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General Relativity: Physics and Mathematics

The inaugural lecture of P. Chrusciel in January 2011 was entitled: "General relativity: Physics or Mathematics?" The above variant of this title provides a good description of the breakthrough progress in gravitational physics in spring 2017: the third detection of gravitational waves with LIGO.

Two middleweight black holes about to collide in this artist's impression of the latest black-hole merger detected by LIGO. Credit: LIGO/Caltech/MIT/Sonoma State (Aurore Simonnet)

A contribution by by Peter C. Aichelburg and Piotr T. Chrusciel

Indeed, this week's issue of Physical Review Letters (dated June 2, 2017) contains a report of the observation of a black hole merger, as seen via direct detection of gravitational waves. The wave  passed through earth on January 4, 2017, and was seen at both detectors of the Laser Interferometric Gravitational Observatory (LIGO), located roughly at opposite geographical ends of the United States. This is the third such event observed by LIGO, following two previous detections in the fall of 2015.

The LIGO experiment is designed to measure minute displacements of mirrors caused by the change of the geometry of space-time associated with the passage of a gravitational wave. Einstein's theory of gravitation predicts in detail how the mirrors should move when a gravitational wave traverses earth. The challenge of the experiment is to measure displacements by a hard-to-imagine distance of a thousandth of the diameter of a proton, and this is what the LIGO experiment is designed for, and does.

The analysis of the signal tells us that two black holes of  20 and 30 solar masses have collided to form a new black hole of about 50 solar masses, releasing an energy of 2 solar masses in a fraction of a second in the form of gravitational waves. The black hole merger took place about 3 billion years ago. Typical black hole collisions are not accompanied by optical signals and so gravitational waves are our only opportunity to observe such events.

This observation provides yet another milestone for gravitational physics. The main breakthrough was the first direct observation of a gravitational waves in September 2015, rewarded meanwhile by many major prizes. However, a reasonable amount of skepticism should be applied when a scientific discovery of extremely rare events, done at the uttermost limit of the existing technology, is announced. A second wave, with very different observation profile, seen in December 2015, confirmed that we are on the right track. The January 2017 event, with overall detection features similar to the first one, is a major step towards a watertight proof of the science involved.

From the new observation we learn that black hole mergers are much more frequent than what we thought. From the lack of observation, so far, of gravitational waves by neutron stars in neutron star-neutron star binary systems, or neutron star-black hole binary system, we learn that such systems are less frequent than what we thought. The detailed wave profile of the signal suggests an unexpected configuration of the spins of the black holes. All this information advances our understanding of the universe, adding on the way some theoretical puzzles which we will need to resolve.

All three black hole collisions took place far away and a long time ago, and it is natural to wonder why we are not seeing  such event nearer to us. Keeping in mind that black hole collisions are extremely rare events, one realises that the larger the volume of space we observe, the larger the probability of witnessing a collision. Equivalently, the probability of observing such a collision near to us is much smaller than observing one far away. So while it would be nice to see such collisions nearer to us, and therefore with much stronger signals, or perhaps from smaller black holes, it is likely that the events that we will continue to detect in the future will keep happening far away.

What about mathematics and general relativity? Well, one of the deep questions arising in Einstein's theory of gravitation is that of the nature of the energy of the gravitational field. Unlike other physical theories, general relativity does not come equipped with an obvious definition of the energy of the field. This problem was one of the puzzles leading to misconceptions concerning the existence of gravitational waves: a  wave needs to transfer energy in order to be seen, so what's the point of a gravitational wave if the gravitational field does not carry energy?  It is fair to say that the gravitational wave detections settled the issue: the mirrors moved, gaining kinetic energy, so the energy must have come from the gravitational wave.

Because of the elusive nature of the energy of the gravitational field, it is not surprising that mathematical studies of its properties turned out to be especially challenging. The basic question of positivity of total energy has been only settled in the early 1980's by Richard Schoen and Shing-Tung Yau, and independently by Edward Witten. That work carried a warning: the proof depended on the dimension of space-time, and the conclusion could only be reached in low dimensions. Since higher-dimensional models of physical reality are bread-and-butter in theoretical physics, this restriction on the dimension is  undesirable for a theoretical physicist, and a challenge for mathematicians in any case. In fact, the question whether or not the total energy is positive was known to have key implications to the to fundamental questions in geometry, including properties of black hole space-times. Many people tried without success to remove the dimension-restriction for the last thirty years until Schoen and Yau struck again: In April this year they released their new monumental work, proving positivity of total energy of the gravitational field in all dimensions.

We are very happy that the Erwin Schroedinger Institute (ESI) for Mathematics and Physics of the University of Vienna will be hosting Richard Schoen and Shing-Tung Yau in July this year, who will be lecturing on their work to the participants of a summer school and research program. The program, entitled "Geometry and Relativity", is centered around mathematical problems of general relativity, and coorganised by Michael Eichmair from the Department of Mathematics, and by Robert Beig and Piotr Chrusciel from  Physics. The LIGO observations, and their implications for gravitational physics, will also be studied at ESI in a later workshop  within the same research program.

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