# Space-time transformations with different shape

I find in the literature the following transformation equations for the space-time coordinates
x'=g(x-vt)
t'=t/g
g=gamma.
Please tell me what do they bring new in the approach to SRT?
Thanks

## Answers and Replies

HallsofIvy
Science Advisor
Homework Helper
How are those different from the usual formulas?

I find in the literature the following transformation equations for the space-time coordinates
x'=g(x-vt)
t'=t/g
g=gamma.
Please tell me what do they bring new in the approach to SRT?
Thanks

The second equation is true only for a body stationary in the ref frame S', so that x' = 0 and x = vt. If you already have the Lorentz transforms, those you have written are just a particular case and don't bring anything new.

I find in the literature the following transformation equations for the space-time coordinates
x'=g(x-vt)
t'=t/g
g=gamma.
Please tell me what do they bring new in the approach to SRT?
Thanks
Are you sure about the t'=t/g?
Because if you are, this is different from a particularization of the Lorentz transforms. These are the so called Tangherlini transforms, a variant of relativity. See here.

Tangherlini has his own theory, the absence of the term in vx/c^2 in his time transform results into absolute simultaneity. There is a claim that his theory is experimentally indistinguishable from SR.

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Dale
Mentor
2020 Award
I find in the literature the following transformation equations for the space-time coordinates
x'=g(x-vt)
t'=t/g
g=gamma.
Please tell me what do they bring new in the approach to SRT?
Thanks
The second equation is not a coordinate transformation for SR since it does not introduce the relativity of simultaneity. It could be that the second one is intended to be an interval transformation (e.g. maybe it means dt'=dt/g)

lorentz transformations

Are you sure about the t'=t/g?
Because if you are, this is different from a particularization of the Lorentz transforms. These are the so called Tangherlini transforms, a variant of relativity. See here.

Tangherlini has his own theory, the absence of the term in vx/c^2 in his time transform results into absolute simultaneity. There is a claim that his theory is experimentally indistinguishable from SR.
Thanks for your help. Please tell me if you aggree with:
1. The Lorentz-Einstein transformations hold only in the case when the times they involve are displayed by clocks synchronized a la Einstein t(E) and t'(E) respectively.
2. They also hold when we replace t(E) and t'(E) with other linear combinations of time and space coordinates as in the case of a radar detection or in the case of the photographic detection.

DrGreg
Science Advisor
Gold Member
I find in the literature the following transformation equations for the space-time coordinates
x'=g(x-vt)
t'=t/g
g=gamma.
Please tell me what do they bring new in the approach to SRT?
Thanks
This transformation is sometimes called the Tangherlini transform and was also studied by Mansouri and Sexl in 1976.

(Note: everything that follows assumes no gravity, i.e. a flat spacetime.)

(x,t) is a standard inertial frame in which light speed is isotropic (the same in all directions), which can be called the "aether frame" in this context. (x',t'), defined by the above, defines a coordinate system for an observer moving at speed v relative to the aether frame. Tangherlini coordinates exhibit the same length-contraction and time-dilation properties as the usual SR inertial coordinates defined by the Lorentz transform, but they use a different definition for simultaneity (or to put it another way, they use a different synchronisation gauge).

In the context of the mathematics of GR (manifolds), Tangherlini coordinates are a valid coordinate chart, but the axes are not orthogonal. The metric, though flat, is not diagonalised in Tangherlini coordinates. Equations converted to Tangherlini coordinates are generally more complicated, the Tangherlini-speed of light is not isotropic, and momentum is not an isotropic function of Tangherlini-velocity. (If I haven't made a silly mistake, I think the metric is ds2 = dx'2/g2 - c2 dt'2 - 2vc dx' dt'.) All of these facts make Tangherlini coordinates rather painful to use compared with the usual SR inertial coordinates.

The main reason why advocates of Tangherlini coordinates like them is because they use absolute simultaneity. Events that are simultaneous in the aether frame are simultaneous in all other Tangherlini frames. So those who have some philosophical objection to relative simultaneity, or who can't grasp the concept, or who have other reasons for postulating the existence of an aether, will find comfort in the Tangherlini approach and the fact that there's nothing demonstrably wrong with it. His theory is "experimentally indistinguishable from SR" because it's actually the same theory transformed into a non-standard coordinate system.

What is unfortunate is that to use Tangherlini coordinates you have to determine an aether frame and you need to know the velocity of your own Tangherlini frame relative to the aether. An observer inside a sealed box, moving inertially relative to a postulated aether, would be unable to determine their speed relative to the aether and therefore unable to construct any Tangherlini coordinates.

But, in fact, it doesn't matter which frame you choose for your aether frame. Just choose any inertial observer and define an inertial frame using the Einstein synchronisation convention, and derive all the other Tangherlini frames from that. The fact that a privileged aether frame is postulated, but you are free to choose any such frame you like, seems a philosophical defect, compared with Einstein's formulation of SR.

Please tell me if you aggree with:
1. The Lorentz-Einstein transformations hold only in the case when the times they involve are displayed by clocks synchronized a la Einstein t(E) and t'(E) respectively.
Yes. That is, t(E) is Einstein-synchronized to one observer's clock and t'(E) is Einstein-synchronized to the other observer's clock.

Please tell me if you aggree with:
2. They also hold when we replace t(E) and t'(E) with other linear combinations of time and space coordinates as in the case of a radar detection or in the case of the photographic detection.
No, if you transform (x,t) and (x',t') into something else, the Lorentz-transformation equations will transform into something else. E.g. try substituting r' = t' + x'/c, s' = t' - x'/c, t = (r + s)/2, x = c(r - s)/2 into the Lorentz transform equations and see what you get for (r',s') in terms of (r,s). (The radar-coordinate version of the Lorentz transform.)

Reference:

R Mansouri & R U Sexl (1977), http://scholar.google.com/scholar?h...nchronization"+author:r-mansouri&btnG=Search", General Relativity and Gravitation, Springer, Vol 8, No 7, pp.497-513.

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Lorentz transformation

No, if you transform (x,t) and (x',t') into something else, the Lorentz-transformation equations will transform into something else. E.g. try substituting r' = t' + x'/c, s' = t' - x'/c, t = (r + s)/2, x = c(r - s)/2 into the Lorentz transform equations and see what you get for (r',s') in terms of (r,s). (The radar-coordinate version of the Lorentz transform.)

Thanks for your help. I have studied a paper devoted to relativistic navigation mimicking the maritime one. The author proposes a navigation convention according to which observers in the I' frame measure distances on a chart devised by observers in I and times using a clock comoving with the ship. What the observers in I' need are the time attributed by observers from I (t) and their space coordinate in their I' frame. They obtain transformation equations which express t(x,t') and x'(x,t'). Do you see there some thing wrong.
Leubner quoted below presents an "everyday clock synchronization procedure" consisting in the fact that an observer in I has at his disposal his wrist watch Eintein synchronized with the other clocks which displays t(E) and the information brought by the radio signal
t(r) equal to the reading of a clock from I located at its origin O when the radio signal was emitted. We have in an one space dimension
t(E)=t(r)+x/c.
Replacing that value in the Lorentz transformations we obtain transformation equations for x',t(E) and t'(r). The result is absolute simultaneity. From a practical point of view observers from I can express x', t'(E) and t'(T) as a function of the physical quantities they can measure x t(E), t(t). Is there something wrong? Is the job well done?
The authors solve the problem in a complicated way using the concept of "base vectors".
Regards
I Elementary relativity with I 'everyday' clock synchronization
C Leubnert, K Aufingert and P KrummS
t Physics Education Group, lnstitut fur Theoretische Physik, Leopold-Franzens-Universitat Innsbruck,
Technikerstr. 25, A-6020 Innsbruck, Austria
$Department of Physics, University of Natal DrGreg Science Advisor Gold Member I have studied a paper devoted to relativistic navigation mimicking the maritime one. The author proposes a navigation convention according to which observers in the I' frame measure distances on a chart devised by observers in I and times using a clock comoving with the ship. What the observers in I' need are the time attributed by observers from I (t) and their space coordinate in their I' frame. They obtain transformation equations which express t(x,t') and x'(x,t'). Do you see there some thing wrong. It's an unusual way to do things (mixing primed and unprimed coordinates) but, assuming the author makes no mistakes, there's nothing actually wrong with this and it could be of some practical use. A "moving" observer may be able to measure their distance on a "static" scale but only have access to their own "moving" clock, for example. (Of course, the words "static" and "moving" are both relative to a single inertial frame.) Dividing "static" distance by "moving" (i.e. proper) time you get something called "celerity" or "proper velocity" (see this post). Leubner quoted below presents an "everyday clock synchronization procedure" consisting in the fact that an observer in I has at his disposal his wrist watch Eintein synchronized with the other clocks which displays t(E) and the information brought by the radio signal t(r) equal to the reading of a clock from I located at its origin O when the radio signal was emitted. We have in an one space dimension t(E)=t(r)+x/c. Replacing that value in the Lorentz transformations we obtain transformation equations for x',t(E) and t'(r). The result is absolute simultaneity. From a practical point of view observers from I can express x', t'(E) and t'(T) as a function of the physical quantities they can measure x t(E), t(t). Is there something wrong? Is the job well done? The authors solve the problem in a complicated way using the concept of "base vectors". Regards I Elementary relativity with I 'everyday' clock synchronization C Leubnert, K Aufingert and P KrummS t Physics Education Group, lnstitut fur Theoretische Physik, Leopold-Franzens-Universitat Innsbruck, Technikerstr. 25, A-6020 Innsbruck, Austria$Department of Physics, University of Natal

For the benefit of other readers I'll reproduce the abstract:

C Leubner said:
(here)

"Elementary relativity with 'everyday' clock synchronization"

C Leubner, K Aufinger and P Krumm 1992 Eur. J. Phys. 13 170-177 doi:10.1088/0143-0807/13/4/004

Abstract. Although the importance of clock synchronization for relativity is discussed from time to time in the educational literature, the fact that different synchronization conventions imply different coordinizations of spacetime with ensuing changes of the form of possibly all coordinate-dependent quantities, has neither entered textbooks nor undergraduate physics education. As a consequence, there is a widespread belief among students that the familiar form of coordinate-dependent quantities like the measured velocity of light, the Lorentz transformation between two observers, 'addition of velocities', 'time dilation', 'length contraction', '$$E = mc^2 \gamma$$', which they assume under the standard clock synchronization, is relatively proper. In order to demonstrate that this is by no means so, the paper studies the consequences of a non-standard synchronization, and it is shown that drastic changes in the appearance of all these quantities are thus induced. For example, the phrases 'moving clocks go slow', and 'simultaneity is relative', which are usually considered as intrinsic features of relativity, turn out to be no longer true, whereas all coordinate-independent quantities remain of course indifferent to such a change in coordinization. Although Einstein's standard convention of clock synchronization enjoys distinct advantages over the 'everyday' method, the message clearly conveyed is that in the teaching of elementary relativity much more stress should be laid on the intrinsic (coordinate-independent) features of spacetime.
It was only a few years ago, soon after I started using this forum, that I realised that some of the standard relativistic effects, as described above, are coordinate-dependent and that other coordinate systems were permissible.

The point the authors are making is that there are lots of different coordinate systems to choose from apart from the standard "Einstein-synced" coordinates. Einstein coords are arguably the best coords but they are not the only coords, and it is educational to consider some of the other possibilities. The "everyday" coords that the authors define are one possibility.

The "radar coordinates" (e,r) which I defined in this post are another possibility. These are unusual because their two axes are not timelike and spacelike respectively, like most systems, but both null. (For mathematicians, radar coordinates are particularly interesting because they diagonalise the Lorentz transform; the two coord directions are eigenvectors and the k and k-1 red- and blue-shift Doppler factors are the eigenvalues.)

Time and space coordinates $$(t(\epsilon), x)$$ can be defined from (e,r) by

$$t(\epsilon) = e + \epsilon(r - e) = (1 - \epsilon)e + \epsilon r$$

$$x = c(r - e)/2$$

(Sorry: you'll need to look closely at the above equations to see the difference between "epsilon" and "e".)

You have a choice of $$\epsilon$$:

• Standard Einstein-synced coordinates result when $$\epsilon$$ = ½. (t(E) in your notation.)
• Leubner et al’s "everyday" coords result when $$\epsilon$$ = 0. (t(r) in your notation.)
• Tangherlini coords (which I discussed in an earlier post in this thread) result when $$\epsilon$$ = ½(1 + v/c), where v is the supposed velocity of the observer relative to a postulated aether.

I think it is useful to be aware of these different coord systems, to help understand which relativistic effects are "intrinsic" (coord independent) – e.g. the twin "paradox" -- and which are not. However I’m not sure whether it’s a good idea to present all this to someone learning relativity for the first time; it might just confuse them.

I have a particular fondness for radar coordinates because of how, with k-calculus, they can be used to derive many results with quite simple proofs, and some without even having to define "simultaneity". The concept of "relative simultaneity" seems to be what most people have most difficulty understanding when learning relativity.

It is not clear to me whether Leubner’s "everyday" coords help with the original navigation problem. Nor am I convinced that using "base vectors" (I would call them "unit basis vectors") is the easiest method. I would think you just need to write down all the relevant equations to convert from one coord system to another and then it’s just maths (algebra) to solve them.

I’m not sure I’ve answered your question. Does any of this help you?

DrGreg
Science Advisor
Gold Member
(If I haven't made a silly mistake, I think the metric is ds2 = dx'2/g2 - c2 dt'2 - 2vc dx' dt'.)
Nobody spotted I did make a silly mistake. The correct formula should be

ds2 = dx'2/g2 - c2 dt'2 + 2v dx' dt'

(I hope.)

lorentz transformations

It's an unusual way to do things (mixing primed and unprimed coordinates) but, assuming the author makes no mistakes, there's nothing actually wrong with this and it could be of some practical use. A "moving" observer may be able to measure their distance on a "static" scale but only have access to their own "moving" clock, for example. (Of course, the words "static" and "moving" are both relative to a single inertial frame.) Dividing "static" distance by "moving" (i.e. proper) time you get something called "celerity" or "proper velocity" (see this post).

For the benefit of other readers I'll reproduce the abstract:

It was only a few years ago, soon after I started using this forum, that I realised that some of the standard relativistic effects, as described above, are coordinate-dependent and that other coordinate systems were permissible.

The point the authors are making is that there are lots of different coordinate systems to choose from apart from the standard "Einstein-synced" coordinates. Einstein coords are arguably the best coords but they are not the only coords, and it is educational to consider some of the other possibilities. The "everyday" coords that the authors define are one possibility.

The "radar coordinates" (e,r) which I defined in this post are another possibility. These are unusual because their two axes are not timelike and spacelike respectively, like most systems, but both null. (For mathematicians, radar coordinates are particularly interesting because they diagonalise the Lorentz transform; the two coord directions are eigenvectors and the k and k-1 red- and blue-shift Doppler factors are the eigenvalues.)

Time and space coordinates $$(t(\epsilon), x)$$ can be defined from (e,r) by

$$t(\epsilon) = e + \epsilon(r - e) = (1 - \epsilon)e + \epsilon r$$

$$x = c(r - e)/2$$

(Sorry: you'll need to look closely at the above equations to see the difference between "epsilon" and "e".)

You have a choice of $$\epsilon$$:

• Standard Einstein-synced coordinates result when $$\epsilon$$ = ½. (t(E) in your notation.)
• Leubner et al’s "everyday" coords result when $$\epsilon$$ = 0. (t(r) in your notation.)
• Tangherlini coords (which I discussed in an earlier post in this thread) result when $$\epsilon$$ = ½(1 + v/c), where v is the supposed velocity of the observer relative to a postulated aether.

I think it is useful to be aware of these different coord systems, to help understand which relativistic effects are "intrinsic" (coord independent) – e.g. the twin "paradox" -- and which are not. However I’m not sure whether it’s a good idea to present all this to someone learning relativity for the first time; it might just confuse them.

I have a particular fondness for radar coordinates because of how, with k-calculus, they can be used to derive many results with quite simple proofs, and some without even having to define "simultaneity". The concept of "relative simultaneity" seems to be what most people have most difficulty understanding when learning relativity.

It is not clear to me whether Leubner’s "everyday" coords help with the original navigation problem. Nor am I convinced that using "base vectors" (I would call them "unit basis vectors") is the easiest method. I would think you just need to write down all the relevant equations to convert from one coord system to another and then it’s just maths (algebra) to solve them.

I’m not sure I’ve answered your question. Does any of this help you?

Thank you very much for your answer which is of big help in finishing my paper in which I try to simplify Leubner's approach making it available to a large number of interested people. I started with the Lorentz transformations in their standard shape, replacing t(E)
wity t(T)+x/c recovering all Leubner's results and even adding to them new ones showing that as expected t(T) and t"(T) are related by the Doppler effect which is clock synchronization independent. Receiving e-mail addresses I could send to solicitors a draft
of it for a critical inspection.
Thanks again DrGrieg. Your help is a good reason for me to visit the Forum learning from people who know more then I do, sharing with generosity their knowledge.

Nobody spotted I did make a silly mistake. The correct formula should be

ds2 = dx'2/g2 - c2 dt'2 + 2v dx' dt'

(I hope.)

I think the metric might still be wrong.I am getting:
From:
$$x=g(x'-vt')$$
$$t=t'/g$$

Producing:
$$ds^2=x^2-(ct)^2=g^2(x'-vt')^2-(ct'/g)^2$$

So:

$$ds^2=g^2(dx')^2-2vg^2dx'dt'+(g^2v^2-(c/g)^2)(dt')^2$$

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DrGreg
Science Advisor
Gold Member
I think the metric might still be wrong.I am getting:
From:
$$x=\gamma(x'-vt')$$
$$t=t'/\gamma$$

Producing:
$$ds^2=x^2-(ct)^2=\gamma^2(x'-vt')^2-(ct'/\gamma)^2$$

So:

$$ds^2=\gamma^2(dx')^2-2v\gamma^2dx'dt'+(\gamma^2v^2-(c/\gamma)^2)(dt')^2$$

In this thread I had the primed and unprimed coordinates the other way round! See post #1.
$$x'=\gamma(x-vt)$$
$$t'=t/\gamma$$

In this thread I had the primed and unprimed coordinates the other way round! See post #1.
$$x'=\gamma(x-vt)$$
$$t'=t/\gamma$$

So, interchanging primed with unprimed you would get:

From:
$$x'=g(x-vt)$$
$$t'=t/g$$

Producing:
$$ds^2=x'^2-(ct')^2=g^2(x-vt)^2-(ct/g)^2$$

So, the metric is:

$$ds^2=g^2(dx)^2-2vg^2dxdt+(g^2v^2-(c/g)^2)(dt)^2$$

No?

DrGreg
Science Advisor
Gold Member
So, interchanging primed with unprimed you would get:

From:
$$x'=\gamma(x-vt)$$
$$t'=t/\gamma$$

Producing:
$$ds^2=dx'^2-(cdt')^2=\gamma^2(dx-vdt)^2-(cdt/\gamma)^2$$

So, the metric is:

$$ds^2=\gamma^2(dx)^2-2v\gamma^2dxdt+(\gamma^2v^2-(c/\gamma)^2)(dt)^2$$

No?
No. Remember what's what.

x & t are the Einstein coords in the "reference" frame. x' and t' are the Tangherlini coords in the "moving" frame. We know $$ds^2=dx^2-c^2dt^2$$ and we want to find the expression in terms of dx' and dt'.

(I added the T and E suffixes in other threads to avoid this confusion.)

No. Remember what's what.

x & t are the Einstein coords in the "reference" frame. x' and t' are the Tangherlini coords in the "moving" frame. We know $$ds^2=dx^2-c^2dt^2$$ and we want to find the expression in terms of dx' and dt'.

(I added the T and E suffixes in other threads to avoid this confusion.)

Yes, in this case you obtain the metric in your original post:

$$ds^2=(x'/\gamma)^2+2vx't'-(ct')^2$$

Setting $$ds=0$$ produces the anisotropic speed of
$$\gamma^2(c-v)$$ and $$-\gamma^2(c+v)$$ (negative, as one would expect)

The two-way light speed in the "Tangherlini frame" comes to $$\gamma^2 v$$ . This doesn't look good, something isn't right

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DrGreg
Science Advisor
Gold Member
Yes, in this case you obtain the metric in your original post:

$$ds^2=(x'/\gamma)^2+2vx't'-(ct')^2$$

Setting $$ds=0$$ produces the anisotropic speed of
$$\gamma^2(c-v)$$ and $$-\gamma^2(c+v)$$ (negative, as one would expect)
Yes.

The two-way light speed in the "Tangherlini frame" comes to $$\gamma^2 v$$ . This doesn't look good, something isn't right
The two-way speed isn't the average of the two one-way speeds. It's the total distance divided by total time; the two distances are the same but the two times are different. If you think about it long enough you will see that in fact the reciprocal of the two-way speed is the average of the reciprocals of the two one-way speeds.

Yes.

The two-way speed isn't the average of the two one-way speeds. It's the total distance divided by total time; the two distances are the same but the two times are different. If you think about it long enough you will see that in fact the reciprocal of the two-way speed is the average of the reciprocals of the two one-way speeds.

Got it, thank you !

$$\frac{L}{c_{two-way}}=.5(\frac{L}{|c_+|}+\frac{L}{|c_-|})$$

Great stuff, can you recommend a book on this material?

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DrGreg
Science Advisor
Gold Member
Got it, thank you !

$$\frac{L}{c_{two-way}}=.5(\frac{L}{|c_+|}+\frac{L}{|c_-|})$$

Great stuff, can you recommend a book on this material?
I'm afraid not. This is stuff I've gradually picked up from a variety of online sources (including discussions on this forum) and my own calculations over the last 2 years.

Did you see this post in another thread? The link given there is well worth reading for a philosophical overview of different synchronisation conventions. It also gives lots of references, none of which I've checked yet. (EDIT: that link seems to be broken; try this instead: Stanford Encyclopedia of Philosophy - Conventionality of Simultaneity)

If you Google our site for "Mansouri Sexl", you'll find some old 2006 discussions around this subject, although there's a lot of stuff to wade through and you might not have the energy for it.

A more general Google (for "Mansouri Sexl", or "Tangherlini", or "Selleri", or "Generalised Galilean Transformation" will find a lot more, but some of that will be "crackpot" stuff claiming that this theory is an alternative (i.e. rival) theory to relativity and proves the existence of an aether. If you've followed this thread you should realise such claims are rubbish and all you are doing is relativity in a weird (viz. non-orthogonal) coordinate system.

There's a mention of Mansouri & Sexl on Wikipedia* here (put epsilon = 0, and a(v), b(v) the values quoted there). (*You can't always rely on Wikipedia, but that section looks OK, today!)

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I'm afraid not. This is stuff I've gradually picked up from a variety of online sources (including discussions on this forum) and my own calculations over the last 2 years.

Did you see this post in another thread? The link given there is well worth reading for a philosophical overview of different synchronisation conventions. It also gives lots of references, none of which I've checked yet.

yes, I read it yesterday :-)

If you Google our site for "Mansouri Sexl", you'll find some old 2006 discussions around this subject, although there's a lot of stuff to wade through and you might not have the energy for it.

A more general Google (for "Mansouri Sexl", or "Tangherlini", or "Selleri", or "Generalised Galilean Transformation" will find a lot more, but some of that will be "crackpot" stuff claiming that this theory is an alternative (i.e. rival) theory to relativity and proves the existence of an aether. If you've followed this thread you should realise such claims are rubbish and all you are doing is relativity in a weird (viz. non-orthogonal) coordinate system.

There's a mention of Mansouri & Sexl on Wikipedia* here (put epsilon = 0, and a(v), b(v) the values quoted there). (*You can't always rely on Wikipedia, but that section looks OK, today!)

Thank you !

It's an unusual way to do things (mixing primed and unprimed coordinates) but, assuming the author makes no mistakes, there's nothing actually wrong with this and it could be of some practical use. A "moving" observer may be able to measure their distance on a "static" scale but only have access to their own "moving" clock, for example. (Of course, the words "static" and "moving" are both relative to a single inertial frame.) Dividing "static" distance by "moving" (i.e. proper) time you get something called "celerity" or "proper velocity" (see this post).

For the benefit of other readers I'll reproduce the abstract:

It was only a few years ago, soon after I started using this forum, that I realised that some of the standard relativistic effects, as described above, are coordinate-dependent and that other coordinate systems were permissible.

The point the authors are making is that there are lots of different coordinate systems to choose from apart from the standard "Einstein-synced" coordinates. Einstein coords are arguably the best coords but they are not the only coords, and it is educational to consider some of the other possibilities. The "everyday" coords that the authors define are one possibility.

The "radar coordinates" (e,r) which I defined in this post are another possibility. These are unusual because their two axes are not timelike and spacelike respectively, like most systems, but both null. (For mathematicians, radar coordinates are particularly interesting because they diagonalise the Lorentz transform; the two coord directions are eigenvectors and the k and k-1 red- and blue-shift Doppler factors are the eigenvalues.)

Time and space coordinates $$(t(\epsilon), x)$$ can be defined from (e,r) by

$$t(\epsilon) = e + \epsilon(r - e) = (1 - \epsilon)e + \epsilon r$$

$$x = c(r - e)/2$$

(Sorry: you'll need to look closely at the above equations to see the difference between "epsilon" and "e".)

You have a choice of $$\epsilon$$:

• Standard Einstein-synced coordinates result when $$\epsilon$$ = ½. (t(E) in your notation.)
• Leubner et al’s "everyday" coords result when $$\epsilon$$ = 0. (t(r) in your notation.)
• Tangherlini coords (which I discussed in an earlier post in this thread) result when $$\epsilon$$ = ½(1 + v/c), where v is the supposed velocity of the observer relative to a postulated aether.

I think it is useful to be aware of these different coord systems, to help understand which relativistic effects are "intrinsic" (coord independent) – e.g. the twin "paradox" -- and which are not. However I’m not sure whether it’s a good idea to present all this to someone learning relativity for the first time; it might just confuse them.

I have a particular fondness for radar coordinates because of how, with k-calculus, they can be used to derive many results with quite simple proofs, and some without even having to define "simultaneity". The concept of "relative simultaneity" seems to be what most people have most difficulty understanding when learning relativity.

It is not clear to me whether Leubner’s "everyday" coords help with the original navigation problem. Nor am I convinced that using "base vectors" (I would call them "unit basis vectors") is the easiest method. I would think you just need to write down all the relevant equations to convert from one coord system to another and then it’s just maths (algebra) to solve them.

I’m not sure I’ve answered your question. Does any of this help you?
Please let me know which kind of clock synchronization leads to the Edward's transformations?
Thanks.

DrGreg
Science Advisor
Gold Member
Please let me know which kind of clock synchronization leads to the Edward's transformations?
Thanks.
I've only just discovered what an "Edward's transformation" is. I'll get back to you once I've read more about it.

DrGreg
Science Advisor
Gold Member
Please let me know which kind of clock synchronization leads to the Edward's transformations?
Thanks.
According to Shen(1), the Edwards transform is

$$X' = K(X - VT)$$
$$T'=K \left[ \left(1 + \frac {\lambda + \lambda'}{c} V \right) T + \left( \frac {\lambda^2 - 1}{c^2} V + \frac {\lambda - \lambda'}{c} \right) X \right]$$
$$K = \frac {1} {\sqrt{(1 + \lambda V / c)^2 - V^2/c^2}}$$​

(When $\lambda = \lambda' = 0$, this is just the Lorentz transform.)

If I have done my calculations correctly, these equations can be derived from Einstein-synced inertial coordinates (t,x) and (t',x') moving at relative velocity v, by setting

$$X = x$$
$$T = t + \lambda x / c$$
$$V = \frac {v}{1 - \lambda v / c}$$​

and similar equations for primed coordinates.

This, in turn, corresponds to setting Reichenbach's $\epsilon = (1 + \lambda)/2$.

I haven't double checked my calculations, I may have made a mistake.

Reference

(1) Shen, J.Q. (2005), "Lorentz, Edwards transformations and the principle of permutation invariance".

George Jones
Staff Emeritus
Science Advisor
Gold Member
The "radar coordinates" (e,r) which I defined in this post are another possibility.

These coordinates (multiplied by $1/\sqrt{2}$) are popular in string theory, where they are called light-cone coordinates, and are denoted $x^+$ and $x^-$ there.

DrGreg
Science Advisor
Gold Member
$$X = x$$
$$T = t + \lambda x / c$$​
I've now seen Edwards' original paper (1). Following his method, I see a very easy way to prove my result: simply put V=0 and $\lambda' = 0$ into the transform itself!

I did make one mistake, though: for consistency with my previous post #9, Reichenbach's epsilon should really be $\epsilon = (1 - \lambda)/2$.

To put this into context, the Edwards' transform is between two coordinate systems which do not necessarily use Einstein's sync convention, where $\lambda = 1 - 2\epsilon$ and $\lambda' = 1 - 2\epsilon'$ parameterise the synchronisation conventions chosen in each of the two coordinate systems, zero representing Einstein synchronisation.

In a private message to me, Bernhard pointed out that some other equations (29) in Edwards' paper correspond to the "Leubner everyday coordinates" mentioned earlier in this thread (and others -- Google bernhard.rothenstein Leubner site:physicsforums.com if you are interested). In reply, I point out these are the special case where $\lambda = \lambda' = 1$, i.e. where Reichenbach's $\epsilon = 0$, and where the one-way coordinate-speed of light in the positive x direction is infinite for both unprimed and primed observers.

As an aside, note quite a significant result of section IV of Edwards' paper: in such circumstances (infinite light speed in the positive x direction), the one-way coordinate-speed of light in the orthogonal yz-plane must be isotropic (=c) in order that the "four-way speed of light" round a non-planar quadrilateral is isotropic as experiment demands. So Leubner's sync convention can be applied only in one dimension.

(Sorry, I will be off-line for the next week and a half, and will not be able to reply for a while.)

Reference

Edwards, W.F. (1963), http://link.aip.org/link/?AJPIAS/31/482/1 [Broken], American Journal of Physics 31, Issue 7, pp. 482-489.

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