Modifying Einstein's field equations
Hello everybody,
Since quite a while I am working on an emergent gravity theory. I have explored it in many ways, but what I haven't done is write down particular field equations for it.
There is something analogous to space time curvature in my model, and basically the left side of Einstein's equation can be left untouched (except of getting rid of the cosmological constant).
What I have to do on the right side is getting rid of all pure energy contributions. Now, I could do that by replacing the stress energy tensor by the Newtonian (Cauchy) stress tensor, but can that even work if the number of components of the tensors are not the same?
The second problem is that although according to my model rotation, pressure and photons don't excert gravity, the frame dragging effect is very real and exists in this model too  otherwise I would get in trouble with observations.
So how can I write down equations which don't deny the frame dragging effect, but at the same time say that angular momentum doesn't cause additional gravity?
The whole question might sound ludicrous to some of you, but you have to keep in mind that gravity must not necessarily be related to mass, but it could be something else. Some say entropy, some say charge. I don't want to get too much into the details here. For the sake of simplicity we can pretend I want it to depend on rest mass only. You might think this doesn't work, but maybe some of you find it challenging to try and write down such equations, just for the fun of it.
Thank you in advance!
