What does adding a scalar field component to the Einstein field equations mean for black holes and string theory?

Question

If a scalar field component has to be added to the Einstein field equations (see below) to solve dark matter/energy, then how would string theory need to be modified and do black holes still exist?

The proposed modified equations are (ignoring physical constants) $$ R_{ij} - \frac12 R g_{ij} = T_{ij} + \nabla_i\nabla_j \varphi $$ where $\varphi$ represents some kind of scalar potential. The conservation law for energy-momentum is proposed to be $$ \nabla^i(T_{ij} + \nabla_i\nabla_j\varphi) = 0 $$

Refs:

• Dark Matter, Dark Energy, and the Fate of Einstein’s Theory of Gravity http://mathgradblog.williams.edu/dark-matter-dark-energy-fate-einsteins-theory-gravity

• UNIFIED THEORY OF DARK ENERGY AND DARK MATTER, TIAN MA, SHOUHONG WANG http://www.indiana.edu/~fluid/paper/report.pdf

Answer 1

It does nothing more than saying that $T_{ij}$ can be decomposed into ''ordinary matter'' and ''scalar field.'' That equation is an assertion that the matter content of the universe contains a scalar field.

Answer 2

I cannot answer about string theory, but here is the answer about black holes. In fact, for $T_{\mu \nu} = 0$ there exists an exact scalar field solution of Einstein's equations, known as the Janis-Newman-Winicour solution. Singularities exist in such a solution, but depending on whether the scalar field is massive and on one another field dependent parameter, they may have distinguishing qualitative features. The following literature may help in your research of such solutions:

• Role of the scalar field in gravitational lensing

• Short description of Janis–Newman–Winicour spacetime

Answer 3

I think that their lower case phi would have to be a function of the curvature scalar, at least where the energy-momentum tensor vanishes.

To see that this is true, take the divergence of their field equation. Then calculate the gradient of the d'Alembertian of phi, keeping in mind that covariant differentiation is not commutative, and using their field equation to eliminate the Ricci tensor. It's inescapable that phi must be a function of the curvature scalar.

I have written to the authors about this.

Andrew Hyman Monroe, CT