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Quasi-local mass in the covariant
Newtonian space time
Yu-Huei Wu 吳育慧
Institute of Astronomy, NCU
Yu-Huei Wu and Chih-Hung Wang
http://arxiv.org/abs/0803.2194
Accept by Classical and Quantum Gravity without correction
Provisionally scheduled to publish for June 2008
2008 May 31
5th Italian-Sino Workshop
Mass in General Relativity
•
Equivalence principle forbids us to have a localized gravitational energymomentum expression in GR.
•
Unlike Newtonian theory, we cannot separate background and dynamics in GR
•
General relativity is not a gauge field theory and highly non-linear.
•
All other fields can have proper energy density definitions in curved space-time.
Gab = 8  G /c^4 Tab
•
Unlike Newtonian theory, there’s no unique definition of quasi-local mass in GR.
Mass is formulated in various ways.
 Dark energy: back to Newtonian --------Lucy Calder and Ofer Lahavn 2008
Quasi-local mass construction
• Quasi-local idea is to define gravitational energymomentum associated to a closed 2-surface
• We examine (1) Komar integral, (2) Brown-York
expression and (3) Dogan-Mason mass expression in
this work
source
sphere
Quasi-local mass energy?
What does it mean physically?
•
•
Bondi's prolate and oblate mechanism.
Gravitational wave can transfer energy locally
(GW carry mass loss and radiate radiate to null infinity.)
•
Bondi put it: 'one can heat water with them'.
•
Tidal heating Io & Jupiter [Both + Creighton 2000]
General Relativity v.s. Newtonian
General Relativity:
Newtonian Gravity:
• 10 field equations and 10
potentials
• Non-linear equations
• Intrinsically geometrical
• Horizons of black holes
• Gravitational waves
• 1 field equation and 1
potential
• Linear equation
• Absolute space and time
• No horizons or no black hole
• No gravitational waves
• Have localized mass and
energy
What does it mean by the
Newtonian limit?
• Post-Newtonian theory depends on the choice of the
coordinate and background Minkowski metric is required. Not a
covariant fashion !
• Can these quasi-local expressions directly go back to the
unique surface integration in Newtonian space time?
• Can spinor field has a Newtonian correspondence?
• Newtonian and post-Newtonian approximations
are asymptotic to general relativity by using the
Newtonian sequence method [Futamase and
Schutz ’83].
• Hence Jeffryes use it to verify the Newtonian
limit of Penrose quasi-local mass. Unfortunately,
it can return to the Newtonian mass and energy
only by using transaction between 2-surface
twistors and 3-surface twistors.
• All these are from GR aspect
• Can a purely Newtonian type of star distort the
geometry of 3-space , i.e., generate pseudoRiemannian curvature on it?
• A more consistent transition between relativistic and
Newtonian theory of gravity is needed.
Neutron star
Outer crust: Newtonian theory or flat background
V.S. Inner core: General Relativity
?
Equation of State (EOS) of inner core ??
1. For the ordinary pulsars, the
magnetic field is sufficiently low.
a purely Newtonian
framework.
2. For the magnetars, the magnetic field
is sufficiently strong.
relativistic treatment will be
indispensable even locally.
Covariant Newtonian Theory

How to write the Newtonian theory in terms of spacetime geometry?

Newtonian space time is handicap, therefore cannot
give an unique connection unless we introduce the
ether frame e.
• The covariant Newtonian theory is developed by
Kunzle ’72, and Carter ’04.
• The purpose is to show how to set up and apply a fully
covariant formulation of the kinds of non-relativistic
multiconstituent fluid dynamical models and relativistic
correspondence.
Newtonian (Galilean) space-time is considered as a direct product of a
flat Euclidean 3-space and one dimensional Euclidean time line t, i.e.,
four dimensional fiber bundle. Each fiber t contains a Euclidean
(contravariant) flat metric γ.

Each fiber t contains a Euclidean (contravariant) flat metric γ and one-form
dt where γis a degenerate metric and cannot upper or lower indices, i.e. no
metric-dual definition of tensor fields. It has been found that conditions ∇γ = 0
and ∇dt = 0 with vanishing torsion cannot give an unique connection ∇ in the
Newtonian space-time.

Milne gauge in the Newtonian
space time
•
•
Equivalence principle in the Newtonian space time [Milne 1934].
In the Milne structure of Newtonian space-time, ∇ is not gauge invariant and has the
following gauge transformations
•
Milne gauge invariant connection D (Newton-Cartan connection)
•
Newtonian gravitational equation
Gauge invariant
Mass expression in the covariant
Newtonian space time
From volume integration to surface integration
Gauge invariant!!
Relativistic correspondence
From Newtonian to GR
Carter et al.'03 considered the following four
dimensional non-degenerate metric
where
The generalization of
can be written as
We further restrict deformations of
as following
It is sufficient to represent a deformation generated by a star.
Einstein field equation gives
(1) Komar integral
• In terms of a asymptotic time-like Killing vector
field , Komar integral yields
• It gives exactly the same covariant expression as
Newtonian surface integration without choosing a
specific closed 2-surface.
(2) Brown-York quasi-local energy
Einstein field equation gives
BY mass in Newtonian limit then becomes
If it is spherical symmetry with
, then
(3) Dogan-Mason mass:
Twistor program
• Twistor is a geometrical object that is more fundamental
than spinors. Space-time themselves can be regarded as
derived objects [Penrose ‘67].
• We can use the spinors to define mass or angular
momentum.
• The Newtonian limit of spinor field is shown here.
• Metric:
Spin frame in Newtonian limit
The asymptotic constant spinor
can be expressed as
They satisfy the holomorphic condition
The key NP coefficients are:
The first order spinor solutions are
The second order spinor
solutions are
The mass expression can be expanded as
Conclusion
• Since general relativity has a well known Newtonian limit, it is
reasonable for us to argue that it should yield the Newtonian
expression for all of quasi-local expressions in the covariant
Newtonian space-time.
• In this work, we verify that Komar integral can yield the
Newtonian quasi-local mass expression without choosing a
specific 2-sphere or referring to spherical symmetry of the
Newtonian potential, however, the Brown-York expression and
Dougan-Mason mass can give Newtonian expression with
spherical symmetry
• Whether this problem is due to the expressions
themselves or some other technical problems requires
a further investigation.
Brown D and York J W Quasilocal energy and conserved charges derived from the gravitational
action 1993 Phys. Rev. D 47 1407 (Preprint gr-qc/9209012)
Carter B, Chachoua E and Chamel N Covariant Newtonian and Relativistic dynamics of
(magneto)-elastic solid model for neutron star crust 2006 Gen. Rel. Grav. 38 83-119
Carter B and Chamel N Covariant analysis of Newtonian multi-fluid models for neutron stars: I
Milne-Cartan structure and variational formulation 2004 Int. J. Mod. Phys. D 13 291-325
(Preprint astro-ph/0305186)
Dougan A J and Mason L J Quasi-local mass constructions with positive energy 1999 Phys. Rev.
Lett. 67 2119-22
Künzle H P Galilei and Lorentz structures on space-time: comparison of the corresponding
geometry and physics 1972 Annales de l’institut Henri Poíncare
Futamase T and Schutz B 1983 Phys. Rev. D 28 2363
Jeffryes B 1986 Class. Quantum Grav. 3 841-852
Escher:
Relativity