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Exact Solutions of the Einstein Equations These Notes Are Not a Substitute in Any Manner for Class Lectures

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Exact Solutions of the Einstein Equations These Notes Are Not a Substitute in Any Manner for Class Lectures Notes from phz 6607, Special and General Relativity University of Florida, Fall 2004, Detweiler Exact Solutions of the Einstein Equations These notes are not a substitute in any manner for class lectures. Please let me know if you ¯nd errors. I. MINKOWSKI SPACE The metric of Minkowski space is a b 2 2 2 2 2 2 2 ¡ 2 2 2¢ gabdx dx = ¡dt + dx + dy + dz = ¡dt + dr + r dθ + sin θdÁ (1) where x = r sin θ cos Á y = r sin θ sin Á z = r cos θ; (2) t and r have ranges ¡1 < t < 1 and r ¸ 0: (3) This same geometry looks rather di®erent in \double null" coordinates, u = t ¡ r v = t + r (4) or 1 r = (v ¡ u) 2 1 t = (v + u); (5) 2 where the u and v have ranges ¡1 < u + v < 1 and v ¡ u > 0 (6) ab ab and are called \null" coordinates because g rau rbu = 0 = g rav rbv. A sketch of Minkowski space in double null coordinates is in Fig. 2. In these double null coordinates 1 ¡ ¢ g dxadxb = ¡dudv + (v ¡ u)2 dθ2 + sin2 θdÁ2 : (7) ab 4 Note that even though this metric might look odd, regularity is determined by the invert- ibility of the metric, and this metric is perfectly regular everywhere except when v ¡ u = 0, which corresponds to r = 0. 1 Minkowskii space ∞ t v = t + r u = 0 r = 0 t = 0 r v = 0 u = t - r −∞ FIG. 1: Minkowski space in double null coordinates. t and r axes are vertical and horizontal, respectively, with ranges ¡1 < t < 1 and 0 · r < 1. The null coordinate axes are on 45± lines and are bounded by ¡1 < u < 1 and ¡1 < v < 1 and the additional constraint that v ¡ u > 0 which implies that r > 0. Note that below the u axis v < 0, and below the v axis u is negative. For reasons which might become clear later, it is useful to de¯ne coordinates p and q from tan p = u tan q = v (8) with ranges ¼ ¼ ¼ ¼ ¡ < p < ; ¡ < q < and q ¸ p (9) 2 2 2 2 so that v ¡ u = tan q ¡ tan p = (sin q cos p ¡ sin p cos q)= cos q cos p = sin(q ¡ p) sec p sec q: (10) 2 Also, sec2p dp = du and sec2q dq = dv: (11) Now in p; q coordinates Minkowski space is 1 £ ¡ ¢¤ g dxadxb = sec2p sec2q ¡4dp dq + sin2(q ¡ p) dθ2 + sin2 θdÁ2 (12) ab 4 Now, the part of the metric in square brackets is similar to the metric for Minkowski space in Eq. (7). So de¯ne a new tensor a b 2 ¡ 2 2 2¢ g¹abdx dx = ¡4dp dq + sin (q ¡ p) dθ + sin θdÁ (13) which is interesting in its own right. The two metrics gab andg ¹ab are an example of \con- formally" related metrics where 2 g¹ab = ­ gab (14) for some scalar ¯eld ­: De¯ne new coordinates T and R by T = q + p and R = q ¡ p (15) and a b 2 2 2 ¡ 2 2 2¢ g¹abdx dx = ¡dT + dR + sin R dθ + sin θdÁ (16) When the coordinates T and R have ranges ¡1 < T < 1 and 0 < R < ¼ (17) g¹ab is the metric of the \Einstein static universe," a homogeneous, isotropic, time indepen- dent solution of the Einstein equations, with a cosmological constant and perfect fluid. The three-volume of each T = constant surface, is a three-sphere. For the Minkowski space that we started with the range of the coordinates is ¡¼ < T < ¼ and 0 < R < ¼: (18) Thus a simple, ¯nite picture of the entire Minkowski space can be drawn on the blackboard using this conformal transformation. A summary of the coordinate relationships is sin T t = ; cos R + cos T sin R r = : (19) cos T + cos R A sketch of the geometry is in Fig. 2. What is notable is that the radial null geodesics of both Minkowski space and the confor- ± mally relatedg ¹ab geometry move along 45 lines. This is the ¯rst example of what is loosely referred to as a \Penrose diagram." Particular parts of the \embedding of Minkowski space" within an extended Minkowski space are labeled in the diagram. Speci¯cally, ² i+, future timelike in¯nity, is \located at" t = 1 for any ¯xed value of r 3 T = π i+ t = ∞ Minkowskii space I+ t + r = const. = ∞ r=const. t = 0 r = 0 i0 r = ∞ R = π t=const. I- t - r = const. = −∞ i- t = −∞ T = −π FIG. 2: Minkowski space in a Penrose diagram. T and R axes are vertical and horizontal, respec- tively, with ranges ¡¼ < T < ¼ and 0 · R < ¼. Null geodesics are at 45± from the axes. Note that the bisector of the angle at the intersection of a t = constant and r = constant line are also at 45± from the axes. 4 ² i¡, past timelike in¯nity, is \located at" t = ¡1 for any ¯xed value of r ² i0, spatial in¯nity, is \located at" r = 1 for any ¯xed value of t ² I+, pronounced \scrI-plus" or future null in¯nity, is \located at" r + t = 1 for any ¯xed value of r ¡ t. This is where all null geodesic go. ² I+, pronounced \scrI-minus" or past null in¯nity, is \located at" r ¡ t = 1 for any ¯xed value of r + t. This is where all null geodesic come from. The understanding of Penrose diagrams is, perhaps, the single most important tool for developing physical intuition in general relativity. A. An aside on conformal transformations The two metrics gab andg ¹ab which are related by 2 g¹ab = ­ gab (20) for some scalar ¯eld ­ are said to be \conformally equivalent" or \conformally related." The Christo®el symbols of these two metrics are related by ¡ ¢ ¹a a ¡1 a a ad ¡bc = ¡bc + ­ ±b @c­ + ±c @b­ ¡ gbcg @d­ : (21) The relationship between the Riemann tensors, Ricci tensors and scalar curvatures for two conformally related metrics may be derived in a straightforward, if tedious, manner from Eq. (21). One result is that ¹ ¡2 ¡3 cd ¡4 cd R = ­ R ¡ 2(n ¡ 1)­ g rcrd­ ¡ (n ¡ 1)(n ¡ 4)­ g rc­ rd­ (22) for an n dimensional manifold; see Hawking and Ellis p 42. A useful and interesting feature of conformally related metrics on the same manifold are that null geodesics for one metric are also null geodesics for the other metric. This is easy a to see: ¯rst, a null vector for gab is trivially a null vector ofg ¹ab. And second, let k be the tangent vector to a null geodesic of gab, then ¡ ¢ b ¹ a b a b ¡1 a a ad c k rbk = k rbk + k ­ ±b @c­ + ±c @b­ ¡ gbcg @d­ k b a a b a b = k rbk + 2k k @b ln ­ = 2k k @b ln ­ (23) where the ¯rst equality follows from the relationship between the Christo®el symbols, the a a second follows from the fact that k is null, and the third from k being a geodesic of gab. b ¹ a a b a However, k rbk 2k k @b ln ­ implies that k is, indeed tangent to a null geodesic, and the fact that the right hand side here is not zero only implies that the parameter along the geodesic is not an a±ne parameter. Note: Only null geodesics are preserved under a conformal transformation|all other geodesics are not necessarily invariant under a conformal transformation. Note: To say that a metric is conformally flat is to say nothing directly about the actual curvature of the metric. For example, in the next subsection it is proven that all two dimensional metrics are conformally flat. In particular this implies that the usual surface of 5 a two-sphere is conformally flat; however, it is quite clear that the two-sphere has plenty of curvature. Note: It is easy to see that the angle between two vectors, which are not null, is preserved under a conformal transformation from a b a b A B gab A B g¹ab a b 1=2 a b 1=2 = a b 1=2 a b 1=2 ; (24) (A A gab) (B B gab) (A A g¹ab) (B B g¹ab) and also that a pair of orthogonal vectors remain orthogonal under a conformal transforma- tion. B. All two dimensional geometries are conformally flat For any metric on a two dimensional manifold, a particular coordinate system exists where the metric is conformally flat, i.e. the metric is the product of a scalar ¯eld (the conformal factor  below), with a flat metric. The components of the metric in one coordinate system (x; y) is related to metric in a second coordinate system (Á; Ã) by a b 2 2 gabdx dx = gxxdx + 2gxydxdy + gyydy = Â(dÃ2 § dÁ2) (25) where the transformation properties of the components of a tensor imply that @XA @XB gAB = gab : (26) @xa @xb Thus, we look for functions Ã(x; y) and Á(x; y) such that µ ¶ @à @Á @à @Á @à @Á @à @Á gÃÁ = gxx + gxy + + gyy = 0; (27) @x @x @x @y @y @x @y @y and "µ ¶ µ ¶ # · ¸ @à 2 @Á 2 @à @à @Á @Á gÃà ¨ gÁÁ = gxx ¨ + 2gxy ¨ @x @x @x @y @y @x "µ ¶ µ ¶ # @à 2 @Á 2 + gyy ¨ = 0: (28) @y @y While being tedious, direct substitution shows that Eq. (27) is satis¯ed if µ ¶ @Á @à @à = · gxy + gyy (29) @x @x @y and µ ¶ @Á @à @à = ¡· gxx + gxy (30) @y @x @y 6 for any arbitrary function ·: µ ¶ @à @Á @à @Á @à @Á @à @Á gÃÁ = gxx + gxy + + gyy @x @x @x @y @y @x @y @y µ ¶ µ ¶ @à @à @Á @à @à @Á = gxx + gxy + gxy + gyy @x @y @x @x @y @y µ ¶ µ ¶ @à @à @à @à = · gxx + gxy gxy + gyy @x @y @x @y µ ¶ µ ¶ @à @à @à @à ¡ · gxy + gyy gxx + gxy @x @y @x @y = 0 (31) Substitution of Eqs.
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