A lame question from someone who barely remembers college-level physics: isn't that "electromagnetic field" just waves in the spacetime metric-tensor?
In a continuous medium like air, water, or a solid, you could describe a "displacement field" d(x) that tells you how far the piece of material that used to be at position x is displaced from its equilibrium position. This is governed by the equations of physical acoustics, which also have wave-like solutions: sound.
The Riemmann curvature tensor R is a function R(x) of spacetime coordinates x, which tells you the curvature of spacetime at location x. Einstein's equations tell you how R evolves, and it also turns out to have wave-like solutions: gravitational waves.
In radio, it's F (or E and B) that's waving. In gravitational waves, it's R.
What's confusing is that R also tells you "the shape of space," and you need to use it to figure out "where" the coordinate x corresponds to.
No, they are completely independent.
Here's another analogous example. Suppose you have a two-dimensional rubber sheet. You put a coordinate system on it, so any position on the rubber sheet can be identified by coordinates (x, y). You deform the rubber sheet so it has a curvy shape - anything you want.
You can have a tensor R(x,y) that tells you the curvature of the rubber sheet as a function of position (x,y). And a scalar T(x,y) that tells you the temperature of the rubber sheet at (x,y). These are completely independent quantities. Different fields.
I'm trying to look at R and F from the signal processing point of view: if we apply Fourier transform to both, we'd see that R is described by low frequency band (planetary scale distortions), while F would be ideally described by a high frequency band.
https://en.wikipedia.org/wiki/Einstein_ring#:~:text=An%20Ein....
It's possible that you could observe a kind of "shimmer" in the position of background stars as a gravitational wave passes between you and them. In fact there is an attempt to detect gravitational waves this way through pulsar timing:
I get that it's an analogy, but I don't think they would be completely independent quantities here. There is likely going to be some correlation with temperature and curvature of the rubber sheet. It would likely pale in comparison to almost everything else though, but on the topic of discussion that might be a large enough correlation.
Also, don't high energy photons curve spacetime?
Yeah, that's a failure of my analogy. Deforming a rubber sheet does create heat, but deforming spacetime does not create light.