Radar distance on DrGreg's Rindler chart?

[Jorrie]

In an old (2008) thread, DrGreg posted this transformation from a Minkowski chart to a Rindler chart

He also wrote:

I haven't drawn the photon worldlines either, but you can imagine them joining the red dots across the diagonals of each distorted "square" of the grid.

My question: using these photon worldlines, does a two-way radar measurement give a valid radar distance between two points in the Rindler frame, as per this zoomed view?

The two vertical dashed lines are Rindler observers and the blue arrows are radar signals send and received by each of them. They attempt to measure the radar distance between the same two inertial particles at different times. If correct, it appears as if the radar distance between the two particles are increasing over time.

-J

PS: I did search for other Rindler threads, but could not find one that quite fits this question. Any links will be welcome.

 

[PAllen]

You are correct (assuming diagram is to scale). Is there any particular reason you find this surprising?

 

[PAllen]

However, when I do the math, unless I've blundered, I come to the opposite conclusion. The radar distance between two constant Minkowski position world lines, as measured by a given Rindler observer, decreases with time (from t=0, t as conventionally defined for Rindler). Of course, by symmetry, for t increasing toward t=0, the distance increases with time.

 

[Jorrie]

However, when I do the math, unless I've blundered, I come to the opposite conclusion. The radar distance between two constant Minkowski position world lines, as measured by a given Rindler observer, decreases with time (from t=0, t as conventionally defined for Rindler).

This was the initial motivation for the question, but is 'Rindler distance' the same as 'radar distance' for Rindler observers? If Rindler distance is a constant Rindler time foilation, then it makes sense that the distance in question decreases. What would 'ruler distance' do in this case?

-J

 

[PAllen]

This was the initial motivation for the question, but is 'Rindler distance' the same as 'radar distance' for Rindler observers? If Rindler distance is a constant Rindler time foilation, then it makes sense that the distance in question decreases. What would 'ruler distance' do in this case?

-J

No, Rindler and radar distance are not the same. For any Rindler observer, radar simultaneity matches Rindler coordinate simultaneity, but Radar distance does not match 'ruler distance' along a line of simultaneity. However, I took that into account in my calculation (but I only considered a small spacetime region, and used a series correction to ruler distance to get radar distance). If more terms of the series correction become significant, the result might change. This doesn't make one distance any more or less valid than another, as coordinate distances. However, the Rindler coordinate distance is the proper distance measured along simultaneity surface.

[Edit: done exactly, rather than approximately, I get the same result. Radar distance measured by a Rindler observer between two constant Minkowski position world lines decreases with time for t increasing from zero.]

 

[Jorrie]

[Edit: done exactly, rather than approximately, I get the same result. Radar distance measured by a Rindler observer between two constant Minkowski position world lines decreases with time for t increasing from zero.]

This then still leaves the question: what distance is represented by the increasing two-way "radar distance" on DrGreg's Rindler diagram (as in my 'zoom' graphic), calculated as 2-way elapsed Rindler time divided by c?

I gathered from his original post that DrGreg's diagram was exactly to scale, but I should ask him about that.

 

[PAllen]

This then still leaves the question: what distance is represented by the increasing two-way "radar distance" on DrGreg's Rindler diagram (as in my 'zoom' graphic), calculated as 2-way elapsed Rindler time divided by c?

I gathered from his original post that DrGreg's diagram was exactly to scale, but I should ask him about that.

I think trying to be precise on such a diagram is hard. Also, I assumed you were aware that you are not showing all the light paths you need for the relevant radar distance? The change in distance when including the missing paths is very small, and I could sort of make it come out either way a the resolution of that diagram (though, my first attempt did seem to show a very slight increase rather than decrease). Specifically, you need a right going signal pair for the earlier case to measure the radar distance to the same pair of Minkowski world lines. When that is included, and you add up the two smaller times below and the bigger time on top, the result is too close to rely on diagram measurements.

To clarify this, I am assuming the upper measurement needs only one signal path because you have chosen a Minkowski world line pair such that at the time midpoint of the signal path, the second M(inkowski) world line coincides with the Rindler observer. Continuing these two M world lines down, you have shown the left signal path, but you need a right signal path, and then you add up the proper times (=Rindler coordinate time). Then, comparing to the top is really too close to draw any conlusion from the diagram.

 

[Jorrie]

To clarify this, I am assuming the upper measurement needs only one signal path because you have chosen a Minkowski world line pair such that at the time midpoint of the signal path, the second M(inkowski) world line coincides with the Rindler observer. Continuing these two M world lines down, you have shown the left signal path, but you need a right signal path, and then you add up the proper times (=Rindler coordinate time). Then, comparing to the top is really too close to draw any conlusion from the diagram.

I understand the inaccuracies of the diagram, but I think we are talking about different radar distances here. My zoomed portion was not very clear, but I think the arrows were basically correct. Attached is a better effort.

Two independent measurements of the radar distance are made between the black Minkowski worldline (apex at 0,0) and the red Minkowski worldline with apex at -2,0 (second one to the left of black, for more clarity). The two measurements are made by the Rindler observer at x=0 and the Rindler observer at x~-1.2 respectively (vertical black arrows). The first (bottom) Rindler radar distance is roughly 2.5 light years and the second (top) is about 4 light years - a large difference, much more than can be attributed to the diagram's inaccuracies. The radar arrows were broken up onto segments to slightly curve them, but it makes little or no difference.

Where did I go wrong?

 

[PAllen]

With that clarification, your radar measurement makes no sense. I was too generous in trying to give it sense. You have two different Rindler observers measuring distance to one Minkowski world line at different Rindler times. What conclusion do you think follow from this?

Your original claim was "They attempt to measure the radar distance between the same two inertial particles at different times. If correct, it appears as if the radar distance between the two particles are increasing over time." That is what I computed and described. What you now clarify you meant does not correspond to this, and I cannot see what it corresponds to at all, except two unrelated measurements.

Note also, that the Rindler coordinate distances corresponding the radar measurements you show ALSO increase. Again, though, I don't see the sense of this: distance measurements by different Rindler observers, at different times??!

What would make sense is:

1) measurements at different times by one Rindler observer, to one Minkowski world line, which, in the case at hand, would show it to be moving away from the Rindler observer (whether Rindler coordinate distance or radar distance was used).

2) What you originally described: Measurements by one Rindler observer of the distance between two Minkowski world lines, at different times. This would show a decrease over time, as would Rindler coordinate distance (but the exact numbers would be different).

 

[DrGreg]

I gathered from his original post that DrGreg's diagram was exactly to scale, but I should ask him about that.

Yes, that's correct, but (of course) it's only accurate to the nearest pixel.

2) What you originally described: Measurements by one Rindler observer of the distance between two Minkowski world lines, at different times. This would show a decrease over time, as would Rindler coordinate distance (but the exact numbers would be different).

...And that means the observer calculates the radar distance to each of the two particles and subtracts them. But the two radar reflections need to occur at the same "radar time" for this to make sense.

 

[Jorrie]

With that clarification, your radar measurement makes no sense. I was too generous in trying to give it sense. You have two different Rindler observers measuring distance to one Minkowski world line at different Rindler times. What conclusion do you think follow from this?

OK, let me state the basic 'experiment' that I tried to perform and see where I went wrong in translating it to the Rindler chart, arguing as follows:

On a rod accelerated Born-rigidly, two particles are released from rest in the rod frame; one at x=0 (black) and one at x=-2 (red), to use the coordinates on the zoomed view. By the equivalence principle, the two particles should undergo tidal separation in the rod frame, due to the Rindler acceleration of red always being larger than black.

Which of the various distance measuring methods would show this separation over time, if any?

[Edit: I should have defined what I meant by 'Rindler acceleration'. The rod suffers larger proper acceleration at (-2,0) than at (0,0), so red would have the largest coordinate acceleration.]

 

[PAllen]

OK, let me state the basic 'experiment' that I tried to perform and see where I went wrong in translating it to the Rindler chart, arguing as follows:

On a rod accelerated Born-rigidly, two particles are released from rest in the rod frame; one at x=0 (black) and one at x=-2 (red), to use the coordinates on the zoomed view. By the equivalence principle, the two particles should undergo tidal separation in the rod frame, due to the Rindler acceleration of red always being larger than black.

Which of the various distance measuring methods would show this separation over time, if any?

[Edit: I should have defined what I meant by 'Rindler acceleration'. The rod suffers larger proper acceleration at (-2,0) than at (0,0), so red would have the largest coordinate acceleration.]

Gotta run, but this bears no resemblance to what you seemed to be saying before.

The first complication is that the Minkowski lines on Dr. Greg's chart are inapplicable to this problem.They are inertial paths in the original rest frame. What you need are two geodesic world lines tangent the Rindler verticals at a Rindler t=constant slice. The 4-velocity of these will be different at the front an back. Thus, you need a whole new computation of these world line's in Rindler coordinates.

I am not going to have time to do this calculation.

[Edit: if you start the dropping at t=0, then you can use the lines on Dr. Greg's chart. You have to decide whether to do radar measurements from the front or the back. You can't mix both. Then, you need one radar measurement for t=0, giving you the radar length of the rod - and the starting radar separation of the dropped particles. Then, at some later point, you need two radar measurements that you subract. Thus, 3 radar measurements is the minimum you need to figure out.]

 

[Jorrie]

...And that means the observer calculates the radar distance to each of the two particles and subtracts them. But the two radar reflections need to occur at the same "radar time" for this to make sense.

OK, makes sense, tx.

But does "same radar time" means "same Rindler time" (horizontal line), or is there a different meaning to the radar time?

 

[PAllen]

OK, makes sense, tx.

But does "same radar time" means "same Rindler time" (horizontal line), or is there a different meaning to the radar time?

I already answered this earlier. Radar simultaneity and Rindler simultaneity match, so the answer is yes. Note that for any acceleration profile more general than uniform, Radar simultaneity and the anaolog of Rindler simultaneity (Fermi-Normal coordinates) will NOT match, but for uniform acceleration they do match.

Anyway, for dropping balls at t=0, I realize my prior calculation applies - the balls will get closer together whether you use Radar distance or Rindler distance (but the number will not be the same). This is simply because they move away from you at increasing speed, and length contraction is the dominant effect.

[Edit: and, of course, this means the balls getting closer by either distance is always true. For the uniformly accelerating observer, just set up the mapping to Minkowski coords of the frame where the rod is momentarily at rest. Then, analysis already done applies.]

 

[PAllen]

On a rod accelerated Born-rigidly, two particles are released from rest in the rod frame; one at x=0 (black) and one at x=-2 (red), to use the coordinates on the zoomed view. By the equivalence principle, the two particles should undergo tidal separation in the rod frame, due to the Rindler acceleration of red always being larger than black.

If the balls are released at the same time as determined by either end of the rod (using Radar simultaneiety = Rindler simultaneity), they have the same 4 velocity (this follows from the definition of Born rigid acceleration - the ends can't be momentarily changing distance from each other as perceived by either end). Then, in an inertial frame, the balls stay the same distance apart. However, in the Rindler frame, they get closer, whichever distance measurement is made.

On the other hand, if balls are arranged to be released at the same time per an inertial frame in which the rod is moving (and rigidly accelerating), then one ball will catch up to the other; and this catchup is obviously frame/coordinate independent.

The effect you describe would occur if SR did not apply [that is, time and simultaneity were absolute], and you had gradient in Newtonian gravity. However, the very fact that there is a gradient in proper acceleration in a Rindler frame is an SR effect. Thus, SR cannot be ignored for any part of the analysis, and it leads to the opposite expectation from the one you suggest.

[Edit: More fundamentally, you cannot use the EP to draw any conclusions about situations with tidal gravity. Part of its formal definition is that tidal gravity can be ignored. ]

 

[Jorrie]

Thanks PAllen, this was very informative.

[Edit: More fundamentally, you cannot use the EP to draw any conclusions about situations with tidal gravity. Part of its formal definition is that tidal gravity can be ignored. ]

Is this similar to viewing Rindler coordinates as equivalent to a uniform gravitational field, where tidal gravity is absent?

 

[PAllen]

Thanks PAllen, this was very informative.
Is this similar to viewing Rindler coordinates as equivalent to a uniform gravitational field, where tidal gravity is absent?

No, Rindler frame is NOT equivalent to uniform gravity except on the scale where difference in proper acceleration is undetectable. I don't know a good term for what you have on larger scales (fictitious tidal gravity??). You have a gradient in proper acceleration, but a cloud of inertial particles in free fall neither changes shape nor volume, so there is no true tidal gravity.

The upshot is you can only use the EP (in its acceleration = gravity formulation) in a uniformly accelerating rocket on a scale such that the gradient in proper acceleration can be ignored.

Modern formulations of the EP are much cleaner, in my view. They say a free fall frame in gravity follows the laws of pure SR to the extent tidal (or global) effects can be ignored. The focus shifts to equivalence of free fall everywhere, rather than equivalence of accelerating frames and gravity.