To test Einstein's equations, you need to kick a black hole.

Original author: Kevin Hartnett
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Two teams of researchers have advanced significantly to the proof of the black hole stability hypothesis, the most important mathematical test of Einstein's General Theory of Relativity.




In November 1915, at a lecture at the Prussian Academy of Sciences, Albert Einstein described an idea that turned humanity’s view of the universe. Instead of taking the geometry of space and time fixed, Einstein explained that we live in a four-dimensional reality called space-time, whose form fluctuates, reacting to matter and energy.

Einstein described in detail this important idea in several equations, called " Einstein's equations " (or gravitational field equations), which form the core of his GRT. This theory was confirmed by all experimental tests that it underwent in the next century.

And although Einstein's theory, it seems, describes the observable world, the underlying mathematics remains mostly mysterious. Mathematicians were able to give very little evidence concerning the equations themselves. We know that they work, but we cannot say exactly why. Even Einstein had to return to approximations, and not to exact solutions, in order to see the Universe through the lenses he had created.

But in the last year, mathematicians brought the mathematics of GR to a clearer focus. Two groups derived solutions related to an important problem in GR, known as the black hole stability hypothesis. Their work proves that the Einstein equations correspond to physical intuition for the behavior of space-time: if you apply a sharp perturbation to it, it will tremble like a jelly, and then calm down in a steady state, from which it all began.

“If solutions were unstable, it would mean that they are not physical. It would be a mathematical ghost, existing in mathematics, but irrelevant from the point of view of physics, ”said Sergiu Kleinerman , a mathematician at Princeton University, and the author, together with Jeremy Szeftel ,one of two results .

To complete the proof, mathematicians needed to solve the basic complexity of the Einstein equations. To describe the evolution of the space-time form, a coordinate system is needed — something like lines of latitude and longitude — telling you where the points are. And in space-time it is very difficult to find a coordinate system that works everywhere.

Shake the black hole


As is well known, GR describes spacetime as something like a rubber sheet. In the absence of matter, the sheet is flat. Start dropping balls on him - stars and planets - and the sheet is deformed. Balls roll towards each other. When objects move, the shape of the rubber sheet also changes in response.

Einstein's equations describe the evolution of the form of space-time. You give them information about the curvature and energy at each point, and they give the form of space-time in the future. In this sense, Einstein's equations are similar to any equations that model any physical phenomenon: here the ball is at time zero, here it is five seconds later.

“This is a mathematically precise quantitative version of the statement that space-time is bent in the presence of matter,” said Peter Hinz, a researcher at the Clay Mathematics Institute at the University of California at Berkeley, who is responsible for the second result with Andras.

In 1916, almost immediately after the GTR, the German physicist Karl Schwarzschild found an exact solution to the equations describing what we now know as the black hole (this term appeared only five decades later ). Later, physicists found exact solutions describing a rotating black hole and a BH with an electric charge.

And these are all exact decisions describing BH. If you add at least a second BH, the interaction of forces becomes so complicated for modern mathematics that it only copes with it in very special cases.

However, we can still ask important questions about this limited group of solutions. One of these questions appeared in 1952 as a result of the work of the French mathematician Yvonne Choquet-Bruhat [Yvonne Choquet-Bruhat]. In fact, it sounds like this: what happens if you shake a black hole?


If you shake the BH, it will create gravitational waves. To prove the stability hypothesis is the same as to prove that these waves will scatter into emptiness, like waves on the surface of a pond after a stone falls.


The space-time changes over time, and the grid used to measure damped waves changes with it. The template defines the grid changes, and it must be selected correctly. Suppose we have a space-time with a grid of 1 cm, associated with a certain pattern. We will perturb the space-time so that gravitational waves appear. Incorrectly chosen pattern can cause the grid distances to change, and it will look as if the waves do not fade. The correct pattern is crucial for measuring the return to sustainability.

This problem is known as the BH stability hypothesis. She predicts that solutions to the Einstein equations will be “stable under perturbations”. Informally speaking, if you shake the BH, the space will also survive first, and then in a toga will calm down in a form that will look very similar to what we started from. “To put it bluntly, sustainability means that if we take special solutions and we disturb them a little, change the data, then the final dynamics will be very close to the initial decision,” said Kleinerman.

The so-called "stability" is an important test of any physical theory. To understand this, it will be useful to present an example more familiar than BH.

Imagine a pond. Now imagine that you have outraged its surface by throwing a stone there. The pond will wallow a little, and then calm down. Mathematically, solutions of the equations used to describe a pond (in this case, the Navier-Stokes equations ) should describe this basic physical picture. If the original solution does not coincide with the solution in the distant future, you can ask yourself about the correctness of your equations.

“An equation can have any properties, it can be mathematically in order, but if it contradicts physical expectations, it cannot be correct,” said Vasya.


Peter Hinz, a mathematician from the University of California

For mathematicians working on Einstein's equations, proof of stability was even harder to find than solutions to the equations themselves. Consider the case of a flat Minkowski space - the simplest of all space-time configurations. This solution of the Einstein equations was discovered in 1908, in the context of Einstein’s earlier special theory of relativity. But only in 1993 mathematicians were able to prove that if you shake a flat, empty space-time, then as a result you will again get a flat and empty space-time. This result, obtained by Kleinerman and Demetrios Christodoulou, is an admired work in this field.

One of the main difficulties with the proofs of stability is associated with tracking what is happening in four-dimensional space-time during the evolution of the solution. You need a coordinate system that allows you to measure distances and determine points in space-time, such as lines of latitude and longitude, used to determine a location on Earth. But it is not easy to find a coordinate system that works at every point in the space of time and continues to work when the form of space-time changes.

“We don’t know the way to do it, suitable for all occasions,” Hinz said in an e-mail. “The universe does not give us our preferred coordinate system.”

Measurement problem


The first thing to understand about coordinate systems is that people invented them. Secondly, not every coordinate system allows you to define all points in space.

Take latitude and longitude: you can assign them arbitrarily. Cartographers could choose any imaginary line as the prime meridian. And although latitude and longitude help determine almost any place on Earth, they cease to make sense at the north and south poles. If you didn’t know anything about the Earth, and you only had readings of latitude and longitude, you could wrongly conclude that something topologically wrong is happening at these points.

This possibility — to make incorrect conclusions about the properties of a physical space due to the inadequacy of the coordinate system describing it — is the essence of why it is so difficult to prove the stability of space-time.

“It may be that sustainability exists, but we use unstable coordinates, and thus we miss the fact that sustainability is true,” said Michalis Dafermos, a mathematician at the University of Cambridge, a leading expert in the study of Einstein's equations.

In the context of the theory of the stability of a black hole, any coordinate system used should evolve in the same way as the shape of spacetime — as a convenient glove adjusts to a change in the shape of a hand. The correspondence between the coordinate system and the space-time should be good at first and remain good all the way. If this is not the case, then two things can happen that interfere with attempts to prove the existence of stability.


Sergiu Kleinerman, a mathematician from Princeton University

First, your coordinate system can thus change shape, which breaks at certain points, just as latitude and longitude stop working at the poles. Such points are called “coordinate singularities” (to distinguish them from physical singularities, for example, black holes). These are undefined points in the coordinate system, which do not allow to fully describe the development of the solution to the very end.

Secondly, a poorly chosen coordinate system can hide the very physical phenomenon that it should measure. To prove that solutions of the Einstein equations come to a quiet state after disturbances, mathematicians need to carefully monitor the space-time ripples caused by disturbances. To understand why this is necessary, it is necessary to return to the analogy with the pond. A stone thrown into a pond generates waves. The long-term stability of the pond follows from the fact that the waves weaken with time - they become less and less, until there is no trace of their presence.

The situation is similar to space-time. The disturbance will cause a cascade of gravitational waves, and to prove stability you need to prove that these waves are attenuated. And for this you need a coordinate system, or "grid", which allows to measure the size of the waves. The correct grid allows mathematicians to see how the waves flatten and eventually disappear forever.

“Weakening needs to be measured relative to something, and this is where the problem with the grid appears,” Kleinerman said. “If we take the wrong grid, then even if stability is present, it cannot be proved, because the grid will not show me weakening.” And if you do not calculate the rate of attenuation of the waves, it is impossible to prove the stability. "

The problem is that although the coordinate system is extremely important, it’s not obvious which system to choose. “There is too much freedom in choosing the conditions for this grid,” said Hinz. “And most of the options will be wrong.”

On the way to the goal


A complete proof of the stability of black holes requires the proof that all known solutions of the Einstein equations for a BH (with a spin of a black hole within certain limits) are stable after a perturbation. Among the known solutions are the Schwarzschild solution describing the space-time of a non-rotating BH, and the Kerr family of solutions describing the space-time configuration, in which there is nothing but one rotating BH (and the properties of this BH — mass and angular momentum — differ within the family of solutions) .

Both new results are partially advanced in the direction of proving a complete hypothesis.

Hinz and Vasya, in workpublished on the website arxiv.org in 2016, proved that slowly rotating BH are stable. But their work does not include BHs rotating at a speed greater than a certain threshold.

Also, their evidence has several assumptions about the nature of spacetime. The initial hypothesis took place in Minkowski space, which is not only flat and empty, but also has a certain size. The proof from Hinz and Vasya takes place in de Sitter space, where space-time grows outward with acceleration, as in the real Universe. Changing the scene simplifies the problem from a technical point of view, and this can be understood by analogy: if you throw a stone into an expanding pond, the expansion will stretch the waves and they will weaken faster than if the pond did not expand.

“We are considering an accelerated expansion of the universe,” said Hinz. “It makes the task a little easier, because this process dilutes the gravitational waves.”

The work of Kleinerman and Sheftel has a slightly different feature. Their proof, the first part of which was published last November, occurs in the Schwarzschild space-time - which is closer to the original, more complex condition of the problem. They prove the stability of a non-rotating BH, but do not concern the decisions in which it rotates. Moreover, they prove the stability of BH only for a narrow class of perturbations - those in which the generated gravitational waves are in a certain way symmetric.

In both results, new techniques are presented for the selection of a suitable coordinate system. Hinz and Vasya start with an approximate solution of equations based on an approximate coordinate system, and gradually increase the accuracy of the answer, until they come to exact solutions and behave well to the coordinates. Kleinerman and Sheftel use a more geometric approach.

Now two teams are trying to build proof of a complete hypothesis based on their methods. Some expert observers believe that the day when it will turn out is not far away.

“I truly believe that now everything is at the stage of technical difficulties,” said Dafermos. “It turns out that to solve this problem, new ideas are no longer required.” He stressed that any of the mathematicians working on the task at the moment could offer final proof.

One hundred years of Einstein's equation served as a reliable experimental instruction to the Universe. Now, mathematicians may be getting closer to demonstrating why they work so well.

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