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# Conserved Quantities in General Relativity

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```									Conserved Quantities in General
Relativity

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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