Revisiting the lesson of the Relativity of Simultaneity

In summary, the conversation discusses the concept of the "relativity of simultaneity" and how it demonstrates that two observers in relative motion have different perceptions of time. However, the problem with this logic is that even two stationary observers at different locations can witness events in a different sequence, solely based on their position. This leads to the conclusion that only position, not motion, dictates one's understanding of time. The conversation also touches on the idea of conventionality of simultaneity and the definition of time in physics.
  • #1
joey_m
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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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  • #2
joey_m said:
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.
 
  • #3
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.
 
  • #4
joey_m said:
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.
 
  • #5
joey_m said:
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.

joey_m said:
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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  • #6
Aether said:
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.
 
  • #7
JesseM said:
"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.
 
  • #8
Aether said:
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?
 
  • #9
joey_m said:
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 occurred 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.
 
  • #10
JesseM said:
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.
JesseM said:
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.

JesseM said:
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"?
 
  • #11
Aether said:
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?
 
  • #12
JesseM said:
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.
 
  • #13
Aether said:
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.
 
  • #14
JesseM said:
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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  • #15
Aether said:
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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  • #16
JesseM said:
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 [itex](t_S(I), x_S(I))[/itex] (parameterised by inertial observer I), which are related to each other via

[tex]x_S(I) = \gamma_{v(I)}(x_S(R) - v(I) t_S(R))[/tex]...(1)
[tex]t_S(I) = \frac{t_S(R)}{\gamma_{v(I)}}[/tex].......(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

[tex]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)[/tex]​

(see this thread) is invariant and that's why it's a "symmetry".
 
  • #17
DrGreg said:
See, for example, this post which defines a family of coordinate systems [itex](t_S(I), x_S(I))[/itex] (parameterised by inertial observer I), which are related to each other via

[tex]x_S(I) = \gamma_{v(I)}(x_S(R) - v(I) t_S(R))[/tex]...(1)
[tex]t_S(I) = \frac{t_S(R)}{\gamma_{v(I)}}[/tex].......(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)
 
  • #18
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.
 
  • #19
Mentz114 said:
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.
 
  • #20
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
 
  • #21
Mentz114 said:
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).
 
  • #22
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.
 
  • #23
Mentz114 said:
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:

[tex]x' = x - vt[/tex]
[tex]y' = y[/tex]
[tex]z' = z[/tex]
[tex]t' = t[/tex]

and

[tex]x = x' + vt'[/tex]
[tex]y = y'[/tex]
[tex]z = z'[/tex]
[tex]t = t'[/tex]

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 [tex]m_1[/tex] at position [tex](x_1 , y_1 , z_1)[/tex] and another mass [tex]m_2[/tex] at position [tex](x_2 , y_2 , z_2 )[/tex] in your reference frame. Then the Newtonian equation for the gravitational force between them would be:

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

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 [tex]m_1[/tex] has coordinates [tex](x'_1 , y'_1 , z'_1)[/tex] and the mass [tex]m_2[/tex] has coordinates [tex](x'_2 , y'_2 , z'_2 )[/tex]. Using the Galilei transformation, we can figure how the force would look in this new coordinate system, by substituting in [tex]x_1 = x'_1 + v t'[/tex], [tex]x_2 = x'_2 + v t'[/tex], [tex]y_1 = y'_1[/tex], [tex]y_2 = y'_2[/tex], and so forth. With these substitutions, the above equation becomes:

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

and you can see that this simplifies to:

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

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 [tex]f(x,y,z,t)[/tex] [of course it may have more than one of each coordinate, like the [tex]x_1[/tex] and [tex]x_2[/tex] above, and it may be a function of additional variables as well, like [tex]m_1[/tex] and [tex]m_2[/tex] above] then for this equation to be "Galilei invariant", it must satisfy:
[tex]f(x'+vt',y',z',t') = f(x',y',z',t') [/tex]

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

[tex]f( \gamma (x' + vt' ), y' , z', \gamma (t' + vx' /c^2 ) ) = f(x' ,y' ,z' , t')[/tex]
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.
 
  • #24
JesseM said:
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.

JesseM said:
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.
 
  • #25
JesseM said:
Mentz114 said:
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.

[tex]x_S(I) = x_E(I)[/tex].....(3)
[tex]t_S(I) = t_E(I) + \frac{v(I) x_E(I)}{c^2}[/tex]...(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 [itex]v(I) x_E(I)/c^2[/itex]. 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.
 
  • #26
JesseM said:
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?
No.

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?
No.

Or are you just saying that a statement which is true in one coordinate system can be false in another?
Yes.

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.
I agreed that in full context the statements are consistent with some more general physical feature of nature, or experiment.

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.
Speeds do imply a coordinate system, so I will try to restate my example in a coordinate-independent way using "wavenumber isotropy" in place of "speed invariance": Two-way light wavenumber isotropy is both a mathematical feature of the laws of physics, and a physical feature of nature; but although one-way light wavenumber isotropy is also a mathematical feature of the (standard) laws of physics, it is not necessarily a physical feature of nature.

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).
Let's try using "the two-way wavenumber of light is isotropic" in place of "the two-way speed of light is constant" and see if we can make the statement coordinate-independent.

But if you agree they are empirically distinguishable, doesn't that mean they must both be physical statements about the universe?
Yes, but these statements are still coordinate-dependent. I would like to see if we could separate the genuine physical content of the statements from their coordinate dependency.

I'm not familiar with the GGT transform, can you give the equations, or a link?
For motion along the x direction the equations are: [tex]x=\gamma(x_0-vt_0), t=\gamma^{-1}t_0, y=y_0, z=z_0[/tex]

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)?
Yes.

If so, wouldn't it be logically impossible for one to be true and the other false?
The two statements are empirically equivalent within the bounds of the nonconventional content of special relativity. I have read that it is still possible for one to be true and the other false, but in that case the nonconventional content of special relativity would be falsified.

DrGreg said:
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).
That's right, thank-you.
 
  • #27
Aether said:
Speeds do imply a coordinate system, so I will try to restate my example in a coordinate-independent way using "wavenumber isotropy" in place of "speed invariance": Two-way light wavenumber isotropy is both a mathematical feature of the laws of physics, and a physical feature of nature; but although one-way light wavenumber isotropy is also a mathematical feature of the (standard) laws of physics, it is not necessarily a physical feature of nature.
Can you define your notion of wavenumber isotropy explicitly? Normally wavenumber is defined as 1/wavelength, and wavelength obviously depends on the distance between peaks, so how your coordinate system defines distance will come into play.
Aether said:
For motion along the x direction the equations are: [tex]x=\gamma(x_0-vt_0), t=\gamma^{-1}t_0, y=y_0, z=z_0[/tex]
Well, as I said to DrGreg, it's clearly not true that the equations of physics will have the same form in this family of coordinate systems. If Maxwell's equations work in one of these coordinate systems (as they should, since the 'aether' frame is supposed to just be one of the frames of the Lorentz transform) then they can't work when you use this transform on the equations (see my example involving Newton's gravitation and the Galilei transform above for an example of using a coordinate transformation on a set of equations), since all the coordinate systems in this family agree on simultaneity and thus cannot all agree that the coordinate speed of light is isotropic.
JesseM said:
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)?
Aether said:
Yes.
They aren't, though. A hypothetical equation which was invariant under the GGT transformation would not be invariant under the Lorentz transformation, and vice versa.
 
  • #28
JesseM said:
Can you define your notion of wavenumber isotropy explicitly? Normally wavenumber is defined as 1/wavelength, and wavelength obviously depends on the distance between peaks, so how your coordinate system defines distance will come into play.
Using "wavenumber" as defined http://en.wikipedia.org/wiki/Wavenumber" [Broken], and taking our unit of space to be the arbitrary two-way length of a hypothetical Michelson interferometer arm, then my notion of wavenumber isotropy for an actual experiment is this: for any wave of arbitrary wavelength making a round-trip along the interferometer arm, the number of repeating units of the propagating wave (the number of times the wave has the same phase) inside the arm is independent of the spatial orientation of the arm.

Wavenumber said:
Wavenumber in most physical sciences is a wave property inversely related to wavelength, having SI units of reciprocal meters (m−1). Wavenumber is the spatial analog of frequency, that is, it is the measurement of the number of repeating units of a propagating wave (the number of times a wave has the same phase) per unit of space.

Well, as I said to DrGreg, it's clearly not true that the equations of physics will have the same form in this family of coordinate systems. If Maxwell's equations work in one of these coordinate systems (as they should, since the 'aether' frame is supposed to just be one of the frames of the Lorentz transform) then they can't work when you use this transform on the equations (see my example involving Newton's gravitation and the Galilei transform above for an example of using a coordinate transformation on a set of equations), since all the coordinate systems in this family agree on simultaneity and thus cannot all agree that the coordinate speed of light is isotropic.

They aren't, though. A hypothetical equation which was invariant under the GGT transformation would not be invariant under the Lorentz transformation, and vice versa.
The equations of physics that are used in SR don't have the same form in the presence of gravity either. The laws of physics in any coordinate system are derived from http://en.wikipedia.org/wiki/General_covariance" [Broken]. I agree with DrGreg that these other coordinate systems such as GGT "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".
 
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  • #29
Aether said:
Using "wavenumber" as defined http://en.wikipedia.org/wiki/Wavenumber" [Broken], and taking our unit of space to be the arbitrary two-way length of a hypothetical Michelson interferometer arm, then my notion of wavenumber isotropy for an actual experiment is this: for any wave of arbitrary wavelength making a round-trip along the interferometer arm, the number of repeating units of the propagating wave (the number of times the wave has the same phase) inside the arm is independent of the spatial orientation of the arm.
That only works if you assume the arm is rigid as it rotates (so as measured in an inertial coordinate system from SR, its length in one orientation is the same as its length in the other), which is a little problematic since rotation involves acceleration. Maybe there'd be a way around this, I'm not sure.
Aether said:
The equations of physics that are used in SR don't have the same form in the presence of gravity either.
No, but it's built into GR that it reduces to SR locally, meaning that as you zoom in on a smaller and smaller region of spacetime, the error in using the SR laws to make predictions in that region alone will go to zero.
Aether said:
The laws of physics in any coordinate system are derived from http://en.wikipedia.org/wiki/General_covariance" [Broken].
General covariance is not really a physical feature of laws at all but more of a feature of the mathematical form the law is stated in, since any law whatsoever can be put into the appropriate tensor form and it will then be generally covariant, even Newtonian laws. See the last paragraph on http://www.bun.kyoto-u.ac.jp/~suchii/gen.GR7.html [Broken] Julian Barbour writes:
In 1907, Einstein realized that the equivalence principle enabled him to extend the restricted relativity principle to include another ‘impotence’ – the inability to detect uniform acceleration. This insight was decisive. Some years earlier, Einstein had read Mach’s critique of Newton’s absolute space and time and was extremely keen to reformulate dynamics along the broad lines advocated by Mach. He wanted to show that absolute space did not correspond to anything in reality – that it could not be revealed by any experiment. Special relativity had shown that uniform motion – relative to the ether or to absolute space – could not be detected by any physical process. The equivalence principle suggested to him that it might be possible to extend his relativity principle further. If he could extend it so far as to show that the laws of nature could be expressed in identical form in all conceivable frames of reference, this requirement of general covariance would “[take] away from space and time the last remnant of physical objectivity”[6]. He would have achieved his Machian aim. Unfortunately, within two years Einstein had been forced by a critique of Kretschmann [7] to acknowledge that any physical theory must, if it is to have any content, be expressible in generally covariant form. He argued [8] that the principle nevertheless had great heuristic value. One should seek only those theories that are simple when expressed in generally covariant form. However, Einstein gave no definition of simplicity. Since then, and especially as a result of quantization attempts, there has been a vast amount of inconclusive discussion about the significance of general covariance [9] and its implications for quantization.
Barbour then discusses his own notion of what it means to say a theory is "simple" in general covariant form, but there doesn't seem to be any widely agreed-upon meaning.

On the other hand, Lorentz-invariance means that when you don't specify the dynamical laws in terms of a metric but just in terms of the position and time coordinates of some inertial coordinate system, the equations when stated in this form will be the same as if you do the same thing in any other inertial coordinate system. This is clearly a real physical feature of the laws of physics, since not all possible laws would be Lorentz-invariant in this sense.
 
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  • #30
JesseM said:
That only works if you assume the arm is rigid as it rotates (so as measured in an inertial coordinate system from SR, its length in one orientation is the same as its length in the other), which is a little problematic since rotation involves acceleration. Maybe there'd be a way around this, I'm not sure.
In a Michelson interferometer, since a split beam travels round-trip along two orthogonal arms before being compared, normally only differential accelerations between the two arms would be an issue.

No, but it's built into GR that it reduces to SR locally, meaning that as you zoom in on a smaller and smaller region of spacetime, the error in using the SR laws to make predictions in that region alone will go to zero.
It's the same for any flat coordinate system that you may choose though, not just for inertial coordinate systems.

General covariance is not really a physical feature of laws at all but more of a feature of the mathematical form the law is stated in, since any law whatsoever can be put into the appropriate tensor form and it will then be generally covariant, even Newtonian laws.
Generally covariant laws of physics have all of the same physical features as any laws of physics which can be derived from them to describe spacetime using a family of inertial coordinate systems. So this is one way to test whether some feature of a coordinate-dependent law is really physical or partly mathematical.

On the other hand, Lorentz-invariance means that when you don't specify the dynamical laws in terms of a metric but just in terms of the position and time coordinates of some inertial coordinate system, the equations when stated in this form will be the same as if you do the same thing in any other inertial coordinate system. This is clearly a real physical feature of the laws of physics, since not all possible laws would be Lorentz-invariant in this sense.
Start with some generally covariant laws of physics, and then choose some arbitrary coordinate system to describe spacetime. Will the resulting coordinate-dependent law of physics always be Lorentz invariant? I agree that any feature that is inherited by all coordinate systems might be a real physical feature, but if Lorentz invariance is unique to inertial coordinate systems then it only represents one of many possible ways to cover a real physical feature with coordinates.
 
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  • #31
Aether said:
In a Michelson interferometer, since a split beam travels round-trip along two orthogonal arms before being compared, normally only differential accelerations between the two arms would be an issue.
But your definition of wavenumber isotropy only works if we include some assumption about the two arms having the same length, and if we want to do that without referring to coordinate distance, it seems like we'd have to talk about taking a single arm and rotating it. I suppose we might also define length in terms of some physical feature like multiples of the distance between atoms in a diamond placed alongside each arm.
JesseM said:
No, but it's built into GR that it reduces to SR locally, meaning that as you zoom in on a smaller and smaller region of spacetime, the error in using the SR laws to make predictions in that region alone will go to zero.
Aether said:
It's the same for any flat coordinate system that you may choose though, not just for inertial coordinate systems.
What is the same? It's certainly not true that in any flat coordinate system in spacetime, the laws of physics in that coordinate system (written in non-tensor form) will be the same as those in inertial frames in SR.
Aether said:
Generally covariant laws of physics have all of the same physical features as any laws of physics which can be derived from them to describe spacetime using a family of inertial coordinate systems.
Sure, but I think some of these "physical features" may only definable be translating the equations into a non-tensor form in some physically-constructed coordinate system. For example, I don't think it's possible to explain what it means to say that GR is locally Lorentz-invariant without actually showing that in a local region, if you translate the tensor equations of GR into their non-tensor form in the family of local inertial coordinate systems defined by the Lorentz transform, the non-tensor form of the equations will be the same in all these coordinate systems.
Aether said:
So this is one way to test whether some feature of a coordinate-dependent law is really physical or partly mathematical.

Start with some generally covariant laws of physics, and then choose some arbitrary coordinate system to describe spacetime. Will the resulting coordinate-dependent law of physics always be Lorentz invariant?
And you think you can decide if it's Lorentz-invariant without transforming from your arbitrary coordinate system to the family of inertial coordinate systems given by the Lorentz transformation? How would you do that?
Aether said:
I agree that any feature that is inherited by all coordinate systems might be a real physical feature, but if Lorentz invariance is unique to inertial coordinate systems then it only represents one of many possible ways to cover a real physical feature with coordinates.
It isn't unique to inertial coordinate systems. If you are using some non-inertial coordinate system, you can still say that the laws of physics are Lorentz-invariant if it's true that, were you to transform the equations in your system to the family of inertial frames given by the Lorentz transform, the resulting equations would be identical for every member of this family. The statement "the laws of physics are Lorentz-invariant" describes a physical feature of the world which is true for everyone, it doesn't depend on whether you are moving inertially, or on what coordinate system you actually choose to use when approaching problems.
 

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