Time dilation vs Differential aging vs Redshift

Hi,
I would like to ask for a clarification about the terms time dilation vs differential aging vs gravitational redshit.

As far as I can tell, time dilation is nothing but the rate of change of an object's proper time ##\tau## w.r.t. the coordinate time ##t## of a given coordinate chart (aka reference frame). Therefore it is not an invariant since by definition depends on the chosen coordinate chart.

Differential aging is an invariant instead since it is evaluated as the difference in proper time along different paths in spacetime passing through two given events (as for instance in twin-paradox).

I think gravitational redshift is an invariant as well, since it involves experiments done using round-trip signals exchanged between two observers.

Does the above make sense ? Thanks.

Just the nomenclature "gravitational redshift" presupposes that there are two stationary observers in a stationary spacetime. In any coordinates that do not use a time coordinate such that the time like Killing field is ##\partial_t## the effect will seem different. What is invariant is the frequency ratio between the emitted frequency and observed frequency. This will be invariant regardless of whether we are talking about gravitational redshift, cosmological redshift, or just Doppler shift.

Yes to what time dilation and differential aging are.

Some care is needed with gravitational redshift. It is always possible to write down the frequency ratio between any pair of emitters and receivers. It is only possible to break this down into separate "gravitational" and "kinematic" redshifts in stationary and static spacetimes because those are the only ones where there's an invariant definition of "space" that does not vary with time, so are the only ones where there's an invariant notion of "not moving in space". In less restricted spacetimes you can still have round-trip communications and the received and emitted frequencies are still well-defined and invariant, but may very with time and this isn't attributable to "receiver moving in space" versus "changing geometry of space" because there isn't an invariant notion of space.

As always, things that are "not invariant" become invariant if you specify which frame you are talking about. The time dilation measured by frame S for clock C is an invariant; I can calculate it using any coordinate system I like. But "the time dilation of clock C" is not invariant because, as you say, it depends which frame I'm measuring against.

cianfa72 said:

As far as I can tell, time dilation is nothing but the rate of change of an object's proper time ##\tau## w.r.t. the coordinate time ##t## of a given coordinate chart (aka reference frame). Therefore it is not an invariant since by definition depends on the chosen coordinate chart.

Correct

cianfa72 said:

Differential aging is an invariant instead since it is evaluated as the difference in proper time along different paths in spacetime passing through two given events (as for instance in twin-paradox).

Correct

cianfa72 said:

I think gravitational redshift is an invariant as well, since it involves experiments done using round-trip signals exchanged between two observers.

See post #2

Ibix said:

Some care is needed with gravitational redshift. It is always possible to write down the frequency ratio between any pair of emitters and receivers. It is only possible to break this down into separate "gravitational" and "kinematic" redshifts in stationary and static spacetimes because those are the only ones where there's an invariant definition of "space" that does not vary with time, so are the only ones where there's an invariant notion of "not moving in space".

Ok however, although in general it were not possible to break down it in an invariant way, the frequency ratio between any pairs of emitters and receivers is an invariant.

cianfa72 said:

Ok however, although in general it were not possible to break down it in an invariant way, the frequency ratio between any pairs of emitters and receivers is an invariant.

Frequency ratio at what time? Yes, if I emit a pulse at frequency ##f## and you receive it at ##f'## both of those quantities are well defined and invariant. If I'm emitting a long duration signal, though, you can try to ask "what is the ratio of the frequency I'm emitting now and you're receiving now" and that is not invariant because "now" is not. And because you don't have a constant-over-time notion of "space" you can't duck the issue by picking transmitters and receivers that are stationary in that space, which you can in a static/stationary spacetime.

Ibix said:

If I'm emitting a long duration signal, though, you can try to ask "what is the ratio of the frequency I'm emitting now and you're receiving now" and that is not invariant because "now" is not. And because you don't have a constant-over-time notion of "space" you can't duck the issue by picking transmitters and receivers that are stationary in that space, which you can in a static/stationary spacetime.

Ah ok, so for example it makes sense in a stationary spacetime (i.e. there is a timelike KVF) in which the notion of "space" is well defined and constant-over-time (i.e. any spacelike hypersurface of constant coordinate time ##t## pointing along the timelike KVF has the same spatial geometry).

cianfa72 said:

Ah ok, so for example it makes sense in a stationary spacetime (i.e. there is a timelike KVF) in which the notion of "space" is well defined and constant-over-time (i.e. any spacelike hypersurface of constant coordinate time ##t## pointing along the timelike KVF has the same spatial geometry).

Better to say that if two observers are following the timelike KVF and we connect them by a null path then we can transport that null path along the KVF without changing it. So if observer A emits a constant frequency observer B will receive a constant frequency, and we can avoid questions about what we mean by "space" entirely.

Ibix said:

Better to say that if two observers are following the timelike KVF and we connect them by a null path then we can transport that null path along the KVF without changing it. So if observer A emits a constant frequency observer B will receive a constant frequency, and we can avoid questions about what we mean by "space" entirely.

You mean basically transport that null path connecting A and B using Lie dragging along the timelike KVF.

How the above is related to the constant frequency of A emitting the radio signal and B receiving it at a different frequency ?

cianfa72 said:

As far as I can tell, time dilation is nothing but the rate of change of an object's proper time ##\tau## w.r.t. the coordinate time ##t## of a given coordinate chart (aka reference frame). Therefore it is not an invariant since by definition depends on the chosen coordinate chart.

Yes, I agree.

cianfa72 said:

Differential aging is an invariant instead since it is evaluated as the difference in proper time along different paths in spacetime passing through two given events (as for instance in twin-paradox).

Yes, I also agree.

cianfa72 said:

I think gravitational redshift is an invariant as well, since it involves experiments done using round-trip signals exchanged between two observers.

I think that gravitational redshift involves one-way signals, not round trip signals. It is an invariant, but the reason for that is because the worldlines are defined.

In geometric terms (please ask for clarification as this may be a bit confusing), the worldlines are specifically the integral curves of a timelike Killing vector field. Those worldlines are connected by a null geodesic. Any null vector is parallel transported along the null geodesic, and the contraction with the tangent to each worldline is calculated. The ratio is the gravitational redshift, and it is invariant because all of the above was defined in purely geometric terms.

Edit: @Ibix for the win! I guess I should have read the rest of the thread.

Dale said:

Those worldlines are connected by a null geodesic. Any null vector is parallel transported along the null geodesic, and the contraction with the tangent to each worldline is calculated. The ratio is the gravitational redshift, and it is invariant because all of the above was defined in purely geometric terms.

You mean the worldlines of observers A and B are actually integral curves (orbits) of the timelike KVF. Then the 'scalar product' between the null vector and the A worldline's tangent at the event of emission is calculated and the same for the null vector w.r.t. the B worldline's tangent at the event of reception. The claim is that the ratio between these 2 scalar products is the gravitational redshift.

cianfa72 said:

You mean the worldlines of observers A and B are actually integral curves (orbits) of the timelike KVF. Then the 'scalar product' between the null vector and the A worldline's tangent at the event of emission is calculated and the same for the null vector w.r.t. the B worldline's tangent at the event of reception.

Yes, where one null vectors is determined by parallel transport from the other.

cianfa72 said:

The claim is that the ratio between these 2 scalar products is the gravitational redshift.

Yes.

Dale said:

Yes, where one null vectors is determined by parallel transport from the other.

As you said in #10 any null vector is actually parallel transported along the null path (geodesic) so the vector we get from the parallel transport process is the same as the null vector tangent to the null geodesic at the event of reception.

cianfa72 said:

As you said in #10 any null vector is actually parallel transported along the null path (geodesic) so the vector we get from the parallel transport process is the same as the null vector tangent to the null geodesic at the event of reception.

I wouldn’t use the word “same”. Each event has a separate vector space. And in curved spacetime you can get different vectors by parallel transporting one vector along two different paths. So “same” is hard to justify.

cianfa72 said:

How the above is related to the constant frequency of A emitting the radio signal and B receiving it at a different frequency ?

If A emits a pulse of frequency ##f## and B receives it with frequency ##f'##, then A later emits another identical pulse and it follows an identical path through identical spacetime, what's the received frequency going to be?

That's why in stationary spacetimes you can have a notion of "frequency ratio" without getting into questions about what events you are comparing: for a pair of hovering observers the answer is a constant. This is not generally the case in non-stationary spacetimes.

Dale said:

And in curved spacetime you can get different vectors by parallel transporting one vector along two different paths. So “same” is hard to justify.

Yes, the point I was trying to make is a bit different. It goes like this.

Consider the tangent vector space at the point of emission (along the A's worldline) and take the null vector pointing in the same direction as the null path connecting A and B worldlines. By parallel transporting it along that null geodesic path up to the reception event (that is along the B's worldline) we get a vector in the tangent vector space at that event. From the definition of geodesic path as "autoparallel" such a vector is the same as the null tangent vector at the reception event pointing in the direction of that null path.

OTOH, in post #10, you were talking of parallel transporting along the null path the entire light cone from the tangent space at emission event up to the tangent space at reception event?

Ibix said:

If A emits a pulse of frequency ##f## and B receives it with frequency ##f'##, then A later emits another identical pulse and it follows an identical path through identical spacetime, what's the received frequency going to be?

With identical paths I believe you mean that the later pulse path in spacetime is congruent with the path obtained from the first by Lie dragging it along the flow of the timelike KVF.

In a chart in which the time coordinate represents the integral curves of the timelike KVF, the Lie dragging of the first pulse path is actually its "translation" along the time coordinate up to the later emission event.

cianfa72 said:

take the null vector pointing in the same direction as the null path connecting A and B worldlines

This may be the disconnect. There are an infinite number of such null vectors. They all represent pulses of light leaving in the same direction at the same time, but they have different frequencies. You can use any of them.

cianfa72 said:

From the definition of geodesic path as "autoparallel" such a vector is the same as the null tangent vector at the reception event pointing in the direction of that null path.

Yes, I see what you mean by “same”. Again, there are an infinite number of null tangent vectors at the reception event pointing in the direction of the null path. The specific vector obtained from parallel transport is the one with the correct frequency.

cianfa72 said:

OTOH, in post #10, you were talking of parallel transporting along the null path the entire light cone from the tangent space at emission event up to the tangent space at reception event?

No, although I see how my wording was confusing. I was only talking about the geodesic, not the whole light cone. My “any” referred to different frequencies.

Dale said:

This may be the disconnect. There are an infinite number of such null vectors. They all represent pulses of light leaving in the same direction at the same time, but they have different frequencies. You can use any of them.

Ah ok, there are an infinite number of such null vectors in a given direction (all pointing in the same direction of the light pulse from the emission event connecting A and B worldlines). Each one is basically an electromagnetic pulse with a different frequency.

Dale said:

Yes, I see what you mean by “same”. Again, there are an infinite number of null tangent vectors at the reception event pointing in the direction of the null path. The specific vector obtained from parallel transport is the one with the correct frequency.

Ok, so the evaluate the gravitation redshift we need to calculate the 'scalar product" between the specific parallel transported null vector (carrying a given frequency) along the null geodesic path w.r.t the B's tangent worldline at the reception event. Of course such scalar product is evaluated in the tangent vector space at reception event (that is along the B's worldline).

cianfa72 said:

Ah ok, there are an infinite number of such null vectors in a given direction (all pointing in the same direction of the light pulse from the emission event connecting A and B worldlines). Each one is basically an electromagnetic pulse with a different frequency.

Yes, exactly!

cianfa72 said:

Ok, so the evaluate the gravitation redshift we need to calculate the 'scalar product" between the specific parallel transported null vector (carrying a given frequency) along the null geodesic path w.r.t the B's tangent worldline at the reception event. Of course such scalar product is evaluated in the tangent vector space at reception event (that is along the B's worldline).

Yes. I think you have it.

I wrote this quite some time ago: https://www.physicsforums.com/insights/coordinate-dependent-statements-expanding-universe/
It is another example of how the split into "different" types of frequency shifts are really a matter of convention.

cianfa72 said:

With identical paths I believe you mean that the later pulse path in spacetime is congruent with the path obtained from the first by Lie dragging it along the flow of the timelike KVF.

In a chart in which the time coordinate represents the integral curves of the timelike KVF, the Lie dragging of the first pulse path is actually its "translation" along the time coordinate up to the later emission event.

Just to be more specific: consider the null path ##\gamma## from the first emission event (on the A's worldline that is an integral curve of the timelike KVF) to the first reception event (on the B's worldline that is an integral curve of the timelike KVF as well).

Since A and B worldlines will not be geodesics in general, the Lie dragging along the KVF flow of the null path ##\gamma## from the first emission event to the later emission event (both along A's worldline) does not give the same answer (evaluated at the tangent space at the later emission event on A's worldline) as the Fermi-Walker transport along A's worldline of the tangent vector of the null path ##\gamma## at the first emission event to the later emission event (we had a thread some time ago on this topic).

Now in this scenario what really matters, I believe, is the Lie dragging since it "encodes" the spacetime symmetry (the spacetime is supposed to be stationary).

In this sense the later null path ##\delta## that connect the A's worldline with the B's worldline is actually "congruent" with the Lie dragging of the first null path ##\gamma##.

Does it make sense ?