Introduction to General Relativity 2

Transcription

1 Intoduction to Geneal Relativity 2 Geneal Relativity Diffeential geomety Paallel tanspot How to compute metic? Deviation of geodesics Einstein equations Consequences Tests of Geneal Relativity Sola system The binay pulsas Gavitational wave detectos Gavitational Pobe B Still Roos chapte 2. IDS 160(99) 5-1

2 Geneal Relativity Cuved space-time: 4 dimensional manifold γ (τ ) descibed locally by chat x µ metic of pope signatue (same as dτ 2 = g µν dx µ dx ν One can change chat o fame η µν x µ ξ ρ dτ 2 = g µν dx µ dx ν = γ ρσ dξ ρ dξ σ x µ γ ρσ = g µν Note that locally we can aange that ξ ρ x ν ξ σ IDS 160(99) 5-2 Diagonalize Renomalize γ µν = η µν g oo dt 2 2 = dt ff This is a fee fall system: only safe way to know meaning of coodinates Cannot in geneal be global x ff µ Wold lines= tajectoies = cuves on the manifold Tangent space at one point: x ν ν x ff { } γ (τ ) : x µ ( τ ) g 11 dx 12 = dx 1 2 ff Space of velocities: v x ff x ν µ x ff u µ = dxµ dτ

3 Paallel Tanspot Let us move along a cuve with velocity u(τ) Stating fom u x u V uµ IDS 160(99) 5-3 γ ( τ ) Vaiation of quantities along cuve? Fo scala no poblem df ( x) dτ f ( x) = u µ f ( x) u = dx µ x µ dτ x µ But fo vectos (in tangent spaces) tangent spaces ae not the same! we have to specify how the bases vay ( ) = velocity and vectov ( x) = V µ ( x) DV Dx µ uµ V ρ ; µ One can show that in ode fo dτ 2 to be invaiant λ 1 Γ µν = 2 g λσ g µσ x ν with ( g λσ ) = ( g µσ ) 1 A vecto is paallel tanspoted along u when Geodesics = autopaallel V { e ρ } ρ = u µ x µ e ρ + u µ V ρ + g νσ x µ g µν x σ u u dτ 2 = 0 GR postulate: Tajectoies of fee paticles! It can be shown that also minimum of pope time e µ ( x) ρ e ρ { x = V µ uµ e x µ ρ + u µ V ρ Γ µρ e σ Γ σ µρ d 2 x f ( x) σ e σ u V = 0

4 Einstein Equations Geodesic deviation equation Let us conside a family of geodesics. Call η thei distance n η j = x j n τ u µ = dxµ dτ In a space of n dimensions u u η = R( u,η)u whee R is the Rieman cuvatue tenso constucted fom metic, and its fist and second deivatives. ( ) 12 n 2 n η λ x µ x ν uµ u ν λ = R µρν u µ u ν η ρ independent components n=2: 1 n=3: in each diection, distance and angle:3x2=6 n=4: 20! Einstein equations Attempt to find a set of equations which elate R and the enegy density and educes to Poisson equation in weak field limit ρ T µν = mu µ u ν = p µ p ν 2 ϕ = m 2 ϕ G µν = g vσ R µ x i x i ρσλ g λρ 1 2 g µν σ R ρσλ g λρ with ( g λσ ) = ( g µσ ) 1 Fundamental emak: elates the deviation of geodesics (cuvatue) to the density of enegy and momentum Enough equations (6 independent) 2nd ode in deivatives of g and poduct of fist deivatives => highly non linea (enegy of gavitational field acts on field itself) Difficult to solve Poblem with quantization IDS 160(99) ϕ = 4πρ G µν = 8π GT µν

5 Consequences Weak field limit We may use a petubative appoach g µν = η µν + h µν h µν << η µν We can show using Γ s ( slide 4-9) dx 2 dt = c2 h oo h oo = 2ϕ c 2 using Einstein s equations 2 hoo = 8π c 2 G ρ + 3 p fo a pefect gas of density ρ and pessue p. Pessue tem is new in GR: influence of momentum Photons ae sensitive to gavity See Chapte 4. Popagation of gavitational waves Analogy with electomagnetism IDS 160(99) 5-5 In Loentz gauge the potential vecto A µ is such that µ A µ = 0 µ µ A ν = 2 t 2 c2 2 A ν = Jν ( Maxwell equations) ε o In vacuum A µ = ε µ i k o t 1 c e 2 k. x with k µ ε µ = 0 and k 2 = 0 This is a tansvese wave popagating at the speed of light! Similaly In tansvese Hilbet gauge (specific choice of vaiables) in vacuum ρ h ρ σ = 0 ρ ρ h µν = 16πGT µν h µν = ε µν e i k o t 1 c 2 k. x with k µ ε µν = 0 and k 2 = 0 Metic wave popagating tansvesely at the speed of light! c 2 Quadupola (fo half peiod expands in 1 diection and contacts in pependicula diection, fo othe half, opposite)

6 Tests of Geneal Relativity Sola system Pecession of Mecuy Geneal elativity intoduce 1/ 3 tem! V eff ( ) GMm + L2 2m 2 s Mecuy advance of peihelion : 43"/centuy Deflection of light ays by the sun 1919 sola eclipse (Eddington) VLA (2 fequencies to coect fo sola coona and eath ionosphee) atio with GR γ = 1± 0.03 Rada time delay measuements Viking:(on Mas) The binay pulsa(s) (Hulse and Taylo -1974) γ = 1± s { G.R. Clean system of 2 neutons stas obiting at close distance ( R Sun ) s 10 6 See all expected effects (J.H. Taylo Rev. Mod. Phys. 66(1994)711) pecession of obit time delay+deflection decay of obit => gavitational waves Most pecise test of GR till ecently! Othe even moe pomising systems have been identified e.g. PSR B (see next slides) δ = 1.75" R Sun b IDS 160(99) 5-6

7 Binay Pulsas PSR B Stais et al. Asto-ph/ IDS 160(99) 5-7

8 Geneal Relativity PSR B IDS 160(99) 5-8

9 New Pojects Testing Geneal Relativity Somewhat contovesial! Detection of gavitational waves Bas Lase intefeomete LIGO (in constuction) Abamovici et al. Science 256(1992) VIRGO Michelson intefeomete δl l sensitivity? Coalescing binaies Much highe sensitivity in space: LISA Test of equivalence pinciple Moe pecise test planned in space Detection of geodetic and fame dagging Gavitational Pobe B 2 effects: failue of paallel tanspot because of cuvatue of space close to the eath geodetic (Thomas) pecession (6.9 /yea) fame dagging (Lense-Thiing effect) (.044 /yea) IDS 160(99) 5-9