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

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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

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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.

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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 Hamiltons principle to find the extremum of
  this action resulting in the Euler-Lagrange
  equations
• H. Goldstein, C. Poole and J. Safko, Classical
  Mechanics, Pearson Education Asia (2002)

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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 a Legendre transformation as
• Hamiltons equations of motion

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Review of Hamiltonian Mechanics Classical Fields

• qi ?F(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
• Hamiltons equations become
• 
  
  same as before

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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
  Diracs 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

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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 ?a and
  extremize
• We now have (nm) equations in (nm) unknowns
  which can be solved.
• Imagine now that the ?as 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
• 
  

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Constrained Hamiltonian Formulation for Dynamical
Systems Example

• Consider the EM lagrangian with no source
• The conjugate momentum densities are
• The Hamiltonian becomes
• same as before

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Constrained Hamiltonian Formulation for Dynamical
Systems Example (contd)

• The Hamiltonian equations of motion are
• Clearly the first one, which is Gausss law is
  the constraint equation and the other two are
  evolution equations.

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GR from Hamiltonian Point of View The ADM
Formalism-The Lagrangian in GR

• The dynamical variable in GR is the metric gµ?
• 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)

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The Hamiltonian in GR

• Again we start with the dynamical variable gµ?.
• 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, gµ?) be a globally hyperbolic
  spacetime. Then (M, gµ?) 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 RS, where S
  denotes any Cauchy surface
• Armed with this we now foliate our spacetime into
  Cauchy hypersurfaces, St, parameterized by a
  global function t.
• R. M. Wald, General Relativity, The University
  of Chicago Press (1984)

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The Hamiltonian in GR (contd)

• Let tµ be a vector field on M such that
• Define
• gµ? ? (hij,N,Nj)

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The Hamiltonian in GR (contd)

• A few definitions
• Lie derivative
• Exterior derivative
• Extrinsic curvature

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The Hamiltonian in GR (contd)

• Using the new variables, the Lagrangian density
  becomes
• The canonical conjugate momentum densities are
• The Hamiltonian density becomes

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The dynamical and constraint equations in GR

• The constraint equations are
• The dynamical equations are
• This completes the derivation.
  as before

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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 Einsteins equations are a set of
  10 non-linear second 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.

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Applications and Misconceptions (contd)

• Myths and Reality
• A 31 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 Diracs 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)

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• Acknowledgements
• Prof. Bernard Whiting, UF for his helpful
  comments and suggestions
• P. Mineault, McGill University for uploading on
  the web his paper on the same subject
• Google, without which this project would never be
  possible!
• Questions and Comments?
• THANK YOU