Sorry, I'm prone to speaking imprecisely, and I'm still learning the correct terminology in GR. I think I'm looking for the metric perturbation caused by the existence of a single photon. Ideally I'd like christoffel symbols, or a line element, or something equivalent which describes the gravity around a photon in otherwise empty flat space.
If there's a better term for the relation between the things causing curvature and the shape of space, it would also be handy to know that.
A photon isn't classical, and GR is a classical theory. But you might be looking for the Aichelburg-Sexl ultraboost. But I might have misunderstood your question. I think the wiki page also talks about how the AS ultraboost can be derived as the limit of the metric of a gaussian pulse.
Typically these solutions are actually for a null dust - which may not have exactly the same stress-energy tensor as a pulse of light, but has many of the same features.
A null dust has the formal definition of ##T_{ab} \propto k^a k^b## where k is a null vector. It's informally described as the stress energy tensor (or the associated space-time geometry) associated with some sort of radiation moving at the speed of light.
What do you mean by "metric for a photon"? The metric is a property of spacetime, not of its contents.
True, but given any metric you can compute the Einstein tensor which is proportional to the stress energy tensor. Thus the OP question can be taken as what is an example of metric producing the stress energy tensor of a photon. Then the answer is the one Pervect gave - there is none because stress energy tensor is a classical construct.
On the other hand there are many metrics that represent a pulse of light. Unfortunately, I am not familiar with any particular example.
