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Gravitational Energy: a quasi-local, Hamiltonian approach

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Gravitational Energy:a quasi-local, Hamiltonian approach

Jerzy Kijowski

Center for Theoretical Physics PAN

Warsaw, Poland

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.

Field energy, if positive, enables a priori estimations.

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energy-momentum

tensor conservation

laws:

relativistic

invariance

Indeed: gravitational energy cannot be additive because of

gravitational interaction:

In special relativity field energy obtained via Noether

theorem:

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!

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Boundary integrals vanish. Volume integrals carry information.

Volume integrals vanish. Boundary integrals carry information.

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!

Trying to mimick these methods in General Relativity

Theory we painfully discover the opposite rule:

General relativity: strong „fall off” conditions trivialize the theory.

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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”.

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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.

We begin with a mathematical analysis of energy as a generator

of dynamics in the sense of symplectic relations.

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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).

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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:

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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

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Legendre transformation: transition from one control mode

to another:

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The (4N-dimensional) „meta- phase space” describes:

positions , velocities , momenta , and forces .

Equipped with the canonical („God given”) symplectic form:

Euler-Lagrange equation in classical mechanics, together with

the definition of canonical momenta may be written as:

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The Hamiltonian

function

The same physics, different

control modes

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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)!

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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.

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Consider scalar field theory derived from a variational principle:

Field equations can be written in the following way:

Hamiltonian formulation: based on a (3+1) decomposition.

The formalism is coordinate-invariant.

Partial derivatives can be organized into invariant geometric

objects: jets of sections of natural bundles over spacetime.

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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.

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Consider scalar field theory derived from a variational principle:

Standard „Legendre manipulations” give us:

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 .

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Integrating energy density over a 3-volume V we obtain:

- transversal to boundary

component of the momentum

amount of energy contained in V

Infinite-dimensional Hamiltonian system

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.

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Another way to kill the boundary term: fix Neuman data :

They are generated by two different Hamiltonians: or .

adiabatic

insulation?

Both evolutions are equally legal.

Which one is the field’s „true energy”?

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Dirichlet data fixed

at the boundary

Neuman data fixed

at the boundary

free energy

energy; can also be obtained

from energy-momentum tensor

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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 .

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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 .

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grounding plug

adiabatic

insulation

generator :

free energy

generator :

true energy

Different control modes at the boundary:

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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:

A „flavour of the proof” for linear Maxwell theory.

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These are the only two dynamical equations.

Initial data:

carry entire information about fields

In spherical coordinates electric fields is decomposed as follows:

Radial component is directly encoded by the function .

( - angular coordinates.)

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 :

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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

Neuman control mode for and Dirichlet for : control of and

. Generator: „canonical energy” .

Free energy

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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).

Gravitational field equations – Einstein equations.

Can be derived from various variational formulations:

Examples:

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.

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Arnowitt-Deser-Misner momentum (1962):

We expect Hamiltonian formula of the type:

Non-unique !

(boundary

manipulations)

expected:

quasi-local

unknown boundary

control

Bulk term –

due to ADM.

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can be obtained directly from

Time derivatives

Einstein equations (e.g.: WMT, page 525).

Pluging these quantities into the bulk term, we can calculate

directly the remaining boundary terms.

No variational formulation necessary!

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Geometry of T in

(2+1) – decomposition.

Gauge invariant canonical

phase space form.

Now invariant!

Non-invariant!

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Gauge transformation:

Now invariant!

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This is not a control mode because are not

independent!

Analogous to 4D phase space of the particle dynamics:

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Given a 2D surface S,

choose the tube T.

Natural choice:

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Examples:

1) „Metric control mode” – complete metric of T is

controlled.

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Examples:

2) „Mixed control mode” – only 2D metric of S and

time-like components of Q controlled.

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Generic 2D-surface S carries a natural „time” and „space”

directions. Control of is natural.

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).

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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 ...

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Example: Klein-Gordon theory. Standard Lagrangian density:

Exchanging role of the field and the momentum we may derive

the same theory from the following variational principle:

Boundary terms added to the Lagrangian are irrelevant!

Both Lanrangians differ by a boundary term.

They define the same field equations. Choosing either Dirichlet

or Neuman control mode we get different Hamiltonian systems.

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