General Relativity Phys4501/6 Second term summary, 2013 Gabor Kunstatter, University of Winnipeg Contents I. Einstein's Equations 3 A. Heuristic Derivation 3 B. Lagrangian Formulation of General Relativity 5 C. Variational Derivation of Einstein's Equations 6 D. Properties of Einstein's Equations 7 II. Energy Conditions 7 III. Spherical Symmetry 9 A. Derivation of solution and general properties 9 B. Geodesics 9 C. Experimental Tests 9 D. Black Holes and Conformal Diagrams 10 E. Charged Black Holes 10 IV. Horizons and Trapped Surfaces 12 A. Stationary vs. Static 12 B. Event Horizons in General 13 C. Trapped Surfaces 14 D. Killing Horizons 15 E. Surface Gravity 16 V. Theorems and Conjectures 17 A. Singularity Theorem 17 B. Cosmic Censorship Hypothesis 18 C. Hawking's Area Theorem 18 D. No Hair Theorem 18 1 VI. Black Hole Thermodynamics 18 VII. Cosmology: Theory 20 A. Symmetry Considerations 20 B. FRW Metric 21 C. Hubble Parameter 22 D. Matter Content 22 E. The Friedmann Equation 23 F. Time evolution of the scale factor and density 25 VIII. Cosmology: Observations 26 A. Redshifts 26 B. Proper Distance 28 C. Luminosity Distance 29 D. Derivation of Luminosity vs Redshift Relation 29 2 I. EINSTEIN'S EQUATIONS A. Heuristic Derivation • So far we have learned how to derive equations of motion for matter fields from a variational principle keeping the geometry fixed (i.e. didn't vary metric). • We now need to understand better the equations of motion for the metric of spacetime, i.e. the geometry. • We know that the energy momentum tensor must be conserved, and that in the New- tonian limit h00 plays the role of the gravitational potential φ which in Newtonian physics is determined by the matter distribution by Poisson equations: r~ 2φ = 4πGρ (1) The equations of motion must therefore be second order in the metric components, tensorial, and guarantee energy-momentum conservation. • Eq. (138) of the February summary gives us a hint, since we can get the vacuum ~ 2 k Poisson equation r φ = 0 by taking the trace R 00k = 0. This is the Newtonian, non- relativistic limit of R00 = 0, so one would be tempted to postulate, as did Einstein, the fully covariant version of the above in the presence of matter: Rµν = κTµν (2) But we know this can't be right, because the energy momentum tensor must be covari- µν 1 µν antly conserved, whereas the LHS is not: rµR = 2 g rµR by the Bianchi identities and does not vanish in general. In fact, (2) implies R = κT so that by using the µν Bianchi identities one sees that rµT = 0 iff rµT = 0, which is way too restrictive. • A better guess for Einstein's equations, which automatically yields energy-momentum conservation, is: 1 G = R − g R = κT (3) µν µν 2 µν µν • In class we used dimensional arguments to show that κ must be proportional to G=c4: 3 2 T00 has dimensions of [ρ] = E=L , whereas G00 has dimensions 1=L . Thus [κ] = L=E. 3 Newton's constant has units [G] = L2=mass2 ∗ (mass ∗ L=T 2) = L3=(Mass ∗ T 2). Since [E] = [mass][c2], this shows that [G=c4] = L3=(Mc2 ∗ T 2) ∗ (T 2=L2) = L=E as required. • We can check more carefully the Newtonian limit and thereby determine whether it works and if so what the constant κ must be: { First, using Eq.(138) in the previous section we see that in the Newtonian limit: 1 1 R ∼ − r~ 2h = r~ 2φ (4) 00 2 00 c2 { Tracing (3) we find 00 R = −κT ∼ −κg T00 = κT00 (5) where we have assumed that in the non-relativistic limit the T00 = ρ is larger than the other components of the stress energy tensor. { In this case Einstein's equation implies that: 1 1 R = κT (1 + g ) = κT 00 00 2 00 2 00 1 1 ! r~ 2φ = κc2ρ (6) c2 2 2 where we have used T00 = ρc , recalling that the energy-momentum tensor is the flow of four-momentum through a surface, i.e. in this case energy, not mass. { which matches Eq.(124) of February summary providing we choose: 8πG κ = (7) c4 • Once we impose general covariance, don't allow higher derivatives of the metric in the equations of motion and require correspondence with Newtonian physics the theory is uniquely determined with no new, tunable parameters!!! As we will see it therefore makes unambiguous testable predictions for corrections to Newtonian physics, such as the perihelion shift of Mercury and light bending around the sun. • But first we will see how to derive the field equations from a variational principle. Given a coordinate invariant action, the energy momentum tensor can be derived (doesn't have to be guessed), and the connection between the energy momentum conservation and symmetry (general covariance) becomes manifest. 4 B. Lagrangian Formulation of General Relativity We now show how to derive Einstein's equations from a variational principle. As always one starts with an action. In this case we require the action to have certain properties: 1. It must be invariant under coordinate transformations 2. It must, at this stage be at most second order in derivatives of the metric. Actually the condition is that the equations of motion resulting from varying this action have no more than second derivatives of the metric.In four spacetime dimensions these two statements are equivalent, but not in higher dimensions. Note also that higher order derivatives are possible in generalized theories of gravity, but we assume that they will not play a significant role until we get to much shorter length scales than we wish to consider at this stage. 3. It should separate into two parts: (a) a matter action, which describes the dynamics of any non-gravitational fields. This part is distinguished by the fact that it cannot have more than first deriva- tives of the metric. Otherwise the equations of motion for the matter field could not be made to be indistinguishable from those of general relativity by going to a locally inertial frame (in which the first derivatives of the metric vanish. This would violate the strong equivalence principle. In general the matter action is constructed by taking the special relativity version and replacing all occurences of the Minkowski metric by gµν and all partial derivatives by covariant derivatives. (b) the gravitational action: assuming the previous conditions this is almost uniquely determined (in four dimensions) proportional the so-called Einstein-Hilbert ac- tion: Z 4 p SEH = d x −gR(g) (8) where R(g) is the Ricci scalar constructed from the metric gµν Thus we start from the total action describing both geometry and matter: 1 I = S + S (9) 2κ EH M 5 where SM denotes a generic matter action constructed as per the rules given above. For example, for a scalar field, the contribution to SM is: Z p 1 1 S = d4x −g − r φgµνr φ − m2φ2 − V (φ) (10) φ 2 µ ν 2 Note that for a scalar field the partial and covariant derivatives are identical. V (φ) is a potential energy term associated with self-interactions of the field. The signs are chosen to make sure that the kinetic energy term is positive, in keeping with the mechanics form of the lagrangian: L = KE − PE. The other matter action we have dealt with, that of the electromagnetic field, is: 1 Z p S = − d4x −gF gµνgαβF (11) EM 4 µα νβ where Fµν is the electromagnetic field strength defined in terms of an electromagnetic potential Aµ by: Fµν = @µAν − @νAµ = rµAν − rνAµ (12) Note that Fµν is by definition anti-symmetric (a two-form) and thus the Christoffel terms on the far right cancel so that it can be equivalently defined either by covariant or partial derivatives. C. Variational Derivation of Einstein's Equations • Definition of stress energy tensor: 2 δS T ≡ −p M (13) µν −g δgµν • Action: c4 Z p S = d4x −gR(g) + S (14) 16πG M • Einstein's Equations: 8πG G = T (15) µν c4 µν • Know how to vary metric in action, for which the following relations are useful: µα µα { g δgµβ = −gµβδg p 1 p µν 1 p µν { δ −g = 2 −gg δgµν = − 2 −ggµνδg 6 α 1 αβ { δfµ νg = 2 g (rµδgβν + rνδgβµ − rβδgµν) R 4 p µ R p µ R p µ { Σ d x jgjrµV = Σ @µ jgjV = @Σ nµ jγjV D. Properties of Einstein's Equations • Degrees of Freedom: qualitative derivation of n(n + 1)=2 − 2n = n(n − 3)=2 • Self-coupled: gravitational binding energy contributions to inertial mass • Attractive for \ordinary matter" • Quantum Gravity: know constants that can be constructed from ~; G; C: r ~G c lpl = 3 = (16) c tpl ~c 2 Epl = = mplc (17) lpl 2 • Cosmological Constant Λ = 1=lΛ: know the problem, its theoretical quantum source and theoretical and experimental orders of magnitudes. II. ENERGY CONDITIONS These are physically motivated constraints on the general form of the stress energy tensor that are useful for proving various theorems and/or motivating conjectures. They are all a bit different, but the basic idea is to define what one considers \reasonable" matter/energy to look like in general relativity. Any or all of these can be violated once quantum mechanics is taken into account. • Strong Energy Condition (SEC): 1 T tµtν ≥ T λ tσt (18) µν 2 λ σ for all timelike tµ. This implies that dθ/dτ ≤ 0 for all timelike geodesics, where θ is their expansion. Thus it implies that strong energy condition restricts to mat- ter/energy for which gravity is purely attractive.