Gravitational Potential Energy: 1/2 Factor Explained

[t_r_theta_phi]

I am currently reading Gravitational Curvature by Theodore Frankel. In the derivation of Einstein's equations in chapter 3, he states that the gravitational potential energy of a blob of fluid is

∫_B½p₀U√g_Vdx

where the integral is a volume integral, p₀ is the rest energy density and √g_vdx is the volume form.

From what I understand p₀√g_vdx is an infinitesimal bit of mass, so why wouldn't the potential energy just be U times that bit of mass? Why ½ that?

 

[mfb]

Without that factor, you double-count the gravitational potential. This is easier to understand with discrete masses: You would sum over the potential of mass A due to B (using GMm/r) and the potential of mass B due to A (using GMm/r again), but the actual potential energy for both together is just one time GMm/r.

 

[t_r_theta_phi]

That makes much more sense. Thank you