A new solution for dilaton-Maxwell gravity (Q1900434)

It is known that the inclusion of a scalar field in the theory of general relativity leads to a violation of the equivalence principle and to a modification of large scale gravitational phenomena. Moreover, some of the scalar fields in tensor multiscalar theories might contain non-positively defined kinetic terms. The purpose of this paper is to explore the behaviour of the static gravity coupled to a scalar field with a negative kinetic term in the presence of the electromagnetic field. In the string theory the action is given by \[ S=- {1\over {16\pi}} \int \sqrt {-g} (R+ 2g^{mn} \nabla_m \varphi \nabla_n \varphi+ e^{-2 \varphi} g^{mn} g^{kl} F_{mk} F_{nl} )d^4 x. \] Applying the variational principle to this functional, a set of field equations is obtained. The solution of this equation is sought under the following requirements: a) covariant de Donder gauge condition; b) assumption that the static spherically symmetric fields are everywhere radial. It is noted that if the scalar field is absent \((\varphi =0)\) the solution resulting from the minimization of the action describes a charged black hole with mass \(\mu_0\) and electromagnetic charge \(Q\). The obtained solution is in general singular, but by choosing a parameter entering in the solution, the author avoids the singularity. In this case the solution is a modification of the Fock solution in the presence of the dilaton and electromagnetic field. It is labeled by means of two parameters, \(\mu\) and \(Q\), and has two regular event horizons \((r_\pm)\) which correspond to the physical mass \(\mu_0\) and the electric charge \(Q\) of the hole \[ r_\pm= \mu_\pm= {\textstyle {1\over 2}} (\mu_0 \pm \sqrt {\mu_0^2- 2Q^2}). \] Other related quantities computed in this paper are: a) the Riemann curvature \(R\); b) the scalar curvature \(R_s\) in the general case and in the extreme regime \((\mu_0= 2Q^2)\). At the end of the paper the author quotes a negative result that he obtained: The negative kinetic term \(-g^{mn} \nabla_m \varphi \nabla_n \varphi\) generally leads to a theory without stable states.