Embed Size (px)
Citation preview
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
dAnM
dAnEQ
ˆ4
1
ˆ0
Extension to General Relativity
Komar Mass, requires existence of Killing vector
ADM Mass, hab is the expansion of gab around Minkowski
S
dcabcdM 8
1
dAnEQ ˆ0
dAnhhM b
S aababa )(161
Extension to General Relativity
Komar Mass, requires existence of Killing vector
ADM Mass, hab is the expansion of gab around Minkowski
S
dcabcdM 8
1
dAnEQ ˆ0
dAnhhM b
S aababa )(161
Small note: These definitions hold in anAsymptotically flat spacetime
“Asymptotically Flat?”
Intuitively,
4
3
2
1
1
1
Rabdc
Rabc
Rab
ababab
h
h
h
hg
“Asymptotically Flat?”
Intuitively, Problems:– Expansion might not be
possible for a general metric
– Exchanging limits and derivatives causes issues
4
3
2
1
1
1
Rabdc
Rabc
Rab
ababab
h
h
h
hg
“Asymptotically Flat?”
Better: Conformal mapping to put “infinity” in a finite place
22222 drdrdtds
Vv
Uu
rtv
rtu
tan
tan
2222
2 )(sin4coscos4
1 dUVdUdV
VUds
“Asymptotically Flat?”
Better: Conformal mapping to put “infinity” in a finite place
UVR
VUT
gVUg
2)coscos2(~
2222
2 )(sin4coscos4
1 dUVdUdV
VUds
22222 sin dRdRdTds
“Asymptotically Flat?”
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
22222 sin dRdRdTds
“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 0gab=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)
