


 * Chad Whitacre (talk) 17:31, 27 July 2009 (UTC)


 * If I could have the reader remember one short sentence, it would be something like: Lorentz transformations precisely describe how, according to the theory of relativity, relationships of distance and time between events vary for observers moving at different velocities. To me, that is far more likely to give an accurate impression than naive readers trying to imagine spinning four-dimensional basketballs.


 * I didn't say Einstein should not be mentioned in the article, but rather that Einstein's name should not be invoked to support arguments or metaphors Einstein did not make.


 * It's true that Einstein and others draw a formal mathematical analogy between Lorentz transformations and rotations. But, as I explained, stating this analogy as the first, most basic, central point of the article (the "one thing" a reader should take away), is likely to lead people astray, because Lorentz transformations are not the four-dimensional generalization of rotations in the usual sense of the word, nor are they rotations at all in the technical criteria of being unitary.  This is an analogy whose appreciation requires sophistication and an understanding of the caveats.  (Even Einstein does not make the analogy until Appendix II of his book, undercutting your implication that he saw it as a central message.)


 * Furthermore, you didn't rest at calling them four-dimensional rotations. You invoked an image of a four-dimensional spinning basketball, bringing the specter of curved manifolds (ala GR) and rotational motion (acceleration, also ala GR) into play.  This is highly likely to give a misleading impression, as I argued above.


 * — Steven G. Johnson (talk) 17:57, 27 July 2009 (UTC)


 * I accept your arguments. I have reworked the first paragraph to begin with a sentence formed both from your language above and the original "conversion between measurements" language (since this gets picked up again in what is now paragraph 2). The new historical note stands.


 * How does that look to you?


 * Chad Whitacre (talk) 20:35, 27 July 2009 (UTC)


 * Looks okay, but I added a second sentence emphasizing the surprising nature of the transformation for observers at different velocities, which I think is necessary to put up-front. — Steven G. Johnson (talk) 21:53, 27 July 2009 (UTC)


 * Yeah, I couldn't find a good way to fit the notion of velocity into the first sentence without bloating it. I agree it needs to be there, and I like your solution. Chad Whitacre (talk) 22:48, 27 July 2009 (UTC)

Four dimensional Rotation???
I for one would be an example of a less sophisticated reader trying to understand this material.

I can understand how a three by three matrix can be used to rotate a vector. I can understand that a three by three rotation matrix is orthogonal which means to me that the dot product of any two different rows or columns of the matrix with each other is zero. I can understand that the dot product of a row with itself is equal to one. To those of us with limited education this is the meaning of orthogonal.

I tried to multiply two rows of the boost matrix, calculating the dot product but the answer I got did not really look like it worked out to zero, so I am having a hard time understanding what is meant my 0(1,3). Is this because I am computing the dot product of two rows incorrectly, because I am not inserting a metric in between the two row vectors? If this is what I am doing wrong, it might help to explain this?

This concept of four dimensional orthogonality is clear as mud to me, but I don't think it really needs to be. I can come up with some good source material on orthogonality conditions, that it would seem to belong in this article? But I am having a hard time understanding what these references are talking about. If there is some parallel between a 3 X 3 orthogonal matrix, and a 4 x 4 O(1,3) matrix, it would be very helpful to explain this in plain English, as simply as possible. A two by two matrix that rotates something has cos(theta) along the main diagonal, but the other two terms are the opposite of each other, ie sin(theta) and -sin(theta), but a boost does not seem to have this opposite sine property for the matrix elements off the main diagonal that are supposed to rotate around the time axis?

So if this orthogonality idea could be explained in plane English, as a four dimensional analog to rotation in three dimensions answering questions like these might be a good place to start? —Preceding unsigned comment added by StressTensor (talk • contribs) 23:53, 14 December 2009 (UTC)

StressTensor (talk) 23:54, 14 December 2009 (UTC)


 * It is only orthogonal if you use the complex Minkowski representation with (√-1 ct, x, y,z) instead of (ct, x, y,z). Otherwise you need to put in a few minus signs.JFB80 (talk) 10:08, 17 November 2010 (UTC)


 * Basically, yes, you must "insert a metric between the two row vectors". The dot product of two vectors is written in index notation as $$A_\mu B^\mu$$ (implied sum over &mu;). Simply multiplying the two covariant expressions gives $$A_\mu B_\mu$$ which is not coordinate-free. If g is the metric tensor, then $$g^{\mu\nu} B_\nu=B^\mu$$ and you can write the dot product $$A_\mu g^{\mu\nu} B_\nu$$. You could say that this is the same process used in Euclidean space, but since the metric tensor is the identity tensor, it is ignored, and the contravariant/covariant distinction is never needed. In spacetime, the metric tensor is not the identity tensor, so you cannot drop the distinction. Orthogonality is defined as the dot product (as defined above) being zero. I think you will get the usual cosine and sine terms if you do it this way, but one or the other becomes a hyperbolic sine or cosine. Also, since this is physics, it is better to think of a 4-vector as a single mathematical object rather than an array of real numbers, and the row or column array is then a list of the components of that vector in a particular coordinate system. In other words, a "row vector" does not represent a physical vector unless a particular coordinate system is specified. PAR (talk) 10:17, 19 November 2010 (UTC)

The change from LaTex to HTML
I think it was not very smart to change from LaTeX to HTML because the contextual information is now lost. I'll change it back unless you can convince me that HTML is better for math notation. Bakken (talk) 00:37, 12 December 2009 (UTC)


 * I change back to LaTeX notation... Bakken (talk) 10:25, 14 December 2009 (UTC)

Primes misplaced?
Consider the following sentence from the article:

"Now suppose that we make a coordinate transformation xμ → x'μ"

The above text is trying to show a transformed general coordinate. But shouldn't the prime follow the μ rather than the x?

Reason: For any given value of μ in 1..4, xμ represents one of the four ordinary (that is, scalar) coordinates which, when combined, constitute the 4-vector X. We could logically write X' because that would refer to a transformed vector, or maybe more accurately to a vector as seen from a transformed coordinate system.

But the symbol xμ represents one of the following: x, y, z or t. Therefore the notation xμ is logically atomic and should not be interrupted by a prime. Rather, (I think) the prime should be appended to the whole symbol, as follows: xμ ', which seems unambiguous as we would never actually want to add a prime to μ itself. (What would that mean? A different component of X?) Dratman (talk) 17:38, 12 December 2009 (UTC)

Lorentz equation for any dimensions in easier, better explanable format?
As such: :$$ \begin{cases}x'_0 = \gamma (x_0 - \mathbf{\beta} \cdot \mathbf{x})\\ \mathbf{x'}= x + \frac{(\gamma -1)}{\beta^2}\mathbf{\beta} \cdot \mathbf{x} \mathbf{\beta} - \gamma \mathbf{\beta} x_0 \end{cases}$$

This is basically separating it into along the speed and the rest. (Note: mathbf doesnt seem to have much effect on beta..82.169.255.79 (talk) 15:44, 11 February 2010 (UTC)

Group properties
In 1905 Henri Poincaré was the first to recognize that the transformation has the properties of a mathematical group

Shouldn't this really say that the set of all Lorentz transformation has the properties of a group? How can one transformation have the properties of a group? 71.178.57.209 (talk) 22:43, 8 April 2010 (UTC)


 * Sure, it's all yours, go ahead. DVdm (talk) 07:26, 9 April 2010 (UTC)

Apparently
Why does it say (italics mine) "Indeed the four group axioms are apparently satisfied" ? RJFJR (talk) 16:37, 14 April 2010 (UTC)


 * "Apparently" as in "as can be seen". We don't need it. This "Lazy Textbook Proof Word" appears 4 times in that subsection. I have removed them all. Good catch. DVdm (talk) 17:07, 14 April 2010 (UTC)

Reworded to fix logical ambiguity
The problem with this sentence
 * "...observers moving at different velocities report different distances, passage of time, and in some cases even different orderings of events"

is that it can be misconstrued to mean that distances and times are in all cases measured differently, which is false. Simply removing the phrase "in some cases" also leads to an ambiguity (whether it applies in all situations or a subset). I therefore removed the prepositional phrase and also added the conditional "may" which applies equally to distance, time, and ordering. Tim Shuba (talk) 19:38, 26 April 2010 (UTC)


 * Yes, that's excellent. Good one. DVdm (talk) 20:40, 26 April 2010 (UTC)

Link to Animated Space-Time Diagram
I would like to add the following text to the list of external links:


 * Special Relativity Space-Time Applet from Jonathan Doolin. A space-time palette where the user can draw events onto a space-time canvas, and perform the Lorentz Transformation by dragging the events along hyperbolic arcs in space-time.

This addition has been blocked, citing WP:EL. However, the policy says, "acceptable links include those that contain further research that is accurate and on-topic." Therefore, under the policy, this web-page is an acceptable link. —Preceding unsigned comment added by JDoolin (talk • contribs) 12:41, 3 May 2010 (UTC)


 * Jonathan, did you also have a look at WP:NOR and WP:COI? DVdm (talk) 12:59, 3 May 2010 (UTC)

Yes, and I don't see any problems. You have to be less ambiguous about what the problem is. (talk) 19:26, 3 May 2010 (UTC))

Hi again, Dirk. You've cited Wikipedia policy articles on external links, conflict of interest, and no original research. Please explain how my link violates any of these policies, or stop removing it. (JDoolin (talk) 20:35, 4 May 2010 (UTC))


 * You don't link to your own work, unless that work happens to be cited by reliable secondary sources. I have reverted your edit again. Please leave it alone. Next time you will be blocked. See a more formal warning on your talk page. Sorry. DVdm (talk) 21:52, 4 May 2010 (UTC)

Well, for the record, I strongly disagree. The COI site says "Editing in an area in which you have professional or academic expertise is not, in itself, a conflict of interest. Using material you yourself have written or published is allowed within reason, but only if it is notable and conforms to the content policies. Excessive self-citation is strongly discouraged. When in doubt, defer to the community's opinion."

So, this is not, in itself, a conflict of interest, according to the policy. As far as the larger community opinion goes, I'm pretty sure they would appreciate the application, as it can be used to make the concepts somewhat clearer. (JDoolin (talk) 23:43, 4 May 2010 (UTC))


 * In WP:EL we say:
 * "8. Direct links to documents that require external applications or plugins (such as Flash or Java) to view the content, unless the article is about such file formats."
 * And, quoting your site: "If you see no applet, you may need to Download Java."
 * "11. Links to blogs, personal web pages and most fansites, except those written by a '''recognized authority".
 * And, quoting your site: "Special Relativity Space-Time Applet by Jonathan Doolin"
 * For that matter, you bring up WP:COI stating:
 * "Editing in an area in which you have professional or academic expertise is not, in itself, a conflict of interest"
 * But there is no way the reader can verify that you indeed have professional or academic expertise. When challenged, you must provide evidence for that, trough a reliable secondary source.
 * The presence of this link forces others to actively go and verify whether every last bit of your work, which by the way is entirely original research, is in fact correct. That is a pretty arrogant thing to do.
 * Read in WP:BRD how this works: If X puts such a link and nobody removes it, it obviously stays. If Y objects to the link, they can remove it. If X insists, they can go to the talk page to propose it (which is what you did after 3 removals - you should have done that after the first). If a consensus is established on the talk page to keep the link, then it can be restored. No such consensus means no link. Your link has been removed by 2 contributors, and up to now (with currently 94 editors watching this talk page), there is no support. By the way, If you think that some of the other links are inappropriate as well, feel free to boldly remove them. Perhaps someone will object and restore. Then you go the talk page for discussion. Again, consensus is needed to keep the link out. If no consensus, the link stays.
 * DVdm (talk) 07:37, 5 May 2010 (UTC)

I still hope the community will consider putting the link back up.

As for verification, there is one VERY simple verification. The Lorentz Transformation is unique--there is only one transformation that preserves the speed of light under a change in velocity. This application does preserve the speed of light (all the red lines stay parallel to the main diagonals). Other than that, if there were any major mathematical errors or finagling in the program, you would see it pretty quick.

I do have the source code linked on the page. It is NOT an elegant piece of work, but it does the job. The central piece of code that performs the Lorentz Transform is the function, applyTransform(Event e), where a double-Lorentz Transformation is applied.



\begin{bmatrix} a \\ b \end{bmatrix} = g1 \begin{bmatrix} -vi&1\\ 1&-vi \end{bmatrix} \begin{bmatrix} e.y\\e.x \end{bmatrix} $$ and



\begin{bmatrix} d \\ c \end{bmatrix} = g2 \begin{bmatrix} vf&1\\ 1&vf \end{bmatrix} \begin{bmatrix} a\\b \end{bmatrix} $$ Though the negative signs on vi and vf were backwards, these are the LT and inverse LT when c=1. The first transform was an undoing of the initial velocity, and the second was performing the LT to the final velocity. vi and vf are determined by the slope of the line from the origin to the initial position and final position of the event, respectively. (JDoolin (talk) 14:56, 6 May 2010 (UTC))


 * Publish this in a scientific journal, and wait for it to be referenced in the literature. Then we can include it here. That is how Wikipedia works. Good old Usenet is over there :-) - DVdm (talk) 15:22, 6 May 2010 (UTC)

Derivation from physical principles
I do not agree with a step in the derivation from physical principles, in the part "Principle of relativity". We say that, in accordance with the principle or relativity, the inverse transformation should be:


 * $$x=\gamma\left(x'+vt'\right)$$
 * $$t=\beta\left(t'-\alpha x'\right)$$.

This is not sufficiently justified, in my opinion. Who says that gamma and beta are the same in the two reference systems? A similar reasoning for alpha becoming -alpha. We know, by relativity, that the functional form of the transformation is the same, but the observer in K sees K' travel with velocity v, whereas the observer in K' sees K travel with velocity -v. Thet are NOT in the same situation! If we have turned alpha into -alpha, for example, who says we shouldn't turn, say, gamma into -gamma?

The inverse transformation that we can safely impose is this:


 * $$x=\gamma'\left(x'+vt'\right)$$
 * $$t=\beta'\left(t'+\alpha' x'\right)$$.

with, if you get my notation here,


 * $$\alpha'=\alpha(-v)$$
 * $$\beta'=\beta(-v)$$
 * $$\gamma'=\gamma(-v)$$

If we knew that alpha is an odd function of v, and beta, gamma are even functions, then we would be done.

--Leo —Preceding unsigned comment added by 93.39.246.56 (talk) 15:37, 12 September 2010 (UTC)


 * I agree. It is of course clear that only factors which are proportional to v should change sign, hence v and &alpha;, but not &gamma;, but this is not justified in the article at all. It looks to me like this entire section is a bit of someone's original research. In any case it is not sourced at all. I have tried to find a source but could not find any. I have added a tag to the section and to the two useless sources. If no source for this derivation can be found, feel free to delete the entire section. DVdm (talk) 16:17, 12 September 2010 (UTC)


 * Well, a better (and easy) derivation is Einstein's own: http://www.bartleby.com/173/a1.html


 * --Leo —Preceding unsigned comment added by 93.39.246.56 (talk) 16:35, 12 September 2010 (UTC)


 * Most derivation found on books are not sufficiently logical, and even Einstein's own isn't either. I found some sources claiming to be derivation from directly the two postulates, but it takes time for me to examine it thoroughly and find additional sources to back up. The common additional assumptions (not necessarily appears together) are:
 * Invariance of relative speed between two inertial coordinates. If A sees B is moving away from him with a constant speed u, then B sees A moving away from him with the same speed u, just in an opposite direction.
 * Lengths measured perpendicular to the relative motion of the two frames are not contracted. This sounds intuitively obvious at first glance, but after you know that the y,z components of velocity ARE contracted, you will ask yourself why it is displacement intead of velocity whose y,z components transform in a Galilien fashion?
 * Linearity of transformation.
 * Space–time is homogeneous and isotropic.
 * The invariance of space-time interval.
 * --Netheril96 (talk) 07:33, 6 October 2010 (UTC)

Netheril196, you said that "..even Einstein's own isn't" sufficiently logical, do you mean the one published 1905-6-30 or the Simple Derivation of the Lorentz Transformation (Supplementary to Section XI) published 1920, or both? ...... John Huang Jh17710 (talk) 02:08, 24 October 2010 (UTC)


 * People, please read the wp:talk page guidelines and discuss the article here - not the subject. Thank you. DVdm (talk) 18:31, 24 October 2010 (UTC)

Derivation
The first paragraph of this section is just a personal opinion of Landau and Lifshitz when they wrote that book "the Course of Theoretical Physics". Since Einstein is the first person who started trying to derive Lorentz Transformation (LT), even if we let this personal opinion to stay as is, we should put both of Einstein's derivation at the end of this section, not the starting point, so that this section is a complete reference for general public. If not the whole derivation, at least a formal link, I think. I also have a comment to the logic of the first paragraph,


 * {The usual treatment (e.g., Einstein's original work) is based on the invariance of the speed of light. However, this is not necessarily the starting point: indeed (as is exposed, for example, in the second volume of the Course of Theoretical Physics by Landau and Lifshitz), what is really at stake is the locality of interactions: one supposes that the influence that one particle, say, exerts on another can not be transmitted instantaneously. Hence, there exists a theoretical maximal speed of information transmission which must be invariant, and it turns out that this speed coincides with the speed of light in vacuum.}

Logically speaking, "can not be transmitted instantly" does not support the existance of "a maximal speed of information transmission", especially if we just consider this issue theoretically. Because in the hyperreal system of mathematics, there is a number equal to infinite but for all other real numbers, not equal to that infinite number, there is no maximum number of them. I think if we remove the "theoretical" or change it to something real or physical then that paragraph will be a little bit more practical.


 * Logically speaking, even if there exist a maximum speed, it will not automatically "must be invariant". Why? Because it may depend on our technology level. The issue of maximum is like the issue of minimum, before people figure out the size of a photon, the minimum is pending; so, after people can see farther than two billion light years, the mximum may change unless we can prove it. Logically speaking, before some one proves it, the maximum speed may change.

Another possible and interesting point of view is the maximum speed could depend on the environment and/or the health status of a photon like in the speed competition of Olympic games, some photon may break the record. You never know, with so limited knowledge about photons that we have acquired so far. Or, someone does know that, it is impossible?

Regards, JohnJh17710 (talk) 21:45, 24 October 2010 (UTC)

The name 'boost'
I wonder if anyone knows who originated the term 'boost'. I believe it was called by Poincaré a pure Lorentz transformation and by Minkowski a special Lorentz transformation. Both are acceptable terms so why then the highly inappropriate misleading term 'boost' which has nothing to do with coordinate transformation and everything to do with rocket propulsion. With such a name it would at least be expected that boost plus boost equals bigger boost but, as everyone knows, this is not so.JFB80 (talk) 17:48, 7 December 2010 (UTC)


 * I am also interested in an answer. Those outside of physics will naturally imagine that "boost" is used as a synonym for "acceleration" (that is, for the application of a force for some period of time). But, since there is apparently no definition of "boost" in WP, it is impossible to determine exactly what is meant. Coordinate transformation is basic to modern physics, and only refers to a renaming of coordinates, not to any actual physical process, such as motion. David Spector (talk) 13:56, 31 December 2011 (UTC)

Merge ? Lorentz transformation under symmetric configuration
I found Lorentz transformation under symmetric configuration while working WikiProject Unreferenced articles where it is one the articles tagged the longest as needing references. Given the breadth of the article here, it seems the content would be best as a sub section on Lorentz transformation. Jeepday (talk) 23:20, 22 December 2010 (UTC)
 * It looks like Lorentz transformation under symmetric configuration was a bit of trivial original research from . I have looked for references but found nothing relevant. I think we can safely tag the article for deletion. DVdm (talk) 10:54, 23 December 2010 (UTC)

inverse transformation
Does the statement "As in the Galilean transformation, the sign of the transport velocity v has to be changed when passing from one frame to the other." in the Principle of Relativity section at the end of the article mean that if, for instance, if v = .6 c for the unprimed frame, that it is - .6 c according to the primed frame? If not, then the statement should be changed to show that v is still considered positive, being the relative speed between frames, not the measured transport speed in the negative or positive direction. If so, however, which is the way I take it to be, then the inverse transformation x = y (x' + v t') is incorrect, so that should be changed. As an example, consider that an event occurs at the origin of the unprimed frame, so that x = 0. According to the primed frame, then, x' is negative since the primed observer Q sees the unprimed observer O travelling in the negative direction and the event occurs in the same place as observer O. If x' is negative and v is now negative also as observer Q measures the speed of observer O in the negative direction, then we do not get x = 0 using the current equation, but rather, that is only achieved with x = y (x' - v t').(Grav-universe (talk) 08:28, 10 January 2011 (UTC))

Upon further examination, the inverse transformation for time should also be t = y (t' - v x' / c^2) for the same reason (really v' should be used considering it is negative while v is positive even though they are otherwise equal, stated as v' = -v). The equations for the transformations from either frame are the same.(Grav-universe (talk) 08:53, 10 January 2011 (UTC))

Okay, I added to the end of the statement I quoted so that it now reads "As in the Galilean transformation, the sign of the transport velocity v has to be changed when passing from one frame to the other, such that v' = -v." That small edition should cover it for the most part, I think. That way, v can continue to be used as it is when considered positive.(Grav-universe (talk) 09:17, 10 January 2011 (UTC))


 * The idea of a mathematical transformation is to have a set of equations transforming a variables tuple (x,y,z,t) into a variables tuple (x',y',z',t') with the property that the mathematical inverse of the transformation converts the tuple (x',y',z',t') into the orginal tuple (x,y,z,t). That is exactly what we have with the standard equations. The parameter v (—not a variable—) that we see in the equations is a quantity that is fixed by the physical situation. This parameter v itself can still be positive or negative in the following sense: when the x and x' axes point in the same direction, then positive (v>0) means that observer O' is travelling in the direction of increasing x as measured by O, whereas negative (v<0) means that observer O' is travelling in the direction of decreasing x as measured by observer O. That is the role of the parameter v in the transformation. The fact that the coefficient of t is -v in the equation x'=&gamma;(x-vt) and the one of t' is +v in x=&gamma;(x'+vt') merely shows that the inverse transformation has the same general form as the original, and there is no reason to introduce or write down a new parameter v', other than perhaps to show that, since obviously v'=-v, indeed they do have the same form. So there is nothing wrong with the standard equations in the article. The edit you made introduced a little problem, since v' is nowhere defined in the text, nor should it be. Note that the parameter v is not transformed into -v or into v'. Actually, v, being the velocity of O' as measured by O, gets transformed into 0 (zero), the velocity of O' as measured by O'. The quantity -v is merely the velocity of O as measured by O', which is an entirely different physical quantity that just happens to be the negative of v. One does not get mathematically transformed into the other, so introducing a primed parameter v' is a bit misleading/confusing.
 * So in order to correct the subtle original problem, I have changed the phrase into "... the transport velocity has to be changed from v to -v when passing from one frame to the other." DVdm (talk) 10:02, 10 January 2011 (UTC)