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

96 P. T. Chruścielr = constant < 2Mr = 2Mi +r = infinityt = constanti 0IIIIISingularity (r = 0)r = 2Mi +It = constantr = infinityi0r = infinityr = constant > 2Mi −r = 2MIVi −Singularity (r = 0)t = constantr = infinityr = constant > 2Mr = 2Mr = constant < 2MFig. 1. The Carter-Penrose diagram for the Kruskal-Szekeres space-time with mass M.There are actually two asymptotically flat regions, with corresponding event horizonsdefined with respect to the second region. Each point in this diagram represents a twodimensionalsphere, and coordinates are chosen so that light-cones have slopes plus minusone.dard maximal analytic extensions can be found in Ref. 67. The isometricembedding, into six-dimensional Euclidean space, of the t = 0 slice in a(5 + 1)–dimensional Tangherlini solution is visualised in Figure 2.One of the features of the metric (1) is its stationarity, with Killingvector field X = ∂ t . A Killing field, by definition, is a vector field the localflow of which preserves the metric. A space–time is called stationary if thereexists a Killing vector field X which approaches ∂ t in the asymptotically flatregion (where r goes to ∞, see below for precise definitions) and generatesa one parameter groups of isometries. A space–time is called static if it isstationary and if the stationary Killing vector X is hypersurface-orthogonal,i.e. X ♭ ∧ dX ♭ =0,whereX ♭ = X µ dx µ = g µν X ν dx µ .A space–time is called axisymmetric if there exists a Killing vector field Y ,which generates a one parameter group of isometries, and which behaves likea rotation: this property is captured by requiring that all orbits 2π periodic,and that the set {Y =0}, called the axis of rotation, is non-empty. Killingvector fields which are a non-trivial linear combination of a time translationand of a rotation in the asymptotically flat region are called stationaryrotating,orhelical.Note that those definitions require completeness of orbitsof all Killing vector fields (this means that the equation ẋ = X has a global