B A question about relativity of simultaneity

[PeterDonis]

And here, I am afraid, you contradict the facts.

I have no idea what you are talking about. Everything I said is perfectly consistent with Einstein's 1905 paper.

 

[Sagittarius A-Star]

Thank you very much for bringing this up! it's exactly what's worrying me.

I think you misunderstood the reply of @PeroK to your posting #38 "to that center in the embankment frame".

Your statement in #38 may be interpreted by a reader in such a way, that you mix-up Einstein's $M$ and $M'$. They are different objects, that move relative to each other. With reference to the embankment rest frame, $M$ and $M'$ meet only at that instance of time, when the lightnings at $A$ and $B$ occur.

This means that this point of simultaneity exists even inside the traincar. It's not the center, but it is there. So, there is at least one observer who belongs to the traincar frame and who thinks that the lightnings are simultaneous.

No! That the lightnings are simultaneous, could only be concluded by an observer at rest in the train, who receives both light-pulses simultaneous, if he is located in the center of the train. But in the center of the train the light-pulses are received not simultaneously.

 

[PeterDonis]

It's from his 1905 article

This article, being a published physics paper, does not discuss underlying motivations much; it focuses on the physics. (That said, as I noted in post #51, the physics discussed is perfectly consistent with my post #49. In Section 2 of the paper, Einstein derives relativity of simultaneity, along with length contraction and time dilation, from the constancy of the speed of light. In Section 3 of the paper, he derives the Lorentz Transformation equations from the constancy of the speed of light. He never derives the LT from relativity of simultaneity. Exactly as I said.)

If you want a better source for Einstein's underlying motivations for the "train and lightning flashes" thought experiment, along the lines I described in post #49, try his popular book Relativity: A Clear Explanation That Anyone Can Understand.

 

[DmitryS]

This article, being a published physics paper, does not discuss underlying motivations much; it focuses on the physics. (That said, as I noted in post #51, the physics discussed is perfectly consistent with my post #49. In Section 2 of the paper, Einstein derives relativity of simultaneity, along with length contraction and time dilation, from the constancy of the speed of light. In Section 3 of the paper, he derives the Lorentz Transformation equations from the constancy of the speed of light. He never derives the LT from relativity of simultaneity. Exactly as I said.)

I don't have this impression from reading Einstein. To make sure we are talking about the same thing -- what's the blueprint of Einstein's derivation of LT in your opinion?

 

[DmitryS]

No! That the lightnings are simultaneous, could only be concluded by an observer at rest in the train, who receives both light-pulses simultaneous, if he is located in the center of the train. But in the center of the train the light-pulses are received not simultaneously.

PeroK said that if something happens in one point of spacetime, that's an event that must exist in any frame of reference. I agree to this, because this is in line with the LT - and that means that inside the train frame of reference there must be a point where the light from both lightnings come simultaneously.
It is not the center, we agree about that. It must be closer to the back of the train, and it is easy to calculate where this point should be based on the concept of closing speed.

 

[PeroK]

PeroK said that if something happens in one point of spacetime, that's an event that must exist in any frame of reference. I agree to this, because this is in line with the LT - and that means that inside the train frame of reference there must be a point where the light from both lightnings come simultaneously.
It is not the center, we agree about that. It must be closer to the back of the train, and it is easy to calculate where this point should be based on the concept of closing speed.

It is the centre of train. In the platform frame the light emission events are not simultaneous.

That a single event in one frame is a single event in all frames is a requirement for physical consistency of the theory.

 

[Sagittarius A-Star]

and that means that inside the train frame of reference there must be a point where the light from both lightnings come simultaneously.
It is not the center, we agree about that. It must be closer to the back of the train, and it is easy to calculate where this point should be based on the concept of closing speed.

The condition, that both light pulses are received by the observer simultaneously, is not sufficient to conclude, that the lightnings happened also simultaneously. Also the distances, the light-pulses travel with $c$, must be equal. In the rest frame of the train, the locations $A'$ and $B'$ of the lightning events are permanently at both ends of the train.

 

[DmitryS]

It is the centre of train. In the platform frame the light emission events are not simultaneous.

That a single event in one frame is a single event in all frames is a requirement for physical consistency of the theory.

It isn't the center. I'm talking about lightnings.
If the light of both lightnings meets at the same time at the place where the observer on the embankment is, then it is a single event. If it is a single event, it must be a single event in the train frame. I agree it's not the centre of the traincar. It must be closer to the tail.

 

[Sagittarius A-Star]

All I am asking is, can you actually derive LT from relative simultaneity test without assumption that the LT must be the correct transformation?

Yes, under a few conditions. A real relative simultaneity test can prove, that the opposite of Newton's assumption of an absolute time is true, that means in general $t' \ne t$. In addition you need SR postulate 1 (principle of relativity) and assuming isotropy of the space, homogeneity of space and time, invariance of causality and that the velocity composition law is commutative.

Under these conditions, SR postulate 2 (invariance of the speed of light in vacuum) is not needed to derive the LT.

You can find such a derivation of the LT here:
https://www.physicsforums.com/threa...rom-commutative-velocity-composition.1017275/

Edit: I must correct my answer: This thought experiment works under the assumption, that the LT is correct. So the answer must be "no".

 

[PeterDonis]

what's the blueprint of Einstein's derivation of LT in your opinion?

If you want a brief summary of Section 3 of the paper, sure:

(1) He assumes we have two inertial coordinate systems, with one (the "moving" system) moving in the positive $x$ direction with speed $v$ with respect to the other (the "stationary" system).

(2) He assumes that the transformation equations between the two systems must be linear.

(3) He assumes a light ray that goes from the spatial origin of the moving system out to some positive $x'$, is reflected there, and returns to the spatial origin of the moving system.

(4) He analyzes the motion of this light ray and uses the fact that the speed of the light ray must be the same in both frames to derive the transformation equations.

 

[PeterDonis]

Under these conditions, SR postulate 2 (invariance of the speed of light in vacuum) is not needed to derive the LT.

You can find such a derivation of the LT here

I think this is a bit misstated.

The units of the constant $\alpha$ in the derivation you refer to are inverse velocity squared. So assuming that $\alpha > 0$ is equivalent to assuming that there is a finite invariant speed. It is true that this, by itself, does not tell you that that finite invariant speed is the speed of light; but since there can be only one finite invariant speed, anything that has an invariant speed must have that invariant speed, including light. With that taken into account, assuming $\alpha > 0$ is really equivalent to assuming invariance of the speed of light.

The interesting part of the derivation is showing that assuming $\alpha > 0$ implies $t' \neq t$. But you only know that by already having the transformation equations in terms of $\alpha$. So I don't think this is quite the same as deriving the LT from relativity of simultaneity without using the invariance of the speed of light; that would imply that relativity of simultaneity by itself, without using the invariance of the speed of light, or something logically equivalent to it (like $\alpha > 0$), could tell you the form of the transformation equations. But the derivation of the transformation equations in terms of $\alpha$ doesn't use relativity of simultaneity at all.

 

[Sagittarius A-Star]

So assuming that $\alpha > 0$ is equivalent to assuming that there is a finite invariant speed. It is true that this, by itself, does not tell you that that finite invariant speed is the speed of light

I don't assume that $\alpha > 0$. This is the only remaining possibility after excluding the cases $\alpha < 0$ (from assumed invariance of causality, as I stated) and excluding $\alpha = 0$ (from experiment disproving absolute time, for example showing relativity of simultaneity = excluding Galileo transformation).

It is true, that this derivation does not also derive, that the invariant speed, called $c$, is equal to the physical speed of light in vacuum. That would need an additional experiment.

The interesting part of the derivation is showing that assuming $\alpha > 0$ implies $t' \neq t$.

It is the opposite direction of reasoning, see above.

But you only know that by already having the transformation equations in terms of $\alpha$.

I get this transformation equations in terms of $\alpha$ from assuming that the velocity composition law is commutative, as stated. This follows from transformation group laws, as stated in the paper "Nothing but Relativity", which I linked at the top of the derivation.

 

[PeterDonis]

I don't assume that $\alpha > 0$.

"Assume" is just a form of words. You can say "exclude all of the other possibilities" if you like that better.

This is the only remaining possibility after excluding the cases $\alpha < 0$ (from assumed invariance of causality, as I stated) and excluding $\alpha = 0$ (from experiment disproving absolute time, for example showing relativity of simultaneity = excluding Galileo transformation).

You exclude $\alpha = 0$ to exclude the Galilean case, but the only reason you can equate that with excluding "absolute time" (which you actually can't really--see below) is by looking at the specific form of the transformation equation for $t'$. However, you can equate $\alpha > 0$ with a finite invariant speed just by looking at the units required for $\alpha$, without making use of any specific form of the transformation equations.

Also, you can't exclude absolute time, or more precisely absolute simultaneity, by experiments, because simultaneity is a convention. You can, however, establish that there is a finite invariant speed by experiments, if you can find a physical phenomenon that moves at that speed, such as light.

It is the opposite direction of reasoning

I disagree. You first derive the transformation equations in terms of a parameter $\alpha$. Then you observe from the transformation equation for $t'$ that $\alpha > 0$ implies $t' = t$. There is no reasoning that let's you go in reverse, because just assuming $t' = t$ doesn't tell you what the form of the transformation equations is in terms of $\alpha$. In fact, even if you take the transformation equations as given, by itself $t' = t$ doesn't require $\alpha > 0$, it just requires $\alpha \neq 0$. Nor did you have to assume $t' = t$ in order to introduce $\alpha$ as a parameter in the equations or derive their form.

 

[vanhees71]

Thank you very much for bringing this up! it's exactly what's worrying me.
It seems that even as we stage the thought experiments, we already know that the relative simultaneity is there. The setup of the experiment already has it.
So, this is either the circular argument, or all those thought experiments are just illustrations and not thought experiments. This brings us back to the question I asked at the beginning: Why do we need them at all?
And, rewording your statement... The light of the two lightnings reaches the observer on the embankment at the same time. This means that this point of simultaneity exists even inside the traincar. It's not the center, but it is there. So, there is at least one observer who belongs to the traincar frame and who thinks that the lightnings are simultaneous. Granting that the clocks of the traincar frame are all synchronous for that frame, we have the time when the lightnings are simultaneous in the traincar frame?

What is unclear with my calculation in #42? You have to use the Lorentz transformation to get the record straight, because it follows from the usual Einsteinian synchronization convention in an inertial frame of reference.

What's missing is, when the light signals reach the observer at the platform. It's easy to calculate that the light signal sent from A reach the observer at $M$ at the time $t_{MA}'=(1+\beta) \gamma L/c$ and the signal from $B$ at $t_{MB}'=(1-\beta) \gamma L/(2c)$.

 

[DmitryS]

(1) He assumes we have two inertial coordinate systems, with one (the "moving" system) moving in the positive $x$ direction with speed $v$ with respect to the other (the "stationary" system).

(2) He assumes that the transformation equations between the two systems must be linear.

(3) He assumes a light ray that goes from the spatial origin of the moving system out to some positive $x'$, is reflected there, and returns to the spatial origin of the moving system.

(4) He analyzes the motion of this light ray and uses the fact that the speed of the light ray must be the same in both frames to derive the transformation equations.

I don't agree that he assumes what you say, or rather, that he only assumes what you say, but let that be for a moment.
Basically, as I was saying, the starting point of his derivation is the same setup as that of the simultaneity test. I already posted the screenshot. And that setup leads to some transforms, which are not LT:

https://www.fourmilab.ch/etexts/einstein/specrel/specrel.pdf

These are very much not the LT. I would be glad if you can explain to me how you can get from those to the LT. I tried, and I failed.
That's, as I am saying, the result of using the simultaneity test as the starting point of derivation. That's why I think the simultaneity test contradicts the LT.

 

[PeroK]

That's, as I am saying, the result of using the simultaneity test as the starting point of derivation. That's why I think the simultaneity test contradicts the LT.

Those are intermediate variables. He ends up with the LT further on in the paper.

Are you trying to learn SR or trying to prove its's wrong, if you don't mind me asking?

 

[Ibix]

These are very much not the LT. I would be glad if you can explain to me how you can get from those to the LT. I tried, and I failed.

They're just intercept calculations between the light ray and a mirror, aren't they?

 

[DmitryS]

Those are intermediate variables. He ends up with the LT further on in the paper.

Are you trying to learn SR or trying to prove its's wrong, if you don't mind me asking?

I don't mind you asking anything as long as we remain within the protocol of mutual respect.
I think I made my point abundantly clear from the start. I find the Lorentz mathematics very much consistent, but I don't like the idea of the physical evidence of relative simultaneity. It seems to me a flop. If I remember correctly, you agreed with me about this point.
Now, to give it a rigorous proof, I refer to Einstein's 1905 paper. Mind that I'm only voicing an opinion without claiming it right. If you think it is wrong, feel free to prove it.
I will put down my points and supporting references in order, so you could tell me where I am wrong.
1) The experiment with the ray going from 0 to x' and its accompanying mathematics is simply an extension of simultaneity test.
2) With that mathematics, Einstein ends up with the transforms

which are not LT. They are pretty much the logical finale of Einstein's setup.
3) You say: "Those are intermediate variables. He ends up with the LT further on in the paper". That's exactly the point that in my opinion needs a proof on your part. What I see from the text - and I provided the link to the online PDF - is that Einstein simply writes down the LT after these transforms. I won't make a snapshot, you can easily find it yourself, here's the link:
https://www.fourmilab.ch/etexts/einstein/specrel/specrel.pdf
To me, that looks like he assumed the LT to be true, whatever his own reasoning might have led to. If you can derive the LT from the relations above, show me how - and I will confess I was wrong.

 

[PeroK]

I find the Lorentz mathematics very much consistent, but I don't like the idea of the physical evidence of relative simultaneity. It seems to me a flop. If I remember correctly, you agreed with me about this point.

Not at all. I just don't like the Einstein simultaneous lightning strikes thought experiment. It's over-complicated and potentially misleading.

I gave a much cleaner example of the incompatibility of universal simultaneity with the invariance of the speed of light in a previous post.

Now, to give it a rigorous proof, I refer to Einstein's 1905 paper. Mind that I'm only voicing an opinion without claiming it right. If you think it is wrong, feel free to prove it.
I will put down my points and supporting references in order, so you could tell me where I am wrong.
1) The experiment with the ray going from 0 to x' and its accompanying mathematics is simply an extension of simultaneity test.
2) With that mathematics, Einstein ends up with the transforms
View attachment 315069

which are not LT. They are pretty much the logical finale of Einstein's setup.
3) You say: "Those are intermediate variables. He ends up with the LT further on in the paper". That's exactly the point that in my opinion needs a proof on your part.

I'm not sure what's to prove.

What I see from the text - and I provided the link to the online PDF - is that Einstein simply writes down the LT after these transforms. I won't make a snapshot, you can easily find it yourself, here's the link:
https://www.fourmilab.ch/etexts/einstein/specrel/specrel.pdf
To me, that looks like he assumed the LT to be true, whatever his own reasoning might have led to. If you can derive the LT from the relations above, show me how - and I will confess I was wrong.

There's a derivation of the LT here:

http://www2.physics.umd.edu/~yakovenk/teaching/Lorentz.pdf

There is an issue if you take Einstein's 1905 paper as some sort of Gospel. It was the first paper on SR, but the theory has been developed and refined for over 100 years since then. I love the 1905 paper, but it's not an ideal source from which to learn SR as a student. Your confusions bear this out to some extent.

I don't want to get into a game of having to justify everything Einstein wrote in 1905. We all see flaws in that paper, when it comes to it.

But, an incompatibility between the LT and RoS? That's just nonsensical. The LT encapsulates the RoS along with time dilation and length contraction. That I can prove, but not by dissecting the 1905 paper.

PS both @vanhees and I have proved the compatibility of the LT and RoS in various posts in this thread.

 

[Sagittarius A-Star]

Also, you can't exclude absolute time, or more precisely absolute simultaneity, by experiments, because simultaneity is a convention. You can, however, establish that there is a finite invariant speed by experiments, if you can find a physical phenomenon that moves at that speed, such as light.

You are right, I made an error in posting #59. This thought experiment works under the assumption, that the LT is correct. Instead, for example, a twin paradox experiment with muons in a particle storage ring could disprove GT / Newtonian physics.

You first derive the transformation equations in terms of a parameter $\alpha$. Then you observe from the transformation equation for $t'$ that $\alpha > 0$ implies $t' = t$.

No, I wrote the opposite:

1. $\alpha < 0$
2. $\alpha = 0$
3. $\alpha > 0$

...
Case 2, the GT, can be excluded when assuming $t' \neq t$, which is the opposite of Newton's assumption of an "absolute time" (see equation 9).

 

[DmitryS]

I'm not sure what's to prove.

To prove your own words, "Those are intermediate variables. He ends up with the LT further on in the paper". To prove that Einstein's model of 1905 leads to LT, and that there's an actual derivation of the LT in 1905.

Instead, you provide a link to another source with the LT derivation - I can derive the LT myself on different grounds. That's not the point I'm making.

There is an issue if you take Einstein's 1905 paper as some sort of Gospel. It was the first paper on SR, but the theory has been developed and refined for over 100 years since then. I love the 1905 paper, but it's not an ideal source from which to learn SR as a student. Your confusions bear this out to some extent.

I just said that in this paper there is no derivation of LT. You said I was wrong. Now you say I was right? I take it for a scientific paper, not for a Gospel.

I don't want to get into a game of having to justify everything Einstein wrote in 1905. We all see flaws in that paper, when it comes to it.

Good.

But, an incompatibility between the LT and RoS? That's just nonsensical.
PS both @vanhees and I have proved the compatibility of the LT and RoS in various posts in this thread.

You're missing my point again. I never said LT and RoS are incompatible. What's more, it's easy to see how RoS works through the LT. I only said that the kind of model of RoS presentedd through the light rays experiments brings us only to the kind of transforms obtained by Einstein in his 1905 paper. And those are incompatible with the LT.

Once again. I take no issue with LT, RoS, or their incompatibility. I take issue with the way RoS is introduced through those experiments.

Are we going to go through all this again? I might retort: "You aren't trolling me, if you don't mind me asking?" But I won't. I will just tell you that it seems to me that instead of the discussion of the relevant points I'm making I sometimes have to struggle with the paternal attitude as if to a would-be renegade.

Maybe points I'm making are not relevant. If so, you can demonstrate that, I hope? So far, I haven't seen in this thread a proof of the light-ray experiments leading to LT which did not contain the assumption of LT in the first place. Einstein so far was the only one who tried to do it - and seems to have failed.

 

[PeroK]

Maybe points I'm making are not relevant. If so, you can demonstrate that, I hope? So far, I haven't seen in this thread a proof of the light-ray experiments leading to LT which did not contain the assumption of LT in the first place. Einstein so far was the only one who tried to do it - and seems to have failed.

I have to say, in all honesty, I doubt I am able to help you understand SR.

 

[DmitryS]

I have to say, in all honesty, I doubt I am able to help you understand SR.

But I do understand it! I'm not after that!

 

[PeroK]

But I do understand it! I'm not after that!

Anyone who posts something like this:

So far, I haven't seen in this thread a proof of the light-ray experiments leading to LT which did not contain the assumption of LT in the first place. Einstein so far was the only one who tried to do it - and seems to have failed.

Is just wasting everyone's time. Every competent student of SR in the last 100 years has derived the LT themselves from only the invariance of the speed of light.

 

[TSny]

To me, that looks like he assumed the LT to be true, whatever his own reasoning might have led to. If you can derive the LT from the relations above, show me how - and I will confess I was wrong.

You appear to be asking how Einstein gets from these equations:

to these equations:

Take, for example, the equation for $\xi$. Recall that Einstein uses the symbol $x'$ to denote $x-vt$. (See near the top of page 6 in your link.) Thus,

$$\xi = a \frac{c^2}{c^2-v^2}x' = a \frac{c^2}{c^2-v^2}\left( x-ct \right) = \frac{a}{1-v^2/c^2}\left( x-ct \right) = \frac{a}{\sqrt{1-v^2/c^2}}\frac{1}{\sqrt{1-v^2/c^2}}\left( x-vt \right)$$

The symbol $a$ represents a yet-to-be-determined function of $v$. Einstein originally used the notation $\phi(v)$ to denote this unknown function. (See near the bottom of page 6 in your link.) However, later when Einstein writes his results at the bottom of page 7, he redefines the notation $\phi(v)$ to denote $\frac{a}{\sqrt{1-v^2/c^2}}$. This is certainly OK, but it could be a bit confusing. $\phi(v)$ is still a yet-to-be-determined function of $v$.

Einstein uses $\beta$ to denote $\frac{1}{\sqrt{1-v^2/c^2}}$. So, using these definitions of $\phi(v)$ and $\beta$, we arrive at Einstein's result $$\xi = \phi(v) \beta \left( x-ct \right).$$The results for $\tau, \eta$, and $\zeta$ can be obtained in a similar manner.

Einstein later argues that the unknown function $\phi(v)$ is actually independent of $v$ and equals 1. At this point, he has finally finished his derivation of the LT equations given at the bottom of page 9.

 

[Ibix]

That's exactly the point that in my opinion needs a proof on your part. What I see from the text - and I provided the link to the online PDF - is that Einstein simply writes down the LT after these transforms.

It's just algebra to derive the Lorentz transforms from the setup Einstein gives (as @TSny has just laid out). I find Einstein's derivation a bit messy because he has these intermediate values with Greek letters, but it's easy enough to do something cleaner.

Assert that everything happens in the $z=0$ plane. Consider two light pulses from the origin, one parallel to $x$ and one parallel to $y$, let them reflect off two mirrors equidistant from the origin and return simultaneously to the origin. Write down the coordinates, $(x,y,t)$ and $(x',y',t')$, of the two reflection events and the return event in two frames, one where the experimental gear is stationary and one where it is moving in the $+x$ direction at constant speed. Equate the primed coordinates of each event to the unprimed coordinates after applying the general linear transform. The general linear transform (a 3$\times$3 matrix) has nine parameters and you have nine constraints. Solve for the parameters. You then have a bit more work to do because you can't assume that the distances to the mirrors in the primed frame are the same as they are in the unprimed one (and, indeed, one turns out not to be the same), but noting that if the forward transform is $\Lambda(v)$ the inverse transform must be $\Lambda(-v)$ and hence $\Lambda(v)\Lambda(-v)=\mathbf{I}$ gets you out of that. Finally you observe from rotational symmetry that the $z$ transform must be the same as the $y$ transform.

It's worth grinding through it once.

 

[DmitryS]

You appear to be asking how Einstein gets from these equations:
View attachment 315075

to these equations:

View attachment 315076

Take, for example, the equation for $\xi$. Recall that Einstein uses the symbol $x'$ to denote $x-vt$. (See near the top of page 6 in your link.) Thus,

$$\xi = a \frac{c^2}{c^2-v^2}x' = a \frac{c^2}{c^2-v^2}\left( x-ct \right) = \frac{a}{1-v^2/c^2}\left( x-ct \right) = \frac{a}{\sqrt{1-v^2/c^2}}\frac{1}{\sqrt{1-v^2/c^2}}\left( x-vt \right)$$

The symbol $a$ represents a yet-to-be-determined function of $v$. Einstein originally used the notation $\phi(v)$ to denote this unknown function. (See near the bottom of page 6 in your link.) However, later when Einstein writes his results at the bottom of page 7, he redefines the notation $\phi(v)$ to denote $\frac{a}{\sqrt{1-v^2/c^2}}$. This is certainly OK, but it could be a bit confusing. $\phi(v)$ is still a yet-to-be-determined function of $v$.

Einstein uses $\beta$ to denote $\frac{1}{\sqrt{1-v^2/c^2}}$. So, using these definitions of $\phi(v)$ and $\beta$, we arrive at Einstein's result $$\xi = \phi(v) \beta \left( x-ct \right).$$The results for $\tau, \eta$, and $\zeta$ can be obtained in a similar manner.

Einstein later argues that the unknown function $\phi(v)$ is actually independent of $v$ and equals 1. At this point, he has finally finished his derivation of the LT equations given at the bottom of page 9.

Thank you very much for talking to the point, but there's a problem.
I cannot see where $\phi(v)$ denotes $\frac{a}{\sqrt{1-v^2/c^2}}$ Maybe you refer to a different edition, or a later issue of the article? As far as I can see, in the article at my link it is universally $\phi(v) = a$
Certainly, $\phi(v)=\frac{a}{\sqrt{1-v^2/c^2}}$ would solve the problem. But how would one justify such definition of $\phi(v)$? I don't see it can be substantiated - maybe you can help me here.

 

[DmitryS]

It's just algebra to derive the Lorentz transforms from the setup Einstein gives (as @TSny has just laid out). I find Einstein's derivation a bit messy because he has these intermediate values with Greek letters, but it's easy enough to do something cleaner.

Assert that everything happens in the $z=0$ plane. Consider two light pulses from the origin, one parallel to $x$ and one parallel to $y$, let them reflect off two mirrors equidistant from the origin and return simultaneously to the origin. Write down the coordinates, $(x,y,t)$ and $(x',y',t')$, of the two reflection events and the return event in two frames, one where the experimental gear is stationary and one where it is moving in the $+x$ direction at constant speed. Equate the primed coordinates of each event to the unprimed coordinates after applying the general linear transform. The general linear transform (a 3$\times$3 matrix) has nine parameters and you have nine constraints. Solve for the parameters. You then have a bit more work to do because you can't assume that the distances to the mirrors in the primed frame are the same as they are in the unprimed one (and, indeed, one turns out not to be the same), but noting that if the forward transform is $\Lambda(v)$ the inverse transform must be $\Lambda(-v)$ and hence $\Lambda(v)\Lambda(-v)=\mathbf{I}$ gets you out of that. Finally you observe from rotational symmetry that the $z$ transform must be the same as the $y$ transform.

It's worth grinding through it once.

That's something, thanks.

 

[DmitryS]

Anyone who posts something like this:Is just wasting everyone's time. Every competent student of SR in the last 100 years has derived the LT themselves from only the invariance of the speed of light.

You really don't see the difference between "deriving the LT from the invariance of the speed of light" and "deriving the LT from Einstein's setup"?

And please, don't bring up that wasting somebody's time thing again. I hope participation in this forum is not mandatory. I wouldn't take offence at all if nobody wrote a reply to my post. That's life. I feel grateful to you for all your effort and to anyone else who participated. Sometimes it seemed to me that you either ignored my questions or took them for something which they weren't - well, maybe it was partly my fault. Maybe I needed to word them in a different way. That's life again. Thank you for your time and effort.

 

[DmitryS]

It's worth grinding through it once.

I never thought of that algebra from this perspective, that you could actually set it as an experiment with light.

 

[Ibix]

I never thought of that algebra from this perspective, that you could actually set it as an experiment with light.

That's why he says "[f]rom the origin of system k let a ray be emitted at the time $\tau_0$ along the
X-axis".

 

[DrGreg]

Thank you very much for talking to the point, but there's a problem.
I cannot see where $\phi(v)$ denotes $\frac{a}{\sqrt{1-v^2/c^2}}$ Maybe you refer to a different edition, or a later issue of the article? As far as I can see, in the article at my link it is universally $\phi(v) = a$
Certainly, $\phi(v)=\frac{a}{\sqrt{1-v^2/c^2}}$ would solve the problem. But how would one justify such definition of $\phi(v)$? I don't see it can be substantiated - maybe you can help me here.

As commented in post #75, $a$ is initially described as "$\text{a function }\phi(v)$", but he later redefines $\phi$ implicitly by the four equations for $\tau$, $\xi$, $\eta$ and $\zeta$ near the bottom of page 7 which you must compare with the previous equations for those four quantities. E.g. compare $$\zeta=a \frac{c}{\sqrt{c^2-v^2}}z$$ with $$\zeta=\phi(v)z.$$It would have been better if he'd used two different symbols instead of using $\phi$ with two different meanings without explicitly saying so.

 

[Sagittarius A-Star]

I cannot see where $\phi(v)$ denotes $\frac{a}{\sqrt{1-v^2/c^2}}$ Maybe you refer to a different edition, or a later issue of the article? As far as I can see, in the article at my link it is universally $\phi(v) = a$

There exists a German site about the Einstein 1905 paper with comments and explanations. If you are using Chrome and right-click on one of the text parts with comments and explanations, then you can get it translated to English.

2. Einstein introduces a new function that is not identical to the function $\varphi( v ) = a$ mentioned on page 899 . Rather is:

$$ \varphi (v)=\frac {a}{\sqrt {1-(\frac {v}{V})^{2}}}$$

Source:
https://de.wikibooks.org/wiki/A._Ei...namik_bewegter_Körper:_Kinematischer_Teil:_§3

 

[TSny]

I cannot see where $\phi(v)$ denotes $\frac{a}{\sqrt{1-v^2/c^2}}$ Maybe you refer to a different edition, or a later issue of the article? As far as I can see, in the article at my link it is universally $\phi(v) = a$
Certainly, $\phi(v)=\frac{a}{\sqrt{1-v^2/c^2}}$ would solve the problem. But how would one justify such definition of $\phi(v)$? I don't see it can be substantiated - maybe you can help me here.

There's nothing much going on here. Suppose Einstein had used the notation $\bar{\phi}(v)$ instead of $\phi(v)$ at the bottom of page 6. That is, suppose he had written:

Then, we are at liberty to define a new function of $v$ which we may denote $\phi(v)$ as follows:

$\large \phi(v) = \frac{a}{\sqrt{1-v^2/c^2}} = \frac{\bar{\phi}(v)}{\sqrt{1-v^2/c^2}}$.

This is the $\phi(v)$ that Einstein uses at the bottom of page 7. Since $\bar{\phi}(v)$ denoted "a function at present unknown", $\phi(v)$ can also be taken to be "a function at present unknown". Einstein proceeds to show that $\phi(v) = 1$. That means that $a$ as a function of $v$ turns out to be

$a = \bar{\phi}(v) = \sqrt{1-v^2/c^2}$.

 

[PeterDonis]

I wrote the opposite

I know what you wrote; I am just saying that I think what you wrote misdescribes the correct reasoning. But this is really tangential to the main topic of the thread.

 

[PeterDonis]

I don't agree that he assumes what you say, or rather, that he only assumes what you say, but let that be for a moment.

No, let's not let that be. If you disagree, then what do you think Einstein's assumptions are? If we can't agree on that initial point, how can you expect to have a useful discussion of the rest of the paper?

as I was saying, the starting point of his derivation is the same setup as that of the simultaneity test

Sort of. But even if it is, that doesn't mean Einstein is assuming anything about simultaneity in order to derive the LT. He just happens to be using a similar scenario.

that setup leads to some transforms, which are not LT

I realize you've been exchanging posts with others who have tried to correct you here, but just to add my input to that discussion, at the end of Section 3, he obtains final transformation equations which most certainly are the LT. I did not spell out every single step of his reasoning because (a) you were asking mainly about his assumptions, and (b) I assumed you would be able to apply your intelligence to see the rest without my having to walk you through every step.

I just said that in this paper there is no derivation of LT. You said I was wrong. Now you say I was right?

No. He is just saying that the derivation of the LT in Einstein's 1905 paper is not the most efficient derivation that we now know of, after more than a century of having other people look at the issue and come up with more efficient derivations. But that in no way means that Einstein's derivation is wrong. It's not.

 

[PeterDonis]

I think I made my point abundantly clear from the start.

I don't. This thread started out with you asking about a common scenario that is used to illustrate relativity of simultaneity; then it morphed into you claiming that Einstein's 1905 paper somehow does not derive the Lorentz transformations. So I think you are mistaken if you believe your point is clear to others who are in this discussion. I think you need to clarify what, exactly, you want to discuss.

 

[vanhees71]

There is an issue if you take Einstein's 1905 paper as some sort of Gospel. It was the first paper on SR, but the theory has been developed and refined for over 100 years since then. I love the 1905 paper, but it's not an ideal source from which to learn SR as a student. Your confusions bear this out to some extent.

I'd say, the kinematic part is a masterpiece. It derives the Lorentz transformation by very plausible physical arguments from the two empirical foundations of RT, i.e., the special principle of relativity and the independence of the phase velocity of em. waves of the motion of their source. Also the electromagnetical part (LT of fields, Maxwell's equations, aberration and Doppler shift for em. waves in vacuo) are ok but hard to read since we are used to the vector notation rather than writing out all equations for components. The only thing I'd say hasn't been fully understood in this paper is point-particle mechanics, which was correctly formulated for the first time (afaik) by Planck in 1906. The unfortunate result is that the idea of "relativistic mass" has been established in the 1905 paper and is still not eliminated from the textbook literature (particularly in high-school textbooks).

I don't want to get into a game of having to justify everything Einstein wrote in 1905. We all see flaws in that paper, when it comes to it.

But, an incompatibility between the LT and RoS? That's just nonsensical. The LT encapsulates the RoS along with time dilation and length contraction. That I can prove, but not by dissecting the 1905 paper.

PS both @vanhees and I have proved the compatibility of the LT and RoS in various posts in this thread.

I think, now we have all three types of train gedanken experiments in this thread:

(a) Signals sent from both ends of the train simultaneously wrt. the embankment rest frame (Einstein's original one in "The meaning of relativity").
(b) Signals sent from both ends of the train simultaneously wrt. the train rest frame.
(c) One signal sent from the center of the train.

I think that (c) is the best one. The most straight-forward way to analyze either of them is the use of the LT, because the LT is the result of the Einstein synchronization convention or simply the realization of the symmetry transformations of Minkowski spacetime (proper orthochronous Poincare transformations) in the most convenient coordinates (Lorentzian ones).

For illustration you can also draw the Minkowski diagrams for all three cases. It's a very good exercise in learning, how to read those diagrams.

 

[Peter Strohmayer]

Einstein's thought experiment is confusing because S is supposed to "observe" two light pulses at the platform, which S' has emitted in the middle of the train. The similarity with tennis balls is too great.
It is better to imagine that S and S' each emit a light pulse in different directions when they meet, so that the two photons at the tips of the light pulses propagate together (no photon can overtake another).
The signal path of the light pulse of S (from the point of view of S) can never be of the same length as the signal path of the light pulse of S' (from the point of view of S'), because although the photons at the tips of the light pulses are always on the same level, the observers S and S' have moved away from each other. For each observer, the length of his signal path is also the length of his signal time (light-like distance).
Thus, if the signals arrive at the ends of the train at the same time from the point of view of S' when the signal paths are of equal length (simultaneous events), then they cannot possibly arrive at the same time from the point of view of S when the signal paths are of unequal length (non-contemporaneous events).
This is sufficient to derive the L-T.

Peter Strohmayer

 

[PeterDonis]

the two photons at the tips of the light pulses propagate together

What do you mean by this? The photons are propagating in different directions so they are never "together" after the point of emission.

the photons at the tips of the light pulses are always on the same level

What do you mean by "on the same level"?

 

[Peter Strohmayer]

At their encounter, both S and S' emit a light pulse to the left along the x-axis. These photons propagate together. The same happens to the right.

 

[PeterDonis]

At their encounter, both S and S' emit a light pulse to the left along the x-axis. These photons propagate together. The same happens to the right.

There is no need to have two photons going in each direction. The meeting between S and S' is a single event, i.e., a single point in spacetime. Emitting a single photon in a given direction from a single event is sufficient. This follows from the postulate of the constancy of the speed of light in any inertial frame, which is being assumed for the problem. Anything you say about a given pair of photons propagating together in your version, can also be said about a single photon in the standard version.

 

[Peter Strohmayer]

That is correct. The difference is in didactics. That seems to have been the problem in this thread.
If only S' emits a photon (event 1) which subsequently arrives at the end of the train (event 2), then from the point of view of observer S, the coordinates of events 1 and 2 can only be determined via the L-T.
If, on the other hand, each of the observers emits a photon at origin coverage, which propagate together, then it becomes clear why the coordinates of an event also differ in time. Then one can derive the L-T instead of presupposing it.

 

[Ibix]

If only S' emits a photon (event 1) which subsequently arrives at the end of the train (event 2), then from the point of view of observer S, the coordinates of events 1 and 2 can only be determined via the L-T.

Huh? It's just an intercept caculation in either frame (albeit a trivial one in one frame). You need to use the invariance of the speed of light, sure, but that's how you derive the Lorentz transforms.

 

[Peter Strohmayer]

I do not dispute the possibility of a purely mathematical derivation of the L-T in the standard situation. However, I think that the intuitive access, which Einstein's thought experiment shall open, is better served with two complementary photons. The connection with the finiteness of the causal propagation becomes clear. The identity of signal path and (signal)time between two events becomes conscious.

 

[Ibix]

I do not dispute the possibility of a purely mathematical derivation of the L-T in the standard situation. However, I think that the intuitive access, which Einstein's thought experiment shall open, is better served with two complementary photons. The connection with the finiteness of the causal propagation becomes clear. The identity of signal path and (signal)time between two events becomes conscious.

Seems like an unnecessary complication to me, but if it works for you that's fine. It's a rather different claim from "[t]hen one can derive the L-T instead of presupposing it", which you made earlier.

 

[Peter Strohmayer]

Yes, that was a bit inaccurate, I should have written "instead of presupposing it mathematically."

 

[Ibix]

Yes, that was a bit inaccurate, I should have written "instead of presupposing it mathematically."

That's still wrong. You aren't pre-supposing the Lorentz transforms whether you use light pulses emitted only by the train observer or pulses emitted by both train and embankment observers. All the latter does is reinforce the consequence of the invariance of the speed of light by making the case that the parallel pulses should travel side-by-side.

 

[Peter Strohmayer]

I have assumed the L-T for the conversion of the coordinates of only one light pulse (#93). The L-T must be derived mathematically before. I doubt like DmitryS that Einstein's thought experiment (the "observation" of a light pulse) is helpful for it. But you are welcome to see it differently.

 

[PeterDonis]

If only S' emits a photon (event 1) which subsequently arrives at the end of the train (event 2), then from the point of view of observer S, the coordinates of events 1 and 2 can only be determined via the L-T.

This is not correct. Whether the photon is emitted by S' or S is irrelevant, since, as I have already pointed out, the emission event is a single event (a single point in spacetime), namely, the event at which the worldlines of S and S' cross. So each photon is traveling at $c$ in a known direction from a known starting point, and the ends of the train are traveling at a known speed $v$ in a known direction from known starting points. This is true in both frames, so the coordinates of the reception events can be calculated in both frames independently, without having to make use of any transformation whatsoever.