Merging Charged BHs: Is A's Observation Continuous or Discontinuous?

[anorlunda]

An earlier thread inspired me to imagine the following scenario.

Consider observer A, black hole B and black hole C. A's past and future remains far enough away from any EH so that A is classical. In the past, A and B were stationary relative to each other. C approaches from infinity on a collision course with B. B and C have equal but opposite electric charges, thus forming an electric dipole. In the diagram, the circles represent the event horizons. My description is 100% classical, because it involves only what A sees in A's local frame. [A might get a wild ride from the gravitational waves, but that's not relevant here.]

A can measure his local electric field gradient and divergence. In A's past, he sees only the effects of B, and C is far away. In A's future, B and C will merge and their charges will cancel, leaving net zero charge.

The question involves the time evolution of A's measurements from distant past to distant future. I can plot that as a line plotting A's observations from A's past to A's future. Label t=0 (in A's local time) as the moment when A sees the two EHs first kiss and merge to become one EH. I depict this time evolution as a straight line. But it need not be straight. If B and C orbit each other before merging the line might be corkscrew shaped. My question is not whether the line is straight, but whether it is continuous or discontinuous at t=0. In the illustration below, The solid line is A's past and the dotted line is A's future relative to t=0.

Now for the question. I can think of three arguments. (1) If the BH's charge can be modeled as a point source singularity, the line should be continuous at t=0. A can infer transiently the distribution of charge inside the EH of the merged BH by observing the time evolution of the dotted line until all measurements become zero. (2) The line is not continuous, A sees the net charge becomes zero at t=0. A can infer nothing about the interior of the merged BH. (3) Space time distortions will make it appear to A that the line is continuous but that it intercepts zero net charge at t=0.

The question: are any of these arguments correct?

 

[mfb]

Everything is continuous. Calculating the detailed field will be complicated, but there is nothing that could cause a discontinuity in the electric fields (which would correspond to an "infinite" magnetic field, with an "infinite" energy density, and other unphysical things).

The field will quickly go towards zero, I would expect that you can still measure a non-zero oscillating field during the ringdown phase after the merger, it will quickly get undetectable.
The same applies to gravitational waves: Directly after the merger you can still observe the ringdown.

 

[anorlunda]

Thanks for the answer @mfb . If that holds up, it points toward a tantalizing special case where in a restricted sense details about the internal structure of a BH could be observed from outside the EH.

I also thought of a second variation of this experiment. Instead of electric charges, let black holes B and C have angular momenta in different directions. Classical observer A measures magnitude and orientation of frame dragging. Compared to the electric charge scenario, the angular momentum scenario has two advantages.

1. It does not have to cope with the complexity of E and B field coupling.
2. BHs with significant net charge may be encountered seldom or never, but every BH should be expected to have angular momentum.

 

[PAllen]

Thanks for the answer @mfb . If that holds up, it points toward a tantalizing special case where in a restricted sense details about the internal structure of a BH could be observed from outside the EH.

I also thought of a second variation of this experiment. Instead of electric charges, let black holes B and C have angular momenta in different directions. Classical observer A measures magnitude and orientation of frame dragging. Compared to the electric charge scenario, the angular momentum scenario has two advantages.

1. It does not have to cope with the complexity of E and B field coupling.
2. BHs with significant net charge may be encountered seldom or never, but every BH should be expected to have angular momentum.

Why do you think this? The aspects of the metric responsible for frame dragging outside each BH are outside each BH. The merger leads to a complicated merger of these external metrics (as well as the internal, which can'd communicate with the external). So I don't see any flavor getting information about the interior.

 

[anorlunda]

Why do you think this?

OK, here is my reasoning. It is largely the same as post #1.

We start with black holes B and C. They have equal and opposite direction angular momenta. We also have observer A, far enough from any EH as to be classical.

C

Initial State: C is far enough away that is plays no role. A observes frame dragging near B because of B's angular momentum.

Final State: B and C have completely merged. Momentum is conserved, so the net angular momentum of the merged BH is zero. A sees zero frame dragging.

Now the question regards the time evolution of the frame dragging observation from Initial State to Final State.

There is necessarily an intermediate time that we label t=0 where the two distinct event horizons kiss and become a single event horizon. The time evolution between t=0 and Final State reveals something about the interior of the merged BH. For example, consider a naive model where both B and C are represented as point source singularities. Then the time evolution is continuous from negative time through t=0 forward to t=Final. If A's observation of frame dragging could be characterized by a single scalar, it's time evolution might look like this. It is my presumption that the time evolution between t=0 and t=Final that gives evidence of the internal structure.

As in post #1, I think that the critical question is, "Is the time evolution continuous or discontinuous at t=0"

 

[PAllen]

GR is a classical field theory that assumes not only continuity but differentiability. The field equations are second order non linear partial differential equations.

 

[PeterDonis]

We also have observer A, far enough from any EH as to be classical.

I'm not sure I understand. We are talking about GR here, which is a classical theory, so everything in your model is classical.

the question regards the time evolution of the frame dragging observation from Initial State to Final State.

First we need to get a clearer understanding of the intial state and observer A's position. Angular momentum is conserved, so the initial state has zero angular momentum. That means that, far enough away from the holes (i.e., much further away than their initial separation), no frame dragging will be observed. So nothing changes at distances far enough away from the holes. So if observer A is initially observing frame dragging from hole B, he must be close enough to hole B (much closer than hole C) to observe it.

Second, we need to be clear that there are no known exact solutions describing spacetimes with more than one black hole; the best we could do would be to numerically simulate the process. I don't know if simulations have been done of this particular scenario. If not, the best we can do here is to make some reasonable heuristic guesses.

Third, the standard heuristic guess for changes of this sort is as follows: during the merger process, observer A will see gravitational waves passing him; those waves carry away the asymmetries in the spacetime curvature that A was seeing as frame dragging effects due to hole B. So what observer A sees is not a smooth change in "frame dragging". What he sees is a basically unchanging frame dragging effect due to hole B, then a chaotic region of gravitational waves passing him, then no frame dragging.

 

[anorlunda]

I am trying to avoid GR in the problem statement. Perhaps I confused you by posting this in the SR/GR forum. I just naturally though of this forum first for any BH related questions.

From post #1

My description is 100% classical, because it involves only what A sees in A's local frame.

A is positioned far enough away from any EH that A sees no significant GR effects locally.

1. Electric charge case. Observer A could measure local electric field and divergence, if he's close enough. If A is very remote, he may be able to see via astronomy distortions in the shapes of accretion disks or jets caused by the electromagetic force.
2. Angular momentum case. Observer A could remotely observe frame dragging effects via astronomy. There are hopes of doing so soon. https://en.wikipedia.org/wiki/Frame-dragging#Experimental_tests_of_frame-dragging and https://en.wikipedia.org/wiki/Frame-dragging#Astronomical_evidence

To try to make it clearer, below is the plot from the earlier posts, but I added a sinusoidal variation about the mean. (My freehand drawing skills with a mouse are very poor, so please excuse the crude artwork.) My reasoning is that if B and C orbit each other before merging, then A's measurements will vary sinusoidally, not because of gravitational waves but rather because of A's varying distance to the centers of B and C.

If the sinusoidal swings persist after t=0, my interpretation is that there must be two singularities orbiting each other inside the EH. At least for a short time until the singularities merge. That would be information about the internal structure of the BH.

I agree that there will be gravitational waves, but I don't see where they play any role in this question other than they could add even more distortion to the time evolution plot.

I stress again, that I have tried to not invoke GR in any part of these scenarios. Observer A remains classical at all times.

 

[mfb]

If the sinusoidal swings persist after t=0, my interpretation is that there must be two singularities orbiting each other inside the EH.

Your interpretation is wrong, as PAllen said already. Everything you measure comes from things that happen outside the black holes.

 

[anorlunda]

Perhaps it was futile to try to stay classical. When it comes to GR effects, I'm in over my head. I don't want to risk crossing the line into personal theories, so I'm going to back off and be silent on this thread hereafter unless asked.

 

[PeterDonis]

I am trying to avoid GR in the problem statement.

What other theory would you use?

Perhaps I confused you by posting this in the SR/GR forum. I just naturally though of this forum first for any BH related questions.

That's because GR is the only confirmed theory we have that describes black holes.