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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
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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
Komar Mass, requires existence of Killing vector ADM Mass, hab is the expansion of gab around Minkowski Extension to General Relativity
Komar Mass, requires existence of Killing vector ADM Mass, hab is the expansion of gab around Minkowski Extension to General Relativity Small note: These definitions hold in an Asymptotically flat spacetime
“Asymptotically Flat?” • Intuitively,
Intuitively, Problems: Expansion might not be possible for a general metric Exchanging limits and derivatives causes issues “Asymptotically Flat?”
“Asymptotically Flat?” • Better: Conformal mapping to put “infinity” in a finite place
“Asymptotically Flat?” • Better: Conformal mapping to put “infinity” in a finite place
“Asymptotically Flat?” • Einstein Universe • i0 “spacelike infinity” R=p, T=0 • i+ “future timelike infinity” R=0, T=p • “future null infinity” T=p – 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 W such that • W( ) = 0 • dW( ) not 0 • gab=W2gab • 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)