Gravitational Potential Energy: 1/2 Factor Explained

In summary, Theodore Frankel discusses the gravitational potential energy of a blob of fluid in chapter 3 of his book Gravitational Curvature. He states that the energy is given by the equation ∫B½p0U√gVdx, where p0 is the rest energy density and √gvdx is the volume form. The factor of ½ is necessary to avoid double-counting the gravitational potential when considering the potential of one mass due to another. This can be better understood by using discrete masses.
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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
 

Related to Gravitational Potential Energy: 1/2 Factor Explained

1. What is gravitational potential energy?

Gravitational potential energy is the energy that an object possesses due to its position in a gravitational field. It is the energy that an object has because of its height above the ground.

2. What is the formula for calculating gravitational potential energy?

The formula for calculating gravitational potential energy is PE = mgh, where m is the mass of the object, g is the acceleration due to gravity, and h is the height of the object above the ground.

3. What is the significance of the 1/2 factor in the formula for gravitational potential energy?

The 1/2 factor in the formula is a result of the work-energy theorem, which states that the work done on an object is equal to its change in energy. In this case, the work done is equal to half the force applied to lift the object multiplied by the distance it is lifted. This is why the 1/2 factor is included in the formula.

4. How does gravitational potential energy relate to kinetic energy?

Gravitational potential energy can be converted into kinetic energy when an object falls due to gravity. As the object falls, its potential energy decreases and its kinetic energy increases. At the bottom of the fall, all of the potential energy is converted into kinetic energy.

5. Can an object have negative gravitational potential energy?

Yes, an object can have negative gravitational potential energy if it is below the reference point (usually the ground). This means that the object would have to do work to be lifted up to the reference point, and therefore has a negative potential energy.

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