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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 G 8T Einstein Tensor Components Energy-Momentum Tensor Components Energy-momentum tensor for a charged perfect fluid: Proper pressure Metric tensor components 1 1 T ( p)u u g p (F F F F g ) 4 4 Proper density of the fluid Components of fourvelocity Electromagnetic field tensor components Due to symmetries, quantities radially dependent: • Matter density • Charge density • Pressure • Electric field Line element describing a static, spherically symmetric spacetime: ds e 2 2(r) g00 e dt e 2 dr r (d sin d ) 2 2 2 2 2(r) grr e 2(r) g r 2 g r sin 2 2(r) 2 Diagonal components of metric tensor 2 Electric field determined by Maxwell’s Equations: E 4 Other three are satisfied trivially q(r) F E(r) 2 r Radial electric field as measured in the orthonormal frame 0ˆ rˆ q(r) r (r)4r e 0 2 (r ) dr Charge inside sphere of radius r 4 r 2e (r)dr Element of proper volume after integration over angles Getting to the energy: 1 2(r) d 2(r) G00 2 e [r(1 e )] r dr T00 e E [(r) ] 8 2(r) • 00-component of Einstein’s tensor related to energy density/00-component of energymomentum tensor (localizable contribution of matter) Boundary of object at r = R: (r) (r) 0 rR Outside of object: q(R) Q R (r)4r e 0 2 (r ) dr Charge Q constant Consider expressions 2 Q m˜ (r) M 2r e 2(r ) Constant M interpreted as total mass (Keplerian motion) 1 1 2 ˜ (r) 2m 2M Q 1 1 2 r r r Substitution into line element - get Reissner-Nordstrøm (RN) Reissner-Nordstrøm (RN) Line Element 2 2 2M Q 2M Q ds2 (1 2 )dt 2 (1 2 )1 dr 2 r 2 (d 2 sin 2 d 2 ) r r r r • Describes spherically symmetric, static gravitational field with radial electric field present • Applies to exterior solution of extended object here Integrating with respect to r : m(r) r r 4r (r)dr 2 0 0 q(r) (r)4 r e r 2 (r ) • Not total mass-energy inside radius r • Method absent for defining total mass-energy with surrounding material (Keplerian orbits indeterminable) dr Can interpret this as total gravitational mass entering external RN solution: m(R) M R (r) 2 (r) e 4 r (r)e dr R 0 0 Contains contribution of gravitational binding energy q(r) (r)4 r 2e (r ) dr r Contains contribution of electrostatic coupling energy What is gravitational binding energy? • Negative of gravitational potential energy • Holds together all components of object m(r) m0 (r) U(r) (r) Total mass-energy contributing to gravitational field Rest mass-energy Internal energy, in our case electrostatic Gravitational potential energy Re-write equation in terms of binding energy: m(r) m0 (r) E GravBinding U(r) E GravBinding r (1 e (r ) )(r)4 r e 2 ( r ) 0 r 0 m r 4 r dr 2 • Gravity binds only localizable part of mass-energy (perfect fluid) • Electrostatic coupling energy does not appear in gravitational binding energy dr 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 Implications • No purely electromagnetic mass • Need gravitational binding energy • Results independent of thermodynamic considerations (i.e., pressure absent from equations) Important: Effects not coordinate dependent r - given by the expression r A 4 A - proper area of an appropriate sphere t - appropriate slicing by proper spaces of comoving observers Schwarzschild coordinates - natural choice i 1 ti1 Lapse i ti i t Unit Normal i Nn N Spacelike hypersurfaces i Shift Vector 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) References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. Misner, Charles W., Kip S. Thorne, John Archibald Wheeler. “Gravitation,” W.H. Freeman and Company, 1973. Bernard F. Schutz, “A First Course in General Relativity,” Cambridge University Press, Cambridge, 1985. A Nduka 1976 Gen. Rel. Grav. 7 493-99 K D Krori and J Barua 1975 J. Phys. A: Math. Gen. 8 508-11 T Singh 1975 J. Phys. A: Math. Gen. 8 1361-3 R N Tiwari, J R Rao, and R R Kanakamedala 1984 Phys. Rev. D 30 489-91 D N Pant and A Sah 1979 J. Math. Phys. 20 2537-9 S Ray and B Das 2002 Astrophys. Space Sci. 282 635-44 R Gautreau 2985 Phys. Rev. D 31 1860-3 W B Bonnor 1960 Z. Phys. 160 59-65 Ya B Zel’dovich 1968 Sov. Phys. - Usp. 11 381-93 O Gron 1985 Phys. Rev. D 31 2129-31 W Heitler “The Quantum Theory of Radiation,” New York: Dover, 1984. J A Wheeler, “Geometrodynamics,” New York: Academic, 1972 J A Wheeler, Phys. Rev. 97 511-36, 1955 C W Misner and J A Wheeler, Ann. Phys. 2 525-660, 1957