Mathematicians have refuted the hypothesis of the existence of a strong principle of cosmic censorship. Their work answers one of the most important questions in the study of the general theory of relativity and changes the way we reason about spacetime.
Almost 40 years after his statement, mathematicians decided on one of the most outstanding questions in the study of the general theory of relativity. In a
paper published on the Internet last fall, mathematicians
Michalis Dafermos and
Jonathan Luck proved that the strong form of the
principle of cosmic censorship relating to the strange structure of black holes is wrong.
“Personally, I consider this work an incredible achievement - a qualitative leap in our understanding of GR,” wrote
Igor Rodnyansky , a mathematician at Princeton University, to me.
The powerful form of the principle of cosmic censorship was proposed in 1979 by the influential physicist Roger Penrose. It was a way to avoid the trap. For decades, the GRT of Albert Einstein rules as the best scientific description of large-scale phenomena of the universe. However, the mathematical achievements of the 1960s showed that Einstein's equations showed unpleasant inconsistencies when applied to black holes. Penrose believed that if his strong principle of cosmic censorship is correct, then the lack of predictability can be ignored, considering it a mathematical feature, rather than a real description of the physical world.
“Penrose came up with a hypothesis that, in fact, tried to magically get rid of such unpleasant behavior,” said Dafermos, a mathematician from Princeton.
New job breaks Penrose's dream. At the same time, she fulfills his ambitions in other ways, showing that his intuitive understanding of the insides of the black hole was correct, simply not for the reasons he suspected.
Mortal sin of relativity
In classical physics, the universe is predictable. If you know the laws that govern the physical system and its initial state, you should be able to track its development to infinity. This maxim works, whether you are trying to use Newton's laws to predict the future position of a billiard ball, Maxwell's equation to describe the electromagnetic field, or Einstein's GTR to predict the evolution of the space-time form. “This is the basic principle of all classical physics, which can be traced down to Newtonian mechanics,” said
Demetrios Christodoulou , ETH mathematician Zurich and lead expert in the study of Einstein's equations. "Evolution can be defined on the basis of initial data."
But in the 1960s, mathematicians discovered a physical scenario in which Einstein's gravitational field equations — forming the core of his GRT — stop describing the predictable universe. Mathematicians and physicists noticed that something was wrong when they modeled the evolution of space-time inside a rotating black hole.
To understand what went wrong, imagine that you are falling into a black hole. First, you cross the event horizon, the point of no return (although for you it is no different from ordinary space). Here, the Einstein equations are still working as they should, giving a single, deterministic prediction of how space-time will change in the future.
Behind the point of no return of the BH lies the second horizon - the Cauchy horizon . Einstein's equations produce many solutions for the Cauchy horizon, which would mean that the Universe is fundamentally unpredictable.
The strong principle of cosmic censorship says that spacetime ends on the Cauchy horizon, so the Einstein equations do not need to describe the world further.
But the new study shows that there is space-time beyond this horizon, but it is not smooth enough to use Einstein's equations — this preserves predictability.If you continue the journey inside the BH, you will eventually cross another horizon, known as the Cauchy horizon. And then everything goes crazy. Einstein's equations begin to produce many variants of space-time. They all differ from each other, but satisfy the equations. The theory can not say which option will be true. For a physical theory, this is a mortal sin.
“The loss of predictability that we see in the GRT was very unpleasant,” said
Eric Poisson , a physicist at Guelph University in Canada.
Roger Penrose suggested a strong principle of cosmic censorship in order to restore predictability in Einstein's equations. He says that the Cauchy horizon is a purely mathematical construction. It could exist in an ideal scenario in which there is nothing in the Universe, except for a single rotating black hole, but it cannot exist in reality.
The reason for this, in his opinion, was that the Cauchy skyline is unstable. He said that any gravitational waves passing through it should provoke its collapse into a singularity — into an area of infinite density that ruptures space-time. Since the real universe is full of such waves, the Cauchy horizon should not appear in nature.
As a result, it makes no sense to ask what happens to the space-time beyond the horizon of Cauchy, since space-time, as described in the framework of GR, ceases to exist. “This is one way out of this puzzle,” said Dafermos.
But this new work shows that the space-time boundary defined by the Cauchy horizon has less to do with the singularity than Penrose imagined.
Save the black hole
Dafermos and Luck, a Stanford mathematician, proved that the situation on the Cauchy horizon is not so simple. Their work in a sly way refutes the letter of Penrose's initial statement about space censorship, but does not completely reject his spirit.
Based on the methods developed by Christodoulou, a former mentor of Dafermos at the institute ten years ago, the couple showed that the Cauchy horizon can indeed form a singularity, but not the one that Penrose expected. The singularity in their work is not as sharp as that of Penrose - they found a weak, “light” singularity, where they expected to find “spatial”. The weaker form of the singularity attracts the fabric of space-time, but does not tear it. “Our theorem says that observers crossing the Cauchy horizon are not torn by tidal forces. They may feel a shot, but they will not break, ”said Dafermos by mail.
Since the singularity that is being formed on the Cauchy horizon is softer than the one that predicts the strong principle of cosmic censorship, no one forbids GR to predict what is happening inside. “It still makes sense to define the Cauchy horizon, since we can, if there is a desire, continuously continue the space-time beyond its limits,” said
Harvey Real , a physicist from the University of Cambridge.
Dafermos and Lac proved that space-time extends beyond the horizon of Cauchy. They also proved that from the same starting point it can go on in different ways. Beyond the horizon, "there are many such continuations that can be considered, and there is no reason to prefer one of them to the other," said Dafermos.
However - and here lies the cunning of their work - these non-unique continuations of space-time do not mean that the Einstein equations break outside the horizon.
Einstein's equations work by measuring changes in space-time with time. Speaking in mathematical language, it is necessary to take derivatives of the initial configuration of space-time. And in order to take a derivative, it is necessary that the space-time be sufficiently “smooth” - free from discontinuities. Dafermos and Lac show that, although space-time exists beyond the Cauchy horizon, this extended space-time will not be smooth enough to satisfy the Einstein equations. Therefore, although the strong principle of cosmic censorship is refuted, the equations are spared the disgrace of issuing non-unique solutions.
“It makes sense to talk about the horizon of Cauchy; however, one cannot pass for him within the framework of solving the Einstein equations, Rial said. “It seems to me that they offered convincing evidence that this is so.”
This result can be imagined as an unpleasant compromise: although you can continue the space-time beyond the Cauchy horizon, the Einstein equations cannot be solved. But it is the fact of the existence of such a compromise that makes the work of Dafermos and Lak so interesting.
"In fact, there was a discovery of a new phenomenon in Einstein's equations," said Rodnyansky.