Gravitational Potential Energy: 1/2 Factor Explained

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SUMMARY

The discussion centers on the derivation of gravitational potential energy as presented in Theodore Frankel's "Gravitational Curvature." The formula for gravitational potential energy is given as ∫B½p0U√gVdx, where p0 represents rest energy density and √gvdx denotes the volume form. The factor of ½ is crucial to avoid double-counting the gravitational potential when considering interactions between masses, as illustrated through the example of discrete masses A and B. This clarification emphasizes the importance of the factor in accurately calculating total gravitational potential energy.

PREREQUISITES
  • Understanding of Einstein's equations and general relativity
  • Familiarity with the concept of gravitational potential energy
  • Knowledge of volume integrals in calculus
  • Basic principles of mass interaction in physics
NEXT STEPS
  • Study the derivation of Einstein's equations in "Gravitational Curvature" by Theodore Frankel
  • Explore the implications of the factor of ½ in gravitational potential energy calculations
  • Learn about volume integrals and their applications in physics
  • Investigate the concept of gravitational potential in multi-body systems
USEFUL FOR

Physics students, researchers in gravitational theory, and anyone interested in the mathematical foundations of general relativity and gravitational interactions.

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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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