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Emergent modified gravity presents a new set of generally covariant gravitational theories in which the space-time metric is not directly given by one of the fundamental fields. A metric compatible with the modified dynamics of gravity is instead derived from covariance conditions for space-time in canonical form. By staying within the canonical setting throughout all the required steps, several assumptions about space-time made implicitly in modified action principles can be relaxed. This paper presents a significant extension of existing vacuum models to the case of a scalar field coupled to emergent modified gravity in a spherically symmetric setting. Unlike in previous attempts for instance in models of loop quantum gravity, it is possible to maintain general covariance in the presence of modified gravity-scalar couplings. In general, however, the emergent space-time metric depends not only on the phase-space degrees of freedom of the gravitational part of the coupled theory, but also on the scalar field. Matter therefore directly and profoundly affects the geometry of space-time, not only through the well-known dynamical coupling of stress-energy to curvature as in Einstein's equation, but even on a kinematical level before equations of motion are imposed. In addition to the covariance condition, this paper introduces further physical requirements that may be imposed in order to reduce modified gravity-scalar theories to more specific classes. In some cases, coupling emergent modified gravity to a scalar field eliminates some of the modifications that would be possible in a vacuum situation. Moreover, certain results about the removal of classical black hole singularities in vacuum emergent modified gravity are found to be unstable under the inclusion of matter fields. However, alternative modifications exist in which singularities are removed even in the presence of matter. Emergent modified gravity is seen to provide a large class of new scalar-tensor theories with second-order field equations.
Bojowald et al. (Tue,) studied this question.