1. Gravitational Energy: a quasi-local, Hamiltonian approach
Jerzy Kijowski
Center for Theoretical Physics PAN
Warsaw, Poland
Institut Mittag-Leffler

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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

10. Energy as a generating function
Legendre transformation: transition from one control mode
to another:
Institut Mittag-Leffler
November 19, 2012
Wolfgang Pauli Institute, Wien

11. 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

12. Mechanics: Hamiltonian formulation
The Hamiltonian
function
The same physics, different
control modes
Institut Mittag-Leffler
November 19, 2012
Wolfgang Pauli Institute, Wien

13. 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

14. 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

15. 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

16. 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

17. 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

18. 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

19. 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

20. 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

21. 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

22. 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

23. 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

24. 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

25. 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

26. 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

27. 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

28. 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

29. 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

30. General relativity theory
Institut Mittag-Leffler
November 19, 2012
Wolfgang Pauli Institute, Wien

31. 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

32. Wolfgang Pauli Institute, Wien
Fundamental identity
Gauge transformation:
Now invariant!
Institut Mittag-Leffler
November 19, 2012
Wolfgang Pauli Institute, Wien

33. 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

34. 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

35. Wolfgang Pauli Institute, Wien
Fundamental identity
Institut Mittag-Leffler
November 19, 2012
Wolfgang Pauli Institute, Wien

36. 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

37. 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

38. Wolfgang Pauli Institute, Wien
Fundamental identity
Institut Mittag-Leffler
November 19, 2012
Wolfgang Pauli Institute, Wien

39. 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

40. 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