The Einstein field equations

Transcription

1 The Einstein field equations Part I: the right-hand side Atle Hahn GFM, Universidade de Lisboa Lisbon, 21st January 2010 Contents: 1 Einstein field equations: overview 2 Special relativity: review 3 Classical and relativistic fluid dynamics 4 The stress energy tensor

2 References: Book by Wald: General Relativity Wikipedia Prerequisites: Special relativity (basic results) General relativity (vague idea) Differential Geometry (basic knowledge) OR Classical Fluid Dynamics (basic knowledge) Aims: Preparation for future lectures (Big bang, Schwarzschild solution, Gödel cosmos) (Hopefully useful) supplement for the study of Wald s book

3 1 Einstein field equations: overview Let us consider a fixed space time (M, g). We use abstract index notation : We write g ab for the type (2,0) tensor g g ab denotes the type (0,2) tensor given by b g abg bc = δa c (where δc a := δ ac Kronecker symbol) R d abc denotes the curvature tensor associated to (M, g) We set R ab := c Rc acb We set R := a,b R abg ab ( Ricci tensor ) ( scalar curvature ) Remark 1 Elementary reformulation for M = R 4 : Pseudo-Riemannian metric g ab on M = R 4 can be considered as a matrix (g ab ) 1 a,b 4 of smooth functions g ab : R 4 R such that for each x R 4 The matrix (g ab (x)) a,b is symmetric (g ab (x)) a,b has three (strictly) positive eigenvalues and one (strictly) negative eigenvalue (g ab ) 1 a,b 4 determined by the 10 functions g 11, g 22, g 33, g 44, g 12, g 13, g 14, g 23, g 24, g 34

4 (g ab ) 1 a,b 4 : Matrix of functions g ab : R 4 R given by (g ab (x)) ab = ( ) 1 (g ab (x)) a,b for each x R 4 Curvature tensor (Rabc d ) 1 a,b,c,d 4: family of functions Rabc d : R4 R given explicitly as Rabc(x) d := b Γ d ac(x) a Γ d bc(x) + ( Γ i ac (x)γ d ib(x) Γ i bc(x)γ d ia(x) ) i where Γ c ab(x) := 1 g cd (x) ( a g bd (x) + b g ad (x) d g ab (x) ) 2 d R ab, a, b 4, and R are functions on M = R 4 given by R ab (x) = c R c acb(x), R(x) = a,b R ab (x)g ab (x) for each x R 4.

5 Convention 1 Index set {1, 2, 3, 4} instead of {0, 1, 2, 3} Einstein sum convention: e.g. we write R ab g ac instead of a R ab g ac R a a instead of a R a a Normal rules for raising/lowering indices: e.g. we write R c b instead of R ab g ac = a v a v a instead of g ab v b v a = a,b R ab g ac g ab v b v a

6 1.1 Vacuum case Basic problem: For given M and Λ R ( cosmological constant ) find g ab such that R ab 1 2 Rg ab + Λg ab = 0 (1) is fulfilled. System of 10 non-linear PDEs of second order! Example 1 A Schwarzschild black hole solution g ab on M = (R 3 \{0}) R): Solution of Eq. (1) which is rotation-invariant and static. It is completely specified by a parameter m R + (the mass of black hole). Digression 1 If g ab fulfills Eq. (1) then, using R = R ab g ab and g ab g ab = δa a = 4 we obtain R 1 2R4 + Λ4 = 0 and therefore Thus R is a constant function and (M, g) is a Einstein manifold. R = 4Λ R ab = Λg ab Special case Λ = 0: R ab = 0, i.e. (M, g) is Ricci flat.

7 1.2 General case Real space-time contains matter & radiation. Most important cases: A perfect fluid is present ( we have 4-velocity field u a and a density field ρ) Electromagnetic radiation is present ( we have a 4-potential A a ) Now consider general situation where (matter or radiation) field Ψ is present, e.g. Ψ = (u a, ρ) or Ψ = A a. Assumption 1: Equations of motions for Ψ are known, i.e. for a known function F. Assumption 2: The tensor F (g, Ψ) = 0 T ab = T ab (g, Ψ) ( the stress energy tensor ) is known explicitly (see Sec. 4.2 for a definition) Example 2 Ψ is a Klein-Gordon field φ on (M, g), i.e. a field φ : M R with equation of motion F (g, φ) = ( a a m 2 )φ = 0, m R The associated stress energy tensor is (see Digression 5 below) ( a : Levi-Civita connection) T (g, φ) = a φ b φ 1 2 g ab( c φ c φ + m 2 φ 2 )

8 Basic problem: For given M and Λ R find (g, Φ) such that R ab 1 2 Rg ab + Λg ab = 8πG c 4 T ab (g, Φ) (2a) F (g, Φ) = 0 are fulfilled where c is the speed of light and G Newton s gravitational constant. (2b) Remark 2 Eqs. (2a), (2b) is considerably more difficult than Eq. (1). We must find g and Φ simultaneously! Convention 2 For most of the talk (exception: Sec. 2) we use physical units such that c = 1 and G = 1 Most interesting situations involve matter/radiation Friedmann-Robertson-Walker model for the big bang (perfect fluid) Schwarzschild solution for a normal star (perfect fluid) Gödel cosmos (perfect fluid) Black hole with electric charge (electromagnetic field)

9 2 Special relativity: review Let c > 0 be speed of light (in this section not necessarily c = 1) Minkowski space: (R 4,, L ) where We set x, y L = x 0 y x i y i i=1 x := x, x L O(1, 3) denotes Lorentzgroup, i.e. automorphism group of (R 4,, L ). Explictly: O(1, 3) = {A GL(R 4 ) Ax, Ay L = x, y L for all x, y R 4 } Let us now work with a concrete basis of R 4, namely the standard basis (e i ) i of R 4. Using the concrete index notation w.r.t. this basis we have x, y L = η ab x a y b = x a y a where (η ab ) ab =

10 For a v = (v 1, v 2, v 3 ) R 3 we define the corresponding 4-vector v a 0 by v a 0 = (c, v) = (c, v 1, v 2, v 3 ) If v < c we introduce the normalized 4-vector v a by Clearly, the definitions above imply v a := c va 0 v a 0 = 1 1 v 2 /c 2(c, v 1, v 2, v 3 ) v a, v a L = v a v a = c 2 (3) For a particle of rest mass m moving with velocity vector v (with respect to the inertial system (e i ) i ) we call the corresponding normalized 4-vector the 4-velocity and introduce the momentum 4-vector P a by Digression 2 From definitions E := cp 0 = P a := mv a (4) 1 1 v 2 /c 2mc2 If particle at rest (in our inertial system) then v = 0 so E = mc 2 Moreover, or, equivalently, P a P a = c 2 m 2 1 c 2 E 2 = (P 0 ) 2 = c 2 m 2 + P i P i

11 3 Classical and relativistic fluid dynamics 3.1 Classical fluid dynamics Consider fluid (liquid or gas) in domain D R 3. For simplicity take D = R 3. ρ(x, t): mass-density of fluid u(x, t): velocity field of fluid T (x, t): temperature distribution of fluid Assume: Equation of state for fluid is known, i.e. where f is a known function. p = f(ρ, T ) (5) Example 3 For an ideal gas we have f(ρ, T ) = c ρ T where c is a constant

12 3.1.1 Special case: perfect fluid (situation) Special case: perfect fluid (situation) fluid is inviscid (= has vanishing viscosity) fluid in thermal equilibrium, i.e. we have T (x, t) = T 0 where T (x, t) is temperature distribution in fluid and T 0 is a constant Mass conservation & momentum conservation where t ρ + (ρ u) = 0 continuity equation (6) ρ ( t u + ( u ) u ) = p Euler equation (7) p(x, t) := f(ρ(x, t), T 0 ) for all x and t (8) 4 PDEs of first order in t for 4 unknown functions ρ, u 1, u 2, u 3 One can expect that there exists a unique solution (ρ, u) for every nice initial configurations ρ(x, 0) and u(x, 0)

13 Digression 3 Relation p = f(ρ, T ) is invertible for fixed T, i.e. for a suitable function g. ρ = g(p, T ) We can take p as unknown function (instead of ρ) and use Euler and continuity equations with Eq. (8) replaced by ρ(x, t) := g(p(x, t), T 0 ) for all x and t Useful when fluid is almost incompressible, i.e. ρ = g(p, T 0 ) ρ 0 where ρ 0 is a constant. Then use idealization: (fluid totally incompressible ). ρ = g(p, T 0 ) = ρ 0 The continuity equation and Euler equation simplify: u = 0 (9) t u + ( u ) u = p/ρ 0 (10) 4 PDEs of first order in t for 4 unknown functions p, u 1, u 2, u 3.

14 3.1.2 The general case Drop condition that fluid is inviscid Generalization of Euler equation with ρ ( t u + ( u ) u ) = T( u, ρ) (11) T( u, ρ) := i,j i T ij ( u, ρ)e j where T ij ( u, ρ) is the corresponding stress tensor, cf. Sec. 4 below. (if p is considered to be the free variable one uses T( u, p) instead of T( u, ρ)). Special case: fluid is Newtonian and incompressible (with constant density ρ = ρ 0 ) where ν > 0 is viscosity (cf. Sec. 4.1). Eq. (11) reads T( u, p) = p ν u ρ 0 ( t u + ( u ) u ) = p + ν u Navier Stokes equation (12) One expects that there exists a unique solution ( u, p) for every nice initial configurations u(x, 0) and p(x, 0)

15 Digression 4 Most general situation: Drop condition of thermal equilibrium (i.e. condition T (x, t) = T 0 ). temperature distribution T (x, t) will be additional unknown function. we need 5 equations, namely the 4 equations above + an additional equation. This additional equation is obtained from energy conservation (This equation will contain additional material constants like specific heat capacity and the thermal conductivity of fluid)

16 3.2 Relativistic fluid dynamics Consider perfect fluid situation above: velocity field u, density ρ, constant temperature T 0, equation of state p = f(ρ, T 0 ). Question: What is the relativistic modification of the continuity and Euler equation? Answer: If u a is normalized 4-vector of u, i.e. u a = 1 1 u 2 (1, u 1, u 2, u 3 ), (so u a u a = 1) then u a a ρ + (ρ + p) a u a = 0 (13) (ρ + p)u a a u b + (η ab + u a u b ) a p = 0 (14) Exercise 1 Show that t ρ+ (ρ u) = 0 is non-relativistic limit of Eq. (13) p ρ in non-relativistic limit). (note Miracle : Eqs. (13) and (14) can be rewritten in amazingly short and symmetric form a S ab = 0, where (15) S ab := (ρ + p)(u a u b ) + p η ab Exercise 2 Show that Eq. (13) and Eq. (14) are equivalent to u c a S ac = 0, and a S ab ( u c a S ac )u b = 0 Later: we explain miracle with the help of stress energy tensor

17 4 The stress energy tensor 4.1 The classical stress tensor The stress tensor is a concept describing the forces inside a continuous body, like a solid body, a liquid or a gas which are caused by outside forces (like gravity or forces on the surface) and movements inside the body (in the case of a liquid or gas). Following general features: Let D R 3 be the space taken by the body. The corresponding stress tensor is a tensor field T ij on D. T ij is in general time-dependent and depends on the variables describing of the body. For example, in the case of a liquid or gas T ij = T ij ( u, ρ, t). Definition of T ij is based on choosing suitable planes H in R 3 (reason, see below) Question: Why do we have to work with planes H?

18 Answer: Assume for simplicity that solid body or liquid considered is in equilibrium Body in equilibrium in each point x D the net force is zero. Now let us assume that, for some plane H through x, we would suddenly remove the part of the body to the right (or to the left ) of H. In this moment the forces would no longer be in equilibrium. We would immediately, obtain a non-zero force F acting on some area element S in H containing x Observation 1 F will depend on the area of S. Observation 2 H can be described uniquely by its normal vector n in x. Convention 3 The length (resp. area resp. volume) of a time interval t (resp. area element S resp. volume element V ) will also be denoted by t (resp. S resp. V ). Definition 1 i) For each unit vector n R 3 we define: T ( n) := lim S 0 F S

19 More precisely: we set F T ( n) := lim ( n, S) S 0 S where F ( n, S) is the force acting on the area element S H after removing the part of the body on the positive side of the plane H (which is the plane through x and orthogonal to n). ii) We generalize this by setting, for arbitrary v, if v = 0 and T ( v) = 0 otherwise. T ( v) := T ( v/ v ) v Observation 3 The map R 3 v T ( v) R 3 is linear. Accordingly, this map is a tensor of type (1,1) and will be denoted by T i j or by T i j (x). Proof is non-trivial, cf. Wikipedia entry stress tensor Definition 2 The stress tensor of the body considered is the tensor field T ij on D, which is given by T ij (x) := g ik T k j (x) (If T ij are the concrete components w.r.t to standard basis (e i ) of R 3 we have T ij (x) = T i j(x) = ( T (e j )) i

20 Observation 4 1. T ij is symmetric. 2. If system in equilibrium then i it ij = 0 3. Let A GL(3, R) and let (e j ) j be the basis of R 3 given by e j = Ae j. Then if (T ij ) ij are the components of T ij w.r.t. the new basis we have T ij = A k i A l jt kl (this follows immediately from the tensor property of T ij ). Proofs of first two statements: see again Wikipedia article.

21 Consider fluid situation above: velocity field u, density ρ, constant temperature T 0, equation of state p = f(ρ, T 0 ). Example 4 (Inviscid fluid) T ij ( u, ρ) = p δ ij = p where p = f(ρ, T 0 ). (This equation is just the definition of inviscid ) Example 5 (Incompressible Newtonian fluid) where ν is the viscosity. T ij ( u, p) = p δ ij ν( i u j + j u i )

22 4.2 The stress energy tensor Aim: Find tensor field T ab, a, b {0, 1, 2, 3} on (R 4,, L ), defined in an analogous way as the classical stress tensor T ij such that subtensor field T ij, i, j {1, 2, 3} coincides with T ij if system is at rest. More precisely: If T (t) ij (x) is classical stress tensor in x R3 at time t T ij (t, x) = T (t) ij (x) for t R, i, j {1, 2, 3} (16) must hold for all x D in which no movement of the body is present at time t. Obvious Replacements: point x R 3 x = (t, x) R 4 plane H through x hyperplane H through x area element S ( x) volume element V ( x) unit vectors n normalized 4-vector n a (i.e. n a = 1). Non-obvious Replacement: Force vector F momentum 4-vector P a. As a motivation observe that for V = S t and F := P / t we have Naive ansatz: P V = P t 1 S = F S T b (n a P b ) := lim V 0 V (17)

23 However: certain difficulties in interpretation of P b = P b (n a, V ) (Not all n a can be treated analogously; sometimes removal argument is necessary and sometimes not) We work with standard basis (e i ) i of R 4 and consider only special case n a {e 0, e 1, e 2, e 3 }. We define P i (e 0, V ), i = 0, 1, 2, 3, to be the amount of P i contained in the (spacial) volume element V. We define P i (e j, V ) for j = 1, 2, 3 and V = S t as the amount of P i which is created in the time interval t inside the spacial volume element S (R + e i ) R 2 (R + e i ) = R 2 R + provided that at time t we have first removed all of the amount of P i in R 2 (R + e i ). (here we assume that t was chosen such that t is its left endpoint)

24 Define T ij (x) by One can see that T ij (x) := T i P i (e j, V ) (e j ) := lim V 0 V (18) T ij for i, j = 1, 2, 3 indeed coincides with the classical stress tensor T i0 is the P i -density T ij for j = 1, 2, 3 is P i -flux (density) in the e j -direction We can summarize this in the following picture:

25 Observation 5 1. T ab is symmetric 2. We have a T ab = 0 (this follows from energy momentum conservation) 3. Let A O(1, 3) and let (e i ) i be the basis of R 4 given by e i = Ae i where (e i ) i is the standard basis of R 4. Let (T ij ) ij denote the family of numbers which we would have got if we had defined the stress energy tensor using (e i ) i as our inertial system instead of (e i ) i. Then T ij = A k i A l jt kl (This implies that T ij really is a tensor) Remark 3 Definition can be generalized to arbitrary space times (M, g). Generalized T ab will have analogous properties. But one exception: the generalization of a T ab = 0 will not hold in general.

26 Example 6 (Perfect fluid) Reconsider situation of Sec. 3.2 (u a is 4-velocity and ρ density function of perfect fluid on (M, g ab ) at temperature T 0 and equation of state p = f(ρ, T 0 )): i) Special case (M, g ab ) = (R 4, η ab ) and fluid at rest, i.e. u a = (1, 0, 0, 0): ρ p 0 0 T ab = 0 0 p p (cf. the last figure). Observe that we can rewrite T ab as T ab = (ρ + p)u a u b + p η ab iii) Special case (M, g ab ) = (R 4, η ab ) but u a arbitrary constant field: T ab := (ρ + p)u a u b + p η ab This follows from i) by applying the principle of relativity and using the behavior of T ab under a change of the inertial system. iii) Special case (M, g ab ) = (R 4, η ab ) but u a arbitrary: T ab := (ρ + p)u a u b + p η ab This follows from ii) by a locality argument iv) General case: (M, g ab ) and u a arbitrary: T ab := (ρ + p)u a u b + p g ab

27 Example 7 (Electromagnetic field) Let (M, g ab ) and let A a be the 4-potential of a given electromagnetic field. i) Special case (M, g ab ) = (R 4, η ab ): where T ab = 1 ( Fac Fb c 1 4π 4 η abf cd F cd) F ab := a A b b A a ii) General (M, g ab ): where now T ab = 1 ( Fac Fb c 1 4π 4 g abf cd F cd) F ab := a A b b A a (where a is the Levi-Civita connection associated to (M, g ab )).

28 Digression 5 In fact, there is a heuristic functional derivative formula for an arbitrary field Ψ, for which a Lagrangian L(g, Ψ) is given explicitly: with T (g, Ψ) = 1 1 δs(g, Ψ) 8π g δg S(g, Ψ) = M L(g, Ψ)dvol g where g = det((g ab ) ab ). For example, the Klein-Gordon field of mass m has the Lagrangian and heuristically we obtain L(g, φ) = 1 2 ( cφ c φ + m 2 φ 2 ) T (g, φ) = a φ b φ 1 2 g ab( c φ c φ + m 2 φ 2 ) where a is the Levi-Civita connection associated to (M, g ab ).