# Does general relativity account for the other 3 forces?

There are suppose to be 4 fundemental forces of nature, einstein described gravity not as a force but, put simply, the result of objects following curves in spacetime. However does it account for acceleration caused by EM force? Does this theory accommodate the other 3 forces?

• – lurscher Sep 6 at 19:11

General Relativity explains gravity, not the other three forces. Physicists have tried to extend General Relativity to higher dimensions to explain other forces, as in Kaluza-Klein theory, but this has not been successful.

GR does “accommodate” other forces in the sense that we know how to write, for example, the Lorentz force law in curved spacetime, so we can describe motion under both gravity and electromagnetic force. But when we do this the EM force is not due to spacetime curvature.

The current state of GR does NOT account for the other forces but one cannot develop a force model without respecting general covariance, so it has left a mark on physics that has not been challenged as far as I know.

Historically there were several attempts to explain the other forces of nature using space-time curvature. These are not mainstream but have not died completely. My thesis was in this field of study.

Very briefly, Kaluza-Klein Theory (circa 1920s) was able (feeble but somewhat able) to describe electromagnetism using an extra dimension of space, 4+1 rather than 3+1. This extra dimension is closed, a tightly curled up circle, and the extra pieces of the metric tensor associated with the hidden geometry transform as a proper 4-vector under GCT in 3+1 dim and as a gauge field w/r to coordinate transforms in the extra dimension. This theory had some problems with it and as it grew in popularity the weak and strong force were beginning to show their presence in experiments and other approaches proved promising.

Einstein demonstrated an internal symmetry on space-time geometry based on what he called the lambda-transform. From this he argued that an anti-symmetry component to the metric had to exist. This gave rise to the torsion field. Torsion has been investigated for about 100 years in terms of its potential impact on nature and possible use as a grand unified field.

So, as I said in the current state of the art GR does NOT explain the SM of particle physics but there is a rich history of the attempt at a GUT based on it. In fact early attempts as gauge theory were modeled as generalizations of K-K theory. The main difference is that the gauge manifold is not part of space time and not subject to any deformation, it is a fiber over the space-time manifold. In K-K theory the extra dimension can grow and shrink and the extra fields couple to space-time geometry in ways not present in QFT on a curved background. There would be a stronger non-linear coupling between the other forces and gravity than just through the stress energy tensor acting as a source. This coupling is another "issue" in such models as one need to explain, preferably by solving the equations, why they are weak or absent and how the extra dimensions "shrunk". String theory has the same problem in a sense. Why are some dimensions special? To the theorist it's because they work to make everything look right. But you have to explain these things or you may as well appeal to magic.

GR does not predict the existence of other forces, if that's what you're asking.

But all other forces can be generalised to the curved spacetime that GR describes.
This is just done by replacing the derivatives $$\partial_i \psi^j$$ in the Standard Model Lagrangians to covariant derivatives:

$$\partial_i \psi^j \rightarrow \nabla_i \psi^j = \partial_i \psi^j + \sum_k \Gamma^j_{ik}\psi^k,$$

where $$\Gamma^j_{ik}$$ is the Christoffel symbol that takes into account derivatives of the metric $$g$$ and hence of the specific space-time configuration.

The quantum gravity problem nowadays is to go "the other way round", i.e. to make a microscopic (quantum) theory of gravity and including it in the Standard Model.