Analytical representation for metrics of scalarized Einstein-Maxwell black holes and their shadows

Abstract

Here we construct approximate analytical forms for the metric coefficients and fields representing the scalarized Einstein-Maxwell black holes with various couplings of the scalar field, once the parameters of the system are fixed. By increasing approximation order, one can obtain the analytic representation with any desired accuracy, what was tested via calculations of shadows for these black holes by using approximate analytical and accurate numerical metric functions. We share the Mathematica R code [1] which allows one to find an appropriate analytical form of the metric for any couplings and values of parameters. Scalarization increases the radius of the black-hole shadow for all the considered coupling functions.

Shadows and deflection angle of charged and slowly rotating black holes in Einstein-Æther theory

“…Alternatively, another approach is to construct a analytical representations based on the numerical solution, as shown in [50][51][52]. Recently, some analytical approximate space-time solutions have been constructed by using the continuedfaction expansion in the Einstein-Scalar-Gauss-Bonnet gravity and Einstein-Scalar-Maxwell gravity with scalarization [73,74]. The shadow of the approximate black holes have also been explored.…”

Einstein-scalar–Gauss-Bonnet black holes: Analytical approximation for the metric and applications to calculations of shadows

“…Then, one is able to perform a fitting of these data in order to obtain analytical expressions for the CFA parameters as functions of . It is then straightforward to write down approximate analytical expressions for the metric functions and the scalar field to the desired order in the CFA via (9) and (18).…”

“…In order to be able to perform the analysis we need to reduce the number of free parameters and so we choose as an indicative value = 1 in (24). The obtained analytical expressions for the parameters of the CFA (9),(13), (18) and (19) up to second order are given below…”

“…This choice leads to superior convergence properties and allows one to approximate a black-hole metric with a much smaller set of coefficients. This approach was used to construct the analytical approximation of numerical blackhole solutions in the Einstein-Weyl [16], Einstein-dilaton-Gauss-Bonnet [17] and Einstein-scalar-Maxwell [18] theories. Further studies of observables in these parametrized spacetimes [19][20][21][22][23][24] showed that usually it is sufficient only two-three orders of the continued fraction expansion in order to achieve reasonable accuracy.…”

“…Recently this approach was extended in [10] to various types of the scalar field function and allowed therefore, to look whether there are some common features for all the considered couplings of the scalar field. A similar problem was attacked numerically for case of the Einstein-scalar-Maxwell theory [25] and the study of its analytical approximation [18] showed that the radius of the black-hole shadow is increased for any of the considered couplings of the scalar field. Scalarization, that is the phenomenon of spontaneous acquiring of a scalar hair by the black hole as a result of the non-minimal coupling of a scalar field to the system, has been actively studied in [26][27][28][29][30].…”