Is length contraction real or just a matter of perspective?

[phyti]

I understand this. If the rod has 100 proper length, then in the observer's rest frame the light takes proper time 100 out and 100 back, for a total of Δt = 200, and proper length of 200/2, or 100.
I cannot see how this method yields a shorter rod length for O'. For example, in an O' frame in relative 0.8c motion, the rod has length 60. The O' time for the forward flash to reach the mirror is 300 [being 60 / (1 - 0.8)], and for the return flash is 33.333 [being 60 / (1 + 0.8)], for a total Δt = 333.333, and Δt/2 = 166.67 (a value greater than 100). So if O' uses this method, he gets a radar method length greater (not lesser) than the proper length.

If O' is measuring the rod located in the O frame, then with a = v/c, g = gamma,
and d = distance to reflection, total time t1 is

t1 = 2d/c(1+a) = 2t/(1+a) and

d' = ct' = ct/g(1+a) = d*sqrt[(1-a)/(1+a)].

d' = 100*sqrt[.2/1.8] = 100/3 = 33.3

In fig.1, applying inverse time dilation the O' clock shows 66.7 at x = 89. The perception of events as recorded by O' are overlaid for comparison. According to O' the reflection would have occurred at x' = 60.
In fig.2, the essential data for O' is transferred from fig.1 to show the perception from the O' frame.
With O' assuming a rest frame, and M moving past at -.8c, the distance at t' = 0 is
33.3*1.8 = 59.9 (60 without rounding errors).

Points of interest.
1. The reflection R occurs at Ox = 100. Since the signal is detected early according to the O' clock, O' concludes the universe has contracted by .6 along the x axis, i.e. an interpretation resulting from his own time dilation. Being an anaut and studying the SR manual, he would obviously be aware of this. He does have a choice.
2. R is the reflection event resulting from the signal path (magenta) as observed by O. R' is the reflection event as defined by the SR convention. With current knowledge there is no known method of verifying either case and so it is irrelevant since the round trips are equal. Considering science history that may change later.

https://www.physicsforums.com/attachments/65543

 

[JVNY]

Let's revert to WBN's earlier response:

I'm not sure if you're looking for some explicit calculation of radar distance, a conceptual explanation of radar distance, or something else entirely.

Perhaps it is best to start with explicit calculations. Let's restrict to straight line motion to simplify. Also, let's use three points (rear, center and front) and accelerate them Born rigidly. We will measure distance (rather than length) and not have to worry about accelerating the many particles within a single object. Since they are points, we can ignore the issues of how to accelerate all parts of an object simultaneously.

Start with the three points at rest in an inertial frame, rear at x=0.5, center at x=0.75, and front at x=1.0. The center is equidistant from the rear and the front: the distance is 0.25 from rear to center, and 0.25 from center to front, so a total distance of 0.5 from rear to front. The three points accelerate simultaneously Born rigidly: rear at proper rate a=2, center at proper rate a=1.33, and front at proper rate a=1. Simultaneously with starting to accelerate, the center flashes light forward and rearward. The light flashes reflect off of front and rear, then return to center.

As illustrated on the attached, the forward flash strikes the front at center time (Tc) 0.216. The rearward flash does not strike the rear until Tc=0.304. Forward flash returns to center at Tc=0.432. The rearward flash does not return to center until Tc=0.608. The hyperbolas show the world lines of the three points, and the dashed lines show the lines of simultaneity for the accelerating points.

1. Is the proper distance between rear and center always 0.25?
2. Is the proper distance between center and front always 0.25?
3. Is the ruler distance between rear and center always 0.25?
4. Is the ruler distance between center and front always 0.25?
5. Is the coordinate distance between rear and center always 0.25?
6. Is the coordinate distance between center and front always 0.25?
7. Is the radar distance between rear and center 0.304?
8. Is the radar distance between center and front 0.216?

The clock rates on each point are different, of course. This presumably affects the radar distance. Following the convention in http://en.wikipedia.org/wiki/Rindler_coordinates, clocks on the rear and center can be programmed to tick at a proportionately faster rate so that they tick at the same rate as the clock on the front, where g=1. Does one do this to calculate the radar distance?

 

[WannabeNewton]

The answers to your other questions are all "yes".

...clocks on the rear and center can be programmed to tick at a proportionately faster rate so that they tick at the same rate as the clock on the front, where g=1.

Where in the article is this stated? If the $g = 1$ world line describes the very front of the rod then the farther back we get from the front of the rod the slower comoving clocks tick relative to the clock comoving with the very front of the rod. It is ruler distance that takes this clock desynchronization into account when determining simultaneity lines for comoving observers.

The one-way speed of light depends on the global time coordinate $t$ which is also the proper time read by the clock comoving with the very front of the rod. Therefore radar distance will only give accurate results if all comoving clocks are synchronized with the comoving clock at the very front of the rod but we know this isn't true as stated above.

 

[JVNY]

Where in the article is this stated?

See the section "Relation to Cartesian chart," which states: "It is a common convention to define the Rindler coordinate system so that the Rindler observer whose proper time matches coordinate time is the one who has proper acceleration g=1, so that g can be eliminated from the equations"

I used the formulas for the relativistic rocket from http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html to generate the numbers on the Minkowski diagram. I am not sure upon further reflection whether the front ship is the correct clock to use. It has proper acceleration a=1, and the relativistic rocket explanation says that for years and light years, 1g is a=1.03. Is it right to use a=1 for the front ship, or do I have to change to a=1.03?

 

[WannabeNewton]

See the section "Relation to Cartesian chart," which states: "It is a common convention to define the Rindler coordinate system so that the Rindler observer whose proper time matches coordinate time is the one who has proper acceleration g=1, so that g can be eliminated from the equations"

Let me just address this quickly. This isn't the same thing as saying that all the comoving clocks have their rates adjusted so as to be synchronized with the clock comoving with the $g = 1$ world line. All this says is that the proper time read by the clock comoving with the $g = 1$ world line is the same as the global time coordinate $t$ of the Rindler chart. However from the relation $d\tau = x dt$ it is clear that all the other comoving clocks are desynchronized with the clock comoving with the $g = 1$ world line because their proper times disagree with the global coordinate time $t$.

 

[Dale]

A real philosopher is not going to use a useless label, any more than a physicist is. One of the main occupations of philosophy is to try to figure which labels are useful and meaningful, and which ones are not.

I have certainly never found any scientific utility in the label "real".

 

[stevendaryl]

I have certainly never found any scientific utility in the label "real".

My point was that there is nothing added by calling that label "philosophical".

 

[Dale]

My point was that there is nothing added by calling that label "philosophical".

I see your point. A useless philosophical label isn't any more useless than any other category of useless labels.

However, "real" is certainly a philosophical term. It is the central term in ontology, which is part of metaphysics, which is part of philosophy. So the "philosophical" might be redundant, but at least it was accurate here. I guess that I began using "philosophy" as a pejorative specifically because of how scientifically useless "real" is. I will have to keep track if there are other philosophical terms that are similarly irritating to figure out if it is philosophy to blame or just coincidence.

 

[JVNY]

Let me just address this quickly. This isn't the same thing as saying that all the comoving clocks have their rates adjusted so as to be synchronized with the clock comoving with the $g = 1$ world line. All this says is that the proper time read by the clock comoving with the $g = 1$ world line is the same as the global time coordinate $t$ of the Rindler chart. However from the relation $d\tau = x dt$ it is clear that all the other comoving clocks are desynchronized with the clock comoving with the $g = 1$ world line because their proper times disagree with the global coordinate time $t$.

I disagree here. For example, let's look at the three points again, this time having the front send a flash toward the rear and the rear send a flash toward the front, both simultaneously with beginning their Born rigid acceleration. According to the relativistic rocket formulas, the rearward flash strikes the rear at rear time (Tr) = 0.346 (less time than that elapsed in the inertial frame, which is t=0.375). The forward flash strikes the front at front time (Tf) = 0.693 (less than the 0.75 time elapsed in the inertial frame). These two strikes are simultaneous for rear and front. They occur on the dashed line of simultaneity. Their instantaneous velocity when they strike is 0.6c. See the attached diagram.

The Tr and Tf times are rounded; in fact Tf is exactly twice Tr. So if you program the rear clock to show two ticks elapsed for every proper tick, Tr will always show the same time as Tf simultaneously (as they agree on simultaneity -- I don't want to say "in their own frame" or "in their own coordinates" or the like, because I am not yet clear on those phrases).

There is more on this in a presentation by Mallinckrodt at http://www.csupomona.edu/~ajm/professional/talks/relacc.ppt . See particularly slide 12 that shows a similar line of simultaneity. He concludes that the front and rear agree on simultaneity of events, their common velocity, and their proper separation, and he concludes that the clock rate of each clock is directly proportional to the vertex distance. Therefore if you just program the rear clock to show a proportionately greater number of ticks (for the rear clock in the attached example, double the proper rate) then you have clocks that always show the same time at the same time (as they agree on simultaneity).

As a result, the speed of light is isotropic even for these accelerating points. You can confirm this two ways. First, if the rear clock is programmed to display twice its elapsed proper time, then the clocks will agree that each flash took 0.693 to reach the other end. Second, the flashes arrive along the line of simultaneity from the vertex, so they must have arrived simultaneously for the front and rear, which means that they must have had the same speed.

 

[WannabeNewton]

Sorry allow me to clarify. When talking about the desynchronization of the comoving clocks I am referring specifically to Einstein synchronization of ideal clocks. A clock is ideal if it is parametrized by proper time i.e. if its 4-velocity satisfies $\vec{u}\cdot \vec{u} = -1$ (in natural units). In our Born rigid rod scenario, a comoving clock will therefore be an ideal clock if, in the Rindler chart, we have $\vec{u} = \frac{1}{x}(1,0,0,0)$ along the world line of the clock. Therefore two comoving ideal clocks will always tick at different rates given by $\frac{d\tau_1}{d\tau_2} = \frac{x_1}{x_2}$ hence we cannot Einstein synchronize two such comoving ideal clocks permanently because even if the clocks are synchronized initially they will desynchronize due to the difference in ticking rates; one of the consistency relations that Einstein synchrony has to satisfy is that two clocks synchronized initially remain synchronized permanently.

So in other words I'm talking about proper time synchronization; if that was unclear I apologize but Einstein synchrony normally refers to proper time synchronization in standard SR texts so I thought it would be clear. Therefore we cannot synchronize the proper times of comoving clocks in our Born rigid rod scenario using the Einstein synchronization convention. It turns out that a congruence of comoving clocks with 4-velocity field $\vec{u}$ is proper time synchronizable (in the exact sense elucidated above) if and only if the congruence has a 4-velocity field of the form $\vec{u} = \vec{\nabla}t$, which immediately implies that the congruence is geodesic and vorticity-free i.e. $\vec{a} = \nabla_{\vec{u}}\vec{u} = 0$ and $\vec{\omega} = \vec{\nabla}\times \vec{u} = 0$ respectively. The congruence of comoving clocks in our Born rigid rod scenario is obviously vorticity-free but it is not geodesic therefore the comoving clocks are not proper time synchronizable, as stated above.

As an aside, note therefore that in SR the only congruences of ideal comoving clocks that can be Einstein synchronized are those of inertial clocks because in SR a congruence of ideal comoving clocks satisfies $\vec{a} = 0$ and $\vec{\omega} = 0$ if and only if the clocks are inertial. This gives us back the rigid coordinate lattice of inertial clocks talked about at the inception of the thread.

The vanishing vorticity $\vec{\omega} = 0$ of this congruence is in and of itself a very important property of the congruence. To start with let's write $\vec{\nabla}\times \vec{u} = 0$ in a mathematically equivalent form for this congruence. Notice that we can write $\vec{u} =\frac{1}{x}(1,0,0,0)$ as $\vec{u} = \frac{1}{x}\vec{\nabla }t$ where $t$ is as usual the global Rindler coordinate time. This means that all of the comoving clocks have world lines which are orthogonal to the surfaces $t = \text{const}$. These $t = \text{const}$ surfaces are exactly the simultaneity surfaces shared by all of the comoving clocks! In other words the fact that $\vec{\omega} = 0$, or equivalently $\vec{u} = \frac{1}{x}\vec{\nabla }t$, means that if we slice this congruence with the surface $t = \text{const}$, that is, we find the intersection of $t = \text{const}$ with the world lines of all the comoving clocks, then the resulting events will be simultaneous for all of the comoving clocks. The $t = \text{const}$ "global" simultaneity surfaces ("global" because all of the comoving clocks share these simultaneity surfaces) are depicted here: http://upload.wikimedia.org/wikipedia/commons/5/56/Rindler_chart.svg

Recall that two comoving clocks are proper time synchronizable if and only if they belong to a congruence of comoving clocks with 4-velocity field $\vec{u} = \vec{\nabla}t$. In our Born rigid rod scenario we instead have $\vec{u} = \frac{1}{x}\vec{\nabla}t$. I hope it's clear to you now what the difference is: while the comoving clocks all share common simultaneity surfaces $t = \text{const}$, their proper times cannot be Einstein synchronized because of the factor $\frac{1}{x}$ that causes ideal clocks at different locations on the rod to tick at different rates.

EDIT: So what I'm trying to say is if you want clock synchronization along the rod then you need to use non-ideal comoving clocks (except for the clock of proper acceleration $g = 1$ of course) i.e. you need to have the comoving clocks read something other than proper time. Exactly what kind of time each one reads is dependent upon the factor by which you need to mechanically readjust the clock rate of each comoving clock so as to tick uniformly with the ideal clock of proper acceleration $g = 1$.

 

[JVNY]

How about this scenario: (a) you set up the lattice in a rest frame, (b) you synchronize the clocks there, and (c) you program each of them to accelerate simultaneously in the rest frame and simultaneously to begin displaying proportionally greater time than elapsed time, inversely proportionally to their distance from the vertex (e.g., double the rate of proper time for the clock at 0.5 in the diagram above). This way you do not have to synchronize them while they are accelerating. They start synchronized at rest, and they should stay synchronized.

 

[WannabeNewton]

How about this scenario: (a) you set up the lattice in a rest frame, (b) you synchronize the clocks there, and (c) you program each of them to accelerate simultaneously in the rest frame and simultaneously to begin displaying proportionally greater time than elapsed time, inversely proportionally to their distance from the vertex (e.g., double the rate of proper time for the clock at 0.5 in the diagram above). This way you do not have to synchronize them while they are accelerating. They start synchronized at rest, and they should stay synchronized.

Yes so this relates directly to what I explained in post #40. The congruence of comoving clocks associated with the Born rigid accelerating rod has a 4-velocity field $\vec{u} = \frac{1}{x}\vec{\nabla}t$. As explained this means that if we want all the comoving clocks to be ideal (meaning they all read proper time) then we cannot Einstein synchronize them. However as noted $\vec{u} = \frac{1}{x}\vec{\nabla}t\Leftrightarrow \vec{\omega} = \vec{\nabla}\times \vec{u} = 0$* so all the comoving clocks share common simultaneity surfaces $t = \text{const}$; these simultaneity surfaces agree with the proper time of the clock with proper acceleration $g = 1$. Hence if we are willing to use the ideal clock of proper accelerating $g = 1$ as a reference clock and readjust the rates of all the other comoving clocks accordingly (so as to make them non-ideal) then we can synchronize all the comoving clocks.

More explicitly, since $\frac{dt}{d\tau} = \frac{1}{x}$ we can readjust the rate of each clock so that instead of reading proper time $\tau$ it reads the time $t = \frac{1}{x}\tau$ where I have set the integration constant so that all the comoving clocks are initially synchronized with the ideal clock of proper acceleration $g = 1$. This will ensure the synchronization of the comoving clocks for all $t = \text{const}$ simultaneity surfaces. But as noted this will make all the comoving clocks non-ideal. For example we will now have $\vec{u}\cdot \vec{u} = -x^2$ which is a very cumbersome property to have for the 4-velocity field to the congruence of comoving clocks; we really want the 4-velocity field to be normalized to unity so that at the least $\vec{a}\cdot \vec{u} = 0$.

*In general, if we have $\vec{u} = \gamma \vec{\nabla}t$ for some scalar field $\gamma$ then this trivially implies that $\vec{\omega} = 0$ but $\vec{\omega} = 0$ doesn't necessarily imply that $\vec{u} = \gamma \vec{\nabla}t$. In fact $\vec{\omega} = 0$ only implies this locally i.e. in a sufficiently small neighborhood of each event in space-time; this result is known as the Poincare Lemma. The only reason I used the equivalence $\Leftrightarrow$ is we are dealing with Minkowski space-time which has a trivial fundamental group and allows for the equivalence $\Leftrightarrow$ to hold.

 

[PAllen]

Sorry allow me to clarify. When talking about the desynchronization of the comoving clocks I am referring specifically to Einstein synchronization of ideal clocks. A clock is ideal if it is parametrized by proper time i.e. if its 4-velocity satisfies $\vec{u}\cdot \vec{u} = -1$ (in natural units). In our Born rigid rod scenario, a comoving clock will therefore be an ideal clock if, in the Rindler chart, we have $\vec{u} = \frac{1}{x}(1,0,0,0)$ along the world line of the clock. Therefore two comoving ideal clocks will always tick at different rates given by $\frac{d\tau_1}{d\tau_2} = \frac{x_1}{x_2}$ hence we cannot Einstein synchronize two such comoving ideal clocks permanently because even if the clocks are synchronized initially they will desynchronize due to the difference in ticking rates; one of the consistency relations that Einstein synchrony has to satisfy is that two clocks synchronized initially remain synchronized permanently.

So in other words I'm talking about proper time synchronization; if that was unclear I apologize but Einstein synchrony normally refers to proper time synchronization in standard SR texts so I thought it would be clear. Therefore we cannot synchronize the proper times of comoving clocks in our Born rigid rod scenario using the Einstein synchronization convention. It turns out that a congruence of comoving clocks with 4-velocity field $\vec{u}$ is proper time synchronizable (in the exact sense elucidated above) if and only if the congruence has a 4-velocity field of the form $\vec{u} = \vec{\nabla}t$, which immediately implies that the congruence is geodesic and vorticity-free i.e. $\vec{a} = \nabla_{\vec{u}}\vec{u} = 0$ and $\vec{\omega} = \vec{\nabla}\times \vec{u} = 0$ respectively. The congruence of comoving clocks in our Born rigid rod scenario is obviously vorticity-free but it is not geodesic therefore the comoving clocks are not proper time synchronizable, as stated above.

As an aside, note therefore that in SR the only congruences of ideal comoving clocks that can be Einstein synchronized are those of inertial clocks because in SR a congruence of ideal comoving clocks satisfies $\vec{a} = 0$ and $\vec{\omega} = 0$ if and only if the clocks are inertial. This gives us back the rigid coordinate lattice of inertial clocks talked about at the inception of the thread.

The vanishing vorticity $\vec{\omega} = 0$ of this congruence is in and of itself a very important property of the congruence. To start with let's write $\vec{\nabla}\times \vec{u} = 0$ in a mathematically equivalent form for this congruence. Notice that we can write $\vec{u} =\frac{1}{x}(1,0,0,0)$ as $\vec{u} = \frac{1}{x}\vec{\nabla }t$ where $t$ is as usual the global Rindler coordinate time. This means that all of the comoving clocks have world lines which are orthogonal to the surfaces $t = \text{const}$. These $t = \text{const}$ surfaces are exactly the simultaneity surfaces shared by all of the comoving clocks! In other words the fact that $\vec{\omega} = 0$, or equivalently $\vec{u} = \frac{1}{x}\vec{\nabla }t$, means that if we slice this congruence with the surface $t = \text{const}$, that is, we find the intersection of $t = \text{const}$ with the world lines of all the comoving clocks, then the resulting events will be simultaneous for all of the comoving clocks. The $t = \text{const}$ "global" simultaneity surfaces ("global" because all of the comoving clocks share these simultaneity surfaces) are depicted here: http://upload.wikimedia.org/wikipedia/commons/5/56/Rindler_chart.svg

Recall that two comoving clocks are proper time synchronizable if and only if they belong to a congruence of comoving clocks with 4-velocity field $\vec{u} = \vec{\nabla}t$. In our Born rigid rod scenario we instead have $\vec{u} = \frac{1}{x}\vec{\nabla}t$. I hope it's clear to you now what the difference is: while the comoving clocks all share common simultaneity surfaces $t = \text{const}$, their proper times cannot be Einstein synchronized because of the factor $\frac{1}{x}$ that causes ideal clocks at different locations on the rod to tick at different rates.

EDIT: So what I'm trying to say is if you want clock synchronization along the rod then you need to use non-ideal comoving clocks (except for the clock of proper acceleration $g = 1$ of course) i.e. you need to have the comoving clocks read something other than proper time. Exactly what kind of time each one reads is dependent upon the factor by which you need to mechanically readjust the clock rate of each comoving clock so as to tick uniformly with the ideal clock of proper acceleration $g = 1$.

Well, for the special case of Born rigid uniform accelerations covered by the Rindler chart, I've worked through proofs that two clocks of the Rindler congruence remain Radar synchronized, and this can be used to derive the Rindler chart. George Jones, a while back, gave such a derivation here on PF (which I can't find now). However, the following paper (leading up to fig. 7) shows radar simultaneity surfaces for a uniformly accelerating observer are the same as Rindler coordinate surfaces of constant time:

http://arxiv.org/abs/gr-qc/0104077

 

[WannabeNewton]

Well, for the special case of Born rigid uniform accelerations covered by the Rindler chart, I've worked through proofs that two clocks of the Rindler congruence remain Radar synchronized...

I don't understand how this can happen if by "Radar synchronized" you mean Einstein synchronization of the proper times of the two clocks since the "time dilation" factor forces ideal clocks at different locations along the rod to tick at different rates. Is the term "radar synchrony" referring to something more general in this context?

EDIT: Are you referring to something similar to what I described in post #42?

However, the following paper (leading up to fig. 7) shows radar simultaneity surfaces for a uniformly accelerating observer are the same as Rindler coordinate surfaces of constant time:

This certainly agrees with a proof given in one of my SR texts of the equality, for Rindler observers, between the Radar simultaneity hypersurfaces and the local rest spaces ($t = \text{const}$ surfaces in the Rindler chart) wherein the Radar simultaneity hypersurfaces are constructed from the Einstein simultaneity convention. I implicitly assumed this in my post but in retrospect it would have been good to point it out explicitly since the two don't agree to all orders of expansion for arbitrarily accelerating observers.

 

[JVNY]

Well, for the special case of Born rigid uniform accelerations covered by the Rindler chart, I've worked through proofs that two clocks of the Rindler congruence remain Radar synchronized, and this can be used to derive the Rindler chart. George Jones, a while back, gave such a derivation here on PF (which I can't find now). . .

Here are some links to George Jones posts that appear to be on point (some just show the highlighting of his name, so you'll have to look through; others are direct to the posts).

https://www.physicsforums.com/showthread.php?t=110742&highlight=george+jones+radar

https://www.physicsforums.com/showpost.php?p=1191512&postcount=22

https://www.physicsforums.com/showthread.php?t=160357&highlight=george+jones+radar&page=3

https://www.physicsforums.com/showthread.php?t=673178&highlight=george+jones+radar

https://www.physicsforums.com/showpost.php?p=4287098&postcount=51

https://www.physicsforums.com/showthread.php?t=421957&highlight=george+jones+radar

 

[PAllen]

I don't understand how this can happen if by "Radar synchronized" you mean Einstein synchronization of the proper times of the two clocks since the "time dilation" factor forces ideal clocks at different locations along the rod to tick at different rates. Is the term "radar synchrony" referring to something more general in this context?

EDIT: Are you referring to something similar to what I described in post #42?



This certainly agrees with a proof given in one of my SR texts of the equality, for Rindler observers, between the Radar simultaneity hypersurfaces and the local rest spaces ($t = \text{const}$ surfaces in the Rindler chart) wherein the Radar simultaneity hypersurfaces are constructed from the Einstein simultaneity convention. I implicitly assumed this in my post but in retrospect it would have been good to point it out explicitly since the two don't agree to all orders of expansion for arbitrarily accelerating observers.

I see the confusion I was having. Even though two Rindler observers will continue to (and symmetrically) agree on which events are simultaneous by radar convention, they will still see each others clocks going out of synch. So, if set to agree on t=0, some time later A will say:

hmm, my t=1 is simultaneous to your t=.5, your clock is running slow.

and B will say:

hmm, my t=.5 is simultaneous to your t=1, your clock is running fast.

I was conflating agreement on simultaneity with agreement on rates.

 

[JVNY]

I see the confusion I was having. Even though two Rindler observers will continue to (and symmetrically) agree on which events are simultaneous by radar convention, they will still see each others clocks going out of synch. So, if set to agree on t=0, some time later A will say:

hmm, my t=1 is simultaneous to your t=.5, your clock is running slow.

and B will say:

hmm, my t=.5 is simultaneous to your t=1, your clock is running fast.

I was conflating agreement on simultaneity with agreement on rates.

There are several issues at play here. The first issue is agreement on clock ticking rates. Both Rindler observers will agree that their clocks tick at different rates when displaying proper time, so the clocks will desynchronize in their line of simultaneity. The second issue is whether clocks can be programmed to maintain synchronization. If the clocks are synchronized while at rest and each is programmed to display elapsed time equal to a specified proportion of elapsed proper time (e.g., double the proper time for the rear clock as described in post 39, and per the diagram attached there), then the clocks' displayed tick rates will be the same, and the clocks will remain synchronized, throughout their Born rigid acceleration.

The third issue is whether one can synchronize the clocks while they are accelerating Born rigidly. Assume that you set up the lattice of rods and clocks while they are at rest and confirm their proper distances (say rear clock at x=0.5 and front at x=1.0, per diagram in post 39). You neglect to synchronize the clocks, and then they begin to accelerate Born rigidly per post 39. Now, can you synchronize the clocks? Yes, you can (although I was not so sure about this before, since it seemed circular).

The front starts its clock. It then sends a radar signal to rear, which reflects and returns to front. The round trip takes front time 0.693. Knowing the acceleration rates and proper distances as we do, we know that the rear clock has to display twice as many ticks as properly elapse in order to stay synchronized with front. So front sends a signal to rear telling it to program its clock to display twice the elapsing proper time. Then, front sends a signal at front time 1 instructing rear to set its clock to 2 upon receiving the signal.

The fourth issue is whether you even need to know in advance the facts of the proper distances and acceleration rates to synchronize the clocks while accelerating. Using the article linked in post 43, one might be able to do this.

Say there are only the two objects, rear and front. Each can feel some proper acceleration, but neither knows their proper distance. Front sends several test signals to rear that reflect and return. If each signal takes the same total elapsed proper time to return, front knows that they are both accelerating Born rigidly. Front uses his accelerometer to determine that his proper acceleration is a=1. Then, he notes that his proper time for a round trip of a rearward signal is 0.693. He should be able to calculate that the rear is accelerating at a=2 (by backsolving using the relativistic rocket formulas). So he can send the signals as described above to instruct rear how to program her clock (tick at twice her proper rate) and what time to set her clock.

Now, does this suggest that one can also synchronize clocks on a rotating rim (which is also accelerating, although change of direction rather than change of speed)? That is for another thread.

The article linked in post 43 raises some questions. For example, the article states that one can assign a unique time to any event with which the observer can exchange signals, and that that time is independent of any choice of coordinates. Then the article uses that time to explain the twin paradox. However, consider the diagram in post 43. Assume that there is another observer who begins at rest at x=0.625, t=0. He remains at rest (call him "inertial observer"), so his world line is the vertical line that intersects the rear's hyperbolic world line at x=0.625, t=0.375. Now define the event at issue as rear and inertial observers crossing paths (being at the same point on the x axis).

If front sends a radar signal that strikes that event (shown in the diagram in post 39) and returns, the radar time as defined in the article is 0.346 (half the round trip time of 0.693). That is the time on rear's clock at the event. However, it is not the time on inertial observer's clock at the event. The time on his clock is 0.375. So the radar method does not identify a unique proper time for everyone who is present at the event. All it does is identify the proper time at the event of someone who is at rest with respect to the signal sender (0.346). The proper time of the inertial observer is greater (0.375), and simultaneously for the Rindler pair the front's proper time is greater yet (0.693).

The two twins in the twin paradox are not at rest with respect to each other, so it is not clear how radar time has anything to do with resolving the twin paradox.

 

[PAllen]

The article linked in post 43 raises some questions. For example, the article states that one can assign a unique time to any event with which the observer can exchange signals, and that that time is independent of any choice of coordinates. Then the article uses that time to explain the twin paradox. However, consider the diagram in post 43. Assume that there is another observer who begins at rest at x=0.625, t=0. He remains at rest (call him "inertial observer"), so his world line is the vertical line that intersects the rear's hyperbolic world line at x=0.625, t=0.375. Now define the event at issue as rear and inertial observers crossing paths (being at the same point on the x axis).

You misunderstand the paper. The radar method allows anyone observer to assign a t coordinate (a corresponding time on their own clock) to any event reachable by two way signals. It does not say (in fact, it says the opposite) that two observer's assignments will necessarily agree. Further, it notes an example where the requirement of two way signals greatly limits the region of spacetime that can be addressed with the method.

 

[WannabeNewton]

Now, does this suggest that one can also synchronize clocks on a rotating rim (which is also accelerating, although change of direction rather than change of speed)? That is for another thread.

Since PAllen has already addressed your other points, let me reiterate that you cannot do this. I've explained why in post #24. You can only Einstein synchronize ideal clocks on a rotating ring (in flat space-time) if they are infinitesimally separated from one another.

 

[JVNY]

OK, I am suitably chastised for the twin paradox and rotating rim comments!

How about the other conclusion? Do you agree that one can synchronize a row of clocks that is already undergoing Born rigid acceleration by using radar signals as described in post 47?

 

[WannabeNewton]

Yes that's perfectly fine if you are willing to use non-ideal comoving clocks everywhere except at the front of the rod (which undergoes a proper acceleration of $g = 1$). We can use the comoving clock at the front of the rod as a reference clock; all the other clocks, by means of radar and the Einstein simultaneity convention, can adjust their clock rates until they read a time other than proper time so as to tick uniformly with our reference clock. The time that they read will just be the global Rindler coordinate time $t$ and we will have achieved synchronization. Of course the downfall of this synchronization is that apart from our reference clock, the comoving clocks will be non-ideal and this will make our formulas more messy.

See post #42 in case you need further clarification.

From a purely aesthetic point of view, ideal clocks are the ones that are determined solely by the space-time metric whereas non-ideal clocks (such as the ones we've used above) can only be employed if we have extra information (such as a choice of coordinate system) which is unfavorable from a geometric point of view.

 

[JVNY]

Thanks. Now then, to the conceptual question -- what is radar distance, conceptually? In the earlier thread referenced in post 1, the group essentially advised that the Shapiro delay is a misnomer. It does not show a delay (slowing) of light speed, but rather results from the longer distance that light travels in curved space time near a mass. But spacetime is not curved in SR per posts 22 and 24. So is radar distance also a misnomer? All of the other measures of distance in post 32, including ruler distance, agree with each other. Only radar distance differs.

 

[PAllen]

Thanks. Now then, to the conceptual question -- what is radar distance, conceptually? In the earlier thread referenced in post 1, the group essentially advised that the Shapiro delay is a misnomer. It does not show a delay (slowing) of light speed, but rather results from the longer distance that light travels in curved space time near a mass. But spacetime is not curved in SR per posts 22 and 24. So is radar distance also a misnomer? All of the other measures of distance in post 32, including ruler distance, agree with each other. Only radar distance differs.

I don't want to go into the background for post #32 (for one thing, coordinate distance can be anything you want). However, for a uniformly accelerating, rigid rod, all 3 of the following measurements of the rod length disagree:


1) apply a bunch of standard short rulers along the rod, add them up (ruler distance = proper distance in hypersurface 4-orthogonal to the congruence defining the rod).

2) radar distance

3) image distance (back holds up a circle of known size measured local to the back; front measures its visual angle and figures what its distance must be assuming normal trigonometry).

Radar distance has a big advantage in practice - it can be measured on scales of the solar system. I was never able to find a supplier of 1 AU rulers.

 

[JVNY]

But radar distance has a major disadvantage, according to WannabeNewton:

Now in flat space-time . . . the radar distance gives a divergent value even for uniformly accelerating observers . . . Thus if we used the same radar distance prescription that works impeccably for inertial observers by averaging over the time it takes the light signal to complete the radar echo experiment then we get a nonsensical measure of distance as determined by the accelerated observer.

https://www.physicsforums.com/showpost.php?p=4613116&postcount=6​

 

[WannabeNewton]

You might find this article of use JVNY: http://arxiv.org/pdf/0708.0170.pdf

 

[PAllen]

But radar distance has a major disadvantage, according to WannabeNewton:

Now in flat space-time . . . the radar distance gives a divergent value even for uniformly accelerating observers . . . Thus if we used the same radar distance prescription that works impeccably for inertial observers by averaging over the time it takes the light signal to complete the radar echo experiment then we get a nonsensical measure of distance as determined by the accelerated observer.

https://www.physicsforums.com/showpost.php?p=4613116&postcount=6​

Only approaching a horizon, who cares (further, radar simultaneity can be combined with proper distance on the simultaneity surface; then this problem disappears)? Also, there has never existed in the universe, an eternally accelerating observer. The real world analog is hovering near a black hole. You want to try lowering a ruler to the horizon? Meanwhile, radar coordinates never diverge and cover all of spacetime for any travel trajectory that begins near inertial and ends near inertial. Fermi-Normal coordinates will fail to cover parts of spacetime for such a case if, in the interim, there are sharp accelerations.

The fact of the matter is, make a list of features of inertial coordinates in flat spacetime.

1) No non-inertial coordinate system will match these features.

2) Pick different features to maintain, and you get different non-inertial coordinate systems.

3) Reasonable non-inertial coordinates systems agree at lab scales (Radar and Fermi-Normal are indistinguishable at lab scales, and rigorously converge locally).

Fermi-Normal coordinates (the main alternative) can't handle sudden changes in direction. Radar coordinates handle it just fine.

There is no clearly preferred non-inertial coordinate system the way there is for inertial coordinate systems.

 

[JVNY]

Interesting, thanks.

Therefore may one conclude that the speed of light is not constant for observers undergoing Born rigid acceleration? Stick with straight line, change of speed acceleration (not rotation as in post 24). If such an observer sends a light flash to another, who reflects it back to the sender, and the sender measures the speed of light as the proper distance that the light traveled divided by the sender's proper time for the round trip, the result will not be c. And it will differ for rearward and return flashes than for forward and return flashes that cover the same proper distance.

The result might differ in other measures, but per post 56 "There is no clearly preferred non-inertial coordinate system" and so none of the others should be preferred over the use of proper distance for the Born rigidly accelerating observer and proper time for him.

 

[PAllen]

Interesting, thanks.

Therefore may one conclude that the speed of light is not constant for observers undergoing Born rigid acceleration? Stick with straight line, change of speed acceleration (not rotation as in post 24). If such an observer sends a light flash to another, who reflects it back to the sender, and the sender measures the speed of light as the proper distance that the light traveled divided by the sender's proper time for the round trip, the result will not be c. And it will differ for rearward and return flashes than for forward and return flashes that cover the same proper distance.

The result might differ in other measures, but per post 56 "There is no clearly preferred non-inertial coordinate system" and so none of the others should be preferred over the use of proper distance for the Born rigidly accelerating observer and proper time for him.

Light signals traveling faster than c in non-inertial coordinates is competely routine. Even in radar coordinates, speed of light is forced to be c only measured to and from a chosen observer.

 

[JVNY]

Thanks. Is there a special meaning to using the word "coordinates" in the response? As opposed to, for example, saying "Light signals traveling faster than c as measured by accelerating observers is completely routine."

I don't see the purpose of the word "coordinates" in the response, because the Rindler observer determines light speed without the use of coordinates. Say there is a row of Rindler observers, and one at one end lays a meter stick along the row and measures the distance to the other end to be one meter. Then he sends a signal to the other end, which reflects and returns. The observer uses time determined by his own wristwatch (proper time) for the only measure of time (the round trip time for the signal), and distance determined by a meter stick for the only measure of distance. So why use the word "coordinates?"

 

[PAllen]

Thanks. Is there a special meaning to using the word "coordinates" in the response? As opposed to, for example, saying "Light signals traveling faster than c as measured by accelerating observers is completely routine."

I don't see the purpose of the word "coordinates" in the response, because the Rindler observer determines light speed without the use of coordinates. Say there is a row of Rindler observers, and one at one end lays a meter stick along the row and measures the distance to the other end to be one meter. Then he sends a signal to the other end, which reflects and returns. The observer uses time determined by his own wristwatch (proper time) for the only measure of time (the round trip time for the signal), and distance determined by a meter stick for the only measure of distance. So why use the word "coordinates?"

One whole point of this discussion is that while there is a preferred, global, coordinate system for inertial observers in SR, there is no such thing for non-inertial observers. Different coordinate choices lead to different speed of light in those coordinates. To show what is meant by coordinate speed of light, consider you have some spatial coordinate x and some time coordinate t. For a particular light ray moving (for simplicity) in the x direction, you have some x(t) describing its path. Derivative by t is the coordinate speed of light.

The experimental analog of this is that different ways of measuring the same thing that agree for inertial observers disagree for non-inertial observers.

In your example measurement, we just finished discussing that 3 methods of measuring distance give 3 different results for Rindler observers. This will lead to 3 different speeds of light.

Note that your ruler suggestion is actually the most problematic to relate to the math because you have to make assumptions about how an accelerating ruler behaves. You can assume it is approximately Born rigid, but then you need to find a way to verify how precisely this is true.