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
–

Momentum
–

Linear translation
Parity
–

Time translation
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  nˆ dA

M  1    nˆ dA
4
Extension to General Relativity

Q   0  E  nˆ dA
M   1   abcd c d
8 S

b
M1
(

h


h
)
n
dA 
b aa
16 S a ab
Komar Mass, requires
existence of Killing
vector
ADM Mass, hab is the
expansion of gab
around Minkowski
Extension to General Relativity

Q   0  E  nˆ dA
M   1   abcd c d
8 S

b
M1
(

h


h
)
n
dA 
b aa
16 S a ab
Komar Mass, requires
existence of Killing
vector
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,

g ab   ab  hab
hab 
1
R2
 c hab 
Problems:
–
–
1
R3
 c  d hab 
1
R4
Expansion might not be
possible for a general
metric
Exchanging limits and
derivatives causes
issues
“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

1
2
2
ds 

4
dUdV

sin
(
V

U
)
d

4 cos 2 U cos 2 V
2

“Asymptotically Flat?”

Better: Conformal mapping to put “infinity” in a finite
place

1
2
2
ds 

4
dUdV

sin
(
V

U
)
d

4 cos 2 U cos 2 V
2
g~  (2 cos U cos V ) 2 g
T  U V
R  V U
ds 2  dT 2  dR 2  sin 2 R d 2

“Asymptotically Flat?”
ds 2  dT 2  dR 2  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)