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. OAS Fall 2008 Technical Meeting 2/13 Outline I. The Field Energy Problem of for Gravitation II. Analogy of GEM with Maxwell EM Theory III. Gravitational Field Energy in GEM Theory IV. Spherical Solution – Event Horizon V. Friedman Cosmology Model with Gravitational Field Energy VI. Summary/Conclusions

3. OAS Fall 2008 Technical Meeting 3/13 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)

4. OAS Fall 2008 Technical Meeting 4/13 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 c 2 mass equivalence relation this gravitational field energy density is equivalent to the negative mass density

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

6. OAS Fall 2008 Technical Meeting 6/13 Maxwell EM Theory Energy FlowGEM Theory Energy Flow Negative Sign

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

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

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

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

11. OAS Fall 2008 Technical Meeting 11/13 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

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

13. OAS Fall 2008 Technical Meeting 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?