Is relativistic mass not an appropriate concept?

[Dale]

how do you motivate that the stress-energy tensor is the generalization of "gravitational mass" when searching for a generalization of Newton's gravity consistent with special relativity?

AFAIK there is no generalization of Newtonian gravity which is consistent with SR. I always make the point that the source of gravity in GR is the stress-energy tensor precisely to encourage the understanding that GR is not merely an extension of Newtonian gravity.

 

[pervect]

I'd agree that there doesn't appear to be any fundamental definition of mass in GR, which gives rise to the large number of non-fundamental definitions we do have (and these definitions require additional assumptions to calculate at all, such as a static metric, or asymptotic flatness, another reason they are not fundamental).

As far as motivation goes, the approach I tend to use (borrowed from the textbooks I learned from) is to initially talk about gravity as the result of geodesic deviation, rather than a force.

I'll also refer readers to Baez's "The Meaning of Einstein's equation" which ties the the second derivative of the volume of a sphere of coffee grounds following geodesics to the enclosed density and pressure.

I'll tend to use the Newtonian idea of mass more to motivate Komar mass based on Wald's treatment - though this is more of an advanced topic than something I'd introduce right away, unlesss the idea of mass was brought up specifically. The Komar mass also ties in nicely with Baez's treatment, however.