# Approximating a Black Hole Merger Using a Schwarzschild Model

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• BlackWhole
In summary, the conversation discusses the possibility of approximating a black hole merger as the creation of a daughter Schwarzschild black hole when two parent black holes come close enough to trigger a local Schwarzschild radius. The masses and event horizons of the parent black holes are discussed, as well as the distance between them at the moment of the "merge" and the energy emission during the merger. It is acknowledged that the situation is not spherically symmetric, unlike Schwarzschild geometry.
BlackWhole
TL;DR Summary
Can a black hole merger be approximated as a daughter Schwarzschild black hole created when two parent Schwarzschild black holes pack close enough together to trigger a local Schwarzschild radius containing their combined mass?
Summary: Can a black hole merger be approximated as a daughter Schwarzschild black hole created when two parent Schwarzschild black holes pack close enough together to trigger a local Schwarzschild radius containing their combined mass?

We know that the radius of the event horizon of a Schwarzschild black hole (SBH) is defined as r_s = 2*G*M/c^2, corresponding to the radius at which the escape velocity is the speed of light. For a binary black hole system, M_1 is the mass of the larger black hole and M_2 is the mass of the smaller. The LIGO event was of the merger of a 36 M_sun (where M_sun is the mass of the sun) black hole and a 29 M_sun black hole, radiating 3 M_sun equivalent energy before resulting in a 62 M_sun black hole. Thus, the larger SBH would have a Schwarzschild radius (SR) of 106 km and the smaller one would have an SR of 86 km.

If we model this super simplistically, then could we say that the black holes "merged" when they approached close enough to trigger an SBH in their vicinity? I attempted to illustrate this in the figure below.

Here, r_s1 is the radius of the larger SBH, r_s2 is the radius of the smaller SBH, x_c1 is the distance from the larger SBH to the system's barycenter, and x_c2 is the distance from the smaller SBH to the system's barycenter. For a sphere centered at the barycenter with the combined system mass of 65 M_sun (the sum of the masses of the two merging SBH's), the two black holes would enter the sphere starting at x_c2 = r_s3, where r_s3 is the SR of the equivalent SBH that would exist as a replacement of the two parent black holes. Here, x_c2 = r_s3 = 192 km. Using the equation for the position of the center of mass of two point masses, we get that x_c1 = M_2/M_1*r_s3, where M_1 is the mass of the larger SBH and M_2 is the mass of the smaller SBH. Thus, the distance between the two black holes at the time of the "merge" would be d_merge = (1+M_2/M_1)*r_s3, where d_merge is the distance between them at that moment. Could we then state that the black holes "merged" when their centers were 347 km apart? We could then model the gravitational energy emission at the moment of the merger as an instantaneous decay of the merged SBH with an SR of 192 km to a "stable" SBH with an SR of 183 km. It is of course simple to note that r_s3 = r_s1+r_s2, and that d_merge > (r_s1+r_s2), meaning that the event horizons of the two merging SBH's do not touch before the merge and are in fact separated by d_separation = d_merge-r_s1-r_s2 = d_merge-r_s3 = 155 km, where d_separation is the minimum distance between the two parent event horizons.

I know that I am neglected the fact that the space occupied by the two black holes is not uniformly dense.

Schwarzschild geometry is spherically symmetric. Your situation is not.

## 1. What is a Schwarzschild model?

A Schwarzschild model is a mathematical model used to describe the gravitational field around a spherical, non-rotating mass. It is named after the German physicist Karl Schwarzschild who first derived the solution to Einstein's field equations for a non-rotating black hole.

## 2. How does the Schwarzschild model approximate a black hole merger?

The Schwarzschild model can be used to approximate a black hole merger by mathematically describing the gravitational interaction between two black holes. This model assumes that the black holes are non-rotating and that they merge in a head-on collision.

## 3. What is the significance of approximating a black hole merger?

Approximating a black hole merger using a Schwarzschild model allows us to better understand the behavior of black holes and their gravitational interactions. This can provide valuable insights into the nature of space and time, as well as the laws of gravity.

## 4. Are there any limitations to using a Schwarzschild model for black hole mergers?

Yes, there are limitations to using a Schwarzschild model for black hole mergers. This model does not take into account the effects of rotation or the presence of matter and energy outside of the black holes. It also assumes that the black holes are perfectly spherical, which may not always be the case in reality.

## 5. How accurate is the Schwarzschild model in predicting black hole mergers?

The Schwarzschild model is a simplified approximation of reality and therefore may not be entirely accurate in predicting black hole mergers. However, it has been extensively tested and used in various scientific studies and has been found to provide reasonably accurate results in many cases.

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