## Different Clock Rates Throughout Accelerating Spaceship

 Quote by PAllen Agreed. Just don't try to generalize this to other situations without understanding the complexities I described.
Sure.

1) for uniform linear motion, gamma is purely "subjective" - nobody's clock is moving slower than anyone else's in a meaningful way. it is all wrapped up with planes of simultaneity.

2) for uniform circular motion, gamma gains a clear "objective" meaning - RF can use it to describe *all* of AO's time dilation, and AO can use the inverse to describe *average* time anti-dilation of RF. It is similar to the gravity situation where there are agreed differences in time dilations.

3) The next part is harder for me to work out -
the linearly accelerating observer - how RF and AO can measure their relative clock speeds.

I gather than RF can calculate AO's relative time dilation by simply integrating ever changing gamma with ever changing velocity?

But to ask how AO determines RF's clock speed, this situation is neither the simple 'subjective' or 'objective' case above.

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 Quote by 1977ub 3) The next part is harder for me to work out - the linearly accelerating observer - how RF and AO can measure their relative clock speeds. I gather than RF can calculate AO's relative time dilation by simply integrating ever changing gamma with ever changing velocity? But to ask how AO determines RF's clock speed, this situation is neither the simple 'subjective' or 'objective' case above.
Yes, RF could just integrate gamma(t) over the accelerating path. RF can do this for any path.

For AO, this simple example raises one of the SR concepts difficult to grasp on first encounter. This is the so called Rindler horizon. If you try to ask about how AO would model RF clocks by factoring out light delay using some simultaneity convention, you face the following observation:

RF clocks that AO is accelerating away from appear to red shift until they disappear, at a fixed distance behind the AO. Any RF clock further away cannot be seen or communicate in any way with the AO, so mutual clock comparison is impossible. Note that there is a one way aspect to this (as for all horizons): any RF clock can eventually receive a signal from any point in AO's history; however, there is a last time, for every clock in RF, after which it cannot send any signals to AO.

Visually, you can certainly say the case is more like your (1): any clock at rest in RF and AO's clock each eventually see the other clock freezing and red shifting to infinity.
 For normal SR we can use gamma for both observers to determine that they see each other's clocks as ticking slower by gamma. This works for clocks they are passing or any other clocks in either frame. I understand that it becomes more complex doing this between RF and AO for remote clocks, so I would just wish to focus on RF clocks being passed by the AO observer, all of which are deemed simultaneous and synchronized by RF. RF uses integrating gamma to watch AO's single clock continually slow down relative to the RF network of clocks as v and gamma increase. What does AO decide passing these RF clocks. How fast does this network of RF clocks tick? The same method?

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 Quote by 1977ub For normal SR we can use gamma for both observers to determine that they see each other's clocks as ticking slower by gamma. This works for clocks they are passing or any other clocks in either frame. I understand that it becomes more complex doing this between RF and AO for remote clocks, so I would just wish to focus on RF clocks being passed by the AO observer, all of which are deemed simultaneous and synchronized by RF. RF uses integrating gamma to watch AO's single clock continually slow down relative to the RF network of clocks as v and gamma increase. What does AO decide passing these RF clocks. How fast does this network of RF clocks tick? The same method?
A direct clock comparison is invariant. If, using RF, you compute (correctly) that AO's clock will be further and further behind each RF clock it passes, then, ipso facto, AO will find each passing RF clock further and further ahead. But please note, the same would be true if AO were moving uniformly (a inertially moving clock passing this sequence of clocks would interpret that each is ticking slow, but they are increasing out of synch with each other per the moving clock).

To try to create an symmetric situation for AO, we need a configuration of co-accelerating clocks, with one RF clock going past them. For this, we have to decide the acceleration profile of each clock, and also how to synchronize them. For the latter, unfortunately, there is no preferred approach (what is special about inertially comoving clocks is that any reasonable synchronization procedure produces the same result; this is not true for a family of accelerating clocks. In particular, Einstein clock synch using light signals, and Born rigidity based simultaneity, disagree.)

I urge you to focus on questions about what AO observes, and stop trying to treat AO as defining a frame of reference.

 Quote by PAllen I urge you to focus on questions about what AO observes, and stop trying to treat AO as defining a frame of reference.
AO has a clock. For me that's most important thing. I'm trying to build up my understanding of the principles necessary for understanding that the two identical accelerating craft along an axis judge themselves to have differently ticking clocks according to both of them.

But now I realize I need to look back at the SR issue you brought up.

 Quote by PAllen inertially moving clock passing this sequence of clocks would interpret that each is moving slow, but they are increasing out of synch with each other per the moving clock.
As a particular clock in a moving frame passes me, and continues along, I decide that it ticks at a steady rate, slower than mine by gamma, a determination which relies on my network of "simultaneous" clocks.

However since I also determine at any moment using my network that gamma-adjusted clocks placed further back on along the moving frame are set later than clocks forward along the frame, that is a different effect. Do they cancel each other out? As a moving frame passes me, and I watch the counts of these different clocks in the frame, passing me, a different clock each time, do I see this time change at my own rate?

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 Quote by 1977ub As a particular clock in a moving frame passes me, and continues along, I decide that it ticks at a steady rate, slower than mine by gamma, a determination which relies on my network of "simultaneous" clocks. However since I also determine at any moment using my network that gamma-adjusted clocks placed further back on along the moving frame are set later than clocks forward along the frame, that is a different effect. Do they cancel each other out? As a moving frame passes me, and I watch the counts of these different clocks in the frame, passing me, a different clock each time, do I see this time change at my own rate?
If a line of inertial clocks synchronized with each other by Einstein convention, passed you, and the first one matched your clock, each successive clock would be further ahead of yours. The clock synch issue dominates the slower rate on each clock. Think of a muon generated in the upper atmosphere passing earth frame clocks on the way to the ground. It sees the last of these clocks, say, a millisecond ahead of the first clock it passed; meanwhile, for the muon less than 2 microseconds have passed.

 Quote by PAllen If a line of inertial clocks synchronized with each other by Einstein convention, passed you, and the first one matched your clock, each successive clock would be further ahead of yours.
I see. There must be a simple expression for how quickly that succession of clocks appears to tick to me - an expression involving gamma - not sure if you can easily find that... ?

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 Quote by 1977ub I see. There must be a simple expression for how quickly that succession of clocks appears to tick to me - an expression involving gamma - not sure if you can easily find that... ?
There is. It is gamma. You are 'seeing' why the other observer thinks you are slower by gamma.

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 Quote by Austin0 Quote by Austin0 Assuming they are physicists: Distance Wouldn't they assume that the string broke because of Lorentz contraction? And so opt for a different method to determine distance , like radar ranging? Assuming constant proper acceleration and the initial synchronization of the launch frame: 1) it appears to me that reflected distances and times would be equivalent at both ends. 2) I see no obvious reason to infer that the measured distances would increase with time. In fact I would think that the raw times would decrease due to the dilation factor due to acceleration between transmission and return and the increased velocity. This would seem to be the case if carted from the initial frame and if we assume frame agreement between observation events it seems like it should hold in the accelerated frame. Time Wouldn't they ,knowing they are accelerating , attribute the difference in received signals to Doppler due to the relative motion from acceleration during transit??. Couldn't they integrate the dilation factor during transit and the effect of relative motion to extract a value solely related to the clock rates at transmission???
I find this post an argumentative distraction in a thread where the OP is genuinely trying to learn. I will not answer in this thread.

More to the point, the OP asked about direct observations, not interpretations. If front and back experience same g force, then string between front and back will break. This is a fact. Similarly, if they compare clock rates by exchanging signals, the front clocks will be observed to be going faster, by both front and back rocket passengers. Again a simple fact.

That is all that was asked - direct observations.

 Quote by PAllen I find this post an argumentative distraction in a thread where the OP is genuinely trying to learn. I will not answer in this thread. More to the point, the OP asked about direct observations, not interpretations. If front and back experience same g force, then string between front and back will break. This is a fact. Similarly, if they compare clock rates by exchanging signals, the front clocks will be observed to be going faster, by both front and back rocket passengers. Again a simple fact. That is all that was asked - direct observations.
I am sorry if you interpreted my query as argumentative. it was out of pure interest in the subject and directly related to stevedaryl's post. Certainly I understand if you want to end it here or move it to another thread.

 Quote by Austin0 How do they determine these effects within the frame??? Measure relative clock rates and distance??
In the same way you might measure the height of a building on Earth. For instance, if you take a long rope, and make a mark every meter or so, and let the rope dangle from the front rocket to the back, that would show the distance constant (according to scenario 1) and increasing (according to scenario 2). Alternatively, you could measure the distance by bouncing a laser off the front rocket and back to the rear rocket, and measure the round trip time (or you could look at interference patterns).

It really is the case that aboard an accelerating rocket, things don't seem much different from a rocket hovering over the Earth, except that the variation of "gravity" with height is different in the two cases (although you have to have a really huge rocket to notice the difference in either case).

 Quote by PAllen There is. It is gamma. You are 'seeing' why the other observer thinks you are slower by gamma.
Interesting.

ok so the problem of AO's "frame"...

RF can easily decide there is a grid of evenly-spaced meters and synchronized clocks using exchanges of light pulses.

we can say that RF has no trouble assigning AO an x coordinate and measuring his even acceleration 'a', and clock speed related to AO's current gamma.

what happens when AO attempts this... Will AO encounter any ambiguity determining an x coordinate of the RF observer at the origin?

If AO can do this, what will he think of RF-origin's clock speed and 'acceleration' with regard to him?

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 Quote by 1977ub Interesting. ok so the problem of AO's "frame"... RF can easily decide there is a grid of evenly-spaced meters and synchronized clocks using exchanges of light pulses. we can say that RF has no trouble assigning AO an x coordinate and measuring his even acceleration 'a', and clock speed related to AO's current gamma. what happens when AO attempts this... Will AO encounter any ambiguity determining an x coordinate of the RF observer at the origin? If AO can do this, what will he think of RF-origin's clock speed and 'acceleration' with regard to him?
AO will not find an unambiguous answer to assigning x and t coordinates. One issue is that there is no such thing as rigid accelerated motion in SR, while there is no problem with rigid inertial motion. You can choose a way to work around this by assuming a mathematical idealization that works for simple acceleration profiles (Born rigid acceleration). The next issue is that different clock synch methods that produce the same result for an inertial observe will produce different results for AO.

While AO would have no difficulty making some choices to set up some coordinate system, the ambiguity means you can't give a single preferred answer to RF origin's clock speed - it depends on which choices you make for setting up the coordinate system.

Note that any choice made by AO will have a different metric that RF. This means that gamma will not apply (gamma as a time dilation factor is a direct consequence of the inertial metric). Further, the standard SR Doppler formula will not apply.

As an interesting aside, a common idealization of the 'rigid framework' coordinates (Fermi-Normal coordinates), and building simultaneity using Einstein's clock synch convention, produce quite different answers for the rate of RF clock (actually, they agree when the RF origin clock is coincident with AO; the further away this clock is, the more they disagree).

 Quote by PAllen AO will not find an unambiguous answer to assigning x and t coordinates. One issue is that there is no such thing as rigid accelerated motion in SR, while there is no problem with rigid inertial motion.
I'm imagining AO as an idealized point (large enough for a "clock" I guess), thus no issues of rigidity... I presume the limit of a point AO will have a sense of the passing of time, and some form of clock, without rigidity questions? So pulses from the origin will take longer and longer to reach him, just as in unaccelerated motion but moreso - but unlike the unaccelerated motion case, he will not have a simple way to decide how far they traveled to reach him, I take it.

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 Quote by 1977ub I'm imagining AO as an idealized point (large enough for a "clock" I guess), thus no issues of rigidity... I presume the limit of a point AO will have a sense of the passing of time, and some form of clock, without rigidity questions? So pulses from the origin will take longer and longer to reach him, just as in unaccelerated motion but moreso - but unlike the unaccelerated motion case, he will not have a simple way to decide how far they traveled to reach him, I take it.
Correct, on all counts. Viewed from AO as a single world line, the difficulty all comes to simultaneity, and distance is defined in terms of a simultaneity convention. Why I brought up rigidity is that rigid, comoving rulers, to the extent you can idealize them, provide one possible answer to distance; as such they also define a simultaneity, because you can't have distance without a simultaneity convention. As such, it is interesting that this ruler based simultaneity will disagree with Einstein clock synch.

In no way am I saying you can't set up coordinates - just that you can't go from there to talking about 'the' rate of a distant clock 'now' for an accelerated observer. You also can't use inertial frame formulas (like gamma).

The approach I think you have in mind leads to AO coordinates with some nice properties for non-inertial motion with rapidly changing accelerations. Using two way light signals you simply define that if event e1 is reached by a signal you sent at t1 on your clock, and you got a return signal at t2, then you define that e1 is simultaneous to (1/2)(t2+t1) on your clock. Then define radial coordinate in polar type coordinates by c(t2-t1). Such coordinates have the following nice properties:

- coordinate speed of light is c for radial paths from the origin
- these coordinates cover larger regions of spacetime than many other accelerated coordinates
- they behave 'smoothly' around sudden changes in acceleration (other common coordinates for accelerated observers do not).

However, you also have to accept that coordinate distance fairly quickly diverges from ruler distance (either idealized, or approximate real rulers), where both are possible. For example, as defined above, someone at the bottom of the (accelerating) rocket using the above coordinates would get a slightly different result than using a tape measure along the length of the rocket. [If the rocket was coasting, these would always agree].

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