Relativity in Action. David Brown (Cornell, 2004)

Transcription

1 Relativity in Action David Brown (Cornell, 2004)

2 Jimmy's papers on Black Hole Thermodynamics Dynamical origin of black hole radiance, Phys. Rev. D28 (1983) What happens to the horizon when a black hole radiates? in Quantum Theory of Gravity edited by Steven M. Christensen, (Adam Hilger, Bristol, 1984) Black hole in thermal equilibrium with a scalar field: The back-reaction, Phys. Rev. D31 (1985) Black-hole thermodynamics and the Euclidean Einstein action, Phys. Rev. D33 (1986) Density of states for the gravitational field in black-hole topologies, Phys. Rev. D36 (1987) (with H.W. Braden and B.F. Whiting) Black holes and partition functions in Proceedings of the Osgood Hill Conference on Conceptual Problems in Quantum Gravity (publisher, 1988) Action principle and partition function for the gravitational field in black-hole topologies, Phys. Rev. Lett. 61 (1988) (with B.F. Whiting) Action and free energy for black hole topologies, Physica A 158 (1989) Additivity of the entropies of black holes and matter in equilibrium, Phys. Rev. D40 (1989) (with E.A. Martinez) Additivity of the entropies of black holes and matter in Proceedings of the 14 th Texas Symposium on Relativistic Astrophysics, (Annals N.Y. Acad. Sci) (with E.A. Martinez)

3 Thermodynamic ensembles and gravitation, Class. Quantum Grav. 7 (1990) (with J.D. Brown, G.L. Comer, E.A. Martinez, J. Melmed and B.F. Whiting) Charged black hole in a grand canonical ensemble, Phys. Rev. D42 (1990) (with H.W. Braden, J.D. Brown, and B.F. Whiting) Thermodynamics of black holes and cosmic strings, Phys. Rev. D42 (1990) (with E.A. Martinez) Title in Conceptual Problems of Quantum Gravity edited by A. Ashtekar and J. Stachel (Birkhauser, Boston, 1991). Rotating black holes, complex geometry, and thermodynamics in Nonlinear Problems In Relativity and Cosmology edited by J.R. Buchler, S. Detweiler and J. Ipser (New York Academy of Sciences, New York, 1991) (with J.D. Brown and E.A. Martinez) Complex Kerr-Newman geometry and black-hole thermodynamics, Phys. Rev. Lett. 66 (1991) (with J.D. Brown and E.A. Martinez) Quasilocal energy in general relativity in Mathematical Aspects of Classical Field Theory edited by M.J. Gotay, J.E. Marsden, and V. Moncrief (American Mathematical Society, Providence, 1992) (with J.D. Brown) Quasilocal energy and conserved charges derived from the gravitational action, Phys. Rev. D47 (1993) (with J.D. Brown) Microcanonical functional integral for the gravitational field, Phys. Rev. D47 (1993) (with J.D. Brown)

4 Temperature and time in the geometry of rotating black holes in Physical Origins of Time Asymmetry edited by J.J. Halliwell, J. Perez-Mercader, and W. Zurek (Cambridge University Press, Cambridge, 1994) (with J.D. Brown) Jacobi's action and the density of states in Directions in General Relativity volume 2, edited by B.L. Hu and T.A. Jacobson (Cambridge University Press, Cambridge, 1993) (with J.D. Brown) The back reaction is never negligible: Entropy of black holes and radiation in Directions in General Relativity volume 1, edited by B.L. Hu, M.P. Ryan, and C.V. Vishveshwara (Cambridge University Press, Cambridge, 1993) Positivity of entropy in the semiclassical theory of black holes and radiation, Phys. Rev. D48 (1993) (with D. Hochberg and T.W. Kephart) Semiclassical black hole in thermal equilibrium with a nonconformal scalar field, Phys. Rev. D50 (1994) (with P.R. Anderson, W.A. Hiscock and J. Whitesell) Energy, temperature, and entropy of black holes dressed with quantum fields in Proceedings Of the 5 th Canadian Conference on General Relativity and Relativistic Astrophysics edited by R.B. Mann and R.G. McLenaghan (World Scientific, Singapore, 1994) Microcanonical action and the entropy of a rotating black hole in Physics on Manifolds edited by M. Flato, R. Kerner, and A. Lichnerowicz (Kluwer, Dordrecht, 1994) (with J.D. Brown)

5 Effective potential of a black hole in thermal equilibrium with quantum fields, Phys. Rev. D49 (1994) (with D. Hochberg and T.W. Kephart) The path integral formulation of gravitational thermodynamics in The Black Hole 25 Years After edited by C. Teitelboim and J. Zanelli (World Scientific, Singapore, 1988). (with J.D. Brown) Semiclassical black hole in thermal equilibrium with a nonconformal scalar field in Proceedings of the 7 th International Marcel Grossmann Meeting edited by R.T. Jantzen and G.M. Keiser (World Scientific, Singapore, 1996) (with J. Whitesell, W.A. Hiscock and P.R. Anderson) Effects of quantum fields on the space-time geometries, temperatures, and entropies of static black holes in Proceedings of the 7 th International Marcel Grossmann Meeting edited by R.T. Jantzen and G.M. Keiser (World Scientific, Singapore, 1996) (with P.R. Anderson, H. Bahrani, W.A. Hiscock, and J. Whitesell) Energy of isolated systems at retarded times as the null limit of quasilocal energy, Phys. Rev. D55 (1997) (with J.D. Brown and S.R. Lau) Canonical quasilocal energy and small spheres, Phys. Rev. D59 (1999) (with J.D. Brown and S.R. Lau) Action and energy of the gravitational field, Ann. Phys. 297 (2002) (with J.D. Brown and S.R. Lau)

6 Jimmy's papers on Black Hole Thermodynamics Dynamical origin of black hole radiance, Phys. Rev. D28 (1983) What happens to the horizon when a black hole radiates? in Quantum Theory of Gravity edited by Steven M. Christensen, (Adam Hilger, Bristol, 1984) Black hole in thermal equilibrium with a scalar field: The back-reaction, Phys. Rev. D31 (1985) Black-hole thermodynamics and the Euclidean Einstein action, Phys. Rev. D33 (1986) Density of states for the gravitational field in black-hole topologies, Phys. Rev. D36 (1987) (with H.W. Braden and B.F. Whiting) Black holes and partition functions in Proceedings of the Osgood Hill Conference on Conceptual Problems in Quantum Gravity (publisher, 1988) Action principle and partition function for the gravitational field in black-hole topologies, Phys. Rev. Lett. 61 (1988) (with B.F. Whiting) Action and free energy for black hole topologies, Physica A 158 (1989) Additivity of the entropies of black holes and matter in equilibrium, Phys. Rev. D40 (1989) (with E.A. Martinez) Additivity of the entropies of black holes and matter in Proceedings of the 14 th Texas Symposium on Relativistic Astrophysics, (Annals N.Y. Acad. Sci) (with E.A. Martinez)

7 Black-hole thermodynamics and the Euclidean Einstein action, Phys. Rev. D33 (1986)

8 Black-hole thermodynamics and the Euclidean Einstein action, Phys. Rev. D33 (1986) Change of boundary terms/boundary conditions in the action {Canonical transformation (dynamical level) Legendre transformation (thermodynamical level) Gravity is the essential ingredient!

9 ...this talk... Action N u m N u n Initial value u problem u Formulations l of Einstein's equations l Numerical k relativity j

10 Usual approach to numerical modeling: Continuum Eqns of Motion Discretize Discrete Eqns of Motion

11 Usual approach to numerical modeling: Action Vary Continuum Eqns of Motion Discretize Discrete Eqns of Motion

12 Usual approach to numerical modeling: Action Variational Integrator approach to numerical modeling*: Action Vary Discretize Continuum Eqns of Motion Discrete Action Discretize Vary Discrete Eqns of Motion Discrete Eqns of Motion *Kane, Marsden, Ortiz, Patrick, Pekarsky, Shkoller, West

13 WHY BOTHER? The action principle provides a unique perspective that can lead to important insights into the physical and mathematical descriptions of a dynamical system.

14 WHY BOTHER? The action principle provides a unique perspective that can lead to important insights into the physical and mathematical descriptions of a dynamical system. Variational integrators typically do a superior job of conserving energy.

15 Hamiltonian mechanics zone centers: n = nodes: n = Time

16 Variation of the action

17 Example: Harmonic oscillators with nonlinear coupling

18 Example: Harmonic oscillators with nonlinear coupling

19 Why is energy conservation so important?

20 Why is energy conservation so important? Because energy conservation in mechanics is analogous to the preservation of constraints in general relativity. This is seen most clearly with the parametrized form of the action...

21 Preservation of the constraints in numerical relativity is crucial. As documented by the Cornell/CalTech group, the run time for free evolution codes is limited by unphysical, exponentially growing, constraint violating modes. Variational integrators are good at staying close to the constraint hypersurface, at least for mechanical systems.

22 Look again at the parametrized form of the action... This system, like GR, is both underdetermined (because the equations of motion don't determine the lapse) and overdetermined (because the variables that are determined by the equations of motion are also constrained). ---JWY What if we apply the variational integrator construction to this action?

23 VI equations of motion: The dot equations do not preserve the value of the constraint. The system is not overdetermined/underdetermined! Five equations for five unknowns, with the constraint explicitly enforced at each timestep.

24 For the coupled oscillators... The time step adjusts itself during the evolution to maintain the constraint.

25 For the coupled oscillators...

26 By discretizing the action for a constrained Hamiltonian system, then varying that action, we obtain discrete equations of motion that explicitly maintain the constraints. The price we pay is a loss of freedom to choose the evolution for the constraint multipliers. The undetermined multipliers of the continuum theory become Lagrange (determined) multipliers in the discrete theory.

27 Is this price too high to pay for numerical relativity? Not necessarily. If needed, we can always guide the time slices by making occasional adjustments using the prior spacetime solution:

28 Let's boldly forge ahead and apply the VI construction to general relativity. Some issues I want to avoid for now: What is the prefered form of the action/equations of motion at the continuum level? Can we write an action for the Einstein-Christoffel system? BSSN? (Answer is yes, with some caveats...) How do we implement spatial boundary conditions that are consistent with the variational principle, and physically appropriate for radiating systems?

29 Consider the ADM action with periodic BC's: The VI equations have the form: to be solved for

30 Comments: The form of the equations depends on the choice of whether the coordinates, momenta, and multipliers are cell centered or node centered in space and in time. In practice this choice is very important! The equations are implicit, making them a challenge to solve. The equation obtained by varying the metric ranges from n = 1,...,N-1, while the other equations range from n = 0,...,N-1. More on this later.

31 Example: GR with plane symmetry described by the model* *Solutions are cosmologies with a big crunch.

32 Metric function g(x) after 0 th and fifth timesteps

33 Lapse function at the second timestep: Lapse at 5, 10, 15,...,40 timesteps:

34 Log(L2 norm of Hamiltonian constraint) versus time for solutions of varying numerical accuracy.

35 However...we can't solve for the shift vector (i.e. we can't solve the momentum constraint). Let's take a closer look:

36 However...we can't solve for the shift vector (i.e. we can't solve the momentum constraint). Let's take a closer look: We want to solve the constraints at each timestep. In particular for the initial timestep (remember the n=0 equations), we need to solve This is a discrete version of the original thin sandwich problem of Baierlein, Sharp and Wheeler: Given the metric on two nearby time slices, and the definition for the extrinsic curvature, solve the Hamiltonian and momentum constraints for the lapse and shift. Sometimes it works, but not generically!

37 We need to make the thin sandwich problem well posed. Jimmy has shown us how to do it with his conformal thin sandwich construction:* split the metric into a conformal factor and a background split the extrinsic curvature (momentum) into its trace and trace-free parts add conformal weights,... implies *Phys. Rev. Lett. 82 (1999) 1350

38 Throw in a canonical transformation to make tau a coordinate with -phi^6 its conjugate momentum... Now discretize the action. The initial timestep (n = 0) equations are discrete versions of

39 These are the extended conformal thin sandwich equations! (Pfeiffer and York, PRD67 (2003) )

40 The essential content of the extended conformal thin sandwich equations is a set of five nice, well behaved elliptic equations for the lapse, shift, and conformal factor. Generically, we expect solutions to exist. The discrete action principle directs us to choose as free initial data. The level n = 0 equations are the conformal thin sandwich equations in extended form. These determine the remaining initial data

41 Question: can we always solve the n = 1, 2,... equations for the coordinates, momenta, and multipliers? Answer: I don't know. There is no continuum version of this question.

42 Question: Is the loss of freedom to choose the lapse and shift too high of a price to pay? If so... Option 1: As mentioned before, guide the slices by using the spacetime solution to construct new initial data during the course of the evolution. Option 2: Don't solve the constraints. The VI discretization of the evolution equations might do a good enough job of staying close to the constraint hypersuface anyway. (Recall the first example of the coupled oscillators.)

43 Question: Is the loss of freedom to choose the lapse and shift too high to pay? If so... Option 3: Apply a studder-step evolution: Specify the lapse and shift Solve the evolution equations for the canonical coordinates and momenta at the next timestep Solve the conformal thin sandwich equations using the canonical coordinates at the two neighboring timesteps This procedure is well defined at each timestep and keeps the constraints satisfied. The lapse and shift are freely specified but then subject to a (perhaps small) correction. The price we pay is that the discrete evolution equations obtained by varying the action with respect to the coordinates (the momentum-dot equations) are not satisfied exactly.