Static gravitational equations of general relativity and “the fifth force”

Abstract

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

$$\begin{aligned} e^{2\omega ({\mathbf {x}})} \,{{\mathbf {\cdot }}}\, (\mathop {\nabla }\limits ^{\circ })^2 \omega ({\mathbf {x}}) = 4\pi G \,{{\mathbf {\cdot }}}\, \big [T^\alpha _\alpha ({\mathbf {x}}) - T^4_4 ({\mathbf {x}})\big ], \end{aligned}$$

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.