If we set the speed of light to be infinite, we lose gravity. Einstein’s field equations, in full dimensional form, read

Rμν−12Rgμν=8πGc4Tμν.Rμν−12Rgμν=8πGc4Tμν.

Taking the limit c→∞c→∞ we end up with the vacuum field equation Rμν=0Rμν=0. Gravity is gone. The connection between matter and spacetime is gone, as the coupling constant went to zero.

If we set the speed of light to be infinite, we also lose electromagnetism. The two Maxwell vacuum field equations that represent the connection between electricity and magnetism, in the Gaussian convention that is the most appropriate here, read

curl E=−1cB˙,curl B=1cE˙.curl E=−1cB˙,curl B=1cE˙.

Thus when c→∞c→∞, we end up with both the electric and the magnetic fields curl-free, and with no connection between the two.

So setting the speed of light to infinity pretty much amounts to losing all the physics that we know.

These results, by the way, are both manifestations of a deeper fact: namely that the fundamental symmetry of both electromagnetism and gravitation is the Lorentz-Poincaré group of spacetime transformations, a group that is characterized by a finite invariant (same for all observers) velocity. Take that velocity to infinity (or equivalently, set its reciprocal to zero), and you lose the topological properties of this group and consequently, you lose the theories that depend on the topological properties of this group.

Credit: Viktor T. Toth