# Revisiting the lesson of the Relativity of Simultaneity

## Main Question or Discussion Point

Revisiting the lesson of the "Relativity of Simultaneity"

So, we have the "relativity of simultaneity", which is meant to show us that two relatively moving observers have different notions of the concept of "time", because (as in chapter 9 of Relativity), the person on the moving train will witness the lightning flashes in a different sequence than the person who is standing on the embankment. (The one who is on the train sees the forward flash before the rear flash, while the one who is on the embankment sees both flashes at the same moment.)

The problem with this logic is that two relatively stationary observers (i.e. they exist within the same reference-frame) at different locations on the embankment may also witness lightning flashes in a different sequence. Say the strikes occur 100 meters apart, and each observer is located one meter away from each strike location. Well, it stands to reason, that, given any non-infinite, constant value for the speed of the propagation of light, each person will say that the flash event to which he is standing the closest happens before the other flash event.

The point that I am trying to make here is that Einstein's thought experiment, used to demonstrate the different "meanings of time" for two relatively moving observers, does not ultimately make use of the relative motions between observers, but rather of the relative positions between observers and events (such as lightning strikes). In other words, in Einstein's thought experiment, the guy on the train only witnesses a different sequence of events because he happens to be closer to the forward flash-event, than is the guy on the embankment, at the moment that the oncoming wavefront crosses his line of sight.

We can also easily rig Einstein's scenario so that both guys see both flashes simultaneously, which also negates the intended lesson of the entire chapter. (I leave this as a challenge to you.)

The only lesson that I've learned from all of this is not that motion dictates how one understands "time", but only position!

Or, perhaps we should stop confusing the concepts of "temporal sequence" and "time"! Only the former concept lies within the realm of scientific scrutiny. The latter is purely a philosophical question. I particulary like Kant's definition of time: it is the form of the internal intuition. I would just say that time is given by nothing other than the sensation of continuousness. That is, time does not change: it only endures.

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JesseM
The problem with this logic is that two relatively stationary observers (i.e. they exist within the same reference-frame) at different locations on the embankment may also witness lightning flashes in a different sequence. Say the strikes occur 100 meters apart, and each observer is located one meter away from each strike location. Well, it stands to reason, that, given any non-infinite, constant value for the speed of the propagation of light, each person will say that the flash event to which he is standing the closest happens before the other flash event.
When you see two events is different from when they happened in your frame. If I see an event in 2005 which occurred 5 light-years away in my frame, and I see another event in 2010 which occurred 10 light-years away in my frame, I retrospectively judge them to have happened "simultaneously" in 2000 in my frame. Two observers at rest with respect to one another will both agree on whether two events happened simultaneously or not once they factor out the signal delays, even if one of them saw the events simultaneously and the other didn't. The point Einstein is making that if observers in relative motion do the same thing, each "factoring out the signal delay" by assuming that the signal moved at c in their own rest frame, they will not necessarily agree on whether two events are simultaneous or not.

The point Einstein is making that if observers in relative motion do the same thing, each "factoring out the signal delay" by assuming that the signal moved at c in their own rest frame, they will not necessarily agree on whether two events are simultaneous or not.
Okay, so the ultimate point here is that two relatively moving observers will always measure the speed of light to be the same, causing us to have to play around with the "speed" of relatively moving clocks. That only leads us back to the original problem that I had with this thought experiment. You didn't answer my most recent post in that previous thread.

JesseM
Okay, so the ultimate point here is that two relatively moving observers will always measure the speed of light to be the same, causing us to have to play around with the "speed" of relatively moving clocks. That only leads us back to the original problem that I had with this thought experiment. You didn't answer my most recent post in that previous thread.
Sorry, I didn't catch that post, you posted it a week when I was away on a vacation and didn't have much computer access. I've responded to that post now.

Aether
Gold Member
Only the former concept lies within the realm of scientific scrutiny. The latter is purely a philosophical question.
Although the reasoning in your original post isn't right, your conclusion is not far wrong. The standard formulation of special relativity really is a mixture of nonconventional concepts (e.g., "lies within the realm of scientific scrutiny") and conventional concepts (e.g., "purely a philosophical question").

See: http://www.science.uva.nl/~seop/entries/spacetime-convensimul/#Rel]
Stanford Encyclopedia of Philosophy: Conventionality of Simultaneity said:
The debate about conventionality of simultaneity seems far from settled, although some proponents on both sides of the argument might disagree with that statement.
I particulary like Kant's definition of time: it is the form of the internal intuition.
In physics, "time" is "what a clock measures".

Okay, so the ultimate point here is that two relatively moving observers will always measure the speed of light to be the same, causing us to have to play around with the "speed" of relatively moving clocks.
The "relativity of simultaneity" arises not by measuring the speed of light to be the same, but "...by assuming that the [one-way] signal moved at c in their own rest frame...". It's okay to make this assumption, and often convenient, but it's wrong to confuse that with making a measurement.

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JesseM
Although the reasoning in your original post isn't right, your conclusion is not far wrong. The standard formulation of special relativity really is a mixture of nonconventional concepts (e.g., "lies within the realm of scientific scrutiny") and conventional concepts (e.g., "purely a philosophical question"). See http://www.science.uva.nl/~seop/entries/spacetime-convensimul/#Rel.
"convention" just means it's a choice you make about how to construct your coordinate systems, it doesn't mean it's a philosophical question, any more than the question of where the place the origin of your spatial coordinate axes is a philosophical question. The key to why this convention is a useful one lies in a symmetry of the laws of physics--all known fundamental laws obey Lorentz symmetry, meaning they will have the same equations in the different inertial coordinate systems which are related by the Lorentz transformation.
Aether said:
The "relativity of simultaneity" arises not by measuring the speed of light to be the same, but "...by assuming that the [one-way] signal moved at c in their own rest frame...".
This is not meant to be a physical assumption, just a choice about how to construct each inertial coordinate system. Again, the utility of this choice lies in the Lorentz symmetry of the laws of physics.

Aether
Gold Member
"convention" just means it's a choice you make about how to construct your coordinate systems, it doesn't mean it's a philosophical question, any more than the question of where the place the origin of your spatial coordinate axes is a philosophical question.
That's true when the choice is made as an informed choice; e.g., when the speed of light assumption is not confused with a measurement.

The key to why this convention is a useful one lies in a symmetry of the laws of physics--all known fundamental laws obey Lorentz symmetry, meaning they will have the same equations in the different inertial coordinate systems which are related by the Lorentz transformation.
Correct, but as you know, Lorentz symmetry is a "symmetry of the laws of physics", and not necessarily a "symmetry of nature".

This is not meant to be a physical assumption, just a choice about how to construct each inertial coordinate system. Again, the utility of this choice lies in the Lorentz symmetry of the laws of physics.
I know that you are careful to teach this, thank-you.

JesseM
Correct, but as you know, Lorentz symmetry is a "symmetry of the laws of physics", and not necessarily a "symmetry of nature".
I wouldn't say I know that at all, in fact I have no idea what you mean by drawing a distinction between symmetries of nature and symmetries of the laws of physics (laws of nature). This does not bear any resemblance to the terminology physicists use when talking about symmetries, as far as I know. Can you elaborate? Would you say the various physical symmetries listed here, such as spatial translation symmetry (the observation that the laws of physics don't vary from one position in space to another), are not "symmetries of nature" for example?

Janus
Staff Emeritus
Gold Member
We can also easily rig Einstein's scenario so that both guys see both flashes simultaneously, which also negates the intended lesson of the entire chapter. (I leave this as a challenge to you.)
Okay, let's do so. We'll arrrange it such that the flashes arrive at the station at the moment that the observer on the train passes the station, so that both observers "see" the flashes simultaneously.

It is obvious that the station observer will determine that the lightning strikes occured simultaneously.

Not so for the train observer. Since it takes a finite time for the light to travel from the point of the strikes to his eyes, and he is moving relative to those points, he will be closer to one "strike point" then he is to the other when the strikes actually occur. If he is closer to one strike than he is to the other, then it takes less time for the light from that strike to reach him than it does for the light of the other strike. (remember, it is his relative position with respect to the strikes when they occur that dtermines how long it takes the light to reach him, not his position when he "sees' the flashes.)

We have already established that he sees the flashes at the same time, so if takes less time for the light from one strike to travel the distance between the strike and his eye then it does for the light form the other strike to travel its distance, then one strike has to happen before the other for both lights to arrive at the same moment.

Thus the station observer determines that the flashes reaching his eyes originated simultaneously, and the observer determines that they orginated at different times.

Aether
Gold Member
I wouldn't say I know that at all, in fact I have no idea what you mean by drawing a distinction between symmetries of nature and symmetries of the laws of physics (laws of nature). This does not bear any resemblance to the terminology physicists use when talking about symmetries, as far as I know. Can you elaborate?
Okay, I withdraw that statement. I was thinking of what you said below, but I didn't notice at first that the transformation you gave is empirically wrong (intentionally so), sorry.
That's just because the laws of physics don't have the property of being symmetric relative to this coordinate transformation, this is not one of the symmetries of nature.
Would you say the various physical symmetries listed here, such as spatial translation symmetry (the observation that the laws of physics don't vary from one position in space to another), are not "symmetries of nature" for example?
This article describes a symmetry of a physical system as not just a physical feature of the system, but as "a physical or mathematical feature of the system (observed or intrinsic)". I would expect that a physical feature of a system might qualify as a "symmetry of nature", but not a non-physical mathematical feature of a system.

Symmetry in Physics said:
A symmetry of a physical system is a physical or mathematical feature of the system (observed or intrinsic) that is "preserved" under some change.
What terminology would you use to distinguish between "a symmetry of a physical system that is a physical feature of the system" and "a symmetry of a physical system that is a mathematical feature of the system"?

JesseM
This article describes a symmetry of a physical system as not just a physical feature of the system, but as "a physical or mathematical feature of the system (observed or intrinsic)". I would expect that a physical feature of a system might qualify as a "symmetry of nature", but not a non-physical mathematical feature of a system.
The article isn't talking about symmetries of physical systems at all (like a solid sphere which will exhibit rotation symmetry along any axis through its center), but rather symmetries in the laws of nature which govern all systems. Lorentz symmetry is one example (the laws of physics follow the same equations in different inertial coordinate systems related by the Lorentz transformation) and spatial translation symmetry is another (the laws of physics follow the same equations in different coordinate systems where the origin of your coordinate system is shifted, which is another way of saying the laws of physics don't vary from one location to another).
Aether said:
What terminology would you use to distinguish between "a symmetry of a physical system that is a physical feature of the system" and "a symmetry of a physical system that is a mathematical feature of the system"?
I don't understand this distinction either. For example, the fact that a solid sphere will appear unchanged under any rotation about an axis that goes through its center appears to be both a "physical feature" and a "mathematical feature". Can you give an example of a symmetry which you would count as one but not the other?

Aether
Gold Member
I don't understand this distinction either. For example, the fact that a solid sphere will appear unchanged under any rotation about an axis that goes through its center appears to be both a "physical feature" and a "mathematical feature". Can you give an example of a symmetry which you would count as one but not the other?
Two-way light speed invariance is both a mathematical feature of the laws of physics, and a physical feature of nature; but although one-way light speed invariance is also a mathematical feature of the (standard) laws of physics, it is not necessarily a physical feature of nature.

JesseM
Two-way light speed invariance is both a mathematical feature of the laws of physics, and a physical feature of nature; but although one-way light speed invariance is also a mathematical feature of the (standard) laws of physics, it is not necessarily a physical feature of nature.
Two-way light speed invariance doesn't depend on your simultaneity convention, but it still seems to depend on the assumption that you're moving inertially and that you're using a coordinate system where the coordinate distance that the light covers (which you need to calculate distance/time) is equal to the distance as measured on an inertial ruler that's at rest relative to you, and the coordinate time for the light to leave you and come back is equal to the time measured on a clock at rest to you and next to you. One could find non-inertial coordinate systems where the two-way speed is not invariant.

It seems to me that any coordinate-dependent statement about the laws of physics can qualify as a genuine physical feature of nature as long as you specify what coordinate system (or family of coordinate systems) the statement is meant to apply to. For example, "in any inertial coordinate system based on local readings of inertial clocks and rigid rulers at rest with respect to one another, the two-way speed of light is constant" is a genuine physical statement about the laws of nature, even though "the two-way speed of light is constant" alone would not be (since we can imagine a universe where this doesn't hold in all inertial coordinate systems but it does hold in one, like the classical universe with an aether theory of electromagnetism, and this universe would clearly be physically different from our own). Likewise, even though "the laws of physics are the same in the family of inertial coordinate systems related by the Lorentz transform" and "the the laws of physics are the same in the family of inertial coordinate systems related by the Galilei transform" are both coordinate-dependent statements, they can't both be true in the same universe, and thus learning that one is true and the other false is actually telling you some physical facts about the universe you live in.

Aether
Gold Member
Two-way light speed invariance doesn't depend on your simultaneity convention, but it still seems to depend on the assumption that you're moving inertially and that you're using a coordinate system where the coordinate distance that the light covers (which you need to calculate distance/time) is equal to the distance as measured on an inertial ruler that's at rest relative to you, and the coordinate time for the light to leave you and come back is equal to the time measured on a clock at rest to you and next to you. One could find non-inertial coordinate systems where the two-way speed is not invariant.
When a Michelson interferometer is used to measure two-way light speed invariance (isotropy), it isn't necessary to assume "that you're using a coordinate system where the coordinate distance that the light covers (which you need to calculate distance/time) is equal to the distance as measured on an inertial ruler that's at rest relative to you, and the coordinate time for the light to leave you and come back is equal to the time measured on a clock at rest to you and next to you".

It seems to me that any coordinate-dependent statement about the laws of physics can qualify as a genuine physical feature of nature as long as you specify what coordinate system (or family of coordinate systems) the statement is meant to apply to.
I agree that any such statement can qualify as being consistent with some more general physical feature of nature, or experiment, but it is also possible that the exact opposite of this statement can also so qualify; so, I don't agree that both statements can simultaneously qualify as a genuine physical feature of nature. For example, both "one-way light speed is isotropic" and "one-way light speed is generally anisotropic" can't both simultaneously qualify as genuine physical features of nature, yet these two statements are empirically indistinguishable in terms of the nonconventional content of special relativity.

For example, "in any inertial coordinate system based on local readings of inertial clocks and rigid rulers at rest with respect to one another, the two-way speed of light is constant" is a genuine physical statement about the laws of nature, even though "the two-way speed of light is constant" alone would not be (since we can imagine a universe where this doesn't hold in all inertial coordinate systems but it does hold in one, like the classical universe with an aether theory of electromagnetism, and this universe would clearly be physically different from our own).
Since a Michelson interferometer can be used to measure two-way light speed invariance (isotropy), we do know that "the two-way speed of light is constant (isotropic)" is a genuine physical statement about the laws of nature. Although we can imagine a universe where "the two-way speed of light is constant" doesn't hold, the Michelson interferometer is well able to rule this out (to a high but still finite precision) as being a genuine physical statement about the laws of nature. Inertial coordinate systems are not relevant in experiments conducted with a Michelson interferometer.

Likewise, even though "the laws of physics are the same in the family of inertial coordinate systems related by the Lorentz transform" and "the the laws of physics are the same in the family of inertial coordinate systems related by the Galilei transform" are both coordinate-dependent statements, they can't both be true in the same universe, and thus learning that one is true and the other false is actually telling you some physical facts about the universe you live in.
These two statements are more than simply both coordinate-dependent statements, they are empirically distinguishable statements; that is why they can't both be true in the same universe. However, "the laws of physics are the same in the family of inertial coordinate systems related by the Lorentz transform" and "the laws of physics are the same in the family of coordinate systems related by the GGT transform" are also both coordinate-dependent statements, but since they are also empirically equivalent statements they can both be true in the same universe; learning that one is true and the other false would also actually tell you some physical facts about the universe you live in, but that would be inconsistent with the nonconventional content of special relativity.

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JesseM
When a Michelson interferometer is used to measure two-way light speed invariance (isotropy), it isn't necessary to assume "that you're using a coordinate system where the coordinate distance that the light covers (which you need to calculate distance/time) is equal to the distance as measured on an inertial ruler that's at rest relative to you, and the coordinate time for the light to leave you and come back is equal to the time measured on a clock at rest to you and next to you".
Sure it's necessary. Do you think a non-inertial coordinate system in which the two waves had different two-way speeds would disagree that the two waves were in phase when they rejoined? All coordinate systems will agree about all local events--after all, different coordinate systems are just different ways of labeling local events, and switching from one coordinate system to another is just a matter of applying the appropriate transformation to these labels.

As a very simple example, if the arms of the interferometer are oriented along the x and y axes of an inertial coordinate system in SR, define a new x',y',t' coordinate system using the following transformation:

x' = x
y' = 2y
t' = t

In this new coordinate system, the light wave moving along the y' axis will have a two-way speed that's twice the speed of the wave moving along the x' axis.
JesseM said:
It seems to me that any coordinate-dependent statement about the laws of physics can qualify as a genuine physical feature of nature as long as you specify what coordinate system (or family of coordinate systems) the statement is meant to apply to.
Aether said:
I agree that any such statement can qualify as being consistent with some more general physical feature of nature, or experiment, but it is also possible that the exact opposite of this statement can also so qualify
Huh? You think even if you specify the coordinate system you are using, both some statement and its exact opposite can be true in that coordinate system? Or are you just saying that a statement which is true in one coordinate system can be false in another? If the latter, as long as you specify the context of the coordinate system for each statement, the two statements are not really "opposites" at all because the context is different in each case, and when you take the context into account you see they are physically consistent.
Aether said:
so, I don't agree that both statements can simultaneously qualify as a genuine physical feature of nature. For example, both "one-way light speed is isotropic" and "one-way light speed is generally anisotropic" can't both simultaneously qualify as genuine physical features of nature
But you didn't specify the context of what coordinate system each statement was supposed to be true in, which was my whole point in saying "any coordinate-dependent statement about the laws of physics can qualify as a genuine physical feature of nature as long as you specify what coordinate system (or family of coordinate systems) the statement is meant to apply to." For example, the statement "in an inertial coordinate system A constructed according to the standard SR procedure (inertial rulers and clocks at rest wrt one another, clocks synchronized by Einstein synchronization convention), the two-way speed of light in the y-direction is the same as the two-way speed of light in the x-direction" would be a genuine physical feature of nature, and the statement "in a coordinate system B constructed by applying the transformation x'=x, y'=2y, t'=t to the previous coordinate system A, the two-way speed of light in the y'-direction is twice the two-way speed of light in the x'-direction" would also be a genuine physical feature of nature, these two statements describe exactly the same physical feature, just from the perspective of two different well-defined coordinate systems.

In fact it's impossible to describe any "physical feature of nature" that involves position, time, or some function of these like speed or acceleration, without referring to some coordinate system or another. If you think otherwise (as you seemed to suggest with your interferometer example), then you're misunderstanding something.
Aether said:
Since a Michelson interferometer can be used to measure two-way light speed invariance (isotropy), we do know that "the two-way speed of light is constant (isotropic)" is a genuine physical statement about the laws of nature. Although we can imagine a universe where "the two-way speed of light is constant" doesn't hold, the Michelson interferometer is well able to rule this out (to a high but still finite precision) as being a genuine physical statement about the laws of nature. Inertial coordinate systems are not relevant in experiments conducted with a Michelson interferometer.
Statements like "the two-way speed of light is constant" are not meaningful without a context of a coordinate system or family of coordinate systems (usually it's just assumed implicitly that we're talking about coordinate systems based on inertial rulers and clocks, and of course in the case of two-way speed we don't have to worry about clock synchronization since a single clock will suffice).
Aether said:
These two statements are more than simply both coordinate-dependent statements, they are empirically distinguishable statements; that is why they can't both be true in the same universe.
But if you agree they are empirically distinguishable, doesn't that mean they must both be physical statements about the universe?
Aether said:
However, "the laws of physics are the same in the family of inertial coordinate systems related by the Lorentz transform" and "the laws of physics are the same in the family of coordinate systems related by the GGT transform" are also both coordinate-dependent statements, but since they are also empirically equivalent statements they can both be true in the same universe; learning that one is true and the other false would also actually tell you some physical facts about the universe you live in, but that would be inconsistent with the nonconventional content of special relativity.
I'm not familiar with the GGT transform, can you give the equations, or a link? And are the two statements "empirically equivalent" in the same sense as my statements about the two-way speed of light in coordinate systems A and B above, where the two statements actually have identical meaning (if you know one is true and you know the coordinate transformation that relates A and B, that automatically tells you the other must be true)? If so, wouldn't it be logically impossible for one to be true and the other false?

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DrGreg
Gold Member
I'm not familiar with the GGT transform, can you give the equations, or a link?
I believe the GGT is the "Generalised Galilean Transform", another (rarely-used) name for the "Selleri" or "Tanglerhini" or "Mansouri-Sexl" coords that have been discussed before (and very recently in threads between Bernhard Rothenstein and myself).

See, for example, this post which defines a family of coordinate systems $(t_S(I), x_S(I))$ (parameterised by inertial observer I), which are related to each other via

$$x_S(I) = \gamma_{v(I)}(x_S(R) - v(I) t_S(R))$$......(1)
$$t_S(I) = \frac{t_S(R)}{\gamma_{v(I)}}$$.................................(2)​

where R is a reference ("ether") coordinate system in which light speed is isotropic.

Equations (1) and (2) are the equivalent of the Lorentz transform re-expressed in the new coordinate system.

Now, in my book, I would say, in this coordinate system, that equations (1) and (2) actually are the Lorentz transform. And I would say that a system in which equations (1) and (2) are true possesses Lorentz symmetry. If you don't like using the word "Lorentz" in this context, you could say it possesses "GGT symmetry" or "Tanglerhini symmetry". It's still, technically, a "symmetry" even though the equations don't look very symmetric in the alegraic sense. The metric

$$ds^2 = \frac{dx_S(I)^2}{\gamma_{v(I)}^2} - c^2 dt_S(I)^2 + 2v(I) dx_S(I) dt_S(I)$$​

(see this thread) is invariant and that's why it's a "symmetry".

JesseM
See, for example, this post which defines a family of coordinate systems $(t_S(I), x_S(I))$ (parameterised by inertial observer I), which are related to each other via

$$x_S(I) = \gamma_{v(I)}(x_S(R) - v(I) t_S(R))$$......(1)
$$t_S(I) = \frac{t_S(R)}{\gamma_{v(I)}}$$.................................(2)​

where R is a reference ("ether") coordinate system in which light speed is isotropic.

Equations (1) and (2) are the equivalent of the Lorentz transform re-expressed in the new coordinate system.
But it seems to me that the laws of physics would not obey the same equations in this family of coordinate systems--after all, from the time coordinate transformation it seems like all frames will agree about simultaneity, but doesn't that mean that if the speed of light is isotropic in the R ('aether') frame, it will have different coordinate speeds in different directions in the I frame? That would mean you could no longer use Maxwell's equations to describe the laws of electromagnetism in the I frame.
DrGreg said:
Now, in my book, I would say, in this coordinate system, that equations (1) and (2) actually are the Lorentz transform. And I would say that a system in which equations (1) and (2) are true possesses Lorentz symmetry.
What do you mean by a system in which (1) and (2) "are true"? They are just coordinate transformations, which could be used regardless of the laws of physics of the universe you're living in. Maybe you mean that if each observer used rigid rulers and local clocks at rest relative to themselves to assign coordinates to events, and then if they used a certain synchronization convention which ensured that all frames would agree on simultaneity, then in that case (1) and (2) would correctly transform between their coordinate systems in a universe with relativistic laws? (but not in a universe with Newtonian ones, since in that case there'll be no length contraction of their rulers or time dilation of their clocks)

Mentz114
Gold Member
DrGreg,
why do you think it is necessary for different observers to agree about 'simultaneity' ? The universe functions perfectly well whether or not two observers agree or disagree whether another two distant events were or were not simultaneous by any definition you care to make.

You seem to think simultaneity is a physically important concept - it's not, it is observer dependent and nothing you have said in this or any other thread indicates otherwise.

JesseM
DrGreg,
why do you think it is necessary for different observers to agree about 'simultaneity' ? The universe functions perfectly well whether or not two observers agree or disagree whether another two distant events were or were not simultaneous by any definition you care to make.

You seem to think simultaneity is a physically important concept - it's not, it is observer dependent and nothing you have said in this or any other thread indicates otherwise.
I don't know if DrGreg was actually saying that, he might just have been responding to my question to Aether about the GGT transform. Nothing wrong with pointing out that you can use a family of coordinate systems which agree on simultaneity, although this will be less "elegant" in the sense that I don't think the equations representing the laws of physics can ever be the same in such a family of coordinate systems.

Mentz114
Gold Member
Nothing wrong with pointing out that you can use a family of coordinate systems which agree on simultaneity, although this will be less "elegant" in the sense that I don't think the equations representing the laws of physics can ever be the same in such a family of coordinate systems.
Thanks, Jesse.

I don't agree with your use of 'less elegant'. Such a system seems plain wrong if different observers see different outcomes to the same event.

M

JesseM
Thanks, Jesse.

I don't agree with your use of 'less elegant'. Such a system seems plain wrong if different observers see different outcomes to the same event.
Why would they see a different outcome? As I said earlier:
All coordinate systems will agree about all local events--after all, different coordinate systems are just different ways of labeling local events, and switching from one coordinate system to another is just a matter of applying the appropriate transformation to these labels.
However, it is true that in order to predict the correct outcome in coordinate systems other than those assumed in the Lorentz transformation, you cannot use the same equations for the laws of physics as you would in the Lorentz transform frames (you could find the correct equations in your new coordinate system just by taking the coordinate transform from a Lorentzian frame to your system, and then applying this same coordinate transformation to the equations of the laws of physics as they're expressed in a Lorentzian frame).

Mentz114
Gold Member
Jesse:
But it seems to me that the laws of physics would not obey the same equations in this family of coordinate systems
I obviously misunderstood this. So would electro-dynamics look completely different but still work ? I'm not sure I understand what this thread is about anymore so I'll stay silent.

JesseM
I obviously misunderstood this. So would electro-dynamics look completely different but still work ? I'm not sure I understand what this thread is about anymore so I'll stay silent.
Yes, the equations would be different, but since all you're doing is re-labeling the same set of events with different coordinates and transforming the equations into the new coordinate system, by construction you're guaranteed to get the same physical predictions.

Maybe it would help to look at an example of transforming equations in one coordinate system into another, using the coordinate transformation. I'll use an example I posted a while ago which is meant to show that Newtonian gravity is invariant (i.e. the equation doesn't change) under the Galilei transform:
Here's an example, involving "Galilei-invariance" rather than Lorentz-invariance because the math is simpler. The Galilei transform for transforming between different frames in Newtonian mechanics looks like this:

$$x' = x - vt$$
$$y' = y$$
$$z' = z$$
$$t' = t$$

and

$$x = x' + vt'$$
$$y = y'$$
$$z = z'$$
$$t = t'$$

To say a certain physical equation is "Galilei-invariant" just means the form of the equation is unchanged if you make these substitutions. For example, suppose at time t you have a mass $$m_1$$ at position $$(x_1 , y_1 , z_1)$$ and another mass $$m_2$$ at position $$(x_2 , y_2 , z_2 )$$ in your reference frame. Then the Newtonian equation for the gravitational force between them would be:

$$F = \frac{G m_1 m_2}{(x_1 - x_2 )^2 + (y_1 - y_2 )^2 + (z_1 - z_2 )^2}$$

Now, suppose we want to transform into a new coordinate system moving at velocity v along the x-axis of the first one. In this coordinate system, at time t' the mass $$m_1$$ has coordinates $$(x'_1 , y'_1 , z'_1)$$ and the mass $$m_2$$ has coordinates $$(x'_2 , y'_2 , z'_2 )$$. Using the Galilei transformation, we can figure how the force would look in this new coordinate system, by substituting in $$x_1 = x'_1 + v t'$$, $$x_2 = x'_2 + v t'$$, $$y_1 = y'_1$$, $$y_2 = y'_2$$, and so forth. With these substitutions, the above equation becomes:

$$F = \frac{G m_1 m_2 }{(x'_1 + vt' - (x'_2 + vt'))^2 + (y'_1 - y'_2 )^2 + (z'_1 - z'_2 )^2}$$

and you can see that this simplifies to:

$$F = \frac{G m_1 m_2 }{(x'_1 - x'_2 )^2 + (y'_1 - y'_2 )^2 + (z'_1 - z'_2 )^2}$$

In other words, the equation has exactly the same form in both coordinate systems. This is what it means to be "Galilei invariant". More generally, if you have any physical equation which computes some quantity (say, force) as a function of various space and time coordinates, like $$f(x,y,z,t)$$ [of course it may have more than one of each coordinate, like the $$x_1$$ and $$x_2$$ above, and it may be a function of additional variables as well, like $$m_1$$ and $$m_2$$ above] then for this equation to be "Galilei invariant", it must satisfy:
$$f(x'+vt',y',z',t') = f(x',y',z',t')$$

So in the same way, an equation that's Lorentz-invariant should satisfy:

$$f( \gamma (x' + vt' ), y' , z', \gamma (t' + vx' /c^2 ) ) = f(x' ,y' ,z' , t')$$
Hopefully you can see that if we instead used a different coordinate transform that Newton's gravitational law was not invariant under, we could still use the same procedure to find the correct form of the equations in the new coordinate system, and assuming this gravitational law gave correct predictions in our original coordinate system, the transformed equations would give the same predictions in our new coordinate system.

DrGreg
Gold Member
But it seems to me that the laws of physics would not obey the same equations in this family of coordinate systems--after all, from the time coordinate transformation it seems like all frames will agree about simultaneity, but doesn't that mean that if the speed of light is isotropic in the R ('aether') frame, it will have different coordinate speeds in different directions in the I frame? That would mean you could no longer use Maxwell's equations to describe the laws of electromagnetism in the I frame.
All of the above is absolutely correct. In this S-family of coords, almost all the equations of physics would be modified, and would include v(I) within them.

But the modified form of Maxwell's equations, expressed in S-coords, (I've never worked out what they are), would still represent Maxwell's theory, so I'm asserting you could still describe them as "Maxwell's Equations" (in a sense).

More generally, any statement you can make in Einstein-synced coords corresponds to a statement in S-coords (which may well be much more complicated), so either statement represents the same concept.

What do you mean by a system in which (1) and (2) "are true"? They are just coordinate transformations, which could be used regardless of the laws of physics of the universe you're living in. Maybe you mean that if each observer used rigid rulers and local clocks at rest relative to themselves to assign coordinates to events, and then if they used a certain synchronization convention which ensured that all frames would agree on simultaneity, then in that case (1) and (2) would correctly transform between their coordinate systems in a universe with relativistic laws?
Yes, that is what I mean, and you've phrased it more eloquently than I did.

DrGreg
Gold Member
DrGreg,
why do you think it is necessary for different observers to agree about 'simultaneity' ? The universe functions perfectly well whether or not two observers agree or disagree whether another two distant events were or were not simultaneous by any definition you care to make.

You seem to think simultaneity is a physically important concept - it's not, it is observer dependent and nothing you have said in this or any other thread indicates otherwise.
I don't know if DrGreg was actually saying that, he might just have been responding to my question to Aether about the GGT transform. Nothing wrong with pointing out that you can use a family of coordinate systems which agree on simultaneity, although this will be less "elegant" in the sense that I don't think the equations representing the laws of physics can ever be the same in such a family of coordinate systems.
JesseM has correctly identified my position on this (which is probably not Aether's).

I agree that simultaneity is a not physically important concept -- it's an artificial consequence of imposing a coordinate system. The reason that these GGT coords deserve any consideration at all is just to illustrate that the idea of absolute simultaneity (within the context of a chosen family of coordinate systems) doesn't actually contradict relativistic physics, although it does contradict almost all relativistic equations that are expressed in the standard family of Einstein-synchronised coordinates, or statements that implicitly rely on such coordinates.

For example "speed" is a coordinate-dependent concept, so the assertion that "the one-way speed of light is constant" makes sense only when you know which coordinate systems it applies to. The assertion is true in Einstein-synced coords, but false in GGT coords. (There is an equivalent assertion in GGT coords, but it takes the form of a much more complicated equation.)

I gave some more background to these coords in the original post that I referred to in post #17.

In that post I also gave the relationship between S-coords and the standard Einstein-synchronised "E-coords", viz.

$$x_S(I) = x_E(I)$$.....................(3)
$$t_S(I) = t_E(I) + \frac{v(I) x_E(I)}{c^2}$$....(4)​

These equations are more revealing that the original transform equations (1) and (2) in my last post #17, as they show that the proper length and the proper time of stationary objects in the I-frame are compatible with S-coords, and the only difference is a synchronising offset $v(I) x_E(I)/c^2$. These are the equations you would use to convert any equation of physics from E-coords to S-coords.

To conclude: it's only the pro-ether lobby who would want to use these coords all the time, for the rest of us they are just a curiosity to help illustrate the difference between coord-dependent ("conventional" in the language of synchronisation gauge theory) and coord-independent ("unconventional") statements.