The rigidity might cause problems in a merger. What guarantees that we can fit the two black holes inside the new horizon? The black holes lose some of the mass energy in the merger. The Schwarzschild radius of the new black hole is less than the sum of the radii of the merging black holes.
The gravity field of a Schwarzschild black hole M plus an external mass m
"line of force"
________
___ \_______ • m
/ \
\___/
M
Here we have a problem. If the "lines of force" of the gravity field of m go extremely close to the horizon of M, then the event horizon should effectively move farther behind M?
That probably is true. If m comes very close to the horizon, we expect it to increase the Schwarzschild radius by
2 G m / c².
It makes sense that even before that, there is an effect on the radius of the event horizon.
Schwarzschild black holes which collide head-on
Daniel Pook-Kolb et al. (2020) used numerical relativity to calculate various horizons in a head-on collision of two non-rotating black holes.
Unfortunately, their Figure 1 does not seem to be in scale. The sum of the individual radii (purple, red) is much less than the final radius (blue).
A crucial question is how close the two black holes can come before the local speed of light in the area slows down to a crawl, as measured by a faraway observer. If the two black holes would freeze in their places, their common event horizon could become elongated.
The Werner Israel theorem (1967)
The Israel theorem speaks about a "static" solution. A black hole probably never becomes completely static. In the Oppenheimer-Snyder (1939) collapse, the surface keeps approaching the event horizon for ever.
Anyway, a static non-rotating system must always have a perfectly spherical event horizon. There cannot be any bulges in it. If we assume that the system quickly approaches a static solution, then this rules away the possibility that a common event horizon can form while the two merging black holes are not yet inside their common Schwarzschild radius.
The merger of a large black hole and a small black hole
In the Schwarzschild metric, the radial metric is stretched by the factor
1 / sqrt(1 - r_s / r),
where r_s is the Schwarzschild radius, and r is the Schwarzschild radial coordinate. Let us assume than the large black hole has
M = 10
and the small
m = 1.
Let us calculate the stretching of the radial metric of the large black hole at
r = 1.2 r_s.
Then
1 / sqrt(1 - 1 / 1.2)
= 1 / sqrt( 1/6 )
= 2.5.
Let us assume that the event horizon of the falling small black hole m remains as is in comoving coordinates, and that the event horizon of M stays as is.
M m
● <---•
<---> 0.08 r_s
<-----------------------------><----------->
2 r_s 0.2 r_s
We see that when the small black hole m falls toward M, it is squeezed into 40% or less in the radial direction in the Schwarzschild coordinate r. It fits easily within the diameter 2.2 r_s of the new, forming black hole.
The "original" event horizons of M and m probably do not touch yet when the common event horizon of the combined system M & m forms.
The event horizon is defined as the border from where a light signal can still reach infinity. It is obvious that the individual, up-to-date, event horizons of M and m have to touch before the common event horizon forms and grows to form a perfect sphere.
Above we have two screenshots from a video by Teresita Ramirez and Geoffrey Lovelace (2018). In the video, both black holes seem to be squeezed in the radial direction from their common center of mass. They suddenly obtain a common event horizon when they are close enough.
Could we "freeze" a non-spherically-symmetric field when two black holes meet?
___
/ \ M
\____/
_O_ O = small black hole
/ \
\____/ M
Let us try to build a "bridge" between the event horizons of two black holes which are not yet inside their common Schwarzschild diameter (diameter = 2 * radius). If we succeed, then the speed of light inside the bridge is essentially zero, and the bridge will "freeze" the black holes to their positions.
We put two black holes of a mass M close to each other, but not within the Schwarzschild diameter of 2 M. The letter O denotes a small black hole which we try to use as a bridge between the two larger ones.
Note a black hole M vertically squeezes the other black hole M if we use global (pseudo-"Schwarzschild") coordinates centered at O.
We can roughly add the perturbations of "weak" gravity fields in the Minkowski metric η. Both black holes M squeeze O vertically.
Let M = 10 and the mass of O be 1. Since all black holes are at a low potential, their Komar mass is considerably less.
Let the Schwarzschild radius for a mass 1 be 1.
Let the potential of O be -0.5. Then O only contributes 0.5 to the common Komar mass of the system. From the Schwarzschild metric formula we see that O is squeezed very roughly to 0.5 vertically.
If the potential of the lower big black hole in the field of the upper one is -1, then it contributes to the Komar mass 9.5, and is squeezed vertically to 9.
The Komar mass of the whole system is 2 * 9.5 + 0.5 = 19.5 and the vertical diameter only 2 * 9 + 0.5 = 18.5. The system is inside its combined Schwarzschild diameter. We failed. Our goal was to keep the system too wide to be inside its Schwarzschild diameter.
Building a bridge between the event horizons of the two black holes M seems to fail because everything is squeezed vertically. The whole system slips inside its common Schwarzschild diameter before a bridge is built.
Our analysis suggests that when the two black holes M merge, clocks in the gap between the black holes do not slow down too much, relative to the global "Schwarzschild" coordinate time. This means that the local speed of light in the gap stays relatively fast. The two black holes can approach each other rapidly, measured in the global "Schwarzschild" coordinates.
Only after a common, spherical event horizon forms around the two black holes, does the local speed of light in the gap go to almost zero. The system freezes when the two black holes are almost entirely inside their common Schwarzschild diameter. This is much like the Oppenheimer-Snyder (1939) collapse where only the surface of the dust ball lingers above the forming event horizon.
Cohen, Kaplan, and Scheel (2011) have made computer simulations of merging black holes. Their paper contains pictures of a thin bridge forming between the individual event horizons of the two black holes.
A microscopic black hole or a particle falling into a black hole
If a microscopic black hole falls into a large black hole, that is still a merger of two black holes. The analysis above suggests that a common event horizon quickly encloses both black holes.
The same is probably true of a particle falling into a black hole. If we use the Schwarzschild time coordinate, the particle quickly reaches a radius r which is the new Schwarzschild radius of the system.
What about a large mass (not a black hole) falling into a black hole? The mass consists of particles. If the hypothesis above is true, then a new horizon quickly encloses all the particles.
Conclusions
LIGO observations show that (spinning) black holes merge quickly. The ringdown phase lasts only for about 10 milliseconds. It cannot be that the merging black holes would freeze in a "dipole" formation in which they would produce large gravitational waves for a long time.
Our analysis above suggests that the merging black holes have separate event horizons until they are inside the Schwarzschild diameter of the entire system. Once they are inside that diameter, a common, spherical event horizon forms very quickly.
In our Minkowski & newtonian gravity model, the two black holes are essentially frozen behind the new common event horizon. Observers very close to the individual event horizons of each black hole never get a signal that anything has changed. This is because the local speed of light is very slow close to the horizon. The individual black holes, in this sense, are infinitely rigid. In our model, there is no further development behind the common event horizon. No collapse toward a singularity happens.
In our next blog posts we should analyze the proof of the Werner Israel theorem (1967), and analyze what happens in a merger of spinning black holes.
Also, we need to check what Richard H. Price wrote in his 1971 thesis about a mass falling into a spinning black hole.
D. C. Robinson (2012) wrote an overview of black hole uniqueness theorems.
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