Once Before Time - Martin Bojowald [97]
HORIZONS: POINT OF NO RETURN
When matter has collapsed completely and no force counteracting gravity can keep the elementary particles apart, does a black hole then appear as a single, infinitely dense point in space? The answer is no, for two reasons: In the final stage, matter has completely collapsed, but the curiosities of space-time transformability in general relativity do not make this happen at a point in space. Instead, one could better describe it as a “point in time,” as will be discussed below, but even this does not reflect all properties. Moreover, one cannot see this point in time at all from outside a black hole; the singularity of matter collapse is shrouded by a horizon far away from it. Only the properties of this horizon and its surroundings reveal the existence of a black hole from outside. No direct insight by observations yet exists; but in about ten years the satellite observatory Constellation-X is scheduled to explore the space-time neighborhood of a horizon around a massive black hole by recording emitted X-rays.
All these properties were already contained in the solution found by Karl Schwarzschild for nonrotating black holes in 1916—just one year after the first publications of general relativity. However, the singularity, as well as the existence of a horizon, were long misunderstood, fully decoded only in the 1960s with progress in the geometric understanding of general space-time properties. Einstein himself never believed in the reality of singularities threatening his theory, from which it must, if it is taken as fundamental, perish. Instead, Einstein, like most physicists of his time, thought that a singularity arises only mathematically through the assumed strong symmetry, but that it would disappear in more realistic solutions. (However, Einstein repeatedly emphasized that he did not consider general relativity to be fundamental, but instead expected necessary extensions arising from quantum theory and the atomic form of matter.)
In an exactly spherical matter distribution, as described by Schwarzschild’s solution, collapse can happen only centrally toward a single point. There, all matter will gather and quickly lead to infinitely high densities. One could assume deviations from the exact spherical symmetry of the initial star, as always occurs in realistic cases, to result in a collapse into a more extended region; the end product might be highly concentrated but nonsingular. But this is not the case. Singularities, as analyzed in what follows, always occur, and they must be considered a real threat to the theory of general relativity.2
Some of the geometric techniques used for an analysis of such bizarre objects, where the intense curvature distorts not only space-time but also our imagination, have an intuitive basis. They go back to a construction introduced by Roger Penrose called conformal completion of space-time. “Completion” here means that all of space-time, extending all the way to infinity, is mapped to a much tidier finite region. “Conformal” means that the mapping, while squeezing space-time into that finite region, does not deform it too strongly and preserves geometric shapes, especially the size of angles. This is particularly important because angles in space-time, as already seen, are to be understood not just spatially but also in a spatial-temporal sense, as velocities. Velocities are respected, too—in particular, that of light. And the speed of light as the absolute top speed in space-time traffic has a special meaning; this important physical law is thus respected by the conformal mapping.
In the case of Schwarzschild’s rotationally symmetrical solution, one fortunately need consider just two of the four space-time dimensions: the radius and time. Rotational symmetry means that properties of space-time, like those of a perfect sphere, do not depend on the orientation