There are various different ways to try to add up the effective total mass of a configuration of masses in GR in such a way that it comes out to be equal to the total of the source mass as measured locally minus the gravitational binding energy. One of these is for example the Komar mass, which effectively treats the pressure within a system in equilibrium as effectively contributing to the gravitational mass. However, this is clearly not valid in a dynamic situation. Also, as recently discussed in another thread there is a recent paper by Ehlers and others which says that when you calculate the details for a spherical source, the effect of the pressure terms on the field does not act like additional rest mass but rather cancels out, although an adjustment of exactly the same magnitude arises for other reasons. (So far I have not made a lot of progress on understanding the implications of that paper). I've only recently spotted that there's another very simple way of calculating the effective total mass which gives the expected Newtonian answer directly, at least in the weak field approximation. The answer is to integrate the local mass or energy density times the square root of the local time dilation factor (relative to the distant time rate), which is equivalent to applying half of the fractional time dilation. This causes the effective energy of each piece of local mass or energy to be adjusted downwards by exactly half of its potential energy relative to every other piece of local mass or energy in the configuration, accounting correctly for the binding energy between every pair of pieces. This method does not need to resort to treating pressure as contributing to the effective rest mass nor does it need to include some sort of "gravitational field energy" to adjust the total. This may admittedly seem an odd thing to do in a GR context. For background, I should mention that I spotted it when investigating the implications of a Machian alternative in which the Einstein field equations are still assumed to hold (at least to post-Newtonian accuracy) around a single dominant central mass but the constant G in those equations is effectively an abbreviated notation for the gravitational effect of all distant masses, excluding the local dominant central mass and other smaller bodies. (I presume that it would not be acceptable under PF rules to give any further details of the Machian alternative theory here, so I'm only mentioning it for context. I will however say it's not the same as Brans-Dicke). For this Machian scheme to match GR theory and experiment in this limited case (especially to the precision needed for the PPN beta parameter to give the correct perihelion precession for Mercury) the effective local value of G (which would apply for example to local lab experiments) has to vary as the square root of the time dilation factor. That means that within this scheme, the total gravitational source strength of a system, taking full account of internal gravitational binding energy, is simply the integral of Gm for all masses, provided that the locally varying value of G is used rather than the constant value which excludes the effect of the local central mass. In GR, G itself is defined as constant, but it appears that the same method of summing the effective total energy using the square root of the time dilation applied to the local mass might be a useful tool. However, I don't at present have a strong mathematical justification for it apart from the fact that it clearly trivially matches the Newtonian result in the weak field case (and so far I don't have an equivalent expression for strong fields). Does anyone know if there is already a known method of evaluating the effective total gravitational mass within standard GR by using the square root of the time dilation factor in this way (or equivalently, half of the fractional time dilation factor), and if so can they point me to further information?