WWGD said:
Thanks, I mean, are light and gravitational waves dual to each other under this equivalence? EDIT: Apologies if I am derailing the thread, I can ask this as a stand-alone if that is best.
The short answer is no.
There are several tensors involved, I would point to the rank 2 Faraday tensor, and it's dual, the rank 2 Maxwell tensor, as the source of electromagnetic radiation in flat space-time in a 4-tensor treatment. Components of these tensors include the electric and mangetic fields, which satisfy the wave equation. See for instance
<<wiki link>>.
You'll note that components of this rank 2 Faraday tensor are the electric and magnetic fields. In the 4-tensor treatment of electromagnetism, which I would call the "realtivistic" treatment, the electric and magnetic fields are not rank 1 tensors (vectors) themselves. This may not be familiar, unfortunately - a detailed explanation of the 4-tensor treatment is beyond the scope of what I want to write, though learning about the Faraday tensor, which I've linked to, would be the first step.
For gravitational waves, what satisfies the wave equation is the metric tensor. This is something completely different than either the Faraday or Maxwell tensors.
These are the most important tensors, but there are a bunch more that one might wish to use. For gravitation, one can derive the rank 4 Riemann tensor from the metric tensor, from the rank 4 Riemann tensor one can derive the rank 2 Ricci tensor and the rank 2 Einstein tensor, which gives Einstein's field equations ##G_{\mu \nu} = 8 \pi T_{\mu \nu}##. Here ##G_{\mu \nu}## is the Einsten tensor, ##T_{\mu \nu}## is the stress-energy tensor.
The electromagnetic contribution to the total stress-energy tensor, the "electromagnetic stress-energy tensor", can be computed from the Faraday tensor, see for instance
<<yet another wiki link>>.
So, electromagnetic fields (incuding electromagnetic waves) have a stress-energy tensor (computable from the Faraday tensor) which is an incomplete part of the total stress energy tensor ##T_{\mu\nu}## in Einstein's field equations.
Einstein's field equations themselves involve ##G_{\mu \nu}##, the Einstein tensor, computed from the metric tensor ##g_{\mu \nu}##, and the stress-energy tensor ##T_{\mu \nu}##. The stress-energy tensor is not very intuitive, but it's the key element as being the "source of gravity", replacing the idea of "mass" as the source of gravity in Newtonian theory.
