Expanding Einstein s Equation

Transcription

1 Expanding Einstein s Equation This section won t really help your understanding of Einstein s equation. It just shows how really complicated the equation is, by expanding out everything in terms of the metric tensor. The equation in its most common form is : Rμν ½gμν R + gμν Λ = k Tμν But So R = g αβ Rαβ Rμν ½gμν g αβ Rαβ + gμν Λ = k Tμν Now, since Rμν = (Γ η μν)/ x η (Γ η ημ)/ x ν + Γ η ηλ Γ λ μν Γ λ μη Γ η νλ (The Γ is a Christoffel symbol and is explained in detail in GR2c. Don t worry about it here, because it s going to disappear in a moment.) Then the equation becomes : (Γ η μν)/ x η (Γ η ημ)/ x ν + Γ η ηλ Γ λ μν Γ λ μη Γ η νλ ½ gμν g αβ (Γη αβ)/ x η + ½ gμν g αβ (Γη ηα)/ x β ½ gμν g αβ Γη ηλ Γ λ αβ + ½ gμν g αβ Γλ αη Γ η βλ + gμν Λ = k Tμν But wait, it s about to get much worse, because the formula for the Christoffel symbol is : Γ γ αβ = ½ g δγ gαδ/ x β + ½ g δγ gβδ/ x α ½ g δγ gαβ/ x δ

2 So the equation becomes : ½ (g δη gμδ/ x ν + g δη gνδ/ x μ g δη gμν/ x δ )/ x η ½ (g δη gηδ/ x μ + g δη gμδ/ x η g δη gημ/ x δ )/ x ν + ¼ (g δη gηδ/ x λ + g δη gλδ/ x η g δη gηλ/ x δ ) (g δλ gμδ/ x ν + g δλ gνδ/ x μ g δλ gμν/ x δ ) ¼ (g δλ gμδ/ x η + g δλ gηδ/ x μ g δλ gμη/ x δ ) (g δη gνδ/ x λ + g δη gλδ/ x ν g δη gνλ/ x δ ) ¼ gμν g αβ (g δη gαδ/ x β + g δη gβδ/ x α g δη gαβ/ x δ )/ x η + ¼ gμν g αβ (g δη gηδ/ x α + g δη gαδ/ x η g δη gηα/ x δ )/ x β ⅛ gμν g αβ (g δη gηδ/ x λ + g δη gλδ/ x η g δη gηλ/ x δ ) (g δλ gαδ/ x β + g δλ gβδ/ x α g δλ gαβ/ x δ ) + ⅛ gμν g αβ (g δλ gαδ/ x η + g δλ gηδ/ x α g δλ gαη/ x δ ) (g δη gβδ/ x λ + g δη gλδ/ x β g δη gβλ/ x δ ) + gμν Λ = k Tμν Now we can finally see that it is really just a (very complicated) 2 nd -order partial differential equation in gμν. Expanding the differentials and multiplying out the ( ) ( ) terms in the above, the equation becomes :

3 ½ g δη / x η g μδ / x ν + ½g δη 2 g μδ / x ν x η + ½ g δη / x η g νδ / x μ + ½g δη 2 g νδ / x μ x η ½ g δη / x η g μν / x δ + ½g δη 2 g μν / x δ x η ½ g δη / x ν g ηδ / x μ ½g δη 2 g ηδ / x μ x ν ½ g δη / x ν g μδ / x η ½g δη 2 g μδ / x η x ν + ½ g δη / x ν g ημ / x δ ½g δη 2 g ημ / x δ x ν + ¼g δη g ηδ / x λ g δλ g μδ / x ν + ¼g δη g ηδ / x λ g δλ g νδ / x μ ¼g δη g ηδ / x λ g δλ g μν / x δ + ¼g δη g λδ / x η g δλ g μδ / x ν + ¼g δη g λδ / x η g δλ g νδ / x μ ¼g δη g λδ / x η g δλ g μν / x δ ¼g δη g ηλ / x δ g δλ g μδ / x ν ¼g δη g ηλ / x δ g δλ g νδ / x μ + ¼g δη g ηλ / x δ g δλ g μν / x δ ¼g δλ g μδ / x η g δη g νδ / x λ ¼g δλ g μδ / x η g δη g λδ / x ν + ¼g δλ g μδ / x η g δη g νλ / x δ ¼g δλ g ηδ / x μ g δη g νδ / x λ ¼g δλ g ηδ / x μ g δη g λδ / x ν + ¼g δλ g ηδ / x μ g δη g νλ / x δ + ¼g δλ g μη / x δ g δη g νδ / x λ + ¼g δλ g μη / x δ g δη g λδ / x ν ¼g δλ g μη / x δ g δη g νλ / x δ ¼g μν g αβ g δη / x η g αδ / x β ¼g μν g αβ g δη 2 g αδ / x β x η ¼g μν g αβ g δη / x η g βδ / x α ¼g μν g αβ g δη 2 g βδ / x α x η + ¼g μν g αβ g δη / x η g αβ / x δ ¼g μν g αβ g δη 2 g αβ / x δ x η + ¼g μν g αβ g δη / x β g ηδ / x α + ¼g μν g αβ g δη 2 g ηδ / x α x β + ¼g μν g αβ g δη / x β g αδ / x η + ¼g μν g αβ g δη 2 g αδ / x η x β ¼g μν g αβ g δη / x β g ηα / x δ + ¼g μν g αβ g δη 2 g ηα / x δ x β ⅛g μν g αβ g δη g ηδ / x λ g δλ g αδ / x β ⅛g μν g αβ g δη g ηδ / x λ g δλ g βδ / x α + ⅛g μν g αβ g δη g ηδ / x λ g δλ g αβ / x δ ⅛g μν g αβ g δη g λδ / x η g δλ g αδ / x β ⅛g μν g αβ g δη g λδ / x η g δλ g βδ / x α + ⅛g μν g αβ g δη g λδ / x η g δλ g αβ / x δ + ⅛g μν g αβ g δη g ηλ / x δ g δλ g αδ / x β + ⅛g μν g αβ g δη g ηλ / x δ g δλ g βδ / x α ⅛g μν g αβ g δη g ηλ / x δ g δλ g αβ / x δ + ⅛g μν g αβ g δλ g αδ / x η g δη g βδ / x λ + ⅛g μν g αβ g δλ g αδ / x η g δη g λδ / x β

4 ⅛g μν g αβ g δλ g αδ / x η g δη g βλ / x δ + ⅛g μν g αβ g δλ g ηδ / x α g δη g βδ / x λ + ⅛g μν g αβ g δλ g ηδ / x α g δη g λδ / x β ⅛g μν g αβ g δλ g ηδ / x α g δη g βλ / x δ ⅛g μν g αβ g δλ g αη / x δ g δη g βδ / x λ ⅛g μν g αβ g δλ g αη / x δ g δη g λδ / x β + ⅛g μν g αβ g δλ g αη / x δ g δη g βλ / x δ + g μν Λ = k T μν Finally, by showing all the summations that are implied by the Einstein notation, where δ means to sum over δ=0,1,2,3, the equation becomes : δ η {½ ( g δη / x η g μδ / x ν + g δη 2 g μδ / x ν x η this would be + g δη / x η g νδ / x μ + g δη 2 g νδ / x μ x η 96 terms if written g δη / x η g μν / x δ + g δη 2 g μν / x δ x η ) out completely! ½ ( g δη / x ν g ηδ / x μ + g δη 2 g ηδ / x μ x ν + g δη / x ν g μδ / x η + g δη 2 g μδ / x η x ν g δη / x ν g ημ / x δ + g δη 2 g ημ / x δ x ν ) 96 terms! + α β [ ¼ g μν (g αβ g δη / x η g αδ / x β + g αβ g δη 2 g αδ / x β x η + g αβ g δη / x η g βδ / x α + g αβ g δη 2 g βδ / x α x η g αβ g δη / x η g αβ / x δ + g αβ g δη 2 g αβ / x δ x η ) 1536 terms! + ¼ g μν (g αβ g δη / x β g ηδ / x α + g αβ g δη 2 g ηδ / x α x β + g αβ g δη / x β g αδ / x η + g αβ g δη 2 g αδ / x η x β g αβ g δη / x β g ηα / x δ + g αβ g δη 2 g ηα / x δ x β ) 1536 terms! + λ ( ⅛ g μν (g αβ g δη g ηδ / x λ g δλ g αδ / x β + g αβ g δη g ηδ / x λ g δλ g βδ / x α g αβ g δη g ηδ / x λ g δλ g αβ / x δ + g αβ g δη g λδ / x η g δλ g αδ / x β + g αβ g δη g λδ / x η g δλ g βδ / x α g αβ g δη g λδ / x η g δλ g αβ / x δ g αβ g δη g ηλ / x δ g δλ g αδ / x β g αβ g δη g ηλ / x δ g δλ g βδ / x α + g αβ g δη g ηλ / x δ g δλ g αβ / x δ ) 9216 terms!

5 + ⅛ g μν (g αβ g δλ g αδ / x η g δη g βδ / x λ + g αβ g δλ g αδ / x η g δη g λδ / x β g αβ g δλ g αδ / x η g δη g βλ / x δ + g αβ g δλ g ηδ / x α g δη g βδ / x λ + g αβ g δλ g ηδ / x α g δη g λδ / x β g αβ g δλ g ηδ / x α g δη g βλ / x δ g αβ g δλ g αη / x δ g δη g βδ / x λ g αβ g δλ g αη / x δ g δη g λδ / x β + g αβ g δλ g αη / x δ g δη g βλ / x δ ) 9216 terms! + ¼ (g δη g ηδ / x λ g δλ g μδ / x ν + g δη g ηδ / x λ g δλ g νδ / x μ g δη g ηδ / x λ g δλ g μν / x δ + g δη g λδ / x η g δλ g μδ / x ν + g δη g λδ / x η g δλ g νδ / x μ g δη g λδ / x η g δλ g μν / x δ g δη g ηλ / x δ g δλ g μδ / x ν g δη g ηλ / x δ g δλ g νδ / x μ + g δη g ηλ / x δ g δλ g μν / x δ ) 576 terms! ¼ (g δλ g μδ / x η g δη g νδ / x λ + g δλ g μδ / x η g δη g λδ / x ν g δλ g μδ / x η g δη g νλ / x δ + g δλ g ηδ / x μ g δη g νδ / x λ + g δλ g ηδ / x μ g δη g λδ / x ν g δλ g ηδ / x μ g δη g νλ / x δ g δλ g μη / x δ g δη g νδ / x λ g δλ g μη / x δ g δη g λδ / x ν + g δλ g μη / x δ g δη g νλ / x δ ) 576 terms! } ] ) + g μν Λ = k T μν And no, I am not going to re-write it one more time with all 22,848 terms! As if the above equation (yes, remember it s all one equation!) isn t complicated enough : repeat this equation 9 times, for the various values of μ and ν = (t,x,y,z)! each gμν g αβ (or similar) term is a matrix multiplication, requiring many individual multiplications and additions (these terms are also one of the things which makes the equation non-linear) gαβ is the matrix inverse of g αβ, which not only requires many multiplications and additions to calculate, but makes the differential equation even more non-linear! if you choose a different coordinate system (such as spherical), the format of the derivatives become even more complicated! Solving the Equation

6 In general, Einstein s equation by itself does not contain enough information to determine gμν. Various additional equations, including equations of state, constraint equations, continuity equations, and/or gauge conditions are needed to provide enough information to uniquely determine gμν. These tend to be problemspecific, and are too complicated to describe here simply, but are discussed in GR3x. Once you find gμν, you calculate the line element, which tells you how long a small piece of stretched spacetime actually is : ds 2 = g μν dx u dx v = μ ν [g μν dx u dx v ] Which in Cartesian coordinates (for example) becomes ds 2 = g 00 dt dt + g 01 dt dx + g 02 dt dy + g 03 dt dz + g 10 dx dt + g 11 dx dx + g 12 dx dy + g 13 dx dz + g 20 dy dt + g 21 dy dx + g 22 dy dy + g 23 dy dz + g 30 dz dt + g 31 dz dx + g 32 dz dy + g 33 dz dz Then you can compute the paths of objects in spacetime by using the geodesic equation, which gives four equations of motion. Various initial conditions and conservation laws may also be needed. The geodesic equation for no non-gravitational forces is : 2 x α / τ 2 + Γ α βγ ( x β / τ) ( x γ / τ) = 0 this gives 4 equations, for α = (t,x,y,z) or 2 x α / τ 2 + ½(g δα g βδ / x γ + g δα g γδ / x β g δα g βγ / x δ ) ( x β / τ) ( x γ / τ) = 0 which becomes 2 x α / τ 2 + ½ β γ δ [g δα ( g βδ / x γ + g γδ / x β g βγ / x δ ) ( x β / τ) ( x γ / τ)] = 0 And then you re done! Simple, huh? BTW, in the Newtonian limit (speeds are small compared to c, and the gravitational field is weak, so g η) : Then for x 0 = t τ, 2 x 0 / τ 2 = 1 and x a / τ 0 And for x 1 = x : 2 x 1 / t 2 + Γ 1 00 = 0 so 2 x/ t 2 = Γ 1 00 But Γ 1 00 = g11( g10/ x 0 + g01/ x 0 g00/ x 1 )/2 = ( g00/ x)/2 (because g11 = η11 = 1) So 2 x/ t 2 = ( g00/ x)/2

7 But in Newtonian physics where φ is the gravitational potential, 2 x/ t 2 = φ/ x So φ = g00/2 + c or g00 = 2φ + c (where c is some constant, not the speed of light)