Why is space dependent on time

In 1994 the authors gave several lectures on general relativity at the Isaac Newton Institute for Mathematical Sciences at Cambridge University (England). The excerpts compiled and commented on here (the entire series of lectures was recently published by Princeton University Press under the title "The Nature of Space and Time"; a German translation is planned by Rowohlt) illustrate the different points of view of the two scientists. Despite common roots - Penrose was on the doctoral committee before which Hawking took his doctoral examination in Cambridge - their views on quantum mechanics and its importance for the evolution of the universe diverge. In particular, they argue about what happens to the information stored in a black hole and why the initial state of the universe is different from its final state.

Hawking discovered in 1973 that a black hole emits particles due to quantum effects. In doing so, it evaporates, so to speak, so that in the end nothing of its original mass may be left. But while black holes are formed, with the matter and energy they swallow, much information about the type, properties and configurations of the incident particles is lost for the rest of the universe. Although quantum theory suggests that such information must be preserved, what ultimately happens to it is hotly debated. Both Hawking and Penrose assume that when a black hole emits particles it loses the information stored inside it; but while Hawking believes the loss is irrecoverable, Penrose claims that this is compensated for by spontaneous quantum state measurements, which again feed information into the system.

Both researchers believe that a quantum theory of gravity must first be developed for a complete description of nature; But they disagree about the details. Penrose thinks that this quantum gravity would not be symmetrical in time, although the elementary forces of particle physics remain unchanged under time reversal. The asymmetry should then explain why the universe was extremely homogeneous when it was formed - which the microwave background radiation speaks for as a relic of the Big Bang. On the other hand, he is facing an extremely messy end.

Penrose tries to explain the temporal asymmetry with his hypothesis of the Weyl curvature. According to Albert Einstein (1879 to 1955, Nobel Prize 1921), space-time in the vicinity of masses is more or less strongly curved; however, it can also have its own distortion, the extent of which is expressed by the curvature named after the mathematician Hermann Weyl (1885 to 1955). For example, gravitational waves and black holes deform space-time even where there is no matter far and wide. While Weyl's curvature was probably zero in the young universe, according to Penrose it will be very high in the end because of the large number of black holes - and this distinguishes the end phase of the universe from its beginning.

Hawking also believes that the big crunch will not be a reflection of the big bang; but he turns against a time asymmetry in the laws of nature themselves. In his view, the cause of the difference between beginning and end lies in the way in which the evolution of the universe is programmed, as it were: He postulates a kind of democracy, according to which none Point in space may be something special - and that is why the universe cannot have any limits. According to Hawking, the postulate of infinity explains why the cosmic background radiation is so uniform.

The two physicists also interpret quantum mechanics differently. For Hawking, a theory should deliver no more than predictions that agree with measured data. Penrose does not think that this is enough to explain reality. He points out that quantum theory operates with superimposed states (superpositions of wave functions); sometimes that could have absurd consequences. In a certain way, the two researchers are thus resuming the historical debate between Einstein and Niels Bohr (1885 to 1962, Nobel Prize in 1922) about the completeness of quantum theory.

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Stephen Hawking on black holes:

The quantum theory of black holes ... seems to create a new degree of unpredictability in physics beyond the previously known quantum mechanical indeterminacy. The reason is that black holes appear to have an internal entropy and are pulling information out of our area of ​​the universe. These claims are of course controversial: Many researchers in the field of quantum gravity - especially almost all those who come from particle physics - instinctively reject the idea that information about the quantum state of a system can be lost. So far, however, they have not yet been able to show how information can escape from a black hole. One day they will probably have to accept my opinion and accept that the information disappears - just as they had to admit that black holes, despite all their prejudices, emit radiation ...

Because gravity is always attractive, it agglomerates matter in the universe to form objects such as stars and galaxies. The former resist further contraction for some time through thermal pressure, the latter through rotation and internal movements. Ultimately, however, heat or angular momentum is exhausted and the object begins to shrink. If the mass is less than about one and a half solar masses, the degenerative pressure of the electrons or neutrons can stop the contraction, and the object becomes a white dwarf or a neutron star. However, if the mass is greater than this limit, nothing can prevent further contraction. If the object has shrunk to a certain critical size, the field on its surface becomes so strong that the light cones are curved inwards ... One can see that even the light rays emitted are curved towards each other and thus converge instead of diverging. That is, there is a closed trapped surface ...

Accordingly, there must exist an area of ​​space-time from which nothing can escape into infinity - a so-called black hole. Its limit is called the event horizon; it is a zero surface, which is formed by those rays of light that can no longer escape to infinity ...

When a celestial body collapses into a black hole, a large amount of information is lost. The collapsing body is described by a large number of parameters - including the types of matter and the multipole moments of the mass distribution. But the resulting black hole is completely independent of the type of matter from which it emerges and very quickly loses all multipole moments except for the first two: the monopole moment for the mass and the dipole moment for the angular momentum.

This loss of information did not play a major role in classical theory. It could be said that all of the information about the collapsing body was still present inside the black hole. It is true that an outside observer would be able to determine the nature of the collapsing body only with great difficulty, but in the classical theory this would not be excluded in principle. The observer would never completely lose sight of the collapsing object. As it approached the event horizon, it would seem to slow down more and more and become very indistinct; but he could still see what it is made of and how its mass is distributed.

But quantum theory fundamentally changes the situation. First, the collapsing body sends out only a limited number of photons before it falls below the event horizon - far too few to carry all of the information about the object to the outside. This means that in quantum theory an outside observer is in principle unable to measure the state of the collapsed body. You'd think that doesn't matter much, because the information is still inside the black hole even if you can't get to it from outside. But this is where the second quantum theoretical effect on black holes comes into play ...

Based on the theory, black holes radiate and lose mass. Apparently they finally disappear completely - and with them the information inside them. I will explain why this information is really lost and does not come back in any form. As I will show, this loss of information creates a new level of physical indeterminacy beyond that familiar from quantum mechanics. Unfortunately, this additional degree - unlike Heisenberg's principle of uncertainty - can only be proven with great difficulty experimentally on black holes.

Roger Penrose on quantum theory and spacetime:

The great physical theories of the 20th century - quantum theory, special and general relativity, and quantum field theory - are not independent of one another: general relativity builds on the special, and quantum field theory includes special relativity and quantum theory.

With an accuracy of 10-11, quantum field theory is considered to be the most exact physical theory of all. I would like to point out, however, that the general theory of relativity has recently been confirmed to an accuracy of 10-14 (the accuracy apparently only being limited by the accuracy of earthly clocks). I mean measurements on the binary pulsar PSR 1913 + 16, a binary star system made up of two neutron stars, one of which is a pulsar. According to the general theory of relativity, the two stars must gradually orbit each other closer and closer (whereby the orbital period is shortened), because energy is lost through the emission of gravitational waves. That has actually been observed, and the entire sequence of movements ... agrees with the general theory of relativity (which includes Newton's theory as a special case) with the remarkable accuracy mentioned for a total of 20 years. The discoverers of this system, Joseph H. Taylor and Russell A. Hulse, were honored for this with the Nobel Prize in 1993 [see Spectrum of Science, December 1993, page 21]. Quantum theorists have always insisted that, in view of the accuracy of their theory, general relativity should be adapted to them; but it seems to me that quantum field theory has some catching up to do.

Although these four theories are very successful, each has its problems ... From the general theory of relativity follows the existence of spacetime singularities. The so-called measurement problem arises in quantum theory, which I will deal with later. Perhaps the solution to the various problems lies in the fact that each theory is incomplete on its own. For example, many expect quantum field theory to somehow smear the singularities of general relativity ...

I would now like to turn to the loss of information in black holes, since I am convinced that it is relevant to the latter topic. I agree with almost everything Stephen said about it. But while he considers the loss of information in black holes to be an additional physical indeterminacy that is added to the quantum theoretical, I consider both to be somewhat complementary ... Possibly a bit of information escapes the moment the black hole evaporates ... but this tiny gain in information is much smaller than the loss of information in the event of a collapse (at least this seems to me to be the only plausible idea of ​​the final disappearance of the hole).

If we enclose the system in a huge box in the thought experiment, we can consider the phase space evolution of matter in the box. In the area of ​​phase space that corresponds to situations in which a black hole is present, the trajectories of physical development converge and the volumes moving along these trajectories shrink. The reason for this is the loss of information in the singularity of the black hole. This shrinkage is in direct contradiction to Liouville's theorem of classical mechanics, according to which volumes in phase space remain constant [work by the French mathematician Joseph Liouville (1809 to 1882) contributed a lot to 20th century physics] ... Thus violates a space-time with blacks Holes this conservation law. But in my model the loss of phase space volume is compensated for by a spontaneous quantum measurement process in which information is obtained and certain phase space volumes increase. That is why I see the uncertainty caused by the loss of information in black holes as complementary to the quantum theoretical: Both are sides of the same coin ...

Let us now consider the thought experiment with Schrödinger's cat. It describes the risky fate of a cat in a box in which - for example - a photon is emitted and hits a semi-transparent mirror. The transmitted part of the photon wave function hits a detector, which automatically triggers a shot that is fatal for the cat when the photon arrives; if the detector does not detect a photon, the cat remains alive. (As I know, Stephen is against the mistreatment of cats, even in a thought experiment!) The wave function of the system is a superposition of these two possibilities ... But why can we never perceive macroscopic superimpositions of such states, but only the macroscopic alternative facts " The cat is dead "and" the cat is alive "? ...

I think something goes wrong with the superpositions of the alternate spacetime geometries that would appear once general relativity began to play a role. Perhaps the superposition of two different geometries is unstable and disintegrates into one of the two. For example, the geometries could be the spacetime of a living cat or that of a dead one. I call this decay into one or the other spacetime geometry objective reduction ... What does this have to do with Planck's length of 10-33 centimeters? The natural criterion for when two geometries differ significantly depends on the Planck scale; it in turn defines the time scale for the disintegration into one of the two alternative geometries.

Hawking on quantum cosmology:

I want to close this talk on a subject that Roger and I have very different views on - the arrow of time. In our part of the universe there is a clear difference between moving forward and backward in time. All you have to do is watch a wrong-way film: cups that normally fall from the table and break, miraculously collapse on the floor and jump onto the table. If only that were the case in reality!

The local laws that physical fields obey are time-symmetric or, more precisely, CPT-invariant (where C stands for charge, P for parity and T for time). Therefore the observed difference between past and future must come from the boundary conditions of the universe. Let us assume that the universe is spatially closed, expands to a maximum size and then collapses again. As Roger pointed out, the universe looks very different at the two endpoints of its history. In what we call the beginning of the universe, everything seems to have been very smooth and regular. But in the event of a general collapse, things will probably be very disorderly and irregular. Because there are so many more disordered configurations than ordered ones, the initial conditions must have been chosen incredibly precisely.

Apparently, therefore, different boundary conditions must apply at the two ends of time. Roger argues that Weyl's tensor only disappears at one of the two time ends. The Weyl tensor is that part of the space-time curvature that is not locally determined by the matter via the Einstein equations. It is said to have been small in the smooth, ordered initial phase, but grows large in the collapsing universe. So this approach would differentiate the two ends of time and thus perhaps explain the arrow of time.

I do not consider Roger's hypothesis to be the ultimate conclusion. First, the approach is not CPT-invariant. Roger sees this as an advantage; but I think one should stick to symmetries as long as nothing compellingly speaks against it. Second, if Weyl's tensor had been exactly zero in the early universe, space itself would initially have been completely homogeneous and isotropic, and that would not have changed for all time. Roger's Weyl hypothesis fails to explain the fluctuations in the radiation background or the disturbances from which galaxies and bodies like ourselves emerged.

Despite all of this, I think Roger pointed out an important difference between the two ends of time.But the fact that Weyl's tensor was small at one end should not be postulated as an ad hoc boundary condition, but rather derived from a more fundamental principle, the no-boundary-proposal ...

Why can the two ends of time differ? Why should the disturbances be small at one end but not at the other? The reason is that there are two possible complex solutions to the field equations ... Obviously one solution corresponds to one end of time and the second to the other ... At one end the universe was very smooth and the Weyl tensor very small. However, it cannot have been exactly zero, as this would be a violation of the principle of indeterminacy. Rather, there were small fluctuations from which galaxies and bodies like us later developed. On the other hand, at the other end of time the universe would have to be very irregular and chaotic - with a correspondingly large Weyl tensor. That would explain the actual arrow of time observed and thus why cups fall from the table and break instead of joining and jumping up.

Penrose on quantum cosmology:

If I understand Stephen correctly, our positions on this [Weyl's curvature hypothesis] do not seem too different. For an initial singularity, the Weyl curvature is approximately zero ... Stephen emphasized that there must be small quantum fluctuations in the initial state, and therefore the hypothesis that the Weyl curvature is zero for the initial singularity belongs to classical physics. Certainly there is still some leeway for the precise formulation of the hypothesis. Small disturbances can be reconciled with my thesis, especially in the area of ​​validity of quantum theory. We just need something that will reduce Weyl's curvature to almost zero ...

Perhaps [James P.] Hartle and Hawking's postulate of infinity offers a good explanation for the structure of the initial state. But in my opinion we need something completely different to explain the final state. In particular, a theory that explains the structure of singularities would have to violate [the CPT symmetry and other symmetries] for something like Weyl's curvature hypothesis to come into play at all. This break in time symmetry could be very subtle; it would have to be implicit in the laws of a theory that goes beyond quantum mechanics.

Hawking on Physics and Reality:

These lectures showed very clearly the difference between Roger and me. He's a platonist and I'm a positivist. He worries that Schrödinger's cat is in a quantum state where it is half alive and half dead. In his eyes, this cannot correspond to reality. But I don't mind. I don't ask that a theory agree with reality, because I don't even know what it is. Reality is not a quality that can be tested on litmus paper. I am only interested in whether the theory predicts the results of measurements. Quantum theory does this very successfully ...

Roger says ... the collapse of the wave function is causing a CPT violation in physics. He sees the effect of such injuries in at least two situations: in cosmology and in black holes. I admit that by asking questions about observations, we can bring about a time asymmetry. But I definitely reject the idea that there is a physical process which corresponds to the reduction of the wave function, or that this has something to do with quantum gravity or consciousness. It doesn't sound like science to me, it sounds like witchcraft.

Penrose on physics and reality:

Quantum mechanics has only been around for 75 years. Compared to Newton's theory of gravity, that's a very short time. So I wouldn't be surprised if quantum mechanics had to be modified for extremely macroscopic objects.

At the beginning of this debate, Stephen said that he thought he was a positivist and that I was a platonist. He may be a positivist for me, but I think the key point is that I am a realist. And if you compare this discussion with the famous debate between Bohr and Einstein around 70 years ago, I see Stephen in the role of Bohr and me in Einstein's. Einstein insisted that there had to be something like a real world that did not necessarily have to be represented by a wave function, while Bohr emphasized that the wave function did not describe a "real" micro-world, but only "knowledge" that was useful for predictions.

Bohr is considered the winner in this dispute. In fact, following the recently published biography of Einstein by [Abraham] Pais, Einstein might as well have gone fishing after 1925. It is true that he made no great progress, although his astute objections were very useful. But I believe the reason Einstein did not continue to do fundamentals in quantum theory was that the theory was missing a critical component. That missing ingredient was black hole radiation, which Stephen discovered 50 years later. The loss of information associated with this radiation opens up new perspectives.

This article is included in Spectrum of Science 9/1996