General geometry realized by four-scalar model and application to f(Q) gravity

In this paper, we propose a model including four scalar fields coupled with general gravity theories, which is a generalization of the two-scalar model proposed in Phys. Rev. D 103 (2021) no.4, 044055, where it has been shown that any given spherically symmetric static/time-dependent spacetime can be realized by using the two-scalar model. We show that by using the four-scalar model, we can construct a model that realizes any given spacetime as a solution even if the spacetime does not have a spherical symmetry or any other symmetry. We also show that by imposing constraints on the scalar fields by using the Lagrange multiplier fields, the scalar fields become non-dynamical and they do not propagate. This tells that there does not appear any sound which is usually generated by the density fluctuation of the fluid. In this sense, the model with the constraints is a most general extension of the mimetic theory in JHEP 11 (2013), 135, where there appears an effective dark matter. The dark matter is non-dynamical and it does not collapse even under gravitational force. Our model can be regarded as a general extension of any kind of fluid besides dark matter. We may consider the case that the potential of the scalar fields vanishes and the model becomes a non-linear $\sigma$ model. Then our formulation gives a mapping from the geometry of the spacetime to the geometry of the target space of the non-linear $\sigma$ model via gravity theory although the physical meaning has not been clear. We also consider the application of the model to $f\left(Q\right)$ gravity theory, which is based on a non-metricity tensor and $Q$ is a scalar quantity constructed from the non-metricity tensor. When we consider the $f\left(Q\right)$ gravity in the coincident gauge where the total affine connections vanish, when $f\left(Q\right)$ is not a linear function of $Q$, spherically symmetric spacetime cannot be realized except in the case that $Q$ is a constant. The situation does not change if we use the two-scalar model, as we show. If we use the four-scalar model in this paper, however, spherically symmetric spacetime can be realized in the framework of $f\left(Q\right)$ gravity with the coincident gauge.