Gravity vs Acceleration: The Impact on Clocks in Reference Frame R

[calinvass]

 

[PeterDonis]

the mistake was that the horizontal line from SRL should've intersected the dotted blue line not the helping acceleration path

Yes.

that path goes beyond c

It shouldn't if you generated it correctly. You need to use the $\cosh$ and $\sinh$ functions.

 

[calinvass]

Yes, I understand. I've used constant Newtonian acceleration in flat spacetime but that is not the trajectory in frame F0.

 

[David Lewis]

It's not an illusion, and it's not symmetrical like kinetic time dilation.

Please explain what you mean by not symmetrical.

 

[Nugatory]

Please explain what you mean by not symmetrical.

Gravitational time dilation is asymmetrical in the sense that if B's clock is running slow relative to A's clock as observed by A, it is also running slow (or equivalently, A's clock is running faster) as observed by B. In physical terms: If B is deeper in a gravity well than A, and they exchange light signals from identically constructed (so the emission frequencies are the same) light sources B will receive a light signal that is blueshifted to a higher frequency than his own source, while A will receive a light signal that is redshifted to a lower frequency than his own source.

The speed-based time dilation is symmetrical: as observed by A, B's clock is running slow relative to A's clock but as observed by B, A's clock is running slow relative to B's clock. If they exchange light signals they will both receive either blue-shifted or red-shifted signals according to whether they are approaching one or another or moving apart.

 

[David Lewis]

Many thanks. Is it correct to say gravitational time dilation is asymmetric under interchange of observers?

 

[stevendaryl]

Many thanks. Is it correct to say gravitational time dilation is asymmetric under interchange of observers?

Yes, it's asymmetric.

But it's actually a little complicated as to what "gravitational time dilation" means. What's definitely true is that if you use a curvilinear, noninertial coordinate system, then in general the value of $\frac{d \tau}{dt}$ (where $\tau$ is the elapsed time on a standard clock, and $t$ is coordinate time) will be position-dependent, as well as velocity-dependent, while in SR, using inertial Cartesian coordinates, it is only velocity-dependent.

 

[David Lewis]

So a satellite clock, having higher gravitational potential, would appear to tick faster than an Earth clock, and (assuming the satellite is moving with respect to the Earth clock) SR time dilation would make the satellite clock appear to tick slower than the Earth clock. I suppose you would add the two effects together to arrive at the net dT/dt.

 

[Nugatory]

So a satellite clock, having higher gravitational potential, would appear to tick faster than an Earth clock, and (assuming the satellite is moving with respect to the Earth clock) SR time dilation would make the satellite clock appear to tick slower than the Earth clock. I suppose you would add the two effects together to arrive at the net dT/dt.

Yes, and we have several older threads that show how to do this calculation. Getting it right is kinda important for the GPS system...

 

[calinvass]

There is also frame dragging effect thus the orbiting rotation direction or whether it is geostationary should influence the difference in tick rates.

 

[cianfa72]

if we have two observers inside a rocket accelerating in flat spacetime ("accelerating" meaning "experiencing proper acceleration"), at rest relative to each other (as measured by them exchanging round-trip light signals and seeing that the round-trip travel time by each of their clocks remains constant), the observer at the bottom of the rocket will be time dilated (clock running slower) relative to the observer at the top (as measured by noting the elapsed time on both of their clocks between two successive round-trip light signals).

Reading this old thread I would ask for a clarification: observer A sends a light signal reflected back from B and using his own clock measures the round-trip travel time; the observer B does the same. If the results of both measurements remain constant then we claim both observers are at rest each other.

Furthermore if in the above procedure each observer sends/encodes in the reflected light signal the reading of his own clock (when the signal is reflected back), the receiving observer can evaluate the difference between the encoded times and the readings of his own clock between the receipts of two successive round-trip light signals.

This way both observers will evaluate a non-zero difference (i.e. the observer at the bottom will be time dilated).

Is the above correct ? Thank you.

 

[PeroK]

Reading this old thread I would ask for a clarification: observer A sends a light signal reflected back from B and using his own clock measures the round-trip travel time; the observer B does the same. If the results of both measurements remain constant then we claim both observers are at rest each other.

Furthermore if in the above procedure each observer sends/encodes in the reflected light signal the reading of his own clock (when the signal is reflected back), the receiving observer can evaluate the difference between the encoded times and the readings of his own clock between the receipts of two successive round-trip light signals.

This way both observers will evaluate a non-zero difference (i.e. the observer at the bottom will be time dilated).

Is the above correct ? Thank you.

You could do it that way. Alternatively, each observer sends out a signal once per second, say. They each receive their own signals back at one second intervals. But, receive the other's signals with a longer or shorter time interval.

 

[cianfa72]

Alternatively, each observer sends out a signal once per second, say.

yes, once per second according their own clocks.

They each receive their own signals back at one second intervals. But, receive the other's signals with a longer or shorter time interval.

Again, as measured by their own clocks.

 

[jartsa]

Reading this old thread I would ask for a clarification: observer A sends a light signal reflected back from B and using his own clock measures the round-trip travel time; the observer B does the same. If the results of both measurements remain constant then we claim both observers are at rest each other.

Quite the opposite. All inertial observers say that there is some relative motion. They might call it "length contraction motion". I am guessing that "we" are inertial observers.

This way both observers will evaluate a non-zero difference (i.e. the observer at the bottom will be time dilated).

Yes, according to those non-inertial people the bottom people are time dilated. Different inertial observers have different opinions about the time dilation.

 

[DrGreg]

Reading this old thread I would ask for a clarification: observer A sends a light signal reflected back from B and using his own clock measures the round-trip travel time; the observer B does the same. If the results of both measurements remain constant then we claim both observers are at rest each other.

Quite the opposite. All inertial observers say that there is some relative motion. They might call it "length contraction motion". I am guessing that "we" are inertial observers

If "we" are inertial observers, you are right, but I suspect what cianfa72 meant to say was "each observer claims the other is at rest relative to themself", and, in that case, that is true.

EDIT: Just to clarify possibly ambiguous language:

• Observer A claims that Observer B is at rest relative to Observer A
• Observer B claims that Observer A is at rest relative to Observer B

Both these claims are true; they can be taken as a definition of what "at rest relative to a non-inertial observer" means.

 

[PeroK]

If "we" are inertial observers, you are right, but I suspect what cianfa72 meant to say was "each observer claims the other is at rest relative to themself", and, in that case, that is true.

That's what we all meant! Starting with:

The precise way of putting it is this: if we have two observers inside a rocket accelerating in flat spacetime ("accelerating" meaning "experiencing proper acceleration"), at rest relative to each other (as measured by them exchanging round-trip light signals and seeing that the round-trip travel time by each of their clocks remains constant), the observer at the bottom of the rocket will be time dilated (clock running slower) relative to the observer at the top (as measured by noting the elapsed time on both of their clocks between two successive round-trip light signals).

 

[jartsa]

EDIT: Just to clarify possibly ambiguous language:

• Observer A claims that Observer B is at rest relative to Observer A
• Observer B claims that Observer A is at rest relative to Observer B

Both these claims are true; they can be taken as a definition of what "at rest relative to a non-inertial observer" means.

Last sentence is ambiguous. It should be said like this: Both these claims are true; they can be taken as a definition of what "at rest relative to a non-inertial observer, according to the observer himself" means.

Relative to what, and according to who.

 

[DrGreg]

Last sentence is ambiguous. It should be said like this: Both these claims are true; they can be taken as a definition of what "at rest relative to a non-inertial observer, according to the observer himself" means.

Relative to what, and according to who.

I think in relativity, when you say "A is at rest relative to B" it is automatically assumed it's "...according to B". Or in other words, it means "A is at rest in B's coordinate system". I think it's a bad idea to say "A is at rest relative to B according to C". I'm not even sure it has a well-defined meaning. It would be better to say "A and B have the same velocity relative to C", if that's what you mean, or "A and B have a constant separation as measured by C". The details depend on exactly how C chooses to do the measurement. It may also depend on whether spacetime is curved or flat. ("Same velocity" doesn't always make sense.)

 

[PeterDonis]

I think in relativity, when you say "A is at rest relative to B" it is automatically assumed it's "...according to B".

Actually, I think the proper meaning of "at rest relative to" should be that it is an invariant: the best way to say it is "A and B are at rest relative to each other", and this is to be taken as meaning that, for example, round-trip light signals between them have a constant round-trip travel time, according to either of their clocks, which is an invariant, independent of any choice of coordinates. That is the meaning @cianfa72 was using, and I think it's correct.

 

[PeterDonis]

All inertial observers say that there is some relative motion.

In coordinate terms, yes. But what invariant indicates "relative motion" in this case? Note that, per my previous post just now, there is an invariant (timing of round-trip light signals between A and B) that says there is no "relative motion".

 

[jartsa]

In coordinate terms, yes. But what invariant indicates "relative motion" in this case? Note that, per my previous post just now, there is an invariant (timing of round-trip light signals between A and B) that says there is no "relative motion".

Velocity reading on an inertial navigation system bolted on the rocket floor minus velocity reading on an inertial navigation system bolted on the rocket ceiling. That number is the same in all frames.

 

[Ibix]

Velocity reading on an inertial navigation system bolted on the rocket floor minus velocity reading on an inertial navigation system bolted on the rocket ceiling. That number is the same in all frames.

You need to compare readings taken at the same time, though. What simultaneity condition are you using? You can get growing, shrinking, or steady distances depending on your choice.

 

[cianfa72]

Velocity reading on an inertial navigation system bolted on the rocket floor

Btw, what does a such inertial navigation system actually measure ?

 

[Ibix]

Btw, what does a such inertial navigation system actually measure ?

The proper acceleration and possibly rotation rates with respect to the Fermi-normal axes, all as functions of the instrument's proper time, and integrals thereof.