Is Analyzing an Accelerated Clock with Inertial Equations Justified?

[JM]

A clock that moves in a polygonal path is an accelerated clock, correct? It appears that such a clock is usually analyzed using equations derived for inertial clocks. 1. What is the justification for using inertial equations to analyze an accelerating clock? 2. Has anyone used general relativity to analyze the accelerating clock?
JM

 

[ghwellsjr]

A clock that moves in a polygonal path is an accelerated clock, correct?

Yes.

It appears that such a clock is usually analyzed using equations derived for inertial clocks. 1. What is the justification for using inertial equations to analyze an accelerating clock?

Einstein stated exactly this scenario in his 1905 paper introducing Special Relativity at the end of section 4. He merely assumed that what is "proved for a polygonal line is also valid for a continuously curved line".

2. Has anyone used general relativity to analyze the accelerating clock?
JM

Why bother?

 

[Nugatory]

Why bother?

To expand on this point, because OP may be misunderstanding something about the difference between SR and GR...

General relativity is the general (well, well of course! why else would we call it "general" relativity) theory that works when there may be gravity and curvature effects. Special relativity works in the case of zero (or at least negligibly small) gravity and curvature; this is an interesting and important special case of the general problem.

So:
1) If a problem can be solved with SR, GR is guaranteed to produce the same results, as SR is just GR when gravity and curvature don't matter.
2) SR works just fine for accelerated motion. You'll sometimes come across pop-sci books and crummy introductory texts that say that you need GR to handle acceleration; that's a common misconception, and it's wrong.

The clock on a polygonal path is accelerated, but it's still in a flat spacetime with no gravitational effects, so SR works just fine.

 

[Dale]

2. Has anyone used general relativity to analyze the accelerating clock?

Yes, when necessary.

Why bother?

Because sometimes you have both motion and gravitational effects in the same experiment. E.g. the famous Hafele-Keating experiments where planes with clocks were flown around the world.

 

[pervect]

A clock that moves in a polygonal path is an accelerated clock, correct? It appears that such a clock is usually analyzed using equations derived for inertial clocks. 1. What is the justification for using inertial equations to analyze an accelerating clock? 2. Has anyone used general relativity to analyze the accelerating clock?
JM

To move in a perfect polygon requires bursts of infinite acceleration. So it's a bit non-physical - an actual object would probably only move in an approximately polygonal path, the corners would be rounded rather than sharp.

Certainly, though, an idealized object moving in an idealized polygon (with the required bursts of infinite acceleration) isn't an object in inertial motion, whatever else you want to say about it.

 

[Passionflower]

To move in a perfect polygon requires bursts of infinite acceleration.

Not really, the clock could stop when it gets to a vertex and starts in the new direction.

Would you call it infinite acceleration when an object accelerates from zero velocity?

 

[pervect]

Not really, the clock could stop when it gets to a vertex and starts in the new direction.

Would you call it infinite acceleration when an object accelerates from zero velocity?

I'm not sure I understand the question - or your apparent disagreement.

An object can accelerate from zero velocity to a non-zero velocity over a period of time, in which case the acceleration is finite.

For an object that starts out at rest to acquire non-zero velocity in zero time would require infinite acceleration.

Truly infinite accelerations are not very physical. It is sometimes used as an approximation to a case where an object acquires a non-zero velocity in an unspecified time which is considered to be "short" - for example, impacts.

 

[Passionflower]

An object can accelerate from zero velocity to a non-zero velocity over a period of time, in which case the acceleration is finite.

So then what is your point?

The clock comes to a complete halt at each vertex, and continues in a different direction? What is infinite about that?

 

[A.T.]

A clock that moves in a polygonal path is an accelerated clock, correct? It appears that such a clock is usually analyzed using equations derived for inertial clocks. 1. What is the justification for using inertial equations to analyze an accelerating clock?

The idea is that it doesn't make a difference, if it is one clock moving in a polygon, or multiple inertial clocks, moving in straight lines and meeting at the polygons corners to synchronize there (pass on the elapsed proper time).

 

[DrGreg]

The clock comes to a complete halt at each vertex, and continues in a different direction? What is infinite about that?

The original post never stated whether this was a polygonal path in space or a polygonal path in spacetime. You're assuming space but other posters have assumed spacetime.

 

[PAllen]

The original post never stated whether this was a polygonal path in space or a polygonal path in spacetime. You're assuming space but other posters have assumed spacetime.

Of course a polygonal path in spacetime is either a closed spacelike path or a CTC, neither of which presumably what the OP really meant ...

 

[DrGreg]

Of course a polygonal path in spacetime is either a closed spacelike path or a CTC, neither of which presumably what the OP really meant ...

Indeed. The term "polygonal path", in a spacetime context, usually refers to an "open ended polygon" rather than a true (closed) polygon.

 

[George Jones]

A clock that moves in a polygonal path is an accelerated clock, correct? It appears that such a clock is usually analyzed using equations derived for inertial clocks. 1. What is the justification for using inertial equations to analyze an accelerating clock?

Restating pervect's point:

If the corners of the polygon are "rounded off" by a region of constant acceleration, and if this region is taken to be arbitrarily, then the difference between this situation involving acceleration and the result for the polygon becomes arbitrarily small.

 

[George Jones]

The clock comes to a complete halt at each vertex

and continues in a different direction? What is infinite about that?

The acceleration.

In more detail, taking a distributional derivative gives a Heaviside step function, and taking a distribution derivative again gives a Dirac Delta function.

 

[A.T.]

Passionflower means slowing down smoothly to stop at the corner, then turning, then accelerating smoothly. This way it is possible to move in a spatial polygon with finite acceleration.

But I don't think this is the OPs issue. I suspect the OP thinks about moving on a polygon with constant speed, and why this can be modeled with inertial clocks, despite the infinite acceleration in this case. The reason is that the durrarion of the acceleration goes towards zero too. So it doesn't make a difference, if it is one clock moving in a polygon, or multiple inertial clocks, moving in straight lines and meeting at the polygons corners to synchronize there (pass on the elapsed proper time).

 

[Dale]

I suspect the OP thinks about moving on a polygon with constant speed, and why this can be modeled with inertial clocks, despite the infinite acceleration in this case.

That is a good way of stating it. Assuming that is what the OP is after, then it leads directly to this:

The time on a clock is given by
$d\tau^2=dt^2-dx^2-dy^2-dz^2$
$(d\tau/dt)^2=1-(dx/dt)^2-(dy/dt)^2-(dz/dt)^2$
$1/\gamma^2=1-v^2$

So acceleration doesn't matter.

 

[JM]

Einstein stated exactly this scenario in his 1905 paper introducing Special Relativity at the end of section 4. He merely assumed that what is "proved for a polygonal line is also valid for a continuously curved line".

Thank you for the comment. I didn't see a proof for a polygonal path. Thats what I'm looking for. Can you point it out to me? How does the light postulate apply to an accelerated frame?
I prefer to work in the context of the 1905 paper.
JM

 

[JM]

2) SR works just fine for accelerated motion.

Thanks for your input. In the context of the 1905 paper the step from inertial to accelerated does not appear to be a trivial matter. But I don't recall ever seeing a specific analysis describing the process. If K is inertial and K' is accelerated how is the light postulate applied to relate the two frames?
JM

 

[JM]

That is a good way of stating it. Assuming that is what the OP is after, then it leads directly to this:

The time on a clock is given by
$d\tau^2=dt^2-dx^2-dy^2-dz^2$
$(d\tau/dt)^2=1-(dx/dt)^2-(dy/dt)^2-(dz/dt)^2$
$1/\gamma^2=1-v^2$

So acceleration doesn't matter.

Thanks, DaleSpam. Can you give me a reference with the derivation for this?
JM

 

[PAllen]

Thanks for your input. In the context of the 1905 paper the step from inertial to accelerated does not appear to be a trivial matter. But I don't recall ever seeing a specific analysis describing the process. If K is inertial and K' is accelerated how is the light postulate applied to relate the two frames?
JM

Einstein never worked with accelerated frames except perhaps in the context of GR. For SR, he worked with accelerated motion described in an inertial frame, which is all you need. Any problem, even what an arbitrary observer sees or measures, can be analyzed in any frame - there is no need or requirement to use a frame in which 'Bob' is at rest in order to analyze what Bob seed.

Accelerated frames in SR are complicated (especially for non-uniform acceleration), and the light postulate does not apply to them. An accelerated observer can measure the speed of light to be different from what an inertial observer measures.

Summary:
- The Lorentz transform and axioms of SR are meant only for inertial frames
- Accelerated motion, including what an accelerated observer sees or measures can be analyzed in any inertial frame.
- It is possible to define and use accelerated frames in SR (though Einstein never did so) using mathematical techniques originally developed for GR. Use of these techniques doesn't mean an accelerated frame requires GR (a theory of gravity) .

 

[PAllen]

Thanks, DaleSpam. Can you give me a reference with the derivation for this?
JM

Every book on Special Relativity must have this. The first line is simply the formula for proper time, or the metric, expressed in standard coordinates for an inertial frame.

 

[JM]

Einstein never worked with accelerated frames except perhaps in the context of GR. For SR, he worked with accelerated motion described in an inertial frame, which is all you need. Any problem, even what an arbitrary observer sees or measures, can be analyzed in any frame - there is no need or requirement to use a frame in which 'Bob' is at rest in order to analyze what Bob seed.

Accelerated frames in SR are complicated (especially for non-uniform acceleration), and the light postulate does not apply to them. An accelerated observer can measure the speed of light to be different from what an inertial observer measures.

Summary:
- The Lorentz transform and axioms of SR are meant only for inertial frames
- Accelerated motion, including what an accelerated observer sees or measures can be analyzed in any inertial frame.
- It is possible to define and use accelerated frames in SR (though Einstein never did so) using mathematical techniques originally developed for GR. Use of these techniques doesn't mean an accelerated frame requires GR (a theory of gravity) .

Very good comments PAllen, thank you. I was not aware of all this. There must be some literature on SR that I haven't seen, if that's possible. So help me out here, what lit do you recommend?
JM

 

[PAllen]

Very good comments PAllen, thank you. I was not aware of all this. There must be some literature on SR that I haven't seen, if that's possible. So help me out here, what lit do you recommend?
JM

Analysis of accelerated motion in inertial frames in SR, and the Minkowski metric which is what Dalespam's calculation started with is in every SR book (college textbook, that is) I've ever seen (even an old one from the 1942 that I have).

Systematic development of accelerated frames in SR is less common. I first encountered in in MTW, a big book on GR. However, the development was exclusively in terms of SR as a lead into GR. I believe some modern books on SR must develop accelerated frames, but I can't point to a particular one.

 

[Dale]

Thanks, DaleSpam. Can you give me a reference with the derivation for this?
JM

Hi JM, as PAllen mentioned, it shoul be in every textbook that claims to be a textbook on relativity. Here are some online resources also.

http://en.wikipedia.org/wiki/Proper_time

http://farside.ph.utexas.edu/teaching/em/lectures/node113.html
http://farside.ph.utexas.edu/teaching/em/lectures/node114.html

http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/vec4.html#c2

 

[Nugatory]

For SR, he worked with accelerated motion described in an inertial frame, which is all you need.

Or as ghwellsjr said, way back in post #2 of this thread... "Why bother?"

 

[JM]

Einstein stated exactly this scenario in his 1905 paper introducing Special Relativity at the end of section 4. He merely assumed that what is "proved for a polygonal line is also valid for a continuously curved line".

George, I see where he says 'If we assume that the result proved for a polygon...'. In the preceeding paragraph he says ' It is at once apparent that this result still holds good if the clock moves from A to B in any polygonal line, ...'. But it isn't apparent to me, so by way of trying to understand, I will make something up. Check this out.
Let A be at x,y,z = 0 and B be at 2,0,0 in the K frame. A clock moving along the x-axis will follow the usual slow clock formula developed in his previous paragraphs. Now let the polygon consist of the line from x,y = 0,0 to x,y = 1,1 , then to 2,0, then to 1, -1, and ending at 0,0, all in the x-y plane. Introduce 'local' coordinate frames with their x axes aligned with each line segment. Let one local frame be at rest in K, and the other move along each segment starting at 0,0 and moving in a clockwise direction. Applying the slow clock formula to each segment in turn and adding gives the total time delay on arrival back at A. The time from A to B along the x-axis will be different from the time along the polygon, but the same formula 'holds good'.
Thats how his analysis makes sense to me. No use is made of acceleration. That suggests that the analysis is purely geometrical. The time delay around the polygon will be the same as along a straight line whose length equals the perimeter of the polygon. I sense that this example is out of context and that further study might provide meaningful explanations.
JM

 

[ghwellsjr]

George, I see where he says 'If we assume that the result proved for a polygon...'. In the preceeding paragraph he says ' It is at once apparent that this result still holds good if the clock moves from A to B in any polygonal line, ...'. But it isn't apparent to me, so by way of trying to understand, I will make something up. Check this out.
Let A be at x,y,z = 0 and B be at 2,0,0 in the K frame. A clock moving along the x-axis will follow the usual slow clock formula developed in his previous paragraphs. Now let the polygon consist of the line from x,y = 0,0 to x,y = 1,1 , then to 2,0, then to 1, -1, and ending at 0,0, all in the x-y plane. Introduce 'local' coordinate frames with their x axes aligned with each line segment. Let one local frame be at rest in K, and the other move along each segment starting at 0,0 and moving in a clockwise direction. Applying the slow clock formula to each segment in turn and adding gives the total time delay on arrival back at A. The time from A to B along the x-axis will be different from the time along the polygon, but the same formula 'holds good'.
Thats how his analysis makes sense to me. No use is made of acceleration. That suggests that the analysis is purely geometrical. The time delay around the polygon will be the same as along a straight line whose length equals the perimeter of the polygon. I sense that this example is out of context and that further study might provide meaningful explanations.
JM

You're very close to what Einstein was saying. There are just two exceptions:

1) There's only one frame to consider. Why do you feel the need to introduce four more 'local' coordinate frames? Einstein specifically did not do that.

2) Einstein's calculation is in terms of the time in the K frame that it takes for the clock to go from A to B rather than the distance. This is the correct way to use Coordinate Time to calculate Proper Time. But rather than use Einstein's approximation, we can use the exact calculation based on the value of gamma which is the ratio of the Coordinate Time interval divided by the Proper Time interval.

So, for example, if the velocity is 0.6c, then gamma is 1.25 so we divide the Coordinate Time interval going from A to B by gamma to get the Proper Time interval on the clock.