Relativity in Action - Wake Forest Universityusers.wfu.edu/cookgb/YorkFest/Talks/Brown_YorkFest.pdfDavid Brown (Cornell, 2004) ﬁDynamical origin of black hole radianceﬂ, Phys.

Relativity in Action

David Brown (Cornell, 2004)

“Dynamical origin of black hole radiance”, Phys. Rev. D28 (1983) 2929-2945. “What happens to the horizon when a black hole radiates?” in Quantum Theory of Gravity

edited by Steven M. Christensen, (Adam Hilger, Bristol, 1984) 135-147. “Black hole in thermal equilibrium with a scalar field: The back-reaction”, Phys. Rev. D31

(1985) 775-784. “Black-hole thermodynamics and the Euclidean Einstein action”, Phys. Rev. D33 (1986)

2092-2099. “Density of states for the gravitational field in black-hole topologies”, Phys. Rev. D36

(1987) 3614-3625. (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) 573-596. “Action principle and partition function for the gravitational field in black-hole topologies”,

Phys. Rev. Lett. 61 (1988) 1336-1339. (with B.F. Whiting) “Action and free energy for black hole topologies”, Physica A 158 (1989) 425-436. “Additivity of the entropies of black holes and matter in equilibrium”, Phys. Rev. D40

(1989) 2124-2127. (with E.A. Martinez) “Additivity of the entropies of black holes and matter” in Proceedings of the 14th Texas

Symposium on Relativistic Astrophysics, (Annals N.Y. Acad. Sci) 336-343. (with E.A. Martinez)

Jimmy's papers on Black Hole Thermodynamics

“Thermodynamic ensembles and gravitation”, Class. Quantum Grav. 7 (1990) 1433-1444. (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) 3376-3385.

(with H.W. Braden, J.D. Brown, and B.F. Whiting) “Thermodynamics of black holes and cosmic strings”, Phys. Rev. D42 (1990) 3580-3583.

(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) 225-234. (with J.D. Brown and E.A. Martinez)

“Complex Kerr-Newman geometry and black-hole thermodynamics”, Phys. Rev. Lett. 66 (1991) 2281-2284. (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) 129-142. (with J.D. Brown) “Quasilocal energy and conserved charges derived from the gravitational action”, Phys.

Rev. D47 (1993) 1407-1419. (with J.D. Brown) “Microcanonical functional integral for the gravitational field”, Phys. Rev. D47 (1993)

1420-1431. (with J.D. Brown)

“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) 465-474. (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) 28-37. (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) 388-395.

“Positivity of entropy in the semiclassical theory of black holes and radiation”, Phys. Rev.

D48 (1993) 479-484. (with D. Hochberg and T.W. Kephart)

“Semiclassical black hole in thermal equilibrium with a nonconformal scalar field”, Phys.

Rev. D50 (1994) 6427-6434. (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 5th Canadian Conference on General Relativity and Relativistic Astrophysics edited by R.B. Mann and R.G. McLenaghan (World Scientific, Singapore, 1994) 42-52.

“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) 23-34.

(with J.D. Brown)

“Effective potential of a black hole in thermal equilibrium with quantum fields”, Phys. Rev.

D49 (1994) 5257-5265. (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 7th International Marcel Grossmann Meeting edited by R.T. Jantzen and

G.M. Keiser (World Scientific, Singapore, 1996) 952-954. (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 7th International Marcel Grossmann Meeting edited by R.T. Jantzen and G.M. Keiser (World Scientific, Singapore, 1996) 952-954.

(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) 1977-1984. (with J.D. Brown and S.R. Lau)

“Canonical quasilocal energy and small spheres”, Phys. Rev. D59 (1999) 064028. (with

J.D. Brown and S.R. Lau)

“Action and energy of the gravitational field”, Ann. Phys. 297 (2002) 175-218. (with J.D.

Brown and S.R. Lau)

“Dynamical origin of black hole radiance”, Phys. Rev. D28 (1983) 2929-2945. “What happens to the horizon when a black hole radiates?” in Quantum Theory of Gravity

edited by Steven M. Christensen, (Adam Hilger, Bristol, 1984) 135-147. “Black hole in thermal equilibrium with a scalar field: The back-reaction”, Phys. Rev. D31

(1985) 775-784. “Black-hole thermodynamics and the Euclidean Einstein action”, Phys. Rev. D33 (1986)

2092-2099. “Density of states for the gravitational field in black-hole topologies”, Phys. Rev. D36

(1987) 3614-3625. (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) 573-596. “Action principle and partition function for the gravitational field in black-hole topologies”,

Phys. Rev. Lett. 61 (1988) 1336-1339. (with B.F. Whiting) “Action and free energy for black hole topologies”, Physica A 158 (1989) 425-436. “Additivity of the entropies of black holes and matter in equilibrium”, Phys. Rev. D40

(1989) 2124-2127. (with E.A. Martinez) “Additivity of the entropies of black holes and matter” in Proceedings of the 14th Texas

Symposium on Relativistic Astrophysics, (Annals N.Y. Acad. Sci) 336-343. (with E.A. Martinez)

Jimmy's papers on Black Hole Thermodynamics

“Black-hole thermodynamics and the Euclidean Einstein action”, Phys. Rev. D33 (1986) 2092-2099.

“Black-hole thermodynamics and the Euclidean Einstein action”, Phys. Rev. D33 (1986) 2092-2099.

Change of boundary terms/boundary conditions in the action

{Canonical transformation (dynamical level)

Legendre transformation (thermodynamical level)

Gravity is the essential ingredient!

...this talk...

NumNunuullkj

Initial value problem

Formulations of Einstein's equations

Numerical relativity

Action

Usual approach tonumerical modeling:

Continuum Eqnsof Motion

Discretize

Discrete Eqnsof Motion


Action


Discretize


Vary


Action


Discretize


Variational Integratorapproach to numerical modeling*:

Vary

Action

Discretize

Vary

Discrete Action


*Kane, Marsden, Ortiz, Patrick, Pekarsky, Shkoller, West

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.

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.

Hamiltonian mechanics

Time

0 1 2 3

1 2 3zone centers: n =

nodes: n =

Variation of the action

Example: Harmonic oscillators with nonlinear coupling

Example: Harmonic oscillators with nonlinear coupling

Why is energy conservation so important?

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...

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.

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 (becausethe variables that are determined by the equations of motion are also constrained). ---JWY

What if we apply the variational integrator construction to this action?

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.

For the coupled oscillators...

The time step adjusts itself during the evolution to maintain the constraint.

For the coupled oscillators...

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 becomeLagrange (determined) multipliers in the discrete theory.

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:

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?

Consider the ADM action with periodic BC's:

The VI equations have the form:

to be solved for

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.

Example: GR with plane symmetry described by the model*

*Solutions are cosmologies with a “big crunch”.

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

Lapse at 5, 10, 15,...,40 timesteps:

Lapse function at the second timestep:

Log(L2 norm of Hamiltonian constraint) versus timefor solutions of varying numerical accuracy.

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

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 metricon 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!

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,...

*Phys. Rev. Lett. 82 (1999) 1350

implies

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

These are the extended conformal thin sandwich equations!(Pfeiffer and York, PRD67 (2003) 044022)

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

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.

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.

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 thecourse 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.)

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.

Relativity in Action - Wake Forest Universityusers.wfu.edu/cookgb/YorkFest/Talks/Brown_YorkFest.pdfDavid Brown (Cornell, 2004) ﬁDynamical origin of black hole radianceﬂ, Phys.

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