Three mathematicians have been working on it for several years, and 5 papers have been published before and after, and the total number of paper pages alone is as much as 2100 pages! The latest of these, the 912-page key paper, has been uploaded to arXiv.

Not only Yau Chengtong, but many colleagues in the mathematics community have expressed appreciation for this result. Demetrios Christodoulou, a professor of mathematics at the Swiss Federal Institute of Technology in Zurich, said: “This is indeed a milestone in the development of general relativity mathematics.”

How hard is this proof?

In 1963, mathematician Roy Kerr found a solution to Einstein’s equations, which precisely describe spacetime beyond what we now call rotating black holes, which is where Kerr’s black holes are named. The center of this type of black hole is a singularity (Singularity), and there are two inner and outer event horizons (Event Horizon). The inner event horizon is the limit of the black hole’s singularity, while the outer event horizon is the inescapable limit.

This means that once you fall into the outer event horizon, you are not immediately destroyed by the strangeness of the black hole, but at this point you will inevitably fall into the inner event horizon.**The two interfaces are tangent only at the poles**. In addition to the two horizons, the outermost boundary of the Kerr black hole is called the Static Limit or the infinite redshift surface (Surfaceof Infinite Redshift）。

In the nearly 60 years since Kerr’s achievement, researchers have been trying to prove that so-called Kerr black holes are stable. But one problem encountered is that most explicit solutions to Einstein’s equations are**Only applies to stationary black holes such as Schwarzschild black holes**。

But many other black holes that can be observed in nature are spinning.

The three mathematicians involved in this study are Elena Giorgi and Jeremie Szeftel of Columbia University and Sergiu Klainerman of Princeton University. To assess the stability of Kerr black holes, they needed to subject the black holes to slight perturbations and then observe how the solutions of these objects changed over time.

As a simple example, when a sound wave hits a wine glass, in most cases the wine glass is stabilized after being shaken lightly. But if someone sings loud enough and the pitch perfectly matches the resonant frequency of the glass, the wine glass may break.

Three mathematicians want to know,**Does a similar resonance phenomenon occur when a black hole is struck by a gravitational wave?**。

They presuppose several possible outcomes:

1. Gravitational waves may pass through the event horizon of the Kerr black hole and enter the interior of the black hole, slightly changing the mass and rotation of the black hole;

2. Gravitational waves may revolve around black holes and dissipate, just as most sound waves do not shatter wine glasses;

3. Gravitational waves may gather outside the event horizon of the black hole, and its energy will be concentrated to a single singularity, the space-time outside the black hole will be severely distorted, and the result will be uncertain.

To this end, they employed the paradoxical approach to reasoning, which has been used in previous related work. The argumentation process goes something like this:

First, the researchers assumed, contrary to what they themselves were trying to prove, that the Kerr solution would not last forever, meaning it would fail after a long time.

They then used some mathematical tricks to extend Kerr’s solution beyond the claimed maximum time by performing an analysis of partial differential equations. In other words,**The researchers proved that this “very long time” is actually infinite, that is, very stable**。

This obviously contradicts the researchers’ original hypothesis, which means that the conjecture itself must be correct.

In particular, Klainerman emphasized that he and his colleagues built on previous research. “It’s just that we happened to be lucky,” he said, and he hopes the new findings will be seen as “a victory for the entire mathematics community.”

**future outlook**

So far, the stability of Kerr black holes has only been slowly rotating —**The ratio of the angular momentum of a black hole to its mass is much less than 1**—— has been proved in the case of , and no research has proved that fast rotating black holes are also stable.

In addition, the researchers have not yet determined exactly how small the ratio of angular momentum to mass must be to ensure stability.

In addition to this problem, there is a larger problem known as the final state conjecture. According to the basic definition of the conjecture, if we wait long enough,**The universe will evolve into a finite number of Kerr black holes far away from each other**. The final state conjecture depends on Kerr stability and other very challenging subconjectures.

This also means that the research of the three mathematicians is just a new beginning, as Klainerman said: In the next few years or even decades, mathematicians will continue to study the stability of Kerr black holes.

I wouldn’t be at all surprised if we don’t have a complete conclusion about the stability of Kerr black holes until the end of this decade.

Reference link:

[1]https://arxiv.org/abs/2205.14808

[2]https://www.quantamagazine.org/black-holes-finally-proven-mathematically-stable-20220804/

[3]https://www.quantamagazine.org/to-test-einsteins-equations-poke-a-black-hole-20180308/