Relativity of simultaneity doubt

[Nugatory]

Yes, I should have been more careful. There is a difference between the strictly mathematical detection of coordinate acceleration versus the physics concept of proper acceleration. The point I was trying to make is that even putting blinders on and only considering the mathematics of the coordinate system, one can define acceleration without reference to motion versus other objects. Or is that still too naive?

One can define coordinate acceleration without reference to any other objects. In fact, we can make the coordinate acceleration come out to be any value we want just choosing coordinates that produce that value. There's no need for any other object (although when we are considering only a single object it would be perverse to choose coordinates in which its velocity and coordinate acceleration are non-zero).

It seems that proper acceleration is can also be defined strictly mathematically with no reference to anything beyond the coordinate systems. (https://en.wikipedia.org/wiki/Proper_acceleration )

Whereas proper acceleration is defined physically by the reading of an accelerometer and mathematically by the deviation from a geodesic worldline - neither definition has anything to do with coordinates and both will come out the same no matter what coordinates we choose.

 

[Dale]

Suppose to choose $\epsilon=0.2$ which is not Einstein's synchronization. Then in SR the standard inertial coordinate chart is not Minkowski and the metric is not in the standard form $s^2=x^2 + y^2 + z^2 - (ct)^2$, right ?

Yes, that is correct. The metric would be: $$ ds^2 = - dt^2 -2 \kappa \ dx \ dt + (1-\kappa^2) dx^2 + dy^2 + dz^2 $$ where $\kappa = 2 \epsilon -1$ so in your case $\kappa = -0.6$

If you go through the trouble you should be able to show that this metric is still flat spacetime (assuming I made no mistake). The curvature is zero, but this is not the metric of a standard inertial frame.

 

[FactChecker]

Whereas proper acceleration is defined ... mathematically by the deviation from a geodesic worldline - neither definition has anything to do with coordinates and both will come out the same no matter what coordinates we choose.

I stand corrected. I guess that a geodesic worldline exists agnostic of the choice of a coordinate system (if any) to specify it.

 

[PeterDonis]

I guess that a geodesic worldline exists agnostic of the choice of a coordinate system (if any) to specify it.

Exactly.

 

[bhobba]

I still think the whole issue is most easily resolved by Landau's definition of an inertial frame, i.e. it is a frame where the laws of physics are the same at any point, direction or instant of time. This means whatever laws govern the speed of light, they are the same regardless of orientation. So one way light speed is the same regardless of direction - which is exactly what Maxwell's equations say. If there were an aether, it would play havoc with the value of that definition as part of the POR, i.e. the laws of physics are the same in any inertial frame. That would be because there would be only one inertial frame - the one where the aether is at rest - all other frames moving at constant velocity relative to that frame would have an aether 'wind' breaking isotropy. And since classical mechanics, when analysed carefully, depends on the POR (as done in Landau's Mechanics), it would pose problems for even classical physics. IMHO what Einstein did was what was necessary to put even classical mechanics on a firm footing.

Indeed measuring the one-way speed of light is an issue in experimentally determining if a frame is inertial. But we know no frame is strictly inertial (except maybe locally), although frames in interstellar space are thought to be very close.

Thanks
Bill

 

[vanhees71]

Within SR Maxwell's equations take the usual form in Galilean coordinates of Minkowski space. Minkowski space is one of the two possibilities following from the special principle of relativity (Galilei's principle of inertia). In modern terms this can be formulated as postulating the existence of inertial reference frames and that for an inertial observer space is described as a Euclidean affine manifold. Under these assumptions the possible symmetry groups are the Galilei or the Poincare group, leading to Galilei-Newton spacetime (a fiber bundle) or Minkowski spacetime (a pseudo-Euclidean affine space with signature (+---) or equivalently (+++-)). This implies the existence of particularly simple coordinates, which for the Minkowski spacetime are Galilean coordinates based on the choice of a Minkowski-orthonormal basis. That's however a pretty abstract mathematical approach.

Characteristically for Einstein's works of his earlier years Einstein was after an operational physical construction of the spacetime description, and that's why he took as postulates the special principle of relativity and from the Maxwell equations the only piece that is relevant for this construction of the spacetime description, i.e., the demand that the speed of light, as measured by an inertial observer, must be independent of the motion of the light source relative to the observer, which follows from the assumption that the special principle of relativity should hold also for electromagnetic phenomena.

From this he derived the Galilean coordinates by his choice of clock synchronization, using light signals. Together with the insight that one should synchronize clocks being at rest relative to each other and using just one reference clock of one inertial observer and assuming the above symmetry principles (particularly the isotropy of the Euclidean space wrt. the inertial observer) you have to use (a) the two-way speed of light being $c$ and then assume (by "convention") that the one-way speeds back and forth between the observers reference clock and each one of the other distant clocks is also $c$, independent of the direction (isotropy) and distance of the other clock (homogeneity). Then he could show that this clock synchronization procedure is transitive, i.e., that then any two clocks within the one inertial frame and being at rest relative to each other within this frame are synchronized. From this the usual Lorentz transformations between different Galilean spacetime coordinates follow, and using these coordinates the Maxwell equations are form invariant, and that was the aim of the paper given in the famous first sentence, i.e., to eliminate the asymmetries implied by the then standard interpretation of the Maxwell equations as distinguishing a preferred reference frame defined as the rest frame of "the aether", which had quite odd properties to begin with anyway.

The conclusion then was that not the until then sacrosanct Newtonian mechanics had to be preserved but the Maxwell equations had to be form invariant wrt. inertial frames, which lead to the Lorentz (Poincare) invariance of the new space-time model rather than the Galilei invariance of the Newtonian space-time model, and this implied that the mechanical laws had to be adapted to the new space-time model. This part is then the weak point of the famous paper, because Einstein at this point didn't find the most simple interpretation and thus he introduced the notion of relativistic (velocity not only speed dependent!) masses, which obscured the mechanics tremendously. This was "repaired" pretty quickly by Planck, who gave an elegant derivation using the action principle (in its (1+3)-dimensional form) leading to the correct interpretation of relativistic momentum with the (Newtonian) mass being within SR what we now call the invariant mass.

Of course the full understanding of the mathematical structure then came in 1908 with Minkowski's famous talk about the four-dimensional spacetime description.

In analogy to Euclidean analytical geometry of course also in Minkowski space Galilean coordinates are just a preferred choice in the sense that when expressing the dynamical laws, compatible with the symmetry properties of Minkowski space, in these coordinates they take the most simple form. You can choose of course any other coordinates you like, and since in Minkowski space there's no more any remnant of an absolute time it's natural that you can use any diffeomorphisms between Galilean coordinates and arbitrary four parameters as "generalized spacetime coordinates", leading to a description of non-inertial reference frames. However these generalized spacetime coordinates do not necessarily have a direct physical meaning but they just parametrize spacetime-point location (usually also covering only a part of Minkowski space).

From this it is only a small step to make inertial reference frames and thus the Poincare group a local symmetry, which leads directly to the spacetime description of general relativity.

 

[cianfa72]

If two events are simultaneous in your reference frame then the proper time that elapses on your watch between the two events is zero, as measured by you (*). In a different frame where the events are not simultaneous, there must be an elapsed proper time on your watch between the coordinate times of those events, as measured in that reference frame.

(*) In which case, your watch is measuring the coordinate time (in your rest frame) for the two events.

Sorry maybe I missed your point: if two events are simultaneous in your rest frame they are spacelike separated hence your (timelike) worldline cannot connect them (i.e. proper time between those two events is not defined along your worldline).

 

[PeroK]

Sorry maybe I missed your point: if two events are simultaneous in your rest frame they are spacelike separated hence your (timelike) worldline cannot connect them (i.e. proper time between those two events is not defined along your worldline).

I think that was my point! That post was an attempt to disentangle confusion over proper and coordinate times.

If events are simultaneous in any frame, then there is no concept of proper time between them along any worldline.

If you read the previous posts, you'll see how the mess developed!

 

[cianfa72]

If events are simultaneous in any frame, then there is no concept of proper time between them along any worldline.

ok my bad, since if there exists a frame in which the two events are simultaneous then they are spacelike separated.

 

[PeroK]

ok my bad, since if there exists a frame in which the two events are simultaneous then they are spacelike separated.

Looking back at the previous posts, the issue was to try to explain that the proper time of a clock keeps coordinate time (in some sense) for its rest frame. But, we can't extend that notion to the proper time between any two events - as someone was trying to do.

In other words, we have two events ($E_0, E_1$) with coordinates in some IRF: $(t_0, x_0)$ and $(t_1, x_1)$. Then, we have a clock at rest in that IRF. At coordinate time $t_0$ that clock reads $\tau_0$ and at $t_1$ it reads $\tau_1$. We know that $\tau_1 - \tau_0$ is the proper time of that clock between two events ($C_0, C_1$) on that clock's worldline. And, because that clock keeps coordinate time in that frame, we have $\tau_1 - \tau_0 = t_1 - t_0$.

However, that does not mean that the proper time between events $E_0, E_1$ is $\tau_1 - \tau_0$.

That is the point I was trying to make.

 

[cianfa72]

However, that does not mean that the proper time between events $E_0, E_1$ is $\tau_1 - \tau_0$.

Right, since it is simply not defined for them.

I would like to point out that the term "frame" here really means 'coordinate chart" and not 'frame field' since the latter does not define any simultaneity convention.

 

[italicus]

But, we can't extend that notion to the proper time between any two events - as someone was trying to do.

Sorry, If you are referring to me as “someone …“ , maybe you have badly interpreted something of what I said. But I don’t want to go back on this topic.

@cianfa72

a 4-interval between two events doesn’t change its nature wrt different inertial observers, with different speed from each other. If it is timelike,it remains timelike. If it is spacelike, it remains spacelike.

 

[cianfa72]

a 4-interval between two events doesn’t change its nature wrt different inertial observers, with different speed from each other. If it is timelike,it remains timelike. If it is spacelike, it remains spacelike.

Yes, of course.

 

[vanhees71]

A more clear statement of "relativity of simultaneity" is that only time-like and light-like intervals have an invariant temporal order (i.e., their temporal order is the same in all IRFs), while space-like intervals don't. There's always an IRF, where the corresponding events are simultaneous and they are not in other IRFs. Also the temporal order differs in these other IRFs.

 

[cianfa72]

A more clear statement of "relativity of simultaneity" is that only time-like and light-like intervals have an invariant temporal order (i.e., their temporal order is the same in all IRFs), while space-like intervals don't. There's always an IRF, where the corresponding events are simultaneous and they are not in other IRFs. Also the temporal order differs in these other IRFs.

Does your claim apply to GR as well ?

 

[PeroK]

Does your claim apply to GR as well ?

The post references (global) IRF's, which are not possible in GR.

 

[PeterDonis]

Does your claim apply to GR as well ?

The claim about only timelike or null separated events having an invariant temporal order, not spacelike separated events, applies to any spacetime, flat or curved, as long as some minimal causality conditions are satisfied. See further comments below.

The post references (global) IRF's, which are not possible in GR.

But the claim itself can be made without having to make use of global IRFs. The only requirement is that the spacetime satisfy some minimal causality conditions (I think "stably causal", as defined in Hawking & Ellis, is sufficient). In any spacetime that satisfies those conditions, one can construct an infinite number of different foliations of the spacetime by spacelike hypersurfaces, and look at the ordering of events as given by that foliation (i.e., the ordering of the surfaces in which the events appear). One will find that the ordering of timelike or null separated events is invariant, but the ordering of spacelike separated events is not.

 

[cianfa72]

But the claim itself can be made without having to make use of global IRFs. The only requirement is that the spacetime satisfy some minimal causality conditions (I think "stably causal", as defined in Hawking & Ellis, is sufficient). One will find that the ordering of timelike or null separated events is invariant, but the ordering of spacelike separated events is not.

As discussed so far in a recent thread, I believe the reason for the causality condition mentioned is to rule out spacetimes admitting CTC (Closed Timelike Curves).

 

[SiennaTheGr8]

There's always an IRF, where the corresponding [spacelike] events are simultaneous and they are not in other IRFs.

A boost perpendicular to the "line" connecting two simultaneous events preserves their simultaneity, though, so there are actually an infinite number of IRFs in which the events are simultaneous (in 3+1 flat spacetime), no?

 

[cianfa72]

A boost perpendicular to the "line" connecting two simultaneous events preserves their simultaneity, though, so there are actually an infinite number of IRFs in which the events are simultaneous (in 3+1 flat spacetime), no?

I believe they are actually all the IRFs that share the same velocity but have spatial axes oriented in all possible directions.

 

[SiennaTheGr8]

I believe they are actually all the IRFs that share the same velocity but have spatial axes oriented in all possible directions.

No, boosts as well.

 

[PeterDonis]

As discussed so far in a recent thread, I believe the reason for the causality condition mentioned is to rule out spacetimes admitting CTC (Closed Timelike Curves).

That's one thing that needs to be ruled out, yes, but I'm not sure it's the only one. There are a number of different causality conditions, with increasing strictness in terms of what they rule out. The definitive discussion of them, AFAIK, is in Hawking & Ellis.

 

[cianfa72]

No, boosts as well.

The spacetime direction orthogonal to the "line" connecting those two spacelike separated events is unique. So do you mean all the Lorentz boost affine transformations that share that direction (velocity) ?

 

[PeterDonis]

The spacetime direction orthogonal to the "line" connecting those two spacelike separated events is unique.

No, it isn't. There is an entire 3-space with one timelike and two spacelike dimensions that is orthogonal to a given spacelike line. To see this, just label the direction of the spacelike line as the $z$ axis. Then any vector with only $t$, $x$, and $y$ components is orthogonal to that line.

 

[cianfa72]

No, it isn't. There is an entire 3-space with one timelike and two spacelike dimensions that is orthogonal to a given spacelike line. To see this, just label the direction of the spacelike line as the $z$ axis. Then any vector with only $t$, $x$, and $y$ components is orthogonal to that line.

Yes, that's true: there is a such 3-space for each event along that line (in previous post I was taking into accout one spatial dimension alone). My point was that all Lorentz boosts for which those given spacelike separated events are simultaneous actually "share" the same velocity in $z$ direction.

 

[PeterDonis]

My point was that all Lorentz boosts for which those given spacelike separated events are simultaneous actually "share" the same velocity in $z$ direction.

In the sense that they can have no velocity in the $z$ direction, yes. Isn't that obvious? If you boost in the $z$ direction, you obviously have a change in simultaneity in the $z$ direction. I'm not sure why you felt the need to point that out, or why you think it is important.

 

[cianfa72]

In the sense that they can have no velocity in the $z$ direction, yes. Isn't that obvious? If you boost in the $z$ direction, you obviously have a change in simultaneity in the $z$ direction.

Consider Minkowski spacetime in standard inertial coordinates and from event A take an event B spacelike separated from it. Consider the spacelike straight line connecting them and name it $z$. Take the 3D-space of directions orthogonal to it. Which is the family of Lorentz boosts such that events on $z$ axis are simultaneous ?

 

[PeterDonis]

Which is the family of Lorentz boosts such that events on $z$ axis are simultaneous ?

All of them that don't include a $z$ component, as I already said. So a boost in any direction in the $x y$ plane.

 

[cianfa72]

All of them that don't include a $z$ component, as I already said. So a boost in any direction in the $x y$ plane.

ok, a boost in any direction in the $(x,y)$ plane however with the appropriate velocity otherwise events along that given spacelike direction $z$ do not result as simultaneous.

 

[PeterDonis]

a boost in any direction in the $(x,y)$ plane

Yes.

however with the appropriate velocity otherwise events along that given spacelike direction $z$ do not result as simultaneous.

I have no idea what you mean by this. Any boost in any direction in the $xy$ plane at any velocity leaves any pair of events separated only along the $z$ direction (i.e., with the same $t$, $x$, and $y$ coordinates) simultaneous.

 

[cianfa72]

Any boost in any direction in the $xy$ plane at any velocity leaves any pair of events separated only along the $z$ direction (i.e., with the same $t$, $x$, and $y$ coordinates) simultaneous.

That's true. However the $z$ axis has been redefined (i.e. it is not the same as the $z$ axis in the initial inertial coordinate system) since it is aligned with the spacelike direction defined by [spacelike separated] events A and B (note that A and B may not be simultaneous in the given inertial frame and may not have $x=0, y=0$ as well).

Then, of course, if events A and B are simultaneous in the given initial inertial frame any boost at any velocity in any direction orthogonal to the spacelike "line" defined from them leaves those events simultaneous. Hence if the two simultaneous events A and B are separated only along the $z$ direction then any velocity in any direction in the $xy$ plane leaves them simultaneous.

 

[italicus]

However the z axis has been redefined (i.e. it is not the same as the z axis in the initial inertial coordinate system) since it is aligned with the spacelike direction defined by [spacelike separated] events A and B (note that A and B may not be simultaneous in the given inertial frame and may not have x=0,y=0 as well).

What do you mean? The names of coordinates don’t matter. Given the events A and B, call the space axis through them as you want . They are simultaneous for any observer whose timeline is “normal “ to that axis, in the sense of Minkowski.

 

[cianfa72]

What do you mean? The names of coordinates don’t matter. Given the events A and B, call the space axis through them as you want . They are simultaneous for any observer whose timeline is “normal “ to that axis, in the sense of Minkowski.

Yes, the Lorentz boost should be at any velocity in any spacelike direction that does not include the spacelike direction defined by A and B events.

 

[Dale]

it is not the same as the z axis in the initial inertial coordinate system

Sure, but it is still an inertial coordinate system. By the first postulate it is equally valid as the original and all of the laws of physics are the same there.

 

[cianfa72]

Sure, but it is still an inertial coordinate system. By the first postulate it is equally valid as the original and all of the laws of physics are the same there.

Sorry, maybe I can't explain my point. Let's take two events A and B spacelike separated even if not simultaneous in the global inertial frame $(t,x,y,z)$. Consider the 'new' $z$ axis (call it $z'$) along them: it is in general a linear combination of $t,x,y,z$ axis.

Any Lorentz boost at any velocity in a direction that is linear combination of $x,y$ axes (i.e. having $t=0,z=0$) does not bring simultaneous those given two events, right ?