# Gravitational time dilation

## Main Question or Discussion Point

I have been working on something and I want to see if you guys get the same result. Let's say we have a hovering observer at r from a large gravitating body such as the sun. The gravitational time dilation there would be z_r = sqrt(1 - 2 G M / (r c^2)), correct? If a clock directly passes the hovering observer, then since SR is valid locally, the hovering observer will measure a kinetic time dilation on the clock of z_kinetic = sqrt(1 - (v_local / c)^2), right? To a distant observer, since there is no simultaneity difference between the two observers, the distant observer will also observe that the clock is ticking z_kinetic slower than the hovering observer's clock, and that the hovering observer's clock is ticking z_r slower than his own, so the distant observer measures a time dilation for the clock of z_r * z_kinetic, is that right so far?

Now let's say that there is another small gravitating mass at r. The hovering observer at r is far from it but there is a second hovering observer a distance of d from the smaller mass, but still at r from the first mass. Since the first and second hovering observers are both a distance of r from the first mass, there is no difference in time dilation between them due to the first mass, but the second hovering observer is time dilating by z_d due to the gravity of the second mass, right? That is according to the second hovering observer, but according to the distant observer, the time dilation of the second hovering observer is z_d slower than the first hovering observer while the first hovering observer is time dilating by a factor of z_r to the distant observer's clock, so the distant observer measures a time dilation of the second hovering observer's clock of z_d * z_r, right?

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I have been working on something and I want to see if you guys get the same result. Let's say we have a hovering observer at r from a large gravitating body such as the sun. The gravitational time dilation there would be z_r = sqrt(1 - 2 G M / (r c^2)), correct? If a clock directly passes the hovering observer, then since SR is valid locally, the hovering observer will measure a kinetic time dilation on the clock of z_kinetic = sqrt(1 - (v_local / c)^2), right? To a distant observer, since there is no simultaneity difference between the two observers, the distant observer will also observe that the clock is ticking z_kinetic slower than the hovering observer's clock, and that the hovering observer's clock is ticking z_r slower than his own, so the distant observer measures a time dilation for the clock of z_r * z_kinetic, is that right so far?
Yes.
Now let's say that there is another small gravitating mass at r.
This small mass is free falling radially, orbiting or hovering?

The hovering observer at r is far from it but there is a second hovering observer a distance of d from the smaller mass, but still at r from the first mass. Since the first and second hovering observers are both a distance of r from the first mass, there is no difference in time dilation between them due to the first mass, but the second hovering observer is time dilating by z_d due to the gravity of the second mass, right? That is according to the second hovering observer, but according to the distant observer, the time dilation of the second hovering observer is z_d slower than the first hovering observer while the first hovering observer is time dilating by a factor of z_r to the distant observer's clock, so the distant observer measures a time dilation of the second hovering observer's clock of z_d * z_r, right?
Possibly to an approximation. The Schwarzschild solution is a symmetrical vacuum solution so it does not allow for significant masses outside the original mass so it is hard to be definite.

Yes.
Okay, good.

This small mass is free falling radially, orbiting or hovering?
Well, I am trying to keep all of the observers stationary to each other to avoid simultaneity differences with the second example, so it would have to be hovering. If it were orbiting at r, then we would have to multiply by the kinetic time dilation as well, gaining z_r * z_d * z_kinetic.

Possibly to an approximation. The Schwarzschild solution is a symmetrical vacuum solution so it does not allow for significant masses outside the original mass so it is hard to be definite.
What would keep it from being exact? The only thing would be if the gravitational fields of each of the masses affect each other somehow, wouldn't it? But even if there is anything resembling gravitational shielding, such as during an solar eclipse, which I do not know of, that would be with the body of the moon blocking the sun. If we consider point masses, however, then no gravitational shielding can take place, and the gravitational fields themselves for each of the bodies should not interact, so each can be considered separately, right?

... What would keep it from being exact? The only thing would be if the gravitational fields of each of the masses affect each other somehow, wouldn't it? But even if there is anything resembling gravitational shielding, such as during an solar eclipse, which I do not know of, that would be with the body of the moon blocking the sun. If we consider point masses, however, then no gravitational shielding can take place, and the gravitational fields themselves for each of the bodies should not interact, so each can be considered separately, right?
The gravitational time dilation factor is given by:

$$\frac{d\tau}{dt} = \sqrt{1-\frac{2GM}{rc^2}}$$

The fraction on the right is very like the Newtonian gravitational potential (GM/r) so it might be a reasonable guess that that time dilation is a function of gravitational potential and that for two large masses that the potentials add in the normal way. A very rough guess of the time dilation due a mass M1 a distance r1 away and another mass M2 a distance r2 away might be something like:

$$\frac{d\tau}{dt} = \sqrt{1-\frac{2G}{c^2}\left( \frac{M_1}{r_1}+\frac{M_2}{r_2} \right) } = \sqrt{1- \frac{2GM_1}{r_1c^2}-\frac{2GM_2}{r_2c^2} }$$

When I say "very rough" I mean really rough :tongue:. As far as I know no one has produced an exact solution for two gravitational bodies. One difficulty is that for a single body we can define a distance r as the circumference of a ring divided by 2 pi because we can have a perfect circle around the gravitational body. With two bodies, a circle is distorted and r is poorly defined. In you example, r for an observer on the side of the original mass far from the second mass might not be the same r for the observer nearest the second body depending on how the measurements are done. Hence the complications.

[EDIT]

Looking at the above equation, if r = 4GM/c^2 for both massive bodies the time dilation goes to zero which seems unlikely. It might well be that a better approximation is to simply multiply the two time dilation factors which gives:

$$\frac{d\tau}{dt} = \sqrt{1-\frac{2GM_1}{r_1 c^2}} * \sqrt{1-\frac{2GM_2}{r_2 c^2}} = \sqrt{1- \frac{2GM_1}{r_1c^2}-\frac{2GM_2}{r_2c^2} +\frac{4G^2M_1M_2}{r_1r_2c^4} }$$

This seems a little better behaved, but the truth is I don't really know the answer and I don't think anyone else currently knows an exact solution for the two massive body problem.

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PeterDonis
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But even if there is anything resembling gravitational shielding, such as during an solar eclipse, which I do not know of, that would be with the body of the moon blocking the sun.
That doesn't change the gravitational effect of the sun on the Earth. The eclipse blocks light from the sun, but not gravity.

If we consider point masses, however, then no gravitational shielding can take place, and the gravitational fields themselves for each of the bodies should not interact, so each can be considered separately, right?
In the general case, no. GR is a nonlinear theory, so you can't add together two solutions to get another solution. That means you can't just add together two "copies" of the solution for a single gravitating body to get the field produced by two gravitating bodies. As yuiop said, there is no general analytical solution for the two-body problem in GR.

However, in the special case where gravity is very weak, we can obtain a good approximate solution for the field due to multiple bodies by adding together the fields produced by each body in isolation. For example, to compute the time dilation experienced by us on the surface of the Earth, we would add together the contributions from the Earth and the Sun. (It turns out that the Sun's contribution is about 20 times the Earth's, but of course this is still a very small effect, about 1 part in 10 million, which qualifies as "very weak".)

Thanks yuiop. Okay well, let's say that two hovering observers are at the same distance r from the large mass, but one of them is near a smaller mass and one of them is far away. Shouldn't the field of the large body apply equally to both observers at r, so that the far away observer watching the clock of the near observer will see the near observer's clock ticking slower only according to the gravitational time dilation of the smaller mass, having absolutely no effect between the clocks of the two observers due to the larger mass because they are at the same place in the field of the large mass? That is, SR is valid locally between the two observers at the same place in the field of the larger mass, leaving only the difference between the tick rates of their clocks due to their proximity of the smaller mass, correct?

PeterDonis
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It might well be that a better approximation is to simply multiply the two time dilation factors
In the weak field approximation, yes, this is the strictly correct thing to do. However, as you can see, for small quantities it doesn't really make much difference whether you multiply time dilation factors or add potentials. See below.

I don't think anyone else currently knows an exact solution for the two massive body problem.
Correct, no one does. However, your approximations are fine if gravity is very weak, as in my last post. If you run the numbers in your equation for us here on the surface of the Earth, with body 1 being the Earth and body 2 being the Sun, you will find that only the terms in M/r are significant--the next order terms are too small to matter. So adding potentials basically gives the same answer as multiplying time dilation factors.

That doesn't change the gravitational effect of the sun on the Earth. The eclipse blocks light from the sun, but not gravity.
Right, okay, thanks.

In the general case, no. GR is a nonlinear theory, so you can't add together two solutions to get another solution. That means you can't just add together two "copies" of the solution for a single gravitating body to get the field produced by two gravitating bodies. As yuiop said, there is no general analytical solution for the two-body problem in GR.

However, in the special case where gravity is very weak, we can obtain a good approximate solution for the field due to multiple bodies by adding together the fields produced by each body in isolation. For example, to compute the time dilation experienced by us on the surface of the Earth, we would add together the contributions from the Earth and the Sun. (It turns out that the Sun's contribution is about 20 times the Earth's, but of course this is still a very small effect, about 1 part in 10 million, which qualifies as "very weak".)
GR is non-linear, right. It would have to be from the formula for gravitational time dilation. In that case, however, SR wouldn't be valid locally, at least not in terms of applying another mass locally, only kinematically, although technically I suppose that would still be GR since SR only deals with kinematics anyway, right? If the moon doesn't disrupt the sun's gravity in any way, however, then the fields should be capable of being considered separately, right? If GR is non-linear, however, then there should be some disruption of the fields, perhaps during an eclipse, to some degree, correct?

I suppose to do what I am suggesting, we would have to hypothesize that gravitational fields do not influence each other in any way, whereby two observers at the same distance in a gravitational field will be affected in the same way by that mass, regardless of whatever is taking place locally. Would that be a reasonable assumption for a thought experiment, or is it too far-fetched? Is there any experimental evidence against it?

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Notwithstanding the valid comments by Peter, I noticed a more serious problem with multiplying the gravitational time dilation factors in the strong field. In the second equation:

$$\frac{d\tau}{dt} = \sqrt{1-\frac{2GM_1}{r_1 c^2}} * \sqrt{1-\frac{2GM_2}{r_2 c^2}}$$

if we set r1 = r2 = GM/c^2 so that each gravitational body is at the Schwarzschild radius of the other body, we get:

$$\frac{d\tau}{dt} = \sqrt{1-\frac{2}{1}} * \sqrt{1-\frac{2}{1}} = i*i = -1$$

which is a real value inside the combined horizons On balance I think the first equation which adds the potentials is a better approximation in the strong field. However, I acknowledge there is no rigorous support for either equation and the problems of defining r remain, so maybe better to stay with the week field approximations as Peter suggested and accept the two body problem is unsolved.

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I suppose to do what I am suggesting, we would have to hypothesize that gravitational fields do not influence each other in any way, that each one can be considered separately. Would that be a reasonable assumption for a thought experiment, or is it too far-fetched? Is there any experimental evidence against it?
In Newtonian physics it is OK to simply add gravitational potentials at a point in the combined fields of two bodies and treat that as a single potential for any calculations. Newtonian predictions are reasonably accurate in the weak field and GR predictions are required to reasonably approximate Newtonian predictions in the weak field, so I think that is the way to go.

In Newtonian physics it is OK to simply add gravitational potentials at a point in the combined fields of two bodies and treat that as a single potential for any calculations. Newtonian predictions are reasonably accurate in the weak field and GR predictions are required to reasonably approximate Newtonian predictions in the weak field, so I think that is the way to go.
Great, excellent idea. We can just keep it in the weak field and find the Newtonian approximation to see what happens. :) Yay. Okay, so let's say we have 5 observers, one distant and the other 4 at a distance r_a from a large mass m_a. Of those 4, 3 are a distance r_b from a smaller mass m_b and the fourth far away. 2 of those are a distance r_c from another mass m_c, and the last r_d from a mass m_d. So in the weak field, the 4th observer says the 5th is time dilating by z_{m_d, r_d}, while the 3rd says the 4th is time dilating by z_{m_c, r_c} and the 5th by z_{m_c, r_c} * z_{m_d, r_d}, and so on. The distant observer, then, measures a time dilation of the 5th observer of z_{m_a, r_a} * z_{m_b, r_b} * z_{m_c, r_c} * z_{m_d, r_d}.

Okay, now let's take away all of the observers except for the 5th observer and the distant observer. The distant observer can still measure the same time dilation of the 5th observer without the intermediate observers, only in relation to each of the individual point masses and their distances from the 5th observer, right? Now let's start again and just take two equal point masses m and put them right next to each other, a distance r from an observer. In that case, the distant observer would observe a time dilation of z_{2m, r} = z_{m, r} * z_{m, r}, correct? There is only one direct relation that can be made here for a function of z, then, and that is j^(2m k) = j^(m k) * j^(m k), where j is some constant and k is a function of r. Does that look right so far?

I'm thinking the distance measured locally for an observer from a mass would also be measured differently by a distant observer due to radial length contraction, so that would probably put a big damper on this thought experiment, although perhaps still close enough to proceed in the weak Newtonian limit.

PeterDonis
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On balance I think the first equation which adds the potentials is a better approximation in the strong field.
No, in the strong field we do not *have* a good approximation, at least not in an analytic formula. The only way we currently have to deal with the strong field, multi-body case is numerical simulation.

PeterDonis
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GR is non-linear, right. It would have to be from the formula for gravitational time dilation. In that case, however, SR wouldn't be valid locally, at least not in terms of applying another mass locally, only kinematically, although technically I suppose that would still be GR since SR only deals with kinematics anyway, right?
Right, if I'm understanding correctly what you mean by "SR only deals with kinematics". We can always find a local freely falling reference frame in which the laws of SR hold to a desired approximation. That's true regardless of the global configuration of the spacetime.

If the moon doesn't disrupt the sun's gravity in any way, however, then the fields should be capable of being considered separately, right? If GR is non-linear, however, then there should be some disruption of the fields, perhaps during an eclipse, to some degree, correct?
No. You're thinking of it backwards. The field does not "change" as a result of the motions of bodies; the field is what *determines* the motions of bodies. Thinking of the "field" due to one body as "disrupting" or changing the "field" felt from another body only makes sense in the linear regime, where you can separate the field into distinct "pieces" due to each body. Once you're in the nonlinear regime (i.e., once gravity is no longer weak enough to ignore the nonlinearities), you can't think of separate "fields" due to separate bodies any longer; you have to think of the entire spacetime, the entire "field", as one thing, which takes into account *all* of the matter and energy present.

I suppose to do what I am suggesting, we would have to hypothesize that gravitational fields do not influence each other in any way, whereby two observers at the same distance in a gravitational field will be affected in the same way by that mass, regardless of whatever is taking place locally. Would that be a reasonable assumption for a thought experiment, or is it too far-fetched? Is there any experimental evidence against it?
All the direct evidence we have about bodies responding to gravitational fields is from the Solar System, where gravity is very weak everywhere. So the only direct tests we can currently make are in the regime where gravity is linear to the accuracy we can test it. In that regime your "hypothesis" is obviously valid.

All the evidence we have from the strong-field regime is indirect (for example, observations of binary pulsar systems) and doesn't include any observations of test bodies responding to the combined strong fields of multiple masses. So we have no way of testing directly the effects of the nonlinearity of gravity at this time. But I'm not aware of any physicists who doubt what standard GR tells us should happen in that regime, which is what I described (briefly) above.

PeterDonis
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I'm thinking the distance measured locally for an observer from a mass would also be measured differently by a distant observer due to radial length contraction, so that would probably put a big damper on this thought experiment, although perhaps still close enough to proceed in the weak Newtonian limit.
In the weak field limit, as long as you are in the vacuum region outside a gravitating body, the "radial length contraction factor" is simply the reciprocal of the "time dilation factor", so if you know one, you know the other. If you think about it, you will see that this makes the effect of "radial length contraction" one order of approximation smaller, so to speak, than the effect of time dilation, so we don't need to worry about it for the scenarios you are describing.

Thanks yuiop. Okay well, let's say that two hovering observers are at the same distance r from the large mass, but one of them is near a smaller mass and one of them is far away. Shouldn't the field of the large body apply equally to both observers at r, so that the far away observer watching the clock of the near observer will see the near observer's clock ticking slower only according to the gravitational time dilation of the smaller mass, having absolutely no effect between the clocks of the two observers due to the larger mass because they are at the same place in the field of the large mass? That is, SR is valid locally between the two observers at the same place in the field of the larger mass, leaving only the difference between the tick rates of their clocks due to their proximity of the smaller mass, correct?
My guess is that what you are getting is roughly correct in very broad terms. However I am a little confused by the references to SR as it does not appear to be very relevant here. The effects due to the smaller mass are purely gravitational and therefore do not belong to SR. In those "broad terms" they should be equally affected by the larger mass that they are 'equidistant' from and affected to different degrees by there different distances from the smaller mass. In the weak field there should be a certain amount of overlaying or superposition of the fields and time dilation effects. In stronger fields, problems with defining equal distances in highly asymmetrical gravitational wells make the computations complex and additional complexities arise in the relative motion of the gravitational bodies or in the energy used to keep them stationary in the fields. Add to that Peter's concerns about the non linearity of gravity in a stronger field and is hard to get even get close to a definitive answer.

Having said all that I still think:

$$\frac{d\tau}{dt} = \sqrt{1-\frac{2G}{c^2}\left( \frac{M_1}{r_1}+\frac{M_2}{r_2} \right) }$$

might be a reasonable approximation in the weak to medium field as long as we understand the limitations.

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PeterDonis
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Having said all that I still think:

$$\frac{d\tau}{dt} = \sqrt{1-\frac{2G}{c^2}\left( \frac{M_1}{r_1}+\frac{M_2}{r_2} \right) }$$

might be a reasonable approximation in the weak to medium field as long as we understand the limitations.
I agree, it's reasonable within the weak field regime. Note that similar approximations are used routinely in astronomy to calculate the motions of the planets; the total field a particular planet sees is computed by adding up contributions from the Sun, the other planets, and any other bodies of significance (for example, you need to include a contribution from the Moon for the Earth's motion). These calculations are computing the "acceleration due to gravity", not the potential or "time dilation" factor, but the principle is the same.

In the weak field limit, as long as you are in the vacuum region outside a gravitating body, the "radial length contraction factor" is simply the reciprocal of the "time dilation factor", so if you know one, you know the other. If you think about it, you will see that this makes the effect of "radial length contraction" one order of approximation smaller, so to speak, than the effect of time dilation, so we don't need to worry about it for the scenarios you are describing.
Thanks, but I was trying to be exact in my original work, and somehow when attempting to find a precise time dilation, focused absolutely on that and completely neglected the radial length contraction. The result would only work out if we also hypothesize no length contraction, which would be too much in the context of GR, of course. I guess I will forego the rest of the result in that case, although I have gained insight from this discussion, so thank you both. The time dilation that a local hovering observer measures for a second observer near another mass might be z_{m_b, r_b}, but to the distant observer, that becomes something like z_{m_b, r_b L_b} where L_b is the length contraction due to the original mass m_a and the angle between m_a and m_b and the observer. Each of the time dilations measured locally would have to depend upon the length contractions associated with all other masses present in respect to the distances to each of the masses that the distant observer measures. This complicates things immensely, and I'm thinking perhaps this is what makes GR non-linear. The relation z_total = z_a * z_b * z_c * z_d should still work out for each of the time dilations measured locally for z_total that the distant observer measures, but how to express that in terms of the distances to each of the masses that the distant observer measures rather than the locally measured distances?

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PAllen
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Measuring distance at distance is impossible without a bunch of conventions, which are somewhat arbitrary. For example:

1) light travel time between them (assuming A sends a signal, and B sends a signal on receipt of A's signal, for example). This is affected by variation in light speed. Factoring this measured delay into light speed and distance is coordinate dependent.

2) Measuring angle between distant objects. Besides accounting for intervening geometry, this can only be interpreted given a known distance to one of the objects, and a known direction from one to the other. Both of these lead to purely conventional coordinate choices.

PeterDonis
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The relation z_total = z_a * z_b * z_c * z_d should still work out for each of the time dilations measured locally for z_total that the distant observer measures
No, it won't, because the actual observed time dilation z_total is not "separable" in the general case into contributions from the individual masses. See below.

, but how to express that in terms of the distances to each of the masses that the distant observer measures rather than the locally measured distances?
In the general case, you would not even try to express it this way. Thinking of the time dilation as depending on "distance from the mass" is itself an approximation that works well when there is only one significant mass in the spacetime, but which doesn't extend to the general case. In the case where there are multiple masses, but gravity is very weak, you can use the linear approximation where the total time dilation is the sum of contributions from each mass; so in that case, you can still think of each individual contribution as depending on "distance" from the mass (and the correction to that "distance" for length contraction is second order, as I noted--the factor L_b in your corrected expression for z will be something like 1 - epsilon, where epsilon is very small, so the adjustment to z, which is already 1 - a very small number, will now be very small squared). But if there are multiple masses and gravity is not weak, you can't separate the total time dilation into individual contributions to begin with.

No, it won't, because the actual observed time dilation z_total is not "separable" in the general case into contributions from the individual masses. See below.
Well, let's say that two observers A and B are at distance r from a large mass but observer B is near another mass. Observer A measures a time dilation for observer B of 1/2, say, compared to the tick rate of observer A's own clock. Since there is no simultaneity difference between stationary observers in a gravitational field, won't a distant observer also measure the tick rate of observer B's clock to be half as great as that of observer A? If so, then if the distant observer measures a time dilation for observer A of 1/3, say, then the distant observer should measure a time dilation for observer B of 1/6, right?

In the general case, you would not even try to express it this way. Thinking of the time dilation as depending on "distance from the mass" is itself an approximation that works well when there is only one significant mass in the spacetime, but which doesn't extend to the general case. In the case where there are multiple masses, but gravity is very weak, you can use the linear approximation where the total time dilation is the sum of contributions from each mass; so in that case, you can still think of each individual contribution as depending on "distance" from the mass (and the correction to that "distance" for length contraction is second order, as I noted--the factor L_b in your corrected expression for z will be something like 1 - epsilon, where epsilon is very small, so the adjustment to z, which is already 1 - a very small number, will now be very small squared). But if there are multiple masses and gravity is not weak, you can't separate the total time dilation into individual contributions to begin with.
Right, okay.

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PeterDonis
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Well, let's say that two observers A and B are at distance r from a large mass but observer B is near another mass. Observer A measures a time dilation for observer B of 1/2, say, compared to the tick rate of observer A's own clock. Since there is no simultaneity difference between stationary observers in a gravitational field, won't a distant observer also measure the tick rate of observer B's clock to be half as great as that of observer A?
In the general case, I don't know the answer. We don't have an analytical solution, and I don't know if anyone has done a numerical simulation of this type of scenario when gravity is not weak. In the case of gravity being weak, of course, what you say above is true; but that's because when gravity is weak, as I said, we can just add the contributions to the field of the two masses separately.

Note carefully that that applies *not* just to time dilation, but to *every* aspect of the field, including the "acceleration due to gravity" that it takes for them to "hover" and remain stationary with respect to each other. In the general case, we do not *know* what that acceleration is--another way of saying that is that we do not know how we would have to program the rocket engines of A and B, for example, to keep them stationary with respect to each other. The simple time translation symmetry of the one body spacetime is no longer there, even approximately, so you can't assume that the states of motion of A and B will even be "close" to what they would be in the weak gravity case.

Note also that the same applies to the paths of light rays that A and B would have to exchange with each other, or with an observer "at infinity", to actually measure their relative time dilation. In the simple one-body case, we can solve exactly for the paths of those light rays, so we can compute exactly what the relative time dilation is--basically, we can compute exactly what it takes to do the equivalent of Einstein clock synchronization. That's how we know how to define "relative time dilation" in the first place.

In the weak gravity case with multiple bodies, once again, we can get a good approximation by just adding together the results of these one-body calculations for the paths of light rays. In the general case, we can no longer do that, so we can no longer predict what A, B, and the observer at infinity would actually measure when they exchange light rays, without doing a complex numerical simulation that as far as I know, nobody has done. We can't assume that the results will be even "close" to those for the weak gravity case.

In the general case, I don't know the answer. We don't have an analytical solution, and I don't know if anyone has done a numerical simulation of this type of scenario when gravity is not weak. In the case of gravity being weak, of course, what you say above is true; but that's because when gravity is weak, as I said, we can just add the contributions to the field of the two masses separately.

Note carefully that that applies *not* just to time dilation, but to *every* aspect of the field, including the "acceleration due to gravity" that it takes for them to "hover" and remain stationary with respect to each other. In the general case, we do not *know* what that acceleration is--another way of saying that is that we do not know how we would have to program the rocket engines of A and B, for example, to keep them stationary with respect to each other. The simple time translation symmetry of the one body spacetime is no longer there, even approximately, so you can't assume that the states of motion of A and B will even be "close" to what they would be in the weak gravity case.

Note also that the same applies to the paths of light rays that A and B would have to exchange with each other, or with an observer "at infinity", to actually measure their relative time dilation. In the simple one-body case, we can solve exactly for the paths of those light rays, so we can compute exactly what the relative time dilation is--basically, we can compute exactly what it takes to do the equivalent of Einstein clock synchronization. That's how we know how to define "relative time dilation" in the first place.

In the weak gravity case with multiple bodies, once again, we can get a good approximation by just adding together the results of these one-body calculations for the paths of light rays. In the general case, we can no longer do that, so we can no longer predict what A, B, and the observer at infinity would actually measure when they exchange light rays, without doing a complex numerical simulation that as far as I know, nobody has done. We can't assume that the results will be even "close" to those for the weak gravity case.
Good idea, light pulses. If observer A measures a time dilation of 1/2 for observer B, then pulses that are emitted once every second in B's frame will be received once every 2 seconds in A's frame. That is how observer A directly knows that observer B's clock is time dilated by 1/2 as compared to observer A's own clock. So then, from A's perspective, if observer B emits pulses once every 2 seconds, then one pulse will travel to the distant observer in some amount of time, and then another pulse, emitted 2 seconds later according to observer A, will travel an identical path to the distant observer in an identical amount of time, so arriving at the distant observer's location exactly 2 seconds after the first pulse, regardless of the direction, curvature of the path, and so forth. Light pulses emitted once every second by observer A will arrive at the distant observer once every second. Observer A says the distant observer will receive pulses from A at twice the rate as from B. The distant observer, then, will also directly measure twice the frequency coming from observer A than from observer B, since those events occur at the same place so all observers must agree, and since this is a direct measure of the time dilation of each of the observers, again because each of the successive pulses travel identical paths in the same time according to any stationary observer (with stationary masses), the distant observer will also say that observer B's clock is ticking at half the rate of observer A's clock, the same as observer A does. How does that sound?

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PeterDonis
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Good idea, light pulses. If observer A measures a time dilation of 1/2 for observer B, then pulses that are emitted once every second in B's frame will be received once every 2 seconds in A's frame. That is how observer A directly knows that observer B's clock is time dilated by 1/2 as compared to observer A's own clock.
This is pretty much ok, although I would not use the terminology "in A's frame". A directly measures the frequency with which B's light pulses arrive; by your specification, they arrive once every 2 seconds.

However, note that A does *not* observe directly how often B *releases* light pulses; he only observes how often he (A) receives them. Conversely, B directly observes how often he releases light pulses, but does not directly observe how often A receives them. This will become important in a moment.

So then, from A's perspective, if observer B emits pulses once every 2 seconds,
But A does not know that B is emitting pulses every second; he only knows that he, A, is receiving them once every 2 seconds. In order to convert that observation into a calculated frequency with which B *emits* pulses, he has to know the configuration of the spacetime between himself and B. And to know what the distant observer would observe, he has to know the configuration of spacetime between himself or B and the distant observer. *And*, those configurations have to be unchanging (static); but they aren't. See below.

then one pulse will travel to the distant observer in some amount of time, and then another pulse, emitted 2 seconds later according to observer A, will travel an identical path to the distant observer in an identical amount of time, so arriving at the distant observer's location exactly 2 seconds after the first pulse, regardless of the direction, curvature of the path, and so forth.
This is only true if the spacetime is static and B is a static observer in it. With multiple masses present, this will not be true. For example, suppose A and B are in a binary star system; one star is a giant star and one is a dwarf star. A is relatively distant from both stars, and at a significantly greater distance from the giant than the dwarf is; B is close to the dwarf, meaning it is also about the same distance from the giant as the dwarf is.

If we assume for simplicity that the dwarf is much, much smaller than the giant, so that the giant can be approximated as being motionless, then A can be static in this spacetime; he can fire rockets to "hover" at a constant distance from the giant, without orbiting it. He will then see the dwarf circling around the giant, passing him once per orbit.

However, B will *not* be static in this spacetime; he can't be, because the dwarf isn't, it's orbiting the giant. And of course the spacetime itself is not static either, since it has at least one significant gravitating body that is moving.

<snipped rest of details> How does that sound?
You are making too many assumptions that are only valid in a static spacetime; only a spacetime with a single significant gravitating body will come close to satisfying those assumptions. I recommend re-thinking your scenario in terms of the binary star system I described briefly above, with A and B's locations as given, and a distant observer very far away from the whole system. You will find that you can't carry through the reasoning you are trying to carry through about the paths of the light pulses.

Right, it would only follow if all of the masses and observers are static. That would not be so with an orbitting smaller body, true, where kinetic properties would also need to be accounted for, but I am not considering that, only static contributions. Perhaps you could think of the smaller body as accelerated upon a constantly accelerating platform. If a smaller body were added to the larger one, say a star, then it would be held up by the same process that holds up the star, the internal pressures. The overall reason I am considering only static bodies is not to determine some multiple body orbital problem, which would be extremely complex, but rather to integrate all of the point masses of a static non-rotating sphere of uniform density to find the overall time dilation at some r by considering the contributions of the time dilations for all of the static point masses that make it up.