Some Cosmological Consequences of Negative Gravitational Field Energy

1. Some Cosmological Consequences of Negative Gravitational Field Energy W. J. Wilson Department of Engineering and Physics University of Central Oklahoma Edmond, OK 73034 wwilson@uco.edu

2. Outline • The Field Energy Problem of for Gravitation • Analogy of GEM with Maxwell EM Theory • Gravitational Field Energy in GEM Theory • Spherical Solution – Event Horizon • Friedman Cosmology Model with Gravitational Field Energy • Summary/Conclusions OAS Fall 2008 Technical Meeting

3. The Field Energy Problem of GR • There is no well-defined gravitational field energy in General Relativity (L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields, Sec. 96) • Standard methods to obtain the energy-momentum tensor in GR do not work. They yield a pseudo-tensor that is not unique. • Usually asserted on the basis of the principle of equivalence that the gravitational energy cannot be localized. (C. W. Misner, K. S. Thorne and J. A. Wheeler, Gravitation) OAS Fall 2008 Technical Meeting

4. The Field Energy Problem of GR • Both standard Newtonian Gravitation and the Maxwell-form Gravitomagnetic (GEM) theory of Gravity have a gravitational field energy (density) – • This gravitational field energy has been discounted because it is negative. • By the E = M c2mass equivalence relation this gravitational field energy density is equivalent to the negative mass density OAS Fall 2008 Technical Meeting

5. Negative Sign Maxwell EM Theory GEM Theory Negative Sign OAS Fall 2008 Technical Meeting 5/13

6. Maxwell EM Theory Energy Flow GEM Theory Energy Flow Negative Sign Negative Sign OAS Fall 2008 Technical Meeting

7. Static GEM Theory Reduces to Newtonian Gravitation (with extra nonlinear term OAS Fall 2008 Technical Meeting 7/13

8. Spherically Symmetry For empty space ρ = 0, so Look for Spherical symmetric solution: OAS Fall 2008 Technical Meeting 8/13

9. Spherically Symmetry (cont) This has the general solution: Where B is an integration constant. Using the boundary condition, OAS Fall 2008 Technical Meeting 9/13

10. Event Horizon This solution: Can be written as where Is the usual Schwarzschild Radius OAS Fall 2008 Technical Meeting 10/13

11. Negative Energy in Cosmological Model This field: For a mass m at the edge of the universe at R(t) satisfies the equation of motion Which has the first integral Friedman-like equation for expanding universe cosmology in this model OAS Fall 2008 Technical Meeting 11/13

12. Negative Energy in Cosmological Model Produces expanding Universe model with positive radial acceleration OAS Fall 2008 Technical Meeting 12/13

13. Summary • Gravitational Field Energy give mechanism explanation of Hubble expansion of the Universe with acceleration – Dark Energy ≡ Gravitational Field Energy ?? • Can gravitational collapse produce effective negative mass black holes? OAS Fall 2008 Technical Meeting 13/13