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Das, A.

Static gravitational equations of general relativity and “the fifth force”. (English) Zbl 1327.83214

Gen. Relativ. Gravitation 47, No. 10, Paper No. 110, 12 p. (2015).
Summary: Einstein’s static field equations are investigated in various coordinate charts. After comparing Newtonian gravitational theory (in a curvilinear coordinate chart) with various charts of Einstein’s static gravitational equations, the most appropriate choice of the coordinate chart for Einstein’s static field equations is made. As a consequence, Einstein’s equations imply the non-linear potential equation \[ e^{2\omega(\mathbf{x})}\cdot(\overset{\circ}{\nabla})^2\omega(\mathbf{x}) = 4\pi G \cdot \left[T^\alpha_\alpha(\mathbf{x}) - T^4_4(\mathbf{x})\right], \] instead of the usual Poisson’s equation of the Newtonian theory. Investigating the non-linear potential equation above in the spherically symmetric cases, the corresponding potentials \(\omega(r)\) yield scenarios comparable to “the fifth force”. Next, static gravitational and electric fields generated by an incoherent charged dust are investigated. The corresponding non-linear potential equation is derived. Finally, the static Einstein-Maxwell-Klein-Gordon equations are explored and again, the corresponding non-linear potential equation is obtained. This potential resembles the static Higgs boson field.

MSC:

83D05 Relativistic gravitational theories other than Einstein’s, including asymmetric field theories
83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
83C10 Equations of motion in general relativity and gravitational theory
83C25 Approximation procedures, weak fields in general relativity and gravitational theory

Keywords:

static general relativity; the fifth force; non-linear potential equation; general relativity; Newtonian gravitational theory
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References:

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