HP_Hamiltonian Formulation of General Relativity

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Hamiltonian Formulation of General

Relativity

Hridis Kumar Pal

UFID: 4951-8464

Project Presentation for PHZ 6607, Special and General Relativity I

Fall, 2008

Department of Physics

University of Florida



Outline

Introduction

Review of Hamiltonian Mechanics

Hamiltonian Mechanics for Point Particles

Hamiltonian Mechanics for Classical Fields

Constrained Hamiltonian Formulation for Dynamical Systems

Formulating GR from a Hamiltonian Viewpoint: The ADM Formalism

The Lagrangian in GR

The Hamiltonian in GR

The Equations in GR

Applications and Misconceptions

Questions, Comments and Acknowledgements



Introduction

• Several alternative formulations of GR exist. Hamiltonian formulation is

just one of them.

• Even for the Hamiltonian formulation, there are more than one ways.

• First attempts towards such a formulation was by Pirani et. al. after

Dirac proposed his idea of constrained dynamics in 1949-Not complete.

• Next Dirac himself visited this problem later.

• Shortly thereafter Arnowitt, Deser, and Misner came up with a

Hamiltonian formulation of GR which was satisfactory and later came to

be called as the ADM formalism*.

We will discuss the ADM formalism of GR.

* Arnowitt, Deser and Misner, "Gravitation: An Introduction to Current Research" (1962) 227.



Review of Hamiltonian Mechanics: Point Particles*

Lagrangian formulation

• Describe the system with n independent degrees of freedom by a set of

n generalized coordinates {qi}.

• Construct the Lagrangian as:

• Define the Action as :

• Use Hamilton’s principle to find the extremum of this action resulting inthe Euler-Lagrange equations:

*H. Goldstein, C. Poole and J. Safko, Classical Mechanics, Pearson Education Asia (2002)



Review of Hamiltonian Mechanics: Point Particles (contd…)

Hamiltonian Formulation

• System defined by 2n generalized coordinates {qi,pi}, where

• Construct the Hamiltonian from the Lagrangian by means of aLegendre transformation as:

• Hamilton’s equations of motion:



Review of Hamiltonian Mechanics: Classical Fields*

• qi →Φ(xµ)

• The lagrangian is related to the Lagrangian density:

• Euler-Lagrange equations of motion, which are covariant in nature:

• Similarly define the Hamiltonian density as:

where is the conjugate momentum density

• Hamilton’s equations become:

*same as before



Constrained Hamiltonian Formulation for Dynamical

Systems*

• Constrained systems are very common in nature. E.g., a simple

pendulum.

• Any field theory with gauge freedom will have in-built constraints.

• The formal theory to tackle constrained system within the Hamiltonian

formulation was first given by Dirac who made use of Poisson

brackets.**

• We will however not go through the details of Dirac’s theory, rather take

the example of the electromagnetic field and learn the the essential

ideas.

• Later, when formulating GR we will follow the same ideas that we learn

in this simple example.

*R. M. Wald, General Relativity, The University of Chicago Press (1984)

**B. Whiting, Constrained Hamiltonian Systems: Notes (unpublished) available now on the course website



Constrained Hamiltonian Formulation for Dynamical Systems

(contd…)

• Consider a system with n generalized coordinates with m constraint

equations of the form:

• Use m lagrange undetermined multipliers λα and extremize:

• We now have (n+m) equations in (n+m) unknowns which can be

solved.

• Imagine now that the λα’s are coordinates too. Take L to be

• Then

• → →

• Reverse the argument now: If conjugate momentum =0, that degree of

freedom is constrained and the constraint is hidden in the lagrangian




Systems: Example*

• Consider the EM lagrangian with no source:

• The conjugate momentum densities are:

• The Hamiltonian becomes:

*same as before




Systems: Example (contd…)

• The Hamiltonian equations of motion are:

• Clearly the first one, which is Gauss’s law is the constraint equation

and the other two are evolution equations.



GR from Hamiltonian Point of View: The ADM Formalism

-The Lagrangian in GR*

• The dynamical variable in GR is the metric gμν

• The Lagrangian density for curved spacetime is:

• The action is given by (called the Hilbert action):

*S. Carroll, Spacetime and Geometry: An Introduction to General Relativity, Addison Wesley (2004)



The Hamiltonian in GR*

• Again we start with the dynamical variable gμν.

• But there is a problem-unlike the Lagrangian formulation, the

Hamiltonian formulation is not spacetime covariant.

• Time is singled out from the space part in Hamiltonian formulation

• Against the ‘spirit’ of GR.

• Way out?

• Theorem: Let (M, g μν ) be a globally hyperbolic spacetime. Then (M, g μν )

is stably causal. Furthermore, a global time function, f, can be chosen

such that each surface of constant f is a Cauchy surface. Thus M can

be foliated by Cauchy surfaces and the topology of M is R ×Σ , where Σ

denotes any Cauchy surface• Armed with this we now foliate our spacetime into Cauchy

hypersurfaces, Σt, parameterized by a global function t.

* R. M. Wald, General Relativity, The University of Chicago Press (1984)



The Hamiltonian in GR (contd…)

• Let tμ be a vector field on M such that

• Define:

• gμν → (hij,N,N j)




A few definitions:

• Lie derivative:

• Exterior derivative:

• Extrinsic curvature:




• Using the new variables, the Lagrangian density becomes:

• The canonical conjugate momentum densities are:

• The Hamiltonian density becomes:



The dynamical and constraint equations in GR*

• The constraint equations are:

• The dynamical equations are:

• This completes the derivation. *as before



Applications and Misconceptions

• Uses*

Canonical quantum gravity: any quantum field theory requires a

Hamiltonian formulation of the corresponding classical field theory to

begin with. The same is true for the quantum theory of gravitation.

The resulting equations are called Wheeler-De Witt equations

Numerical GR: Einstein’s equations are a set of 10 non-linearsecond order partial differential equations which are difficult to

handle both analytically and numerically. The ADM formalism which

breaks the equations into constraints and evolution equations is

well-suited for numerical simulations

*For more details, see J. E. Nelson, arXiv: gr-qc/0408083.



Applications and Misconceptions (contd…)

• Myths and Reality*

A 3+1 decomposition of space and time is not an absolute

necessity for Hamiltonian description of GR

The claim that the canonical treatment invariably breaks the space-

time symmetry and the algebra of constraints is not the algebra of

four-dimensional diffeomorphism is not true

Common wisdom which holds Dirac’s analyses and ADM ideas

about the canonical structure of GR to be equivalent is

questionable

*N. Kiriushcheva and S. V. Kuzmin, arXiv: 0809.0097v1 [gr-qc]

Kiriushcheva, et. al., Phys. Lett. A 372, 5101 (2008)



Acknowledgements:

Prof. Bernard Whiting, UF for his helpful comments andsuggestions

P. Mineault, McGill University for uploading on the web hispaper on the same subject

Google, without which this project would never be possible!

Questions and Comments?

THANK YOU

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