Non-Localizability of Electric Coupling and Gravitational Binding of Charged Objects Matthew Corne Eastern Gravity Meeting 11 May 12-13, 2008

Purpose of Investigation To determine whether or not static, spherically symmetric extended objects comprised of a charged perfect fluid possess non-localizable contributions to their total gravitational mass Does electrostatic coupling energy contribute to gravitational binding energy?

Historically, known that purely electromagnetic classical models unstable (hence Poincaré stress) Conjectured (and assumed) instead, electrostatic coupling energy contribution to gravitational binding E.g., Lorentz-Abraham model of electron, certain models of spherical stars

Know in general, total mass-energy not localizable Gravitational energy-momentum not localizable Exceptions: static, spherically symmetric objects composed of neutral perfect fluid, gravitational waves up to wavelength

Einstein’s Equations Einstein Tensor Components Energy-Momentum Tensor Components

Energy-momentum tensor for a charged perfect fluid: Electromagnetic field tensor components Metric tensor components Proper density of the fluid Proper pressure Components of four- velocity

Due to symmetries, quantities radially dependent: Matter density Charge density Pressure Electric field

Line element describing a static, spherically symmetric spacetime: Diagonal components of metric tensor

Electric field determined by Maxwell’s Equations: Other three are satisfied trivially Radial electric field as measured in the orthonormal frame Charge inside sphere of radius r Element of proper volume after integration over angles

00-component of Einstein’s tensor related to energy density/00-component of energy- momentum tensor (localizable contribution of matter) Getting to the energy:

Boundary of object at r = R: Outside of object: Charge Q constant

Constant M interpreted as total mass (Keplerian motion) Consider expressions Substitution into line element - get Reissner-Nordstrøm (RN)

Reissner-Nordstrøm (RN) Line Element Describes spherically symmetric, static gravitational field with radial electric field present Applies to exterior solution of extended object here

Integrating with respect to r : Not total mass-energy inside radius r Method absent for defining total mass-energy with surrounding material (Keplerian orbits indeterminable)

Can interpret this as total gravitational mass entering external RN solution: Contains contribution of gravitational binding energy Contains contribution of electrostatic coupling energy

Negative of gravitational potential energy Holds together all components of object What is gravitational binding energy? Total mass-energy contributing to gravitational field Rest mass-energyInternal energy, in our case electrostatic Gravitational potential energy

Re-write equation in terms of binding energy: Gravity binds only localizable part of mass-energy (perfect fluid) Electrostatic coupling energy does not appear in gravitational binding energy

Vanishing electrostatic coupling energy (pure EM mass) at every point inside of object Charge: integral of density Contribution to total mass only outside of object - not inside What we notice:

No purely electromagnetic mass Need gravitational binding energy Results independent of thermodynamic considerations (i.e., pressure absent from equations) Implications

r - given by the expression A - proper area of an appropriate sphere t - appropriate slicing by proper spaces of co- moving observers Schwarzschild coordinates - natural choice Important: Effects not coordinate dependent

Spacelike hypersurfaces Shift Vector Lapse Unit Normal

Further Results Removing charge - same result as a neutral spherical star with external Schwarzschild solution Introducing rotation (angular momentum) such as in Kerr-Newman - still have charge Electron (or any such particle) cannot be modeled as a field that holds itself together Vacuum fluctuations inadequate

Conclusions Existence of non-localizable contributions to total mass-energy Must always check configuration - no assumptions Cannot model electrons this way! Cannot model stars this way!

Acknowledgments NCSU Department of Mathematics REG program (supported by NSF MTCP grant)

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