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
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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
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The Lagrangian in GR
The Hamiltonian in GR
The Equations in GR
 Applications and Misconceptions
 Questions, Comments and Acknowledgements
Introduction
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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:
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Define the Action as :
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Use Hamilton’s 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)
Review of Hamiltonian Mechanics: Point Particles (contd…)
Hamiltonian Formulation
• System defined by 2n generalized coordinates {qi,pi}, where
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Construct the Hamiltonian from the Lagrangian by means of a
Legendre transformation as:
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Hamilton’s equations of motion:
Review of Hamiltonian Mechanics: Classical Fields*
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qi →Φ(xµ)
The lagrangian is related to the Lagrangian density:
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Euler-Lagrange equations of motion, which are covariant in nature:
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Similarly define the Hamiltonian density as:
where
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is the conjugate momentum density
Hamilton’s equations become:
*same as before
Constrained Hamiltonian Formulation for Dynamical
Systems*
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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…)
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Consider a system with n generalized coordinates with m constraint
equations of the form:
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Use m lagrange undetermined multipliers λα and extremize:
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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
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→
→
Reverse the argument now: If conjugate momentum =0, that degree of
freedom is constrained and the constraint is hidden in the lagrangian
Constrained Hamiltonian Formulation for Dynamical
Systems: Example*
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Consider the EM lagrangian with no source:
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The conjugate momentum densities are:
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The Hamiltonian becomes:
*same as before
Constrained Hamiltonian Formulation for Dynamical
Systems: Example (contd…)
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The Hamiltonian equations of motion are:
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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*
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The dynamical variable in GR is the metric gμν
The Lagrangian density for curved spacetime is:
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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*
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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, gμν) be a globally hyperbolic spacetime. Then (M, gμν)
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…)
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Let tμ be a vector field on M such that
Define:
gμν → (hij,N,Nj)
The Hamiltonian in GR (contd…)
A few definitions:
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Lie derivative:
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Exterior derivative:
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Extrinsic curvature:
The Hamiltonian in GR (contd…)
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Using the new variables, the Lagrangian density becomes:
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The canonical conjugate momentum densities are:
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The Hamiltonian density becomes:
The dynamical and constraint equations in GR*
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The constraint equations are:
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The dynamical equations are:
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This completes the derivation.
*as before
Applications and Misconceptions
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Uses*
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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-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.
Applications and Misconceptions (contd…)
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Myths and Reality*
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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 spacetime 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 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