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. Extension to General Relativity Komar Mass, requires existence of Killing vector ADM Mass, h ab is the expansion of g ab around Minkowski

7. Extension to General Relativity Komar Mass, requires existence of Killing vector ADM Mass, h ab is the expansion of g ab around Minkowski Small note: These definitions hold in an Asymptotically flat spacetime

8. “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

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

12. “Asymptotically Flat?” Einstein Universe – i 0 “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

13. “Asymptotically Flat?” Asymptotically simple: – (M,g ab ) is an open submanifold of (M,g ab ) with smooth boundary – There exists a smooth scalar field  such that  ( ) = 0 d  ( ) not 0 g ab =  2 g ab – Every null geodesic in M begins and ends on – Asymptotically flat: Asymptotically simple R ab =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)