Can dark matter and energy be formulated as local perturbations of the metric

Question

Note, my formal physics education ended over ten years ago so I may be missing some obvious piece of understanding.

The relationship between space-time and matter/energy distribution is described by Einstein's field equations:

$$G_{\mu\nu}=8\pi T_{\mu\nu}$$

G is a tensor that describes the geometry of space time, and T describes the distribution of matter, how it's moving, etc. When I read about Dark Matter I presume these are inserted as terms in the stress energy tensor on the right and are then interpreted as mysterious missing matter. Similarly, Dark Energy terms may be added as extra terms on the left, such as the cosmological constant, or further terms in T.

My question is, then, why these cannot be captured as local perturbations to the global curvature. Ie, scale-dependent terms in the Ricci tensor? This would allow for localized gravitational lensing effects and apparent extra mass around massive objects such as galaxies and would just be a property of space-time.

Answer 1

Of course dark energy and dark matter are described by adding perturbations to "the left", the question is what perturbations to add! The utility of Einstein's equations is that it gives you a precise way to determine the effect of some matter configuration on the Ricci tensor and hence the metric. Essentially, you're always adding the perturbation to both sides simultaneously. Your question, as worded, asks why one cannot "capture" the effect of dark matter/energy by perturbing the metric, which, I presume to mean is, "by only perturbing the metric", i.e., not perturbing the matter terms. The problem is that nearly any other equation you could write breaks general covariance, which has all kinds of bad consequences. In particular, conservation laws will be broken and you'll implicitly define a preferred reference frame in the process. One can add higher order terms to the left hand side, (in fact, some are expected to exist) but these won't have the effect of mimicking dark energy. For example, calculations involving gravitational lensing and what-not merely assume a linearized gravity theory; they generally don't even solve the full equations as written, yet this is enough to mathematically reproduce what we see. Including higher order terms would have consequences which are unobservable at the moment.

One might rephrase your question as follows: "How do we know if a phenomenon is gravitational or matter based?" The answer is simply almost everything is both. It's not that dark energy or dark matter is a "gravitational phenomenon" and so described by the Ricci tensor or a "matter phenomenon" and so described by the stress energy tensor. It is both and thus simultaneously described by both. Einstein's equation just tells us how to relate the metrical properties of a phenomenon to the stress-energy properties. This relation is uniquely determined (up to higher order terms) by symmetry and so can't be tweaked with without severe consequences.