Modern Cosmology Solutions 2: Relativistic Gravity Max Camenzind October 0, 2010 1. Special Relativity Relativity principle Lecture Notes. Doppler shift and the stellar aberration of light given on my Homepage. Quaatic Doppler effect? Doppler formula for θ 90 deg. 2. Calculus on Manifolds The number of independent Cristoffel symbols for a 2 sphere : 6. Γ 1 11 0, Γ 1 12 0 (1 Γ 1 22 sinθ cosθ, Γ 2 11 0 (2 Γ 2 12 cosθ, Γ 2 22 0. ( sinθ Riemann tensor: only one component: R 1212 a 2 sin 2 θ In Riemannian geometry, the Gaussian curvature is given by K ( 2 1 1 2 e 1,e 2 det(g, (4 where i ei is the covariant derivative and g is the metric tensor. Gaussian Curvature for the 2 sphere K R 1212 g a2 sin 2 θ a 4 sin 2 θ 1 a 2. (5 The metric for a sphere embedded in a four dimensional Euclidean space given by the metric ds 2 dx 2 +dy 2 +dz 2 +dw 2. (6 The equation defining a sphere is By differentiation you will get x 2 +y 2 +z 2 +w 2 a 2. (7 2xdx+2ydy +2zdz +2wdw 0. (8 1
This can be solved for dw ds 2 dx 2 +dy 2 +dz 2 + (xdx+ydy +zdz2 a 2 (x 2 +y + z 2 and transforming to spherical polar coordinates (r, θ, φ (9 gives the line element of a sphere x r sinθ cosφ (10 y r sinθ sinφ (11 z r cosθ (12 ds 2 a2 a 2 r 2 2 +r 2 (dθ 2 +sin 2 θdφ 2. (1 Singularity at the radius r a? coordinate singularity which can be transformed away.. Gravity as the Basis of Modern Cosmology Einstein s equivalence principle EEP? LN. Strong equivalence principle SEP? LN. Spacetime Klein Gordon equation for Φ(t,x i, using Γ µ µρ ρ g/ g, Φ µ µ Φ 1 g µ ( gg µσ σ Φ. (14 The energy momentum tensor in spacetime Lecture Notes. The line element of a spherical star in Schwarzschild coordinates ds 2 exp(2φ(rc 2 dt 2 +exp( 2Λ(r 2 +r 2 (dθ 2 +sin 2 θdφ 2. (15 The Tolman Oppenheimer Volkoff equations for the hyostatic equilibrium of spherical stars (see derivation in Camenzind 2007: Compact Objects dm(r 4πρ(rr 2 (16 dp(r GM(rρ(r ( r 2 1+ P(r ρ(rc 2 ( ( 1+ 4πr P(r M(rc 2 1 2GM(r 1 c 2 (17 r e 2Λ(r 1 2GM(r c 2 (18 r ( dφ(r 1 GM(r 1 2GM(r/c 2 r c 2 r 2 + 4πGrP c 4. (19 As in the Newtonian case, the total mass M(r inside a spherical shell of radius r also determines the hyostatic equilibrium, but four corrections occur 2
the mass-density ρ 0 has tobereplaced interms of the total mass energy density ρ, which includes the internal energy; the inertial mass density is given by ρc 2 + P (see also equations of motion; this is the first correction factor on the rhs; pressure gives an active volume correction (second factor; themetricof spaceentersintermsofthelastfactor; thisfactorisofparticular importance, since it determines the stability properties of the solutions. The surface of the object always has to be far outside the Schwarzschild surface. It is important to note that the curvature of space is entirely given in terms of the total mass, while the gravitational potential satisfies its Newtonian analogue, except for the inertial factor ρc 2 + P. It is then obvious that these structure equations goe over into the Newtonian analog for P ρc 2, i.e. roughly speaking for sound velocities much less than the velocity of ligth, for low compactness 2GM(r/c 2 r and for low pressure mass 4πr P(r M(rc 2. The compactness parameter has a particular influence on the hyostatic equilibrium (the last factor in the TOV equation. In this limit, space is flat, i.e. exp(λ 1 for all radii, and expφ(r 1 + Φ(r with the following structure equations, usually derived in the theory of stellar structure, dm(r dp(r dφ(r 4πρ 0 (rr 2 (20 GM(rρ 0(r r 2 (21 GM(r r 2. (22 Mass radius relation for White Dwarfs see Camenzind: Compact Objects. Mass radius relation for Neutron Stars see Camenzind: Compact Objects. Which tests of General Relativity can be done in the Solar System? Gravitational redshift on Earth (GPS! and Sun. Light bending on Sun (Eddington to Hipparcos and Jupiter (GAIA. Perihelion precession of Mercury. Shapiro time delay (Cassini. Light bending on Jupiter: M J 0.001M, R J 71,492km 0.1R φ 16.28 marcsec. Is this important in Astronomy? GAIA satellite (resolution of 20 µarcsec! 4. The Spatially Flat Universe A flat expanding Universe is simply given by stretching space in each direction by the same amount a(t (isotropic expansion, c 1 ds 2 dt 2 +a 2 (tδ ik dx i dx k η ab Θ a Θ b. (2 The factor a(t is called expansion factor. There are at least three different methods to calculate the Einstein tensor for this spacetime. Two methods are discussed in the following.
Method 1: Christoffel Symbols Inthecoordinatesystem(t,x,y,z, themetrictensorhasthefollowingcomponents(c 1 1 0 0 0 g αβ 0 a 2 (t 0 0 0 0 a 2 (t 0 (24 0 0 0 a 2 (t with its inverse, g αβ 1 0 0 0 0 1/a 2 (t 0 0 0 0 1/a 2 (t 0 0 0 0 1/a 2 (t. (25 Fist, we calculate the various partial derivatives for this metric t g αβ 0 2aȧ 0 0 0 0 2aȧ 0 (26 0 0 0 2aȧ x g αβ y g αβ z g αβ With the definition of the Christoffel symbols (27 (28. (29 Γ µ αβ 1 2 gµρ (g ρα,β +g ρβ,α g αβ,ρ. (0 we get the following expressions given as symmetric matrices Γ t αβ 1 2 gtt( g tα,β +g tβ,α g αβ,t Γ x αβ 1 2 gxx( g xα,β +g xβ,α g αβ,x 4 0 aȧ 0 0 0 0 aȧ 0 0 0 0 aȧ 0 ȧ/a 0 0 ȧ/a 0 0 0 (1 (2
Γ y αβ 1 2 gyy( g yα,β +g yβ,α g αβ,y Γ z αβ 1 2 gzz( g zα,β +g zβ,α g αβ,z 0 0 ȧ/a 0 ȧ/a 0 0 0 0 0 0 ȧ/a ȧ/a 0 0 0. ( (4 In fact, this is quite a simple connection, where the space part is vanishing, since space is flat. For the Riemann tensors we obtain now in the coordinate frame R t itk t Γ t ki kγ t ti +Γ t tργ ρ ki Γt kρ Γρ ti tγ t ki Γt kk Γk ti t (aȧδ ik ȧ 2 δ ik aäδ ik (5 R 1 212 1 Γ 1 22 2 Γ 1 12 +Γ 1 1ρΓ ρ 22 Γ1 2ρΓ ρ 12 Γ1 1tΓ t 22 ȧ 2 (6 R 1 1 R 1 212 R 2 2. (7 All other components vanish. Method 2: Cartan s Equations The one frame basis is given by Θ 0 dt and Θ i a(tdx i (i1,2,. The exterior derivatives are simply given by, using dθ µ df Θ µ for a 1 form Θ µ f(xdx µ, dθ 0 0 ω 0 j Θ j (8 dθ i da dx i ȧdt dx i ȧ a Θ0 Θ i ȧ a Θi Θ 0 ω i 0 Θ 0 ω i j Θ j. (9 The solution to these equations gives us the connection one forms with respect to orthonormal frames ω i 0 ȧ a Θi, ω i0 ω 0i +ω 0 i (40 ω 0 j ȧ a Θj (41 ω i j 0. (42 Since space is still flat, the connection ω i j also vanishes. For the curvature two form we first calculate the exterior derivatives dω i 0 ä a Θ0 Θ i (4 dω 0 j ä a Θ0 Θ j (44 dω i j 0. (45 5
The wedge products are simply ω 0 i ω i 0 0 (46 ω i c ω c 0 0 (47 (ȧ 2 ω i c ω c j ω i 0 ω 0 j Θ i Θ j. (48 a With this we get the following curvature two forms Ω 0 j ä a Θ0 Θ j 1 2 R0 jab Θa Θ b (49 Ω i 0 ä a Θ0 Θ i (50 (ȧ 2 Ω i j Θ i Θ j 1 a 2 Ri jab Θa Θ b. (51 From this we can read off the Riemann tensors 1, with H(t ȧ/a being the Hubble parameter, R 0 i0k ä a δ ik (52 R 1 212 H 2 (t R 1 1 R 2 2. (5 With this we get the Ricci tensors (note, these expressions are in orthonormal frames R 00 R i 0i0 η ij R 0 j0i ä/a (54 R 11 R 0 101 +R 2 121 +R 11 ä/a+2h 2 (55 R 22 R R 11 (56 (ä R R 0 0 +R i i 6 a +H2. (57 This finally leads to Einstein s equations including a cosmological constant Λ G 00 R 00 1 2 Rη 00 +Λη 00 H 2 Λ 8πGρ (58 G 11 R 11 1 2 Rη 11 +Λη 11 2ä a H2 +Λ 8πGP, (59 or, when coupled to matter, to the two Friedmann equations for flat spaces H 2 (t 8πG ρ+ Λc2 (60 ä a 4πG Λc2 (ρ+p+. (61 1 Despite the fact that space is flat, the curvature does not vanish. This is the effect of the Gauss Codazzi decomposition. 6
Method : Use an algebra software, e.g. Mathematica, Maple etc. Remark: Curvature can easily be included, since spaces of constant curvature are conformally flat. In this sense, the general Robertson Walker Friedmann metric can be given in the following form (see Lecture notes, Chapter 5 ds 2 dt 2 + a 2 (t [1+K(x 2 +y 2 +z 2 /4] 2 (dx2 +dy 2 +dz 2. (62 K represents the curvature of space and can be positive for spheres S, zero for Euclidean space E, and negative for hyperboloids H. The calculation of the Einstein tensor exactly follows in same manner as for the flat space. The only change occurs in the first Friedmann equation which now has a contribution from space curvature H 2 (t+ kc2 a 2 R 2 0 8πG ρ+ Λc2, (6 where the curvature radius has been scaled as R(t a(tr 0 with a(t 0 1 and a cosmological constant Λ has been included. 7