On quasi-local energy and the choice of reference

1. On quasi-local energy and the choice of reference 劉建良 Liu Jian-Liang 2008 01/08 Taidung school

2. Outline • Variation principle • Energy • Reference • Examples • The choice of reference • Conclusion

3. Variation principle • Least action principle : variation of the Lagrangian density  field equations. • Local diffeomorphisms : replace variation to Lie derivative  (field eq. terms)  define Hamiltonian density as • when field equations satisfied.

4. Variation principle • Least action principle : variation of the Lagrangian density  field equations. • Local diffeomorphisms : replace variation to Lie derivative  (field eq. terms)  define Hamiltonian density as • when field equations satisfied. conserve w.r.t vector N

5. Energy • Hamiltonian associated with time-like displacement vector field N. • Energy is defined by the conserved quantity with respect to time translation. • Quasi-local energy : energy associated with a spatial region which has a closed two-boundary . energy for time-like N

6. Reference configuration • Well-defined requirement : make the vanishing boundary term of at spatial infinityand, • Boundary condition : the vanishing boundary term of gives the suitable boundary conditions (for finite spatial region)  modify by introducing referenceconfiguration,  change the value of energy.

7. Reference configuration • Well-defined requirement : make the vanishing boundary term of at spatial infinityand, • Boundary condition : the vanishing boundary term of gives the suitable boundary conditions (for finite spatial region)  modify by introducing referenceconfiguration,  change the value of energy. Energy is reference dependence

8. Modification of boundary term • Application to GR, using the modification boundary term C. M. Chen, J. Nester, R. S. Tung, Phys. Rev. D72, 104020 (2005).

9. Examples • Schwarzschild metric • The reference frame • The displacement vector • Energy

10. Examples • Eddington-Finkelstein metric • The reference frame • The displacement vector • Energy

11. Examples • Painleve-Gullstrand metric • The reference frame • The displacement vector • Energy

12. The choice of reference • Spherical symmetric coframe : • Choose the flat reference : • and the unit time-like displacement vector : where, and are dual of and .

13. The choice of reference • Quasi-local energy then :

14. The choice of reference • Quasi-local energy then : arbitrary choices of N and reference will affect the value of energy

15. The choice of reference • Quasi-local energy then : • Find a condition to restrict the choices : arbitrary choices of N and reference will affect the value of energy

16. The choice of reference • Quasi-local energy then : • Find a condition to restrict the choices : arbitrary choices of N and reference will affect the value of energy and also where ,

17. The choice of reference • The quasi-local energy then : • Obtain the same energy value in the previous three cases of Schwarzschild geometry Both the quasi-local energy and the displacement vector depends only on the physical frame.

18. Conclusion • The quasi-local energy determined by the Hamiltonian boundary term depends on (1) unit time-like displacement vector N, (2) the reference. • Arbitrary choices of them affect the value of energy. • The restrictions of N and reference, which determine the value of energy with respect to the physical frame only.