100 Years of Relativity Space-Time Structure: Einstein and Beyond ...

The Physical Basis of Black Hole Astrophysics 127The condition for this constant T slice to be spatial, and hence for T to bea time coordinate, is that the value of ds 2 be positive for any displacement{dX, dY, dZ}. But in Eq. (4) we see that this is the case only for T >0. Thus T is a time coordinate only for T > 0. (One might object thatnegative T is prohibited by the form of the transformation in Eq. (2).But here we should view Eq. (3) as a metric given to us for analysis. Theconnection to the Minkowski formula is the ultimate clarification of Eq. (3),but such connections are not always available and, if available, are usuallynot obvious.)The example in Eq. (3) has prepared us to understand the Schwarzchildgeometry which has the formulads 2 = −(1 − 2GM )rc 2 c 2 dt 2 +(1 − 2GMrc 2 ) −1dr 2 + r 2 dθ 2 + r 2 sin 2 θ dφ 2 .(5)Here G is the universal gravitational constant and M is a parameter withthe dimensions of mass. If the M parameter is set to zero, Eq. (5) takesthe form of the Minkowski metric expressed with spherical polar spatialcoordinates, a simple transformation from Eq. (1).For M > 0, it turns out that Eq. (5) cannot be transformed to theMinkowski metric, and therefore does not represent gravity-free spacetime.It does, however, represent a very fundamental solution in Einstein’s theory.That theory, general relativity, consists of partial differential equations thatmust be satisfied by the metric functions, the functions, such as 1−2GM/rc 2in Eq. (5), that appear in the metric. The partial differential equations, thefield equations of general relativity, connect these functions to the nongravitationalenergy and momentum content of spacetime. The Schwarzschildmetric is a solution of Einstein’s equation discovered almost 90 years ago 3 ,for vacuum, i.e. , for spacetime in which there is no matter, no energy andno fields, except for the gravitational field itself which is encoded in the geometry.In Einstein’s theory, furthermore, the Schwarzschild metric turnsout to be the unique spherically symmetric vacuum solution that is asymptoticallyflat. (It approaches the Minkowski metric as r → ∞.) It represents,therefore, the spacetime outside a spherically symmetric gravitating body.In that sense it plays the same role as Φ = 1/r in electromagnetic theory.This solution appears to be stationary. That is, there exist coordinates (thet, r, θ, φ coordinates) in which the metric is independent of the time coordinate.This is intuitively satisfying; like electromagnetic waves, gravitationalwaves are transverse, and like electromagnetic waves, gravitational wavescannot be spherically symmetric. There would then seem to be nothing that