Conserved Quantities in General Relativity

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

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Measurement

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

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Measurement

• Direct
• Indirect
• Fields due to the conserved quantity

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Extension to General Relativity

• Komar Mass, requires existence of Killing vector
• ADM Mass, hab is the expansion of gab around
  Minkowski

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Extension to General Relativity

• 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
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Asymptotically Flat?

• Intuitively,

9
Asymptotically Flat?

• Intuitively,

• Problems
• Expansion might not be possible for a general
  metric
• Exchanging limits and derivatives causes issues

10
Asymptotically Flat?

• Better Conformal mapping to put infinity in a
  finite place

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Asymptotically Flat?

• Better Conformal mapping to put infinity in a
  finite place

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Asymptotically Flat?

• Einstein Universe
• i0 spacelike infinity Rp, T0
• i future timelike infinity R0, Tp
• future null infinity Tp R
• Weve thus taken infinity and placed it in our
  extended spacetime

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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
• gabW2 gab
• Every null geodesic in M begins and ends on
• Asymptotically flat
• Asymptotically simple
• Rab0 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)