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Gravitational Energy: a quasi-local, Hamiltonian approach
Jerzy Kijowski Center for Theoretical Physics PAN Warsaw, Poland Institut Mittag-Leffler
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Energy estimates in field theory
Field energy, if positive, enables a priori estimations. In most cases the Cauchy problem is „well posed” due to the existence of a positive, local energy. Example 1: The (non-linear) Klein-Gordon field: where is the momentum canonically conjugate to the field variable ; (linear theory for ). Example 2: Maxwell electrodynamics: where – electric and magnetic field. Institut Mittag-Leffler November 19, 2012 Wolfgang Pauli Institute, Wien
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Wolfgang Pauli Institute, Wien
Gravitational energy In special relativity field energy obtained via Noether theorem: energy-momentum tensor conservation laws: relativistic invariance But: no energy-momentum tensor of gravitational field! No space-time symmetry or too many symmetries! Various „pseudotensors” have been proposed. They do not describe correctly gravitational energy! Indeed: gravitational energy cannot be additive because of gravitational interaction: Institut Mittag-Leffler November 19, 2012 Wolfgang Pauli Institute, Wien
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Wolfgang Pauli Institute, Wien
Bulk versus boundary Standard „Hamiltonian” approaches to special-relativistic field theory is based on integration by parts and neglecting surface integrals at infinity („fall of” conditions). Paradigm: functional-analytic framework for description of Cauchy data must be chosen in such a way that the boundary integrals vanish automatically! Boundary integrals vanish. Volume integrals carry information. Trying to mimick these methods in General Relativity Theory we painfully discover the opposite rule: Volume integrals vanish. Boundary integrals carry information. General relativity: strong „fall off” conditions trivialize the theory. Institut Mittag-Leffler November 19, 2012 Wolfgang Pauli Institute, Wien
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Wolfgang Pauli Institute, Wien
Quasi-local approach Gravitational energy contained in a three-volume has to be assigned to its boundary ( instead of ) R. Penrose (1982, 83) proposed a quasi-local approach: Gravitational energy asigned to closed 2D surfaces. Meanwhile, many (more than 20) different definitions of a „quasi-local mass” (energy) have been formulated. In this talk I want to discuss a general Hamiltonian framework, where boundary integrals are not neglected! This approach covers most of the above 20 different definitions: each related to a specific „control mode”. Institut Mittag-Leffler November 19, 2012 Wolfgang Pauli Institute, Wien
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Wolfgang Pauli Institute, Wien
Symplectic relations We begin with a mathematical analysis of energy as a generator of dynamics in the sense of symplectic relations. Symplectic space : a manifold equipped with a non-degenerate, closed differential two-form . Basic example: co-tangent bundle of a certain „configuration space” . Cotangent bundle carries a canonical one-form: where are coordinates in and are the corres- ponding „canonical coordinates” in the co-tangent bundle. Institut Mittag-Leffler November 19, 2012 Wolfgang Pauli Institute, Wien
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Wolfgang Pauli Institute, Wien
Generating function Any function on the base manifold generates a Lagrangian (i.e. maximal, isotropic for ) submanifold of the co-tangent bunle, according to the formula: Every lagrangian submanifold which is transversal with respect to fibers of the bundle is of that type, i.e. has a generating function. Also non-transversal manifolds can be described this way in a slightly broader framework (constraints, Morse functions). Institut Mittag-Leffler November 19, 2012 Wolfgang Pauli Institute, Wien
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Energy as a generating function
Thermodynamical properties of a simple body can be encoded by the following formula: The proper arena for describing these properties is a four-dimensional phase space (pressure, temperature, volume and entropy). This „meta-phase space” is equipped with the („God given”) symplectic structure: describing any simple thermodynamical body. A specific body is described by a specific generating function , as a two-dimesional submanifold satisfying equations: Institut Mittag-Leffler November 19, 2012 Wolfgang Pauli Institute, Wien
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Energy as a generating function
Interpretation: have been chosen as „configuration” or „control parameters”, whereas are „momenta” or „response parameters”. The same submanifold (collection of all physically admissible states) can be described by the Helmholz free energy function, via the formula: The same physics, but different control modes Institut Mittag-Leffler November 19, 2012 Wolfgang Pauli Institute, Wien
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Energy as a generating function
Legendre transformation: transition from one control mode to another: Institut Mittag-Leffler November 19, 2012 Wolfgang Pauli Institute, Wien
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Mechanics: variational formulation
Euler-Lagrange equation in classical mechanics, together with the definition of canonical momenta may be written as: The (4N-dimensional) „meta- phase space” describes: positions , velocities , momenta , and forces Equipped with the canonical („God given”) symplectic form: Institut Mittag-Leffler November 19, 2012 Wolfgang Pauli Institute, Wien
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Mechanics: Hamiltonian formulation
The Hamiltonian function The same physics, different control modes Institut Mittag-Leffler November 19, 2012 Wolfgang Pauli Institute, Wien
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Wolfgang Pauli Institute, Wien
Scalar field theory Consider scalar field theory derived from a variational principle: Field equations can be written in the following way: At every spacetime point we have a 10-dimensional symplectic „meta-phase space” describing: field strength , its 4 derivatives , corresponding 4 momenta , and their divergence - exterior derivative within this space („vertical” in contrast to spacetime derivatives)! Institut Mittag-Leffler November 19, 2012 Wolfgang Pauli Institute, Wien
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Wolfgang Pauli Institute, Wien
Scalar field theory Consider scalar field theory derived from a variational principle: Field equations can be written in the following way: At every spacetime point we have a 10-dimensional symplectic „meta-phase space” describing: field strength , its 4 derivatives , corresponding 4 momenta , and their divergence Symplectic structure give by: Submanifold : field equations. Institut Mittag-Leffler November 19, 2012 Wolfgang Pauli Institute, Wien
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Wolfgang Pauli Institute, Wien
Scalar field theory Consider scalar field theory derived from a variational principle: Field equations can be written in the following way: The formalism is coordinate-invariant. Partial derivatives can be organized into invariant geometric objects: jets of sections of natural bundles over spacetime. Hamiltonian formulation: based on a (3+1) decomposition. Institut Mittag-Leffler November 19, 2012 Wolfgang Pauli Institute, Wien
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Wolfgang Pauli Institute, Wien
Scalar field theory Consider scalar field theory derived from a variational principle: Field equations can be written in the following way: index „k=1,2,3” denotes three space-like coordinates, index „0” denotes time coordinate. We denote Hamiltonian formulation: based on a (3+1) decomposition. Institut Mittag-Leffler November 19, 2012 Wolfgang Pauli Institute, Wien
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Wolfgang Pauli Institute, Wien
Scalar field theory Consider scalar field theory derived from a variational principle: Field equations can be written in the following way: index „k=1,2,3” denotes three space-like coordinates, index „0” denotes time coordinate. We denote Standard „Legendre manipulations” give us: Institut Mittag-Leffler November 19, 2012 Wolfgang Pauli Institute, Wien
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Wolfgang Pauli Institute, Wien
Scalar field theory Integrating energy density over a 3-volume V we obtain: - transversal to boundary component of the momentum amount of energy contained in V Energy generates dynamics as a relation between three objects: initial data, their time derivatives and the boundary data. Field dynamics within V can be made unique (i.e Cauchy problem „well posed”) if we impose boundary conditions annihilating the boundary term. Infinite-dimensional Hamiltonian system Institut Mittag-Leffler November 19, 2012 Wolfgang Pauli Institute, Wien
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Wolfgang Pauli Institute, Wien
Scalar field theory Another way to kill the boundary term: fix Neuman data : Both evolutions are equally legal. They are generated by two different Hamiltonians: or adiabatic insulation? Which one is the field’s „true energy”? Institut Mittag-Leffler November 19, 2012 Wolfgang Pauli Institute, Wien
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Dirichlet vs. Neuman evolution
Dirichlet data fixed at the boundary Neuman data fixed at the boundary energy; can also be obtained from energy-momentum tensor free energy Institut Mittag-Leffler November 19, 2012 Wolfgang Pauli Institute, Wien
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Wolfgang Pauli Institute, Wien
Electrodynamics Classical elecrodynamics (also non-linear, like Born-Infeld) can be derived from a variational principle: Variation with respect to the electromagnetic potential Linear Maxwell theory if: Standard (texbook) procedure leads to the „canonical” en-mom. tensor and the corresponding „canonical” energy Adding a boundary term one can construct the „symmetric” en-mom. tensor and the corresponding „symmetric” energy Institut Mittag-Leffler November 19, 2012 Wolfgang Pauli Institute, Wien
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Wolfgang Pauli Institute, Wien
Electrodynamics It can be proved that „symmetric energy” generates a Hamiltonian field evolution based on controlling magnetic and electrix flux at the boundary : For linear Maxwell theory we have: But: „canonical energy” (usually claimed to be „unphysical”) generates an equally good Hamiltonian field evolution based on controlling magnetic flux and electostatic potential at Institut Mittag-Leffler November 19, 2012 Wolfgang Pauli Institute, Wien
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Wolfgang Pauli Institute, Wien
Energy vs. free energy Different control modes at the boundary: grounding plug adiabatic insulation generator : true energy generator : free energy Institut Mittag-Leffler November 19, 2012 Wolfgang Pauli Institute, Wien
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Dirichlet vs. Neuman in Electrodynamics
A „flavour of the proof” for linear Maxwell theory. Field configuration described by two 3-dimesional vectors: B, D. Dynamics given by Maxwell equations: wave operator Consider two functions describing radial components of D and B. Hence, Maxwell equations imply: Institut Mittag-Leffler November 19, 2012 Wolfgang Pauli Institute, Wien
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Dirichlet vs. Neuman in Electrodynamics
These are the only two dynamical equations. Initial data: carry entire information about fields In spherical coordinates electric fields is decomposed as follows: ( angular coordinates.) Radial component is directly encoded by the function On each sphere the tangent part can be further decomposed into a sum of a gradient and co-gradient: The co-gradient part can be reconstructed from : Institut Mittag-Leffler November 19, 2012 Wolfgang Pauli Institute, Wien
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Dirichlet vs. Neuman in Electrodynamics
The gradient part can be reconstructed from the constraint equation , namely: Quasi-local reconstruction: on each sphere separately. Dirichlet control mode for both and : control of and Generator: „symmetric energy” True energy Free energy Neuman control mode for and Dirichlet for : control of and . Generator: „canonical energy” Institut Mittag-Leffler November 19, 2012 Wolfgang Pauli Institute, Wien
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General relativity theory
Gravitational field equations – Einstein equations. Can be derived from various variational formulations: Examples: 1) metric (variation with respect to ). 2) Palatini (variation with respect to both and treated as independent fields). 3) affine (variation with respect to , metric arises as a momentum conjugate to connection). Hamiltonian theory based on (3+1) decomposition is universal and does not depend upon a particular variational formula. Cauchy data on a hypersurface: 3-metric and exterior curvature. Institut Mittag-Leffler November 19, 2012 Wolfgang Pauli Institute, Wien
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General relativity theory
Arnowitt-Deser-Misner momentum (1962): We expect Hamiltonian formula of the type: Bulk term – due to ADM. Non-unique ! (boundary manipulations) expected: quasi-local unknown boundary control Institut Mittag-Leffler November 19, 2012 Wolfgang Pauli Institute, Wien
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General relativity theory
Pluging these quantities into the bulk term, we can calculate directly the remaining boundary terms. Time derivatives can be obtained directly from Einstein equations (e.g.: WMT, page 525). No variational formulation necessary! Institut Mittag-Leffler November 19, 2012 Wolfgang Pauli Institute, Wien
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General relativity theory
Institut Mittag-Leffler November 19, 2012 Wolfgang Pauli Institute, Wien
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Wolfgang Pauli Institute, Wien
Fundamental identity Gauge invariant canonical phase space form. Geometry of T in (2+1) – decomposition. Non-invariant! Now invariant! Institut Mittag-Leffler November 19, 2012 Wolfgang Pauli Institute, Wien
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Wolfgang Pauli Institute, Wien
Fundamental identity Gauge transformation: Now invariant! Institut Mittag-Leffler November 19, 2012 Wolfgang Pauli Institute, Wien
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Wolfgang Pauli Institute, Wien
Fundamental identity This is not a control mode because are not independent! Analogous to 4D phase space of the particle dynamics: Institut Mittag-Leffler November 19, 2012 Wolfgang Pauli Institute, Wien
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Wolfgang Pauli Institute, Wien
Fundamental identity choose the tube T. Given a 2D surface S, Natural choice: Institut Mittag-Leffler November 19, 2012 Wolfgang Pauli Institute, Wien
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Wolfgang Pauli Institute, Wien
Fundamental identity Institut Mittag-Leffler November 19, 2012 Wolfgang Pauli Institute, Wien
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Wolfgang Pauli Institute, Wien
Fundamental identity Examples: 1) „Metric control mode” – complete metric of T is controlled. Institut Mittag-Leffler November 19, 2012 Wolfgang Pauli Institute, Wien
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Wolfgang Pauli Institute, Wien
Fundamental identity Examples: 2) „Mixed control mode” – only 2D metric of S and time-like components of Q controlled. Institut Mittag-Leffler November 19, 2012 Wolfgang Pauli Institute, Wien
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Wolfgang Pauli Institute, Wien
Fundamental identity Institut Mittag-Leffler November 19, 2012 Wolfgang Pauli Institute, Wien
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Wolfgang Pauli Institute, Wien
Conclusions Fundamental formula – natural framework for quasi-local analysis of gravitational energy. Valid also for interacting system: „matter fields + gravity” (if supplemented by appropriate matter terms in both the Hamiltonian and the control parts). Generic 2D-surface S carries a natural „time” and „space” directions. Control of is natural. Institut Mittag-Leffler November 19, 2012 Wolfgang Pauli Institute, Wien
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Wolfgang Pauli Institute, Wien
Conclusions Generic 2D-surface S carries a natural „time” and „space” directions. Control of is natural. Any choice of the remaining control leads to a „quasi-local energy. Which one is „the true energy” and not a „free energy”? Criteria: positivity, „well posedness”, linearization ... Institut Mittag-Leffler November 19, 2012 Wolfgang Pauli Institute, Wien
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