Undergrad Gravitational Potential Energy: 1/2 Factor Explained

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The discussion focuses on the derivation of gravitational potential energy in Einstein's equations, specifically addressing the factor of 1/2 in the expression for a blob of fluid. The integral presented includes terms for rest energy density and volume form, leading to confusion about why the potential energy isn't simply the product of potential U and mass. The explanation clarifies that the 1/2 factor prevents double-counting the gravitational potential when considering interactions between two masses. This is illustrated by comparing the potential energy calculations for two discrete masses, where the total potential energy is correctly represented without duplication. Understanding this concept is crucial for grasping the underlying principles of gravitational potential energy in the context of general relativity.
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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√gVdx

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

From what I understand p0√gvdx is an infinitesimal bit of mass, so why wouldn't the potential energy just be U times that bit of mass? Why ½ that?
 
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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.
 
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That makes much more sense. Thank you
 

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