Conformal Compactification It is often desirable to draw a diagram that shows the causal structure of Lorentzian manifolds. In 2d, or when there is enough symmetry that one can focus on a 2d part of the metric (as in the case of Schwarzschild) there is a way to do this. Let's start from the example of 4d Minkowski metric, which in spherical reads ds2 = −dt2 + dr2 + r2dΩ2; −∞ < t < 1; 0 < r < 1: (1) The idea is to suppress the spherical part and focus on the t − r part of the metric. Then we use the same trick of transforming to null coordinates u = t − r; v = t + r; (2) with the range −∞ < u; v < 1 subject to the constraint v ≥ u: (3) To compactify the u − v part of the metric, we define u¯ = tanh u; v¯ = tanh v (4) which range over −1 < u;¯ v¯ < 1 subject to the constraint (3) which impliesv ¯ > u¯. The metric now reads dud¯ v¯ ds2 = − + r2dΩ2: (5) (1 − u¯2)(1 − v¯2) As long as we are interested in radial motion (i.e. motion inu; ¯ v¯) the causal structure is the same as the 2d Minkowski space ds2 = −dud¯ v¯ except for the constraint on the range of the variables. So we obtain 1 Every point of this diagram should be thought of as a 2-sphere. At every point the lightcone (in radial direction) is made ofu ¯ = constant andv ¯ = constant lines that pass through that point. The interesting parts of this diagram are u = v, corresponding to r = 0, u = v = ±1 corre- sponding to future and past timelike infinity (i±), v = 1, which is future null infinity I +, and u = −1, which is past null infinity I −. 1. (a) Mark the point corresponding to r ! 1 and t = constant. (b) Draw the worldline of an observer at constant r. (c) Draw the full trajectory of a null ray that passes through the origin. Schwarzschild. Now we apply the same logic to the U −V part of the extended Schwarzschild metric u¯ = tanh U; v¯ = tanh V; (6) with the range −1 < u;¯ v¯ < 1, but subject to the constraint 2 UV < 4rg; (7) to avoid the r = 0 singularity. The resulting Penrose diagram is The significance of Penrose diagrams is to illustrate the causal structure. The timelike and spacelike curves can be deformed by change of variablesu ¯ ! f(¯u),v ¯ ! g(¯v), as long as f and g are monotonic functions in the range (−1; 1). Hence in the above diagram the r = 0 singularities, which are 2 spacelike hypersurfaces, are drawn horizontally even though from (6) we would obtain curves that bulge out. 2. Alice lives in region I and Bob in region III of the extended Schwarzschild geometry. How can they meet? How big should be rg for this to be a safe trip? The Penrose diagram of a black hole formed from the collapse of spherically symmetric matter looks like 5 3. The surface temperature of a neutron star of radius 5rg is 6 × 10 K. As the star starts collapsing into a black hole, the surface falls freely. What is the temperature measured by a distant observer as a function of time t? 3.