2008 May 315th Italian-Sino Workshop Yu-Huei Wu and Chih-Hung Wang Accept by Classical and Quantum Gravity without correction Provisionally scheduled to publish for June 2008 Yu-Huei Wu and Chih-Hung Wang Accept by Classical and Quantum Gravity without correction Provisionally scheduled to publish for June 2008 Quasi-local mass in the covariant Newtonian space time Yu-Huei Wu 吳育慧 Institute of Astronomy, NCU

Mass in General Relativity Equivalence principle forbids us to have a localized gravitational energy- momentum 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. G ab = 8  G /c^4 T ab 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 energy- momentum 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: 10 field equations and 10 potentials Non-linear equations Intrinsically geometrical Horizons of black holes Gravitational waves Newtonian Gravity: 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 pseudo- Riemannian 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 space- time 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 d t where γ is 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 ∇ d t = 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 (Preprint gr-qc/ ) 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 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 (Preprint astro-ph/ ) Dougan A J and Mason L J Quasi-local mass constructions with positive energy 1999 Phys. Rev. Lett 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 Jeffryes B 1986 Class. Quantum Grav

Escher: Relativity