Following up on my previous post, here’s an even more convincing argument that it is possible to construct a toy universe in which the curvature of spacetime is due to electromagnetism and not gravity and that I was unjustly vilified over at nForum (remember while reading the rest of my post below that it was asserted that I lacked a rudimentary knowledge of general relativity).
Consider again a charged massless universe, that is a universe in which there are electromagnetic fields but no gravitational fields. In such a case the complete stress-energy tensor only includes the electromagnetic portion. Einstein’s field equations are
In order to prove my point I need to show that is solely determined by (which I always thought was the standard interpretation, but after the tongue-lashing I received I’ll prove it just to be on the safe side).
In my toy universe I will assume that Riemannian geometry still exists (since, as a mathematical tool, it is independent of anything physical anyway). The definition of is
where the s may be written in terms of . Thus it boils down to determining whether , which is the metric, can be independently determined (i.e. from conditions not present in Einstein’s field equations).
As a simple case, let’s take the weak field approximation,
where and is the Minkowski metric for flat spacetime. If we further restrict ourselves to the linearized theory, the linearized, weak field Einstein equations are
where is the d'Alembertian. It is easier to explain the next step by showing what is done in normal GR, i.e. not in my toy universe. In such a case, for example,
and this is compared to the Newtonian
where is a scalar potential identified with Newtonian gravity. Thus, we choose
in order to force this to match the Newtonian gravity!
Let’s switch back to my toy universe now. The energy density, , is given by which, in normal GR is the density of the gravitational field. But in my toy universe,
where we have employed units such that . Further, in the electrostatic case, we note that,
where is the charge density and where we again are employing units with . Thus in the electrostatic case of my toy universe we may choose
where is the charge density! The metric and thus the curvature of spacetime in my toy universe has absolutely nothing to do with gravity!
Addendum: If the full stress-energy tensor is employed in general relativity, i.e. with both gravitational and electromagnetic portions, one could presumably do a similar weak field construction in which depends on both gravitational and electromagnetic fields which means I’m even right in this universe: the metric encodes the curvature of spacetime due to field sources and is not necessarily due solely to gravity.