# Theoretical solution to binary black hole merger based on Hawking and Ellis

Following Hawking and Ellis, Chapter 9, Fig. 60, Pg. 322, the following figure is meant to illustrate the contrast between apparent horizons and event horizons in the case of a binary black hole merger.

I understand the merger of two black holes in the long run settles into a Kerr black hole, and I more or less understand why this is correct. But, I wonder if there exists any simple solution of the binary black hole merger that can actually illustrate this phenomenon? Does there exist any simple metric solution to the Einstein's Field equation that models the black hole merger to illustrate the above concept in explicit form?

• No, there does not. – G. Smith Jun 5 '19 at 20:44

The first explicit example describing black hole coalescence is the Kastor–Traschen class of dynamical multi–black hole solutions:

This is a family of exact solutions to the Einstein–Maxwell equations with a cosmological constant representing an arbitrary number of charged black holes, with $$Q=M$$ for each, in an otherwise closed universe. Charge to mass ratio equal to $$1$$ means that the net force that black holes exert on each other is precisely zero and so it is possible to analytically describe the merger. Some aspects of these coalescing solutions have been investigated here:

• Brill, D. R., Horowitz, G. T., Kastor, D., & Traschen, J. (1994). Testing cosmic censorship with black hole collisions. Physical Review D, 49(2), 840. arXiv:gr-qc/9307014.

• Nakao, K. I., Shiromizu, T., & Hayward, S. A. (1995). Horizons of the Kastor–Traschen multi-black-hole cosmos, Physical Review D, 52(2), 796, doi:10.1103/PhysRevD.52.796.

• McNutt, D., & Coley, A. (2018). Geometric horizons in the Kastor-Traschen multi-black-hole solutions. Physical Review D, 98(6), 064043, doi:10.1103/PhysRevD.98.064043, arXiv:1811.02931.

Another situation amenable to analytic description is extreme mass ratio mergers, when one of the merging black holes is much larger than the other. While most of the literature focuses on obtaining gravitational waves from such extreme mass ratio inspiral, there are some works dedicated to the dynamics of event horizon during such process:

• I am very happy with this answer! – Sandesh Jr Jun 6 '19 at 6:57