# Are accelerating lines of simultaneity correct?

1. Aug 6, 2010

### Austin0

Are accelerating lines of simultaneity correct??

This is an idea and question that I have been considering for a long time but put on hold while I sought a firmer grasp of the geometry of Minkowski spacetime graphs.

My current understanding is this:

Lines of simultanieity wrt inertial frames are a graphing of the interrelationship between the clocks and positions of one frame with another.
As such the points of intersection represent accurate colocations of clocks and ruler points. These are both quantitatively and geometrically (with tranformation) valid both spatially and temporally.
They are equivalent to the clocks and rulers of the frames themselves , whether actual or virtual.
This can be taken as consistent with reality as observers from both frames at these points of intersection will agree on the spatial coordinates and times of these events.

But in the case of accelerating frames this appears to no longer be valid.

The spacetime locations of a point in the accelerating frame as graphed as the worldline, is of course accurate in the coordinates of the rest frame, but the resulting lines of simultaneity no longer conform. They represent a dynamic non-uniform metric mapped onto what is essentially a Euclidean matirx (with single transform).
Taken in sum they perhaps represent varying degrees of curvature into the z plane.

SO the spatial distance between the point of the worldline and a point of intersection is no longer geometrically valid. Neither is the direction.
This is most glaringly obvious where these lines intersect. This represents the simultaneous colocation of two temporally separated points of a single frame.
As these lines can be taken to represent an extension of the frame itself this would also mean colocation of disparate clocks and observers. Clearly this can not be consistent with reality.
Looking outward past the intersection at the diverging lines it is clear that there is the representation of temporal reordering and causality reversal.

It may be suggested that this simply means that the lines are only accurate up to the points of intersection but I think this is not the case. I think they are spatially and temporally inaccurate throughout, with the degree of error a function of spatial and temporal distance from their origens at the worldline and their temporal separation on that worldline.

I have not done the math to confirm this for two reasons
1) Time
2) I unfortunately lack the calculus to derive instantaneous velocities from the slope of worldline tangents.

I am aware that due to this lack of mathematical corroboration , many will dismiss this out of hand but I am hoping that someone with the math skills will find the question interesting enough to run some numbers and put it to rest (or not).

If anybody either does not understand or disagrees with my idea of the equivalance of hyperplanes of simultaneity and the frame itself there is a recent thread addressing this and I welcome all objections and criticisms https://www.physicsforums.com/showthread.php?t=415501"

Thanks

Last edited by a moderator: Apr 25, 2017
2. Aug 6, 2010

### yossell

Re: Are accelerating lines of simultaneity correct??

I agree there's an issue about how accelerating frames should be defined.

I agree that it is not particularly coherent to extend lines of simultaneity arbitrarily far in both the +x and -x directions in an accelerating frame, because there will be places at which these lines cross, and one and the same event will be assigned two different coordinates. (However, Starthaus seemed to think this was incorrect). This makes interpreting these coordinates as time in a frame' and distance in a frame' problematic when the frame is accelerating.

It may be that accelerating frames only make physical sense locally. That if our frames are too big, the concept of a well defined acceleration doesn't apply to the frame as a whole.

But when you say that the accelerating frame is inaccurate as a whole, what do you have in mind? After all, three dimensional notions of length and simultaneity make sense only with respect to a frame - there isn't a frame independent issue. But if we are after accuracy for 4-dimensional invariant quantities, proper time, minkowski separation, then, at least where the frame is still 1-1 and not degenerate, as long as we use the correct formulas for this accelerating frame, we should still be able to work them out properly and there is nothing that is misleading.

3. Aug 6, 2010

### Mike_Fontenot

Re: Are accelerating lines of simultaneity correct??

Here are a few comments that might help:

If you draw vertical and horizontal axes corresponding to the x and t coordinates of some inertial observer (with the location of that observer being at x = 0, and his age being equal to t), you can then plot the world line of any particular accelerating traveler. (I like to plot x vertically, and t horizontally, but most people like to follow convention and make the opposite choice...whenever I need to refer to the geometical picture in this posting, I'll use my convention...be forewarned). When plotting that world line, one can also put in "tic" marks along the curve to indicate the current age of the traveler at that point.

If units are chosen so that c = 1, then any such worldline will be such that the slope of the line will everywhere be between +1 and -1. Otherwise, any (continuous) curve is allowable, although with any realizable accelerations, the curve will be "smooth" (the slope will be continuous).

For the original inertial observer, the line of simultaneity anywhere will always just be a vertical line (t = constant).

If you pick any point on the worldline being plotted, you can draw a different line of simultaneity, which has the slope 1/v, where v is the velocity of the traveler, relative to the original inertial frame, according to the original inertial frame. The slope of the worldline, at the given point, is equal to v. (v is positive when the traveler is moving away from the original observer).

The relationship between those two straight lines (the tangent to the worldline, and the (different) line of simultaneity) is easy to visualize: if alpha is the angle the worldline tangent makes with the t (horizontal) axis (positive when counter-clockwise), then the angle that the line of simultaneity makes with the x (vertical) axis (positive when clockwise) is ALSO alpha.

At that point on the world line, there is a unique inertial frame that is momentarily stationary wrt the traveler at that instant...I call that inertial frame the "MSIRF". The straight line, tangent to the worldline at the given point, is the time axis of the MSIRF. The line of simultaneity described above is the line of simultaneity, for the MSIRF, passing through that point on the world line. The point where that line of simultaneity intersects the t axis gives the current age of the original inertial observer, according to an observer in the MSIRF present at the given spacetime point on the world line.

It is possible to prove that the accelerating observer must adopt that line of simultaneity as his OWN line of simultaneity, if he wishes to avoid contradicting his own elementary measurements and elementary calculations.

That proof basically involves asking, and answering, the question: "If the traveler were to stop accelerating at that given point (and thereafter remain stationary in that MSIRF), how long would it take before his measurements and conclusions about simultaneity agreed with a perpetually-inertial observer in the MSIRF who is co-located now with the traveler"?

The answer is that he will ALWAYS agree with that MSIRF observer, at ALL times after he stops accelerating. As soon as he stops accelerating, he will immediately agree with the MSIRF.

I give this proof, in detail, in my paper:

"Accelerated Observers in Special Relativity",
PHYSICS ESSAYS, December 1999, p629.

Mike Fontenot

Last edited: Aug 6, 2010
4. Aug 7, 2010

### Austin0

Re: Are accelerating lines of simultaneity correct??

Hi yossell
I am not talking about accelerating frames per se. I have no doubt that the Lorentz math works just fine as applied to ICMIF's . I don't know what starthaus is refering to but I wouldn't be surprised if it was exactly that.
I am talking about the geometric representation of simultaneity lines in a Minkowski graph.
I am proposing that if you do the math for a particular line, a particular velocity of the co-moving inertial frame you will find that the spatial and temporal positions mathematically derived will not be the same as the positions indicated by the lines on the graph.
That mathematically those intersections of S lines will not occur. No temporal reordering will occur.
That these are artifacts of the Minkowski convention of sloping lines which do not cause a problem with inertial systems because they remain parallel in the quasi-Euclidean space.
That in the real world those lines are congruent with the path of motion and parallel through time. They are the frame and its clocks and ruler which cannot possibly be colocated when separated by a time-like interval. Nor can they possibly change that interval which is what is indicated by convergence,yes??
I agree that the degree of error is least, close to the worldline, but it does not just start at the points of intersection, it is only most obvious there.
Besides which, in principle both frames and hyperplanes are extended indefinitely in space and until you get to truly cosmic distances they should be accurate or there is some fundamental problem.
As I said I would do the math myself if I had the ability to draw an accurate hyperbolic worldline and then derive instantaneous velocities from the tangent slope to be able to do the math for various points to check against the geometry.
But just logically , it is clear if the geometry is accurate for one line it cannot be accurate for the next line that has a different spatial and temporal metric, with a different simultaneity , yes???

5. Aug 7, 2010

### Austin0

Re: Are accelerating lines of simultaneity correct??

Thanks Mike but I already was familiar with everything you have said here but it doesn't really help me to quantify MSIRF velocities from a graph . Unless I had an accurate graph and actually mechanically drew in tangents for a rough approximation.
AS for your premise that there will always be agreement with th ICMIF that is also exactly my premise . That the calculated simultaneity and spatial positions of the ICMIF will not agree with the geometric representation in a diagram.

6. Aug 7, 2010

### yossell

Re: Are accelerating lines of simultaneity correct??

Austin0,

I can't really follow your worry. Is your problem with the relationship between the Minkowski DIAGRAM and Minkowski space, or between Minkowski something and genuine physical clocks and rods? When you talk about the error, what is in error about what, in your view?

7. Aug 7, 2010

### Al68

Re: Are accelerating lines of simultaneity correct??

Lines of simultaneity are based on the SR simultaneity convention. And it's just that: a convention for assigning time coordinates based on a constant light speed of c.

The physical meaning of the time coordinate assigned to a distant event is the relationship between event occurrence and observation in the inertial frame in which it was assigned. It doesn't have that same physical meaning for observers who are not at rest in that same inertial frame for the observation of the event.

This is because the SR simultaneity convention is based on the assumption of a constant light speed of c, and light speed is not c (globally) in accelerated reference frames.

8. Aug 7, 2010

### Mike_Fontenot

Re: Are accelerating lines of simultaneity correct??

I'm sorry...I'm not following you.

If you plot the diagram that I described above, but for the special case where the traveler is always inertial, then the resulting plot follows directly from the Lorentz equations relating the two inertial frames.

Lines of simultaneity, according to the original inertial frame (whose coordinates x1 and t1 correspond to the perpendicular axes of the plot), are just lines parallel to the x1 axis (vertical lines, with my convention).

Lines of simultaneity, according to the traveler's inertial frame, are just lines parallel to the x2 axis (where x2 is the spatial coordinate of the traveler's inertial frame).

We are free to choose the traveler's frame so that he is stationary at x2 = 0, and so that the coordinate t2 corresponds to his age.

And we are free to choose the original frame so that the traveler's twin Sue is stationary at x1 = 0, and so that the coordinate t1 corresponds to her age.

When the traveler is at some (arbitrary) point on the t2 axis, his conclusion about Sue's current age at that instant is just given by the intersection of his line of simultaneity with the t1 axis.

The answer you get, for the traveler's conclusion about Sue's current age at some instant of his life, when you carry out this geometric construction, is exactly what you get (much quicker and easier) with my CADO equation (which I've described in several other threads).

But both the geometric construction, and the CADO equation, come directly from the Lorentz equations.

I don't understand what "failure to agree" you're talking about above.
The only "failure to agree" that I'm aware of, is the fact that Sue and the traveler won't agree about the correspondence between their ages. But that's just special relativity...that's inherent in the Lorentz equations, and it's unavoidable, just like quantum weirdness is unavoidable.

Mike Fontenot

9. Aug 8, 2010

### Austin0

Re: Are accelerating lines of simultaneity correct??

Both of the above.

Actually in both cases it is inconsistency between the MINKOWKI DIAGRAM for an accelerating frame and either Minkowski space or the real world it graphically represents.

Looking at an accelerating frame over time we can say that it is represented by a series of ICMIF's and that abstractly these are different reference frames.
But in reality this is one physical system/frame which alters over time.
Every point and every clock is moving forward through time throughout the frame.
There is no possible acceleration that can violate this in the real world,, that can lead to two points and clocks from succeding spacetime points of the frame being colocated or even change the time-like separation between them.
If this is not absolutely true we have to rethink the fundamental concept of time.
My assumption is that these intersections and colocations are due to the problem of graphing a non-uniform metric into the essentially flat (single transform) Euclidean/Pythagorean plane and that they would not show up with direct application of the Lorentz transformation for the points in question.
If I am wrong about this it would seem to indicate and even bigger problem with the application of the math to accelerating systems in general.

This is why I hoped to reach an understanding with you in the other thread.
That these intersections are not just abstractions but represent actual observers and clocks from the same frame that are temporally separated yet colocated simultaneously at a single event,( spacetime point), I.e. face to face.

Would you disagree that this was an error , big time????

DO you think this can be dismissed with a shrug and " Oh ,its just a coordinate effect"

10. Aug 8, 2010

### yossell

Re: Are accelerating lines of simultaneity correct??

ICMIF = Instantaneous CoMoving Inertial Frame?

I suppose at a first attempt I would say that this is the problem. An accelerating frame can't be identified with a series of ICMIF's, as these are in conflict with each other. What we can do, is somehow try and patch various sections of these together to cover a part of space-time, and work from there.

But I agree that it can't make sense to accelerate a large physical frame so that lines of simultaneity cross, and suppose that, at the end of the process, you still have something that corresponds to anything physical. My guess is that this is because it doesn't really make sense to talk of accelerating all the parts of the frame by an equal amount.

I'm not sure about this - in SR, Minkowski space-time is always flat - so I tend to think that the metric - which I think of as tracking the underlying geometry - stays the same.

Minkowski space is only a kind of geometrisation of the Lorentz Transforms, so I'd be very surprised to see them coming apart. The Lorentz transformations are only valid for inertial frames too. They too say that lines of simultaneity from different inertial frames will cross.

I'm not sure whether I'd shrug - but I'd like to resolve it in terms of the idea that accelerating frames aren't that well defined in SR - that doesn't mean that you can't solve problems involving acceleration in SR - rather, the idea of an extended rigid system of rods and clocks accelerating doesn't make sense in SR.

11. Aug 8, 2010

### Austin0

Re: Are accelerating lines of simultaneity correct??

Yes

Well if we are going to go the Born rigid route, it's obviously going to get complicated and it becomes questionable out front, if the drawn lines of simultaneity have any real meaning at all. Even with Born rigid acceleration, all points of the system are still moving forward in time with no possibility of looping around and intersecting with any other point from a different time.
As far as accelerating all parts an equal amount, I would say that this is obviously impossible until we have a new physics that we can use to create inertialess drives.
On the other hand actual implementation of Born acceleration is just as much an abstract ideal , impossible to implement. SO if the basic premise is correct, all our attempts are doomed to stretch apart and decompose.
If we view an accelerating system as a bunch of different ICMIF's at a single point on the worldline then this makes the lines of simultaneity for any point meaningless as far as a temporal relationship with far off spatially separated points in other frames.
With inertial frames the lines of simultaneity tell you what a clock from one frame will read relative to a colocated clock at that position from the other frame.
Minkowski version of JesseM's clocks and rulers. Just more easily accessable except that to derive position values for the primed frame you have to apply the gamma transform , whereas the clocks and rulers paradigm shows you exactlyt what observers at those positions will read on their clocks and the other frames.
Useful and valid information. But does this tell you anything about any actual temporal relationship between x' and some disparate x.????

My point exactly. The space maintains a uniform geometry but the lines of simultaneity do not. With a line for an inertial frame you can take a segment, literally an inch in the physical drawing space and this will have a discrete geometric length interpretation. A definite dx' with definite x' coordinates at each end. This holds true for any line .
This is not true for accelerating lines. Successive segments one inch from the world line do not represent the same x' , so their intersections with other points of the unprimed frame will not be geometrically valid.
Of course lines from different inertial frames can cross, this is not a problem. A given point can have any number of clocks and observers from different frames colocated and not conflict with reality whatsoever as they will all agree on these events and the lines of simultaneity in a diagram retain their useful coorespondence with actuality.

I agree , and was not suggesting that SR could not handle acceleration problems, only that the information in diagrams was not a valid representation of the mathematical results regarding simultaneity. Unless you tell me that direct application of the fundamental Lorentz transformation can indicate these intersections and temporal reorderings. If that is the case then it becomes a whole different question , doesn't it???

As far as accelerating " an extended rigid system of rods and clocks"

1) there are no systems of completely rigid rods and Born rigidity is a pure abstraction.

2) What else can we possibly accelerate except a system that is as close as we can get to rigid rods???? I.e. ship or whatever.

Thanks

12. Aug 8, 2010

### Austin0

Re: Are accelerating lines of simultaneity correct??

Well the first part I certainly agree with as it is so elementary I have to wonder why you brought it up. The second part regarding accelerating frames I am unsure of the relevance you are trying to explain. Are you saying the lines of simultaneity are not corresponding to reality because c is not invariant in such frames??

13. Aug 8, 2010

### Al68

Re: Are accelerating lines of simultaneity correct??

I'm saying that the lines of simultaneity in any frame correspond to a convention, not any deeper physical reality.

The physical reality is that a distant event cannot be compared directly to a local clock. A distant event must be assigned a local time coordinate by convention, and the result is "correct" if that convention is used.

14. Aug 8, 2010

### yossell

Re: Are accelerating lines of simultaneity correct??

Does this imply that it's a convention that the speed of light is always the same in any frame?

15. Aug 8, 2010

### Austin0

Re: Are accelerating lines of simultaneity correct??

So you dont think that lines of simultaneity wrt inertial frames represent a physical reality as far as frame agreed clock colocations??

16. Aug 8, 2010

### DrGreg

Re: Are accelerating lines of simultaneity correct??

Inertial frames using standard Minkowski coordinates can be pictured as a grid of rulers with clocks attached to them. They have the following properties
1. each ruler measures the "correct" local distance
2. each clock measures the "correct" local time
3. the clocks are synchronised by Einstein's synchronisation convention
If you want to picture a non-inertial frame in the same way, you run into a problem: you find that you can't satisfy all three of the above conditions and at least one of them must be sacrificed. And because you have a choice, we can't really talk about the frame of a non-inertial observer, only one choice of a frame. Note that (3) now has to be interpreted as the clocks are synchronised to the clocks of a co-moving inertial frame.

For example in Rindler coordinates for a uniformly accelerating observer, we can keep conditions (1) and (3), but (2) gets thrown away: most of the "coordinate clocks" have to be deliberately adjusted to run too fast or too slow (relative to "correct" proper clocks) in order to meet condition (3). As you move behind the observer, the clocks run slower and slower until you reach a point where they stop. In fact it's even worse than that: at that point, the clock ought to show all possible times simultaneously! And if you went beyond that point, the clocks would have to go backwards! This ugliness is usually avoided simply by stipulating that our coordinate system doesn't extend that far. (For a mathematically rigorous definition of coordinates, this is essential: we can't have the same coordinates for two different events.) This is a convention, but all coordinate systems and all frames are conventions anyway.

For what it's worth, in a rotating coordinate system, it turns out that (3) is very problematic; you can't choose a system where (3) is true in all directions relative to every point.

17. Aug 8, 2010

### DrGreg

Re: Are accelerating lines of simultaneity correct??

No. Simultaneity is a human-invented convention: very useful for doing calculations but of no experimentally-verifiable physical significance.

If you refer to the "one-way" speed of light from A to B, yes it's a convention determined by the definition of simultaneity you choose. But the "two-way" speed of light, from A to B and back to A again, is no convention, it's experimentally measurable and constant.

(To be pedantic, the last statement hasn't been strictly true since 1983, when the two-way speed of light became constant by definition. So I'm thinking in terms of the old definition of the metre.)

18. Aug 8, 2010

### yossell

Re: Are accelerating lines of simultaneity correct??

I'm surprised that the one-way speed of light turns out to be a convention. Is this the mainstream view? I've seen a lot of stuff about the conventionality of coordinates in the textbooks, but not about one way speed of light being a convention. I'm usually told that the speed of light is a constant in an inertial frame is an experimental fact.

What about the Lorentz transformations? I understood these to be an empirically discovered fundamental symmetry in our laws of nature. But they say the one-way speed of light is a constant. I'm not sure how this could be if the one-way speed of light is really a convention.

19. Aug 8, 2010

### DrGreg

Re: Are accelerating lines of simultaneity correct??

If the books don't discuss the conceptual difference between the one-way & two-way speed then they are brushing the issue under the carpet. You could interpret Einstein's two postulates as an implicit assumption that we are going to use "Einstein-synchronised" coordinates (and indeed Einstein's original paper says something on this in the paragraphs before he states his postulates).

Any "one-way" measurement needs two clocks to measure the start & finish of the journey, and those clocks have to be synchronised somehow. Einstein's method assumes a two-way light trip takes equal times for both legs of the journey, so it's inevitable that Einstein-synced clocks measure the one-way speed equal to the two-way speed.

Another practical way of syncing clocks is to use "slow transport": move a third clock C slowly from A to B and sync C to A at the start, and B to C at the end. Anyone who understands the twin paradox will appreciate this won't work if C is moved quickly, but we can consider the mathematical limit as the speed of C tends to zero. It turns out this method gives exactly the same synchronisation as Einstein's method.

If you don't use Einstein synchronisation (= slow transport) you find that the one-way speed of light varies with direction and many of the time-dependent equations of the laws of physics look a lot more complicated. You could argue this invalidates such a coordinate system from consideration under Einstein's first postulate.

I'm not sure what is standard terminology, but I think you could argue that a coordinate system that uses non-standard synchronisation is not an "inertial frame" (even if the observer is inertial). Under that terminology, "the (one-way) speed of light is a constant in an inertial frame is an experimental fact" would be correct, because your definition of "inertial frame" implies the one-way and two-way speeds are equal.

Again, there is an implicit assumption that Einstein-synced coordinates are being used. If you used non-standard coords, you get a more complicated transformation than Lorentz's. (One version of this is called Edwards' transformation.)

________

As a final thought, it is worth pointing out that in an accelerating frame, even the two-way speed of light needn't be constant.

Last edited: Aug 8, 2010
20. Aug 8, 2010

### yossell

Re: Are accelerating lines of simultaneity correct??

Thanks, DrGreg, these are interesting posts.

I'm still not clear on whether the view that the one way speed of light is constant is just a convention, like choice of coordinate system or inertial frame, or whether it is something that is factual, which we may reasonably believe based on some kind of inductive grounds, a reasonable and justified extrapolation from our measurements of the two way speed.

Poincare believed that geometry of space was merely conventional: one could explain the behaviour of rods either by supposing that space was curved, or by working in a flat space in which new forces acted that deformed all bodies, whatever their composition, to the same degree. He thought which system we used was a matter of convention. This seems too strong and the kinds of forces we'd have to postulate to give the same results look ad hoc, giving rise to a more complex theory which complicated laws.

It's not clear to me that the kind of system that results on the supposition that the one-way speed of light can vary are like the choice of a complex and ugly coordinate system, or form a theory with a complicated and strange law (how come the speed of light can vary, yet always vary by the right amount to make the two-way speed a constant?) which should be rejected in favour of the simpler theory.

But I agree I need to think about it more! Thanks for the information.