iuvalclejan said:
Why doesn't this work if the field is strong? Or does it work as long as there are no singularities?
Given a coordinate system (or, alternatively, a congruence of worldlines, if you are familiar with that concept), you'll get a number this way. But as ##T_{00}## is not a tensor, but just a piece of a tensor, there is no guarantee that you will get a number independent of your choice of coordinate, as the expression does not follow the tensor rules for generating an invariant quantity.
I don't have any more detailed specific illustrative worked out (such as specific coordinate choices giving different results) however.
I do have another relevant example, originating in MTW's text, "Gravitation". I don't have the specific page/section, it'd take some work to dig it out. My wording is probably not the same as MTW's and is probably less rigorous, be warned.
Consider a solid planet, that you cut up into smaller blocks. Then you pull all the blocks to infinity, dissassembling the planet. The result you find is that the mass of the dissassembled planet is different from the mass of the blocks. It takes work from an external source to dissassemble the planet - thus, the dissasembled planet has more mass than the assembled planet. The difference between these two masses is the Newtonian "gravitational binding energy" of the planet, the energy it would pull the planet apart and transport its pieces to infinity.
This illustrates that in the Newtonian limit, the idea of integrating ##T_00## is different from the idea of finding the system mass, because the system mass includes the gravitational binding energy, while the integral of ##T^00## does not. Again, one is motivated to find some tensor expression for the so-called "gravitational binding energy". Unfortunately, in general, there isn't one. I don't recall the details, but I do recall that MTW has a section on why you can't localize the gravitational binding energy as a tensor. MTW also disccusses the "pseudo-tensor" approach to mass in General Relativity.
Concepts of mass that do exist in GR require additional restrictions, such as the asymptotic flatness required for the ADM mass, and the stationary (or static) geometry for the Komar mass. As I recall, the pseudo-tensor approach also winds up requiring asymptotic flatness, but I could be wrong on this - my memory on this point is hazy.
Wald's text "General Relativity" has a detailed and highly mathematical workthrough for the Komar mass (though he doesn't use this name), and a brief discussion of the ADM mass. MTW covers the pseudo-tensor approach.