Review of General Relativity

Transcription

1 Lecture 3 Review of General Relativity Jolien Creighton University of Wisconsin Milwaukee July 16, 2012

2 Whirlwind review of differential geometry Coordinates and distances Vectors and connections Lie derivative Curvature Formulation of General Relativity Weak gravity and slow motion Matter Einstein field equations Linearized gravity Newtonian limit Plane wave solution

3 Coordinates and distances t = 0 t = ½ Δt t = Δt ½ Δy Δx = v Δt Measure the speed of light Observer in rocket: c = y τ Observer on Earth: c 2 = ( x)2 + ( y) 2 ( t) 2 = ( x)2 + (c τ) 2 ( t) 2 c 2 ( τ) 2 = c 2 ( t) 2 ( x) 2

4 Coordinates and distances Let ds be the distance between neighboring events Curved spacetime: ds 2 = g µν (x α )dx µ dx ν Flat spacetime: ds 2 = η µν dx µ dx ν = c 2 dt 2 + dx 2 + dy 2 + dz 2 (rectilinear coordinates) Equivalence principle: A freely-falling frame defines locally-flat neighborhood of an event in spacetime.

5 Coordinates and distances General coordinate transformation: x α (x µ ) dx α = x α x µ xµ ds 2 = g µν dx µ dx ν = g αβdx α dx β g αβ = g µν x µ x α x ν x β Infinitesimal coordinate transformation: x α = x α + ξ α (x µ ) dx α = dx α + ξα x µ dxµ g αβ ξ µ = g αβ g αµ x β g ξ µ µβ x α

6 Vectors and connections u t = 0 x α (t ) t = Δt On a curve x α (t) define v a function f(t) = F ( x α (t) ) v u t = 0 x α (t ) t = Δt where F(x α ) is some function on spacetime. df dt = dxµ F dt x x µ µ uµ F u = d is the tangent vector to the curve with dt components u α = dxα dt along curve

7 Vectors and connections v Parallel transport v along curve defined by u: v u t = 0 t = Δt 1. Go into a locally flat reference frame 2. Take an infinitesimal step by parallel transport in the usual sense of flat space 3. Repeat steps 1 and 2 until final point is reached x α (t )

8 Vectors and connections v For each step, require v u t = 0 t = Δt v α (t + t) v α (t) 0 = lim t 0 t = dvα dt = u µ µ v α µ is the covariant derivative x α (t )

9 Vectors and connections In locally inertial reference frame v µ = x µ v u t = 0 t = Δt In other coordinates µ v α = vα x µ + Γ α µνv ν Γ α µν are connection coefficients x α (t ) Γ α µν = xα x β 2 x β x µ x ν where x α (x µ ) is the transformation to an inertial reference frame

10 Vectors and connections v Parallel transport preserves the inner product v w = g αβ v α w β v u t = Δt 0 = d (v w) dt = u µ µ (g αβ v α w β ) t = 0 so µ g αβ = 0 x α (t ) This gives Γµν α = 1 ( gβν 2 gαβ x µ + g µβ x ν g ) µν x β

11 Vectors and connections A curve is a geodesic if its tangent vector is parallel-transports itself along the curve: 0 = u µ µ u α = dxµ dt ( dx α x µ dt + Γ µν α dx ν ) dt Geodesic is defined by four coupled ordinary differential equations: d 2 x α dt 2 = Γµν α dx µ dx ν dt dt Specify dx µ /dt at a point x µ and integrate to generate a geodesic

12 Vectors and connections u( ) Δt v( ) Δs v( ) Δs w Δt Δs Vector fields do not necessarily u( ) Δt commute: Move t along u and then s along v to get to B Move s along v and then t along u to get to A w α = lim lim {[u α (P) t + v α (R) s] [v α (P) s + u α (Q) t]} s 0 t 0 t s v α (R) v α (P) u α (Q) u α (P) = lim lim t 0 t t 0 t }{{}}{{} u µ µ v α v µ µ u α

13 Vectors and connections w = [u, v] is the commutator of u and v w α = u µ vα uα vµ x µ x µ If two vectors commute, [u, v] = 0, then going t along u and then going s along v arrives at the same point as going s along v and then going t along u We can use (s, t) to label the points ( coordinates ): u and v are coordinate basis vectors

14 Lie derivative Consider a fluid flow with velocity field u(x) A scalar field ρ(x) is dragged along by the flow if the value remains constant on a fluid element: where u = dx/dt 0 = d dt ρ( x(t) ) = u ρ The scalar field ρ is then said to be Lie-derived by the vector field u; the Lie derivative of ρ, defined by L u ρ = u ρ is the rate of change of ρ measured by a co-moving observer

15 Lie derivative xʹ1 xʹ2 xʹ3 uʹ εv1 εv2 εv3 u x1 x2 x3 Consider now a vector ϵv that joins two nearby fluid elements, ϵv = x (t) x(t). This vector is dragged along by the fluid flow. u = d dt x (t) = d ( x(t) + ϵv ( x(t) )) = u(x) + ϵu v(x) dt u = u(x ) = u(x + ϵv) = u(x) + ϵv u(x) + O(ϵ 2 ) so [u, v] = u v v u = 0

16 Lie derivative The Lie derivative of the vector v along the vector u is the commutator: L u v = [u, v] L u v α = u µ µ v α v µ µ u α = u µ vα uα vµ x µ x µ

17 Curvature w! P w P w,w! In flat space: parallel transport of a vector about a closed path leaves the vector unchanged In curved space: vector is generally different after parallel transport about a closed path

18 Curvature t + Δt t u V V ʹ V v s s + Δs V V Use: u µ x µ Vα = ΓµσV α σ u µ 0. Start at point A with vector V α A 1. V α B = V α A 2. V α C = V α B (s+ s,t) (s,t) (s+ s,t+ t) (s+ s,t) Γ α µσv σ v µ ds Γ α µσv σ u µ dt (s,t+ t) 3. V α D = V α C ΓµσV α σ v µ ds (s+ s,t+ t) (s,t) 4. V α A = Vα D ΓµσV α σ u µ dt (s,t+ t)

19 Curvature V α = V α A Vα A = (s,t+ t) Γ α µσv σ u µ dt (s,t) (s+ s,t+ t) + (s,t+ t) t+ t (s+ s,t+ t) (s+ s,t) (s+ s,t) Γ α µσv σ v µ ds Γ α µσv σ u µ dt (s,t) s+ t Γ α µσv σ v µ ds s v ν t x (Γ µσv α σ )u µ dt + t u ν ν s x (Γ µσv α σ )v µ ds [ ν s t u µ v ν x (Γ µσv α σ ) + v µ u ν ] ν x (Γ µσv α σ ) ν [ = s t x Γ α ν µσ + ΓµρΓ α νσ ρ + ] x Γ α µ νσ ΓνρΓ α µσ ρ u µ v ν V σ } {{ } α R µνσ

20 Curvature V t + Δt t u V V ʹ V v s s + Δs V Change in V α parallel transported A B C D A is Also: V α = ( s t)r µνσ α u µ v ν V σ ( µ ν ν µ )V α = R µνσ α V σ R α µνσ = x µ Γ νσ α + x ν Γ µσ α ΓµρΓ α νσ ρ + ΓνρΓ α µσ ρ Riemann curvature tensor

21 Bianchi identity Q w u P v V α = ϵ 2 R µνσ α (Q)u µ v ν V σ }{{} top face ϵ 2 R µνσ α (P)u µ v ν V σ }{{} bottom face = ϵ 3 (w ρ ρ R µνσ α )u µ v ν V σ 0 = V α (all faces) = ϵ 3 ( ρ R µνσ α + µ R νρσ α + ν R ρµσ α )u µ v ν w ρ V σ ρ R µνσ α + µ R νρσ α + ν R ρµσ α = 0 Bianchi identity

22 Geodesic deviation Let ζ = d/dx be the separation vector between two geodesics Relative velocity of the two bodies v = dζ dt = u ζ = ζ u Relative acceleration of the two bodies a = dv dt = u (ζ u) u( ) u( ) Use geodesic equation and definition of Riemann tensor to obtain ζ a α = R µσν α u µ ζ σ u ν geodesic deviation equation

23 Whirlwind review of differential geometry Coordinates and distances Vectors and connections Lie derivative Curvature Formulation of General Relativity Weak gravity and slow motion Matter Einstein field equations Linearized gravity Newtonian limit Plane wave solution

24 Weak gravity and slow motion Linear perturbation to flat spacetime: g αβ = η αβ + h αβ + O(h 2 ) 4-velocity of slowly moving particle, v c: Geodesic equation: u = dx dt = [1, 0, 0, 0] + O(v/c) d 2 x i dt 2 Γ 00 i 1 2 x i h 00 where we assume a nearly stationary background, h αβ / t 0 Geodesic deviation equation: Identify h 00 = 2Φ d 2 ζ i dt 2 R 0i0jξ j 1 2 h 00 2 x i x j ζj

25 Matter Perfect fluid stress energy tensor Locally-inertial frame: ρc 2 T = P P P Generally: T αβ = (ρ + P/c 2 )u α u β + Pg αβ

26 Einstein field equations G αβ = 8πG c 4 T αβ Einstein field equations where the Einstein tensor, Ricci tensor, and Ricci scalar are G αβ = R αβ 1 2 g αβr R αβ = R αµβ µ R = g µν R µν The Bianchi identity implies µ G µα = 0 which yields the equations of motion for matter µ T µα = 0

27 Whirlwind review of differential geometry Coordinates and distances Vectors and connections Lie derivative Curvature Formulation of General Relativity Weak gravity and slow motion Matter Einstein field equations Linearized gravity Newtonian limit Plane wave solution

28 Linearized gravity Define trace-reversed metric perturbation h αβ = h αβ 1 2 η αβh Choose Lorenz gauge (harmonic coordinates) x µ h µα = 0 via gauge transformation x α new = x α old + ξα where ξ α = µα h x µ old h αβ = 16πG c 4 T αβ linearized field equations

29 Newtonian limit Leading order in 1/c 2 : ρc2 T P P P and = 2 c 2 t Non-trivial field equation is 2 h 00 = 16πGρ Identify h 00 = 4Φ where Φ is the Newtonian potential 2 Φ = 4πGρ Poisson equation

30 Newtonian limit c 2 2Φ 1 2Φ Newtonian metric: g = c 2 1 2Φ c 2 1 2Φ c 2 Geodesic equation: d 2 x i dt 2 = Φ x i Geodesic deviation equation: d 2 ζ i dt 2 = 2 Φ x i x j ζj

31 Plane wave solution Linearized vacuum field equations: h αβ = 0 Plane wave solution travelling in z-direction is h αβ = h αβ (t z/c) Lorenz gauge condition imposes four constraints: h 00 = c h 03 = c 2 h 33, h 01 = c h 31, h 02 = c h 32 and thereby reduces independent degrees of freedom to six h 00, h 01, h 02, h 11, h 12, h 22

32 Plane wave solution Remaining freedom within Lorenz gauge: x α new = x α old + ξα where ξ α = 0 is used to set h 00 = h 01 = h 02 = 0 and h 11 = h 22 Resulting metric perturbation has two degrees of freedom: h = h + h 0 0 h h where h + = h + (t z/c) and h = h (t z/c) are the two transverse polarizations of the plane wave