Can we describe all forces as a curvature in space-time?

Question

If we have Einstein's field equation $$R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R=kT_{\mu\nu}$$ could we generalize it to $$R_{\mu\nu} - \frac{1}{2}m_{\mu\nu}R=kS_{\mu\nu}$$ where $S_{\mu\nu}$ is the source of the curvature and $m_{\mu\nu}=\eta_{\mu\nu}+f_{\mu\nu}$ where $f_{\mu\nu}$ is the perturbation caused by the force. Could we write one of these equations for each force, solve the equation for $f_{\mu\nu}$, sum up all of the $f_{\mu\nu}$ into one perturbation and add it to $\eta_{\mu\nu}$ to get one metric $g_{\mu\nu}$.

For example, since the electromagnetic field is extremely similar to the gravitational field, we could solve for the acceleration and find Poisson's equation. Using Einstein's "derivation" of general relativity one could find that

$$R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R=\frac{2q}{m\epsilon_0 c^2}Q_{\mu\nu}$$

Where $$g_{\mu\nu}=\eta_{\mu\nu}+A_{\mu\nu}$$ $A_{\mu\nu}$ is the electromagnetic pertubation and $$Q_{00}=\rho_Q$$ $$Q_{ij}=\frac{1}{c}v^i v^j\rho_Q$$ $$Q_{i0}=\frac{\vec{J}}{c}$$ and $\rho_Q$ is the charge density

These are all the components since the tensor is symmetric.

With this you can derive Maxwell's equations. But is this a valid approach, to describing how electromagnetism curves space-time?

There is a related question https://physics.stackexchange.com/qu/148028/ but this is about describing the forces using a Yang-Mills approach i.e. a curvature in the connection. In this question I am excluding gauge curvature and only talking about space time curvature.

Answer 1

There are some reasons why this will not work.

The obvious point is that we can write gravity as curvature because gravity is universal, i.e. gravitationaly interactions act in the same way on everything (roughly speaking). In the end, that's a reason why it makes sense to consider gravity as a property of spacetime itself.

For example, all free particles, that is, particles that only feel gravity, move on geodesics determined by the metric, so all particles "pervceive" the curvature. Furthermore, the energy-momentum tensor on the right-hand side of your first equation contains contributions from all fields and/or particles in your theory, charged an uncharged.

On the other hand, electromagnetism is not universal: The path of a particle in an electromagnetic field depends on the charge. Hence, you shouldn't have an energy-momentum tensor of uncharged matter influnce the electromagnetic force on a chraged particle.

Second, gravity is always attractive -- for example,there are no "repulsive" geodesics in Schwarzschild spacetime.

Third, upon quantisation, the electormagnetic (also weak, strong) force become spin-1 fields, while the metric is (presumably) described by a spin-2 field (graviton). Also, you would expect that there can be only one graviton field in a consistent theory.

Finally, note that there is a formulation of electordynamics in a manner very similar to gravity in a certain way via bundles -- here, the correspondence is between, e.g., the electromagnetic field strength and the Riemann tensor, not between photon and graviton field directly.

Answer 2

No, this doesn't work. There are several reasons - first, electric charge is a Lorentz scalar, which means that electric charge density transforms like the time component of a 4-vector, namely $J^\mu = (c\rho,\mathbf J)$. Contrast this with energy density; since energy is not a Lorentz scalar, energy density transforms like the time-time component of a rank 2 tensor (the stress-energy tensor on the right hand side of Einstein's equations).

Furthermore, electromagnetism is not universal in the same way that gravity is. The motion of a test mass in the presence of a gravitational field is independent of its mass, while the motion of a test charge in the presence of an electromagnetic field is not at all independent of its charge.

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There have been more exotic attempts to unify electromagnetism and general relativity. The Kaluza-Klein theory postulates the existence of a compact $4^{th}$ spatial dimension; under suitable assumptions (in particular the assumption that $\partial_4g_{\mu\nu}=0$, the so-called cylindrical condition), one can show that this predicts the existence of two additional fields - one 4-component field $B_\mu$ and one scalar field $\Phi$. If the latter is ignored (i.e. set equal to a constant) and the compact extra dimension is integrated over, then the remaining $B_\mu$ field and the 4-metric $g_{\mu\nu}$ obey the laws of electromagnetism and general relativity, respectively.

This isn't good enough, however. By coupling this 5-dimensional version of GR to a Dirac field (like an electron field), we obtain a relationship between the electric charge and mass of the electron which is off by something like 30 orders of magnitude. Kaluza-Klein is a fairly elegant idea but it falls apart under closer inspection. Other theories of this type have been (and continue to be) investigated, but as yet all have been problematic.