Non-Localizability of Electric Coupling and Gravitational Binding of Charged Objects

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

2. 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?

3. 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

4. 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

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

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

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

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

9. 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

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

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

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

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

14. 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)

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

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

17. 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

18. What we notice: • 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

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

20. Important: Effects not coordinate dependent 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

21. Spacelike hypersurfaces Lapse Unit Normal Shift Vector

22. 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

23. 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!

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

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