general relativity - Source term of the Einstein field equation

My copy of Feynman's "Six Not-So-Easy Pieces" has an interesting introduction by Roger Penrose. In that introduction (copyright 1997 according to the copyright page), Penrose complains that Feynman's "simplified account of the Einstein field equation of general relativity did need a qualification that he did not quite give." Feynman's intuitive discussion rests on relating the "radius excess" of a sphere to a constant times the enclosed gravitational mass $M$: for a sphere of measured radius $r_{\mathrm{meas}}$ and surface area $A$ enclosing matter with average mass density $\rho$ smoothly distributed throughout the sphere, $$\sqrt{\frac{A}{4π}}−r_{\mathrm{meas}}=\frac{G}{3c^2}\cdot M,$$ wherein $G$ is Newton's gravitational constant, $c$ is the speed of light in vacuum, and $M=4π\rho r^3/3$. I don't know what $r$ is supposed to be, but it's presumably $\sqrt{\frac{A}{4\pi}}$. Feynman gratifyingly points out that $\frac{G}{3c^2}\approx 2.5\times 10^{−28}$ meters per kilogram (for Earth, this corresponds to a radius excess of about $1.5$ mm). Feynman is also careful to point out that this is a statement about average curvature.

Penrose's criticism is: "the 'active' mass which is the source of gravity is not simply the same as the energy (according to Einstein's $E=mc^2$); instead, this source is the energy plus the sum of the pressures". Damned if I know what that means -- whose pressure on what?

So, taking into account Penrose's criticism but maintaining Feynman's intuitive style, what is the active mass $M$?

Answer

I'll try and answer in an intuitive way as best I can (as you asked for on the crosslink).

The relation between the true, or physical, surface area of a sphere with radius $r_{meas}$ and the surface area one expects from standard Euclidean space is a measure (as you say) of the average curvature (more precisely it is a measure of the scalar curvature $R$: http://en.wikipedia.org/wiki/Scalar_curvature). In GR this curvature is generated by "mass-energy" (again I'm sure you know this), or the stress-energy tensor $T_{\mu \nu}$. The above relation that Feynman gives for a average mass density $\rho$ is valid if you are talking about cold matter or "dust" where in the average rest frame of the matter, there are no internal motions (i.e. pressures). If there are internal motions of the matter then there is kinetic energy in addition to rest mass energy which you should intuitively expect to also contribute to curvature. This additional energy plus the rest mass energy is what I think you mean by "active" mass.

So it seems to me that Penrose's criticism is really that Feynman didn't use a realistic model of matter, since the original relation is true if the matter you are talking about is ultra cold.