Lecture XVI: Symmetrical spacetimes

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1 Lecture XVI: Symmetrical spacetimes Christopher M. Hirata Caltech M/C , Pasaena CA 91125, USA (Date: January 4, 2012) I. OVERVIEW Our principal concern this term will be symmetrical solutions of Einstein s equations that are of astrophysical relevance. These are relativistic stars, white warfs an neutron stars, which we will approximate as spherical; black holes, the ultimate enpoint of evolution for some massive stars in aition to being foun in the centers of galaxies; an the Universe itself. Before we stuy these solutions, it is worth a igression to examine the role of symmetry in GR. We will consier here the case of spacetimes with continuous symmetries. (We won t investigate iscrete symmetries in any formal way in this class, although we will use some of their intuitive properties later.) We will nee to o this both to efine a spherically symmetric object, an later on to stuy its perturbations. The (optional) reaing for this lecture is: MTW II. SOME PRELIMINARIES Suppose we want to consier a system that is time-inepenent (or time translation invariant or stationary). In Newtonian physics or even in SR, it is clear what this means: one can choose a reference frame in which all quantities epen only on x i an not on t, e.g. the ensity is ρ(x i, t) = ρ(x i ). But in GR the coorinates are free for us to choose, so we cannot simply transfer over this efinition. In linearize GR, we circumvente the problem by making reference to the Minkowski backgroun aroun which we perturb; but in full GR we must begin with a notion of symmetry that is not tie to a particular coorinate system. A. Infinitesimal coorinate transformations an Lie erivatives In general, a symmetry of any object is a mapping of the object to itself that preserves certain relevant properties (in our case, the metric). So let s consier a mapping W(ǫ) : M M, where ǫ parameterizes the continuous symmetry. We will force this mapping to be ifferentiable (with respect to both ǫ an P) as many times as neee, an we will assume that at zero parameter the mapping is simply the ientity: [W(0)](P) = P. (1) The prototype of this is the time translation invariance of Minkowski spacetime, where translating by a time ǫ takes a point (t, x i ) to [W(ǫ)](t, x i ) = (t + ǫ, x i ). (2) A continuous symmetry is often escribe by its behavior for infinitesimal values of the parameter; thus we efine the vector fiel ξ(p) = ǫ [W(0)](P). (3) In our example of Minkowski space, the contravariant components are simply ξ µ = (1, 0, 0, 0). Electronic aress: chirata@tapir.caltech.eu

2 Any such mapping can be thought of as a change of coorinates: given the mapping W(ǫ), an an unprime coorinate system, we can efine the prime coorinates of any point P as the unprime coorinates of [W(ǫ)](P). The prime coorinate system epens on ǫ; in our example above, we have x i = x i an t = t + ǫ. (4) Now let s consier a tensor fiel S α1...αm β 1...β N. In the prime system, it now has components: M S α 1...α M β 1...β N (xµ ) = xα 1 x β1... xα α1 x x αm x β 1... xβn S α1...αm x β β 1...β N (x ν ). (5) N Note that the numerical value of any component here may in general epen on ǫ since the coorinate transformation epens on ǫ. But if W(ǫ) really escribes a symmetry an S is invariant uner that transformation, there shouln t be any such epenence. To make use of this fact for infinitesimal symmetries, we take the erivative of Eq. (5) with respect to ǫ at ǫ = 0: ǫ Sα 1...α M β 1...β N (xµ ) = ξ α 1,γS γα 2...α M β 1...β...,γS N ξα M α 1...α M 1 γ β 1...β N +ξ γ,β 1 S α 1...α M γβ 2...β N... + ξγ,β N S α 1...α M β 1...β N 1 γ + ξ γ S α 1...α M β 1...β N,γ. (6) Here we have ifferentiate each term in Eq. (5), an note that at ǫ = 0 all the Jacobians are Kronecker eltas: x α 1 / x α 1 = δ α 1 α 1, etc. The erivatives are simply x α 1 x α1 = ξ α 1 ǫ x α1,α 1 an ǫ = ξ α1,α. (7) 1 We have replace the ummy inices in Eq. (6) with γs. Note that there is no confusion here of comparing tensors in the prime or unprime system since at ǫ = 0 they coincie. The variation in Eq. (6) is so important that it gets a name: we call it the Lie erivative: x α 1 2 L ξ S α1...αm β 1...β N = ξ α1,γs γα2...αm β 1...β N... ξ αm,γs α1...αm 1γ β 1...β N +ξ γ,β 1 S α1...αm γβ 2...β N... + ξ γ,β N S α1...αm β 1...β N 1γ + ξ γ S α1...αm β 1...β N,γ. (8) It escribes how the tensor S changes uner an infinitesimal coorinate transformation ξ. The rules for computing it are straightforwar one takes the partial erivative an then appens a correction term for each inex but note that it contains the erivative of ξ. The geometry of spacetime is invariant uner a continuous symmetry with erivative ξ if an only if the Lie erivative of the metric is zero, In this case, we say that ξ is a Killing fiel. L ξ g = 0. (9) B. Properties of the Lie erivative Note that we have mae no use of the covariant erivative or metric in efining the Lie erivative. Inee, the Lie erivative makes perfect sense on a manifol with no metric structure. But let us suppose that we have a metric. Since L ξ S is a tensor, we may evaluate Eq. (8) in a local Lorentz frame, where all Christoffel symbols vanish, an conclue that L ξ S α1...αm β 1...β N = ξ α1 ;γs γα2...αm β 1...β N... ξ αm ;γs α1...αm 1γ β 1...β N +ξ γ ;β 1 S α1...αm γβ 2...β N... + ξ γ ;β N S α1...αm β 1...β N 1γ + ξ γ S α1...αm β 1...β N;γ. (10) [You can also prove by brute force expansions of Christoffel symbols that this reuces to Eq. (8).] A consequence is that the Lie erivative of a scalar is simply the irectional erivative, L ξ f = ξ f. (11)

3 3 The Lie erivative of a vector is often calle the commutator: The name arises because of the following easily verifie rule: L u v α = u β v α,β v β u α,β = [u, v] α. (12) L u L v S α1...αm β 1...β N L v L u S α1...αm β 1...β N = L [u,v] S α1...αm β 1...β N. (13) Clearly [u, v] = [v, u]. Finally one can show that the prouct rule hols: L ξ (A α βb γ δ) = (L ξ A α β)b γ δ + A α βl ξ B γ δ, (14) an that the Lie erivative commutes with contraction. WARNING: The Lie erivative oes not in general commute with the raising an lowering of inices, e.g. L ξ A µ is not the same vector as one obtains by raising the inex of L ξ A µ. Therefore it is necessary to specify whether the inices are up or own. The exception is when ξ is a Killing fiel. III. SYMMETRIES We may now escribe a spacetime as having a continuous symmetry if Eq. (9) hols. This is most often escribe by expaning Eq. (10) an noting that g µν;γ = 0; then we have Therefore ξ is a Killing fiel if an only if L ξ g µν = ξ γ ;µg γν + ξ γ ;νg µγ = ξ ν;µ + ξ µ;ν. (15) ξ (µ;ν) = 0. (16) It is easy to see that the Killing fiels form a vector space if ξ an η are Killing fiels, then aξ +bη is also a Killing fiel. This is no surprise: if e.g. t-translation is a symmetry, an x 1 -translation is a symmetry, then a iagonal translation in both space an time must also be a symmetry. We may therefore obtain a escription of all of the symmetries of a spacetime by fining a basis for the Killing fiels an exploring the properties of each of the basis Killing fiels. This is useful because all of the spacetimes we examine in practice have a finite number of linearly inepenent Killing fiels. A. Example: Minkowski spacetime As a first example, let us classify all of the continuous symmetries of Minkowski spacetime. We work in stanar Minkowski coorinates x µ in which the Christoffel symbols vanish. Then the requirement of a Killing fiel is Therefore the tensor f µν = ξ µ,ν is antisymmetric. Moreover, we have ξ µ,ν + ξ ν,µ = 0. (17) f µν,σ = ξ µ,νσ = ξ µ,σν = f µσ,ν = f σµ,ν. (18) Thus f µν,σ flips sign if we cyclically permute the 3 inices. Repeating the above argument 3 times gives f µν,σ = f σµ,ν = f νσ,µ = f µν,σ, (19) so we conclue that f µν,σ = 0 an f µν is a constant. We can then integrate to fin ξ µ = f µν x ν + b µ, (20) where b is any 4-vector. Their components are all affine functions of x µ. Equation (20) then implies that the Killing fiels form a 10-imensional vector space (4 components of b µ an 6 of f µν ). These can be written in the following basis: The 4 translations, which are simply the basis vectors: π α = e α.

4 The 3 rotations, (J 1 ) µ = (0, 0, x 3, x 2 ), (J 2 ) µ = (0, x 3, 0, x 1 ), an (J 3 ) µ = (0, x 2, x 1, 0). [Note that the subscripts on J i inicate which Killing vector, not which component.] The 3 boosts, (K 1 ) µ = (x 1, x 0, 0, 0), etc. These are in fact the familiar transformations of SR between inertial frames. Most of the symmetries we will stuy in GR are appropriate generalizations of these. Their esignation as a particular type of symmetry will epen on other mathematical properties (specifically their commutators). You know from introuctory physics that rotations o not commute: an infinitesimal rotation aroun the 1-axis followe by an infinitesimal rotation aroun the 2-axis is not the same as rotating aroun 2 an then 1. Mathematically, this is incorporate in the commutator, I ll prove just the 1-component of this equation, Generally we have [J 1, J 2 ] = J 3. (21) [J 1, J 2 ] 1 = (J 1 ) β (J 2 ) 1,β (J 2 ) β (J 1 ) 1,β = (J 1 ) 3 (J 2 ) 1,3 = x2 (x 3 ),3 = x 2. (22) [J i, J j ] = ǫ ijk J k. (23) This relation is probably familiar from quantum mechanics, asie from the factor of i that istinguishes the infinitesimal rotation from the quantum angular momentum operator. It is possible to establish commutation relations of this sort for all of the Minkowski Killing fiels; for example an [J i, π 0 ] = 0 (24) [J i, π j ] = ǫ ijk π k. (25) 4 B. Commutation relations The existence of commutation relations is generic to continuous symmetries (in GR an otherwise). Given any two Killing vectors ξ an η, it is easily seen that their commutator must be a Killing fiel, since Furthermore, the commutator satisfies a Jacobi ientity, L [ξ,η] g = L ξ L η g L η L ξ g = 0 0 = 0. (26) [ξ, [η, ζ]] + [η, [ζ, ξ]] + [ζ, [ξ, η]] = 0 (27) (each combination such as L ξ L η L ζ appears twice with opposite signs). A vector space that satisfies Eq. (27) is sai to be a Lie algebra. Clearly a Lie algebra is efine by the commutators of the basis fiels. The familiar angular momentum operators of quantum mechanics satisfy a Lie algebra, given by Eq. (23) (again with the is locate in ifferent places). We are now reay to make a efinition: A spacetime is stationary if it possesses a Killing fiel π 0 that is timelike in some portion of spacetime. (Normally we will use this term to escribe nonrotating stars an black holes; in the latter case, π 0 becomes null on the horizon an is spacelike insie the hole, so we o not want to impose a restriction of π 0 being timelike everywhere.) A spacetime is spherically symmetric if it possesses 3 Killing fiels J 1, J 2, an J 3 satisfying Eq. (23). A spacetime is homogeneous an isotropic if it possesses 3 rotation Killing fiels J i an 3 translation Killing fiels π i satisfying Eq. (23) an (25). Normally situations with more Killing fiels are easier to analyze since they are more symmetrical. Thus our major efforts will be evote to spherical stars (4 Killing fiels) an cosmology (6 Killing fiels). The most ifficult solution this term is the Kerr black hole (2 Killing fiels).

5 5 C. Conservation laws In orinary mechanics, we know that a continuous symmetry is associate with a corresponing conservation law. The same is true in GR. If we take a Killing fiel ξ, an then trace a freely falling particle of momentum p along its trajectory, we fin Therefore p ξ is conserve along a trajectory. The examples from Minkowski spacetime are that Dp (p ξ) = λ λ ξ + p pξ = 0 + p α p β ξ β;α = 0. (28) The π α Killing fiels tell us that p π α = p e α = p α is conserve thus we have conservation of energy an 3-momentum. The J i Killing fiels tell us that e.g. L 1 p J 1 = p 2 x 3 + p 3 x 2 is conserve thus we have conservation of angular momentum. The K i Killing fiels tell us that Y i p K i = p 0 x i + p i x 0 = Ex i p i t is conserve thus the particle s position moves at a rate x i /t = p i /E. In less symmetrical spacetimes, subsets of these conservation laws are vali. For example, in stationary an spherically symmetric spacetimes (such as the exterior of a nonrotating spherical star) we have a conserve energy E = p ξ (where ξ is the time-translation Killing fiel) an 3 conserve angular momenta L i. D. Electroynamics In the case where there is also an electromagnetic fiel obeying the symmetry, it is possible to exten the conservation laws to a charge particle that is not following a geoesic. Specifically, we consier a fiel tensor F µν. Since this fiel satisfies the Maxwell equation F [µν,σ] = 0, it is possible at least locally to write it as the exterior erivative of a potential, F µν = A ν,µ A µ,ν = A ν;µ A µ;ν. (29) Note that A is not unique, as we may always transform A µ A µ + µ Λ for any scalar Λ with no effect on F µν. If a particle has charge e an accelerates only uner the influence of the electromagnetic force, so that Dp α /λ = ef αβ p β, then However λ (p ξ) = ef αβ p β ξ α + p p ξ = ef αβ p β ξ α. (30) λ (A ξ) = Aα p β ξ α;β + ξ α p β A α;β = A α p β ξ α ;β + ξ α p β A β;α ξ α p β F αβ = p β L ξ A β ξ α p β F αβ. (31) Therefore if the electromagnetic potential is also symmetrical uner the transformation generate by ξ i.e. if L ξ A β = 0 then the first term vanishes an we have an so [(p + ea) ξ] = 0 (32) λ (p + ea) ξ = constant. (33)