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Conserved Quantities in General Relativity

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

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  1. Conserved Quantities in General Relativity A story about asymptotic flatness

  2. Conserved quantities in physics • Charge • Mass • Energy • Momentum • Parity • Lepton Number

  3. Conserved quantities in physics • Energy • Time translation • Momentum • Linear translation • Parity • Inversion • Charge • Phase of the gauge field

  4. Measurement • Direct • Scales, meter sticks • Indirect • Fields due to the conserved quantity

  5. Measurement • Direct • Indirect • Fields due to the conserved quantity

  6. Komar Mass, requires existence of Killing vector ADM Mass, hab is the expansion of gab around Minkowski Extension to General Relativity

  7. 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

  8. “Asymptotically Flat?” • Intuitively,

  9. Intuitively, Problems: Expansion might not be possible for a general metric Exchanging limits and derivatives causes issues “Asymptotically Flat?”

  10. “Asymptotically Flat?” • Better: Conformal mapping to put “infinity” in a finite place

  11. “Asymptotically Flat?” • Better: Conformal mapping to put “infinity” in a finite place

  12. “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

  13. “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

  14. “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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