Ivancevic_Applied-Diff-Geom

970 Applied Differential Geometry: A Modern Introductionbe a region of space–time from which it is not possible to escape to infinity.This region is said to be a black hole. Its boundary is called the eventhorizon and it is a null surface formed by the light rays that just fail to getaway to infinity. As we saw in the last subsection, the area A of a crosssection of the event horizon can never decrease, at least in the classicaltheory. This, and perturbation calculations of spherical collapse, suggestthat black holes will settle down to a stationary state.Recall that the Schwarzschild metric form, given byds 2 = −(1 − 2M r )dt2 + (1 − 2M r )−1 dr 2 + r 2 (dθ 2 + sin 2 θdφ 2 ),represents the gravitational field that a black hole would settle down to ifit were non rotating. In the usual r and t coordinates there is an apparentsingularity at the Schwarzschild radius r = 2M. However, this is justcaused by a bad choice of coordinates. One can choose other coordinatesin which the metric is regular there.Now, if one performs the Wick rotation, t = iτ, one gets a positivedefinite metric, usually called Euclidean even though they may be curved.In the Euclidean–Schwarzschild metric( ) 2 ( ) dτ rds 2 = x 2 2 2+4M 4M 2 dx 2 + r 2 (dθ 2 + sin 2 θdφ 2 )there is again an apparent singularity at r = 2M. However, one can definea new radial coordinate x to be 4M(1 − 2Mr −1 ) 1 2 .The metric in the x − τ plane then becomes like the origin of polarcoordinates if one identifies the coordinate τ with period 8πM. Similarly,other Euclidean black hole metrics will have apparent singularities on theirhorizons which can be removed by identifying the imaginary time coordinatewith period 2π κ .To see the significance of having imaginary time identified with someperiod β, let us consider the amplitude to go from some field configurationφ 1 on the surface t 1 to a configuration φ 2 on the surface t 2 . This will begiven by the matrix element of e iH(t2−t1) . However, one can also representthis amplitude as a path integral over all fields φ between t 1 and t 2 whichagree with the given fields φ 1 and φ 2 on the two surfaces,∫< φ 2 , t 2 |φ 1 , t 1 >=< φ 2 | exp(−iH(t 2 − t 1 ))|φ 1 >= D[φ] exp(iA[φ]).One now chooses the time separation (t 2 − t 1 ) to be pure imaginary andequal to β. One also puts the initial field φ 1 equal to the final field φ 2 and