Time Dilation at Moving Black Hole Event Horizon

[sha1000]

TL;DR

Time dilation at the event horizon of a black hole in movement

Hello everyone,

I have a hard time to conceptualize the case of a moving black hole.

We know from SR that time slows down for moving objects; but time dilation at the event horizon is already equal (tends) to zero. It seems that it can create some sort of conflict for the black hole movement.

- What happens with the event horizon of a moving black hole?
- What happens if one tries to accelerate a black hole? Will it resist more to acceleration compared to another non-blackhole object with similar mass?

 

[PAllen]

1) Time dilation is never experienced by a body itself, it is measured by someone else observing it.
2) The 'infinite time dilation' at the horizon of a BH is a limit for a sequence of ever closer stationary hovering observers [as observed by someone far away]. There is no such time dilation at all for an observer falling into a BH.
3) To see a moving BH, just fly by it. No need to do anything to the BH, just change your own state of motion. That's what relativity is all about.
4) In any case, BH have no problem moving at all. In fact, we see gravitational waves from BH rapidly co-rotating and then merging.
5) The inertia (resistance to motion from a force) is the same as for any other body with the same mass. Technical caveats: it is not so easy to define mass in general relativity, and hard to apply any force to BH unless it were charged - noting gravity is not considered a force in general relativity. However, the main point remains that the mass parameter of a BH is fully equivalent for all phenomenon at distance as for a neutron star of the same mass. You know, it's also rather difficult to push a star.

 

[Ibix]

One additional comment - you're making a mistake by trying to take SR concepts like kinematic time dilation and trying to "bolt on" gravitational effects. It won't work.

There are very limited classes of spacetimes (static and stationary ones) where it's possible to define a notion of "not moving" and then analyse relative clock rates in terms of clock rate differences for non-moving observers ("gravitational time dilation") multiplied by SR kinematic effects for moving observers. You often see GPS analysed this way. But it won't work in general - in particular, spacetime at or below the event horizon of a black hole cannot be analysed this way.

 

[PAllen]

I one way to sort of push a BH would be to examine the result of its interaction with a neutron star having high speed relative to the BH. The resulting motion of the merged body would be as expected from the mass parameter of the BH

 

[Dale]

It seems that it can create some sort of conflict for the black hole movement.

You will need to be more specific.

What happens with the event horizon of a moving black hole?

The exact same as what happens with the event horizon of a stationary black hole. There is no physical difference between a moving black hole and a stationary one

What happens if one tries to accelerate a black hole? Will it resist more to acceleration compared to another non-blackhole object with similar mass?

Accelerated black holes are important in gravitational waves of the size that can be detected with modern instruments. So these spacetimes are well studied. Here is a recent paper on the topic

https://journals.aps.org/prd/abstract/10.1103/PhysRevD.102.044005

 

[sha1000]

Thank you all for your responses.

Ok, I understand that a moving black hole can be viewed as stationary.

But what about acceleration (which is absolute)? Dale mentioned the gravitational waves which are produced as the massive objects get accelerated.

1) Is the intensity of gravitational waves higher for a black hole (when accelerated) than any other object with similar mass?

2) Why is it considered as a mistake to couple SR and GR time dilations for black hole acceleration analysis?

 

[PAllen]

Thank you all for your responses.

Ok, I understand that a moving black hole can be viewed as stationary.

But what about acceleration (which is absolute)? Dale mentioned the gravitational waves which are produced as the massive objects get accelerated.

1) Is the intensity of gravitational waves higher for a black hole (when accelerated) than any other object with similar mass?

No. For example, two neutron stars versus two BH, both having similar mass, would produce similar gravitational waves. (Note: with currently understood stellar processes, a neutron star and a BH could not have the same mass. But, in principle, a BH can have any mass.). While the fine details of the GW would be different because of slightly different sized, the overall pattern would be the same.

2) Why is it considered as a mistake to couple SR and GR time dilations for black hole acceleration analysis?

Because they have completely different theoretical basis, and completely different mathematical models. For starters, one is inherently symmetric while the other is inherently asymmetric. That is, SR time dilation has the feature that each observer thinks the other is time dilated. Meanwhile, for stationary observers at different distances from a BH, if one measures the other's time as slow, the other measures the first one to be fast. This is the exact opposite behavior of the phenomenogically different SR time dilation.

 

[PeterDonis]

acceleration (which is absolute)?

Proper acceleration (i.e., feeling weight) is absolute, but objects moving solely under the influence of gravitational waves have zero proper acceleration; they are in free fall.

The "acceleration" @Dale referred to with the term "accelerated black holes" is not proper acceleration, however; it is coordinate acceleration, for example in black holes that are part of binary systems and are on elliptical orbits, so their "acceleration" is due to gravity. (In this case also the hole will be in free fall and will have zero proper acceleration.) This kind of acceleration is not absolute; it can be made to vanish by a suitable choice of reference frame. The gravitational waves produced in such a case are due to a time-varying quadrupole moment for the system as a whole; that is the relevant absolute (i.e., invariant) quantity.

 

[PeterDonis]

two neutron stars versus two BH, both having similar mass, would produce similar gravitational waves.

More precisely, they would produce similar gravitational waves if they are in similar orbital configurations, i.e., similar semi-major axis and eccentricity.

However, black holes that merge can produce much stronger gravitational waves than neutron stars of similar mass that merge, because the black holes can get much closer together before they merge, since a typical neutron star has a radius that is some significant multiple of the Schwarzschild radius for its mass, and the effects that contribute to the strength of gravitational waves from mergers are highly nonlinear in the range of distances very close to the Schwarzschild radius.

 

[Ibix]

For example, two neutron stars versus two BH, both having similar mass, would produce similar gravitational waves. (Note: with currently understood stellar processes, a neutron star and a BH could not have the same mass. But, in principle, a BH can have any mass.). While the fine details of the GW would be different because of slightly different sized, the overall pattern would be the same.

Not sure if you meant this under your "fine details" caveat, but isn't the end of the signal quite different? Once the neutron stars make contact (earlier than identical mass black holes would) they're going to deform and mix, for want of a better word. So I would think that there'd be losses to EM radiation, heating, etc, where black holes have to radiate only gravitational waves. So I'd expect a higher peak GW intensity from a black hole.

Edit: Peter beat me to it, I see.

 

[PeterDonis]

I'd expect a higher peak GW intensity from a black hole.

The effects you mention will reduce the GW intensity from a neutron star merger, but that is in addition to the effect I mentioned in post #9.

 

[PAllen]

Not sure if you meant this under your "fine details" caveat, but isn't the end of the signal quite different? Once the neutron stars make contact (earlier than identical mass black holes would) they're going to deform and mix, for want of a better word. So I would think that there'd be losses to EM radiation, heating, etc, where black holes have to radiate only gravitational waves. So I'd expect a higher peak GW intensity from a black hole.

Edit: Peter beat me to it, I see.

Yes, the end stage would be different. But when the bodies are at similar distance, the GW would be similar, and the end stage difference for BH would be mostly due to their smaller size.

 

[Ibix]

2) Why is it considered as a mistake to couple SR and GR time dilations for black hole acceleration analysis?

Because SR assumes flat spacetime and black hole spacetimes are not flat.

 

[PeterDonis]

the end stage difference for BH would be mostly due to their smaller size.

In terms of the fraction of the total original mass that gets converted to energy of some sort, I would expect that that fraction would be much larger for a BH merger vs. a neutron star merger, because of the closer approach distance and the highly nonlinear dependence on that distance that I mentioned in post #9.

In the BH case, of course all of the energy that is emitted is emitted in GWs. In the neutron star case, only a small fraction of it will be, because of the other effects @Ibix mentioned. So the GW energy emitted in the neutron star case will be smaller than the BH by two significant factors, one for the approach distance and one for the non-GW energy. I'm not sure off the top of my head which of those factors would be larger (in terms of multiplicative factors).

 

[PAllen]

In terms of the fraction of the total original mass that gets converted to energy of some sort, I would expect that that fraction would be much larger for a BH merger vs. a neutron star merger, because of the closer approach distance and the highly nonlinear dependence on that distance that I mentioned in post #9.

In the BH case, of course all of the energy that is emitted is emitted in GWs. In the neutron star case, only a small fraction of it will be, because of the other effects @Ibix mentioned. So the GW energy emitted in the neutron star case will be smaller than the BH by two significant factors, one for the approach distance and one for the non-GW energy. I'm not sure off the top of my head which of those factors would be larger (in terms of multiplicative factors).

Look, my claim is simply that before two neutron stars make contact, their GW would be very similar to two equal mass BH with the same mass, distance, and orbital pattern. Also, that if the neutron stars could somehow be prevented from generating a kilonova and similar effects (suppressing all non-gravitational emissions), then any remaining difference is due to their sizes. In fact, two neutron stars merging under such an assumption would always produce a BH. Of course, this is not possible, in reality.

 

[PeterDonis]

my claim is simply that before two neutron stars make contact, their GW would be very similar to two equal mass BH with the same mass, distance, and orbital pattern.

Yes. I agreed with that in the first part of post #9.

You also, though, talked about the "end stage difference", and that seems to me to refer to the "end stage" just before a merger. I am simply pointing out that the "end stage" for a BH merger allows much closer approach distances than it does for a neutron star merger, so the "end stage difference" between the two cases needs to take that into account.

 

[sha1000]

Thank you all again.
Got the answers I needed and even more.