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

126 R. H. Price2. Stationary Black Hole SpacetimesFor many purposes special relativity is best understood with theMinkowskian 4-dimensional metric 1,2 . Coordinates x, y, z, t are laid outin space and time in a way that can very precisely be dictated, and thegeometry of spacetime is described by the “distance” formula, or “metric,”a spacetime equivalent of the Pythagorean formula for differentialdisplacements,ds 2 = −c 2 dt 2 + dx 2 + dy 2 + dz 2 . (1)It is crucial to understand that a coordinate transformation can lead to avery different appearance for the metric. In the case of Minkowski spacetime(gravity-free spacetime) there are preferred types of coordinates, theMinkowski coordinates, in which the metric takes the simple form in Eq. (1).Tranformations can be carried out from one Minkowski system to another,but can also be made to a nonpreferred system in which the metric takeson a very different appearance. As an example of this we could take therelatively simple transformationt = T x = X(T/T ) 1/2 y = Y z = Z , (2)where T is a positive constant with the dimensionality of time (and isintroduced to maintain dimensional consistency). In these new T, X, Y, Zcoordinates the metric becomesds 2 = −[1 − X24c 2 T T]c 2 dT 2 + X T dT dX + 4 T T dX2 + dY 2 + dZ 2 . (3)This formula has a completely different character from that in Eq. (1). Inparticular the formula suggests that the geometry it describes is dependenton time T . There is another, more subtle ugliness in Eq. (3): the dT dXcross term. This term means that motion in the +X direction is physicallydifferent from motion in the −X direction.We feel intuitively, of course, that the simple spacetime of special relativitydoes not change in time, that it is stationary (unchanging in time)and isotropic (so that +x is the same as −x), but Eq. (3) shows that wemust be careful about how we state this. The correct way is: There existsa coordinate system in which the metric is stationary and isotropic.As long as we are being careful, it is good to be careful about thedifference between a “time” coordinate and a “space” coordinate. If weconsider a slice of spacetime with dT = 0, we getds 2 = 4 T T dX2 + dY 2 + dZ 2 . (4)