Kinematic time dilation in black holes

[jartsa]

Black hole A moves at slow velocity, and there's an Einstein light clock hovering near the event horizon.

Black hole B moves at high velocity, and there's an Einstein light clock hovering near the event horizon.

The black holes are identical. And the light clocks are at the same altitude, and comoving with their black holes.

Do the light clocks run at different rates?



(I mean, velocity addition is more of the ballistic kind in this case, right? So the normal time dilation does not apply?)

 

[PeterDonis]

Do the light clocks run at different rates?

Relative to what? See further comments below.

(I mean, velocity addition is more of the ballistic kind in this case, right? So the normal time dilation does not apply?)

I'm not sure what "ballistic kind" means. To an observer very far away from both black holes, the holes are just isolated regions in an asymptotically flat spacetime, and each one can be assigned a 4-velocity vector in the asymptotically flat spacetime. These 4-velocity vectors could be applied to the light clocks as well, since they are comoving with their respective black holes.

The only difference between this and a scenario where we have two isolated light clocks moving in flat spacetime is the additional gravitational time dilation due to the holes; but that is the same for both light clocks so we can just factor it out. Then we just have two light clocks with different 4-velocities in (asymptotically) flat spacetime, so yes, they will run at different rates, to the extent that phrase has meaning; since the clocks are in relative motion and their state of motion does not change, there is no invariant answer to the question of which one "runs slower", just as in standard SR. And there will be some observer (roughly speaking, the one whose 4-velocity is "in between" that of the two clocks) who will see both light clocks' rates to be the same (because they both have the same magnitude of velocity relative to him).

 

[Nugatory]

Black hole A moves at slow velocity...
Black hole B moves at high velocity...

Moves relative to what?

 

[jartsa]

Moves relative to what?

Relative to an observer O. O will observe the clock rates too.

 

[jartsa]

Relative to what? See further comments below.
I'm not sure what "ballistic kind" means.

This kind of thing:

A black hole changes position by 10 m in one second according to observer O. At the same time in the gravity well of the black hole a photon changes position by 1 m, relative to the black hole, according to O. So photon changes position by 11 m in one second according to O.

 

[PeterDonis]

A black hole changes position by 10 m in one second according to observer O. At the same time in the gravity well of the black hole a photon changes position by 1 m, relative to the black hole, according to O. So photon changes position by 11 m in one second according to O.

First of all, none of this applies to your example in the OP, because in that example each of the hovering light clocks is motionless with respect to the black hole whose horizon it's hovering above. In other words, no objects in your scenario are in orbit around either of the black holes.

That said, the logic you're using to add the velocity of the black hole, with respect to an observer far away, to the orbital velocity of an object in orbit about the hole, is not correct, because it assumes spacetime is flat, and it isn't. In the scenario in your OP, we can ignore that, because, as I just noted, the light clocks are motionless with respect to the black holes they are hovering near, so we can just assign the same 4-velocity, in the asymptotically flat spacetime the holes are moving in, to the holes and their respective light clocks. But this is just an approximation in which we are ignoring all effects of the spacetime curvature produced by the holes.

 

[jartsa]

First of all, none of this applies to your example in the OP, because in that example each of the hovering light clocks is motionless with respect to the black hole whose horizon it's hovering above. In other words, no objects in your scenario are in orbit around either of the black holes.

I'm analysing the motion of the light in the light clock.

That said, the logic you're using to add the velocity of the black hole, with respect to an observer far away, to the orbital velocity of an object in orbit about the hole, is not correct,

I don't care :) The photon had no difficulty traveling that "11 m". It traveled from a black hole point A to a black hole point B without any delay, according to O, although the black hole was moving, according to O.

 

[PeterDonis]

I'm analysing the motion of the light in the light clock.

Then you can't do that in the asymptotically flat Lorentz frame in which the holes (and the light clocks) have a 4-velocity. You could do an analysis using the Schwarzschild metric to figure out how a distant observer would see the light inside the light clock moving, but that analysis will certainly *not* be just a matter of doing relativistic velocity addition of the black hole's 4-velocity and the light's 4-velocity relative to the hole. It will also not be just a matter of assuming that the motion of the hole has no effect; it does, the effect is just not simple relativistic velocity addition in flat spacetime.

I don't care :)

You should. You can't do a correct analysis based on incorrect assumptions.

The photon had no difficulty traveling that "11 m". It traveled from a black hole point A to a black hole point B without any delay, according to O, although the black hole was moving, according to O.

This is not correct, because of that "without any delay, according to O". You can't just wave your hands and assume that; you have to prove it. And you won't be able to, because it's wrong.