Equations of motion in general relativity by krs20830

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									Gravitation and Cosmology
Lecture 27: Equations of motion in general relativity




Equations of motion in general relativity

Reading: Adler, Bazin & Schiffer, Introduction to General Relativity, Chapter 10.


How many equations?
We recall that in the theory of stellar structure, the equations for the gravitational field were
         −B’ ’      B’  B’     A’         1 B’
  Rtt =
          2A
                +
                    4A  B + A  − r A = −4πG (3p + ρ) B                              (25.7t)
                                     
            −B’ ’   B’  B’  A’   1 A’
    Rrr =         +     B + A  + r A = 4πG (ρ − p) A                                (25.7r)
             2B     4B         
                    r  B’  A’   1                 2
    Rθθ = −1 +         B − A  + A = −4πG (ρ − p) r                                  (25.7θ)
                   2A         

    Rϕϕ = sin2θ Rθθ                                                                   (25.7ϕ)
We also had an equation of hydrodynamic equilibrium
  dp     B’
      +     (p + ρ) = 0 .                                                             (25.12)
  dr     2B
and an equation of state
              ρ(r) 
  p(r) = ρc F        .
                ρ 
              c 
Now, Eq. 25.7ϕ is redundant with Eq. 25.7θ because of rotational invariance. But it might seem as
though we had 4 unknowns (A(r), B(r), ρ(r) and p(r)) and 5 equations to determine them. Are these
unknowns overdetermined? Is it possible to solve them at all?


Clearly, the equation of hydrodynamic equilibrium is an identity that follows from
  Rµν − 1 gµν R = Tµν ; ν = 0                                                    (27.1)
           2     ;ν
and so is not independent from it.
Einstein was the first to remark on the rather striking fact that the equations of motion describing
the motion of particles under gravitational forces follow from the field equations----rather than being
distinct from them. The subject was first elaborated by Einstein, Infeld and Hofmann† and then
simplified and extended by Einstein and Infeld‡. In this lecture I can only give you some of the flavor
of the theory, rather than the whole subject.


†    Einstein, Infeld and Hofmann, Ann. of Math. 39 (1938) 66.




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Gravitation and Cosmology
Motion of particles


Motion of particles
The basic equation is Tµν ; ν = 0 .

Isolate a piece of matter in a world-tube D: Clearly,
                                    µ
   √g Tµν ; ν = ∂ν ( √g Tµν) + √g   Tσν = 0 (27.2)
                               
                                    σ ν
Thus,
                                                  µ
    ∫ d4x ∂ν (√g Tµν)
                               +    ∫ d4x √g σ ν Tσν
                                                                  = 0         (27.3)
    D                               D                  

Now, since the matter is inside D, Tµν is zero on S2 , hence

                                                                                           Diagram of the world-tube D




    ∫ d x ∂ν (√g T
         4
                      µν
                            ) =      ∫ d4x ∂ν (√g Tµν)
                                                              =   ∫ d3x √g Tµν nν
                                                                                                 (27.4)
    D                           S1 + S3                        S1 + S3




where on S1 the unit normal is nν = −1, 0, 0, 0 and on S3 it is nν = (1, 0, 0, 0) . Now, suppose we
                                                 
employ locally freely-falling coordinates to describe the barycenter of the particles, for which
     dt
  √g
        = 1.                                                                          (27.5)
     dτ
Then if we ignore pressure (that is, Tµν refers to isolated, non-interacting particles----dust)
  Tµν = ρ Uµ Uν                                                                        (27.6)
and
                                                   dt                            dt 
    ∫ d x √g T
         3
                 µν
                       nν =     ∫ d3x ρ Uµ  dτ               −   ∫ d3x ρ Uµ  dτ 
S1 + S3                                                t=b                       t=a

                                    t=b
                                                 d2xµ
                            ≡   ∫t=a      dτ m
                                                 dτ2
                                                                                                  (27.7)

             df
where m =         ∫ d3x ρ .




‡       Einstein and Infeld, Can. J. Math. 1 (1949) 209.




                                                                         128
Gravitation and Cosmology
Lecture 27: Equations of motion in general relativity


Similarly we use Eq. 27.5 to write
                                                              t=b            σ   ν
             µ                     µ                             µ  dx dx
   ∫ d4x √g σ ν Tσν = ∫ d3x dτ σ ν ρ Uσ Uν ≈ ∫t=adτ
                                                                   σ ν m dτ dτ       (27.8)
  D                       D                                      
so that
                     µ  dx dx 
    t=b       d2xµ         σ  ν
   ∫t=adτ   m 2 +              = 0.                                                 (27.9)
              dτ   σ ν dτ dτ 


That is, we recover the equation of motion for a test particle in a gravitational field,
  d2xµ               σ
            µ  dx dx
                         ν
         +               = 0
  dτ2      σ ν dτ dτ
since the range of integration over τ may be as restricted as much as we like. We have therefore shown
that the conservation of the energy-momentum tensor in the presence of a gravitational field (as
represented by the geometry of space-time), which follows from the Bianchi identity
   Rµν − 1 gµν R = 0 ,
           2       ;ν
is equivalent to Newton’s Second Law with gravitation treated as an external force. It is this possibility
that has led J.A. Wheeler to coin the term ‘‘geometrodynamics’’----and that led Einstein in the last
years of his life to seek a geometrical description of all of physics.




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Gravitation and Cosmology
Motion of particles




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