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Conserved Quantities in General Relativity

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					Conserved Quantities in General
          Relativity



               A story about
               asymptotic flatness
Conserved quantities in physics

   Charge
   Mass
   Energy
   Momentum
   Parity
   Lepton Number
Conserved quantities in physics

   Energy
    –   Time translation
   Momentum
    –   Linear translation
   Parity
    –   Inversion
   Charge
    –   Phase of the gauge field
Measurement

   Direct
    –   Scales, meter sticks
   Indirect
    –   Fields due to the conserved quantity
Measurement

   Direct
   Indirect
    –   Fields due to the conserved quantity
                    
          Q   0  E  n dA
                        ˆ
                         
          M  1    n dA ˆ
                  4
Extension to General Relativity

                                  
                        Q   0  E  n dA
                                      ˆ



           8 S
  M 1           abcd  c d              Komar Mass, requires
                                             existence of Killing
                                             vector
   16 S
M1       ( a hab   b haa ) nb dA        ADM Mass, hab is the
                                             expansion of gab
                                             around Minkowski
Extension to General Relativity

                                  
                        Q   0  E  n dA
                                      ˆ



           8 S
  M 1           abcd  c d              Komar Mass, requires
                                             existence of Killing
                                             vector
   16 S
M1       ( a hab   b haa ) nb dA        ADM Mass, hab is the
                                             expansion of gab
                                             around Minkowski
                Small note: These definitions hold in an
                Asymptotically flat spacetime
“Asymptotically Flat?”

   Intuitively,

      g ab   ab  hab
      hab    1
              R2

       c hab     1
                   R3

       c  d hab      1
                        R4
“Asymptotically Flat?”

   Intuitively,                Problems:
                                 –   Expansion might not be
      g ab   ab  hab              possible for a general
                                     metric
      hab    1
              R2
                                 –   Exchanging limits and
                                     derivatives causes
       c hab     1
                   R3
                                     issues

       c  d hab      1
                        R4
“Asymptotically Flat?”

   Better: Conformal mapping to put “infinity” in a finite
    place
             ds  dt  dr  r d
                2        2          2    2   2

                             u tr
                             vtr
                             u  tan U
                             v  tan V



    ds 
       2        1
         4 cos2 U cos2 V
                             
                          4dUdV  sin 2 (V  U )d 2   
“Asymptotically Flat?”

   Better: Conformal mapping to put “infinity” in a finite
    place
       ds 
          2        1
            4 cos2 U cos2 V
                            
                             4dUdV  sin 2 (V  U )d 2   
                            ~
                            g  (2 cosU cosV ) 2 g
                            T  U V
                            R  V U


              ds 2  dT 2  dR2  sin 2 R d2
“Asymptotically Flat?”

    ds 2  dT 2  dR2  sin 2 R d2

   Einstein Universe
    –   i0 “spacelike infinity”       R=, T=0
    –   i+ “future timelike infinity” R=0, T=
    –        “future null infinity”   T= – R


   We’ve thus taken infinity and placed it
    in our extended spacetime
“Asymptotically Flat?”

   Asymptotically simple:
    –   (M,gab) is an open submanifold of (M,gab) with
        smooth boundary
    –   There exists a smooth scalar field  such that
            ( ) = 0
            d( ) not 0
            gab=2 gab
    –   Every null geodesic in M begins and ends on
    –   Asymptotically flat:
            Asymptotically simple
            Rab=0 in the neighbourhood of
“Asymptotically Flat?”

   What is asymptotically flat:
    –   Minkowski
    –   Schwarzchild
    –   Kerr
   What is not:
    –   De Sitter universe (no matter, positive
        cosmological constant)
    –   Schwarzschild-de Sitter lambdavacuum
    –   Friedmann – Lemaître – Robertson – Walker
            Homogenous, isotropically expanding (or contracting)

				
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posted:8/28/2012
language:Latin
pages:14