# Potential energy

Potential energy is rest mass.[1] The self-gravitational involution of a mass can be visualized as a series of concentric shells. The higher a shell, the lower its rotational frequency. If we sufficiently extend the series of concentric shells, then the outermost shell's rotational frequency will be zero, so that the shell will have the lowest (i.e., zero) actual energy (nonzero‑frequency angular momentum) and the highest (i.e., zero) potential energy (zero‑frequency angular momentum).

Potential energy is the energy that an object has because of its position on a gradient of potential energy called a potential field.[2][3]

The potential fields are irrotationally radial ("electric") fluxes of the vacuum[5][6] and divide into two classes:

• The gravitoelectric fields;[7]
• The electric fields.[8]

The potential energy is negative.[9] It is not a mere convention but a consequence of conservation of energy in the zero-energy universe—as an object descends into a potential field, its potential energy becomes more negative, while its actual energy becomes more positive.

In accordance with the minimum total potential energy principle, the universe's matter flows towards the minimum total potential energy. This cosmic flow is time.

## Simple examples

Bringing a rock uphill increases (i.e., makes less negative) its gravitoelectric potential energy. Stretching a rubber band increases its elastic potential energy, which is a form of the electric potential energy. A mixture of a fuel and an oxidant has a chemical potential energy, which is another form of the electric potential energy. Batteries too have chemical potential energy.

## Gravitational potential energy

### Self‑gravitating sphere

The gravitational potential energy of a massive sphere is proportional to its radius[10] and causes the sphere to fall towards its own centre. In an idealized model built without taking into account the concomitant generation of collapse‑impeding actual energy, the free-fall timescale tff is inversely proportional to the square root of the sphere's density ρ:

${\displaystyle t_{\mathrm {ff} }=\left({\frac {3\pi }{32G\rho }}\right)^{1/2}.}$ (13.8)
Equation (13.8) implies that the denser clumps collapse more rapidly, which may lead to separation and fragmentation. As cores are observed to be densest near their centers, the interiors cave in most quickly, producing an inside-out collapse.

—Pater, Imke de; Lissauer, Jack J. Planetary Sciences CUP, 2010, p. 518

Simply put, when a sphere's density increases a hundredfold, the sphere's free‑fall collapse becomes ten times quicker (the square root of 100 is 10).

At that, the sphere's density increases as the inverse cube of the sphere's radius,[11] so a free‑falling sphere whose radius has shrunk tenfold will have its density increased a thousandfold and will collapse 31.6228 times quicker (the square root of 1000 is 31.6228).

In reality, a massive sphere's self‑gravitational collapse is accompanied by the intensification of the sphere's gravitational potential energy (zero‑frequency angular momentum, which exerts a zero resistance to shrinkage) into the sphere's actual energy (nonzero‑frequency angular momentum, which exerts a nonzero resistance to shrinkage):

After each infinitesimal step of collapse the star has to wait until it has radiated away a half of the released gravitational energy before it can continue to contract.

—Böhm-Vitense, Erika. Introduction to Stellar Astrophysics CUP, 1992, p. 29

But the collapse‑impeding actual energy becomes radiated away, so that the self‑gravitational collapse of a sphere whose radius has shrunk tenfold becomes 1000 times quicker:

1. Lane's law dictates that the temperature of a self‑gravitating perfect-gas sphere is inversely proportional to its radius: rT(r) = constant.[12] So, when the sphere's radius (r) decreases tenfold, the sphere's temperature (T) increases tenfold.[11]
2. The Stefan–Boltzmann law dictates that the rate at which a unit surface area of the self‑gravitationally condensing sphere radiates away heat is proportional to the fourth power of the sphere's temperature. So, even after taking into account that the sphere's surface area decreases a hundredfold (as the square of the radius), Lane's law implies that when the self‑gravitating sphere's radius shrinks tenfold, the sphere's total radiative heat loss per unit time increases a hundredfold.
3. At the same time, the tenfold decrease in the radius implies that the sphere's potential energy has decreased tenfold, so that the next tenfold decrease in the sphere's radius will generate ten times less collapse‑impeding actual energy.[10] Therefore, when the self‑gravitating sphere's radius shrinks tenfold, the speed of the sphere's collapse increases a thousandfold, i.e., as the inverse cube of the radius, or, which is the same, as the density.

So, when the radiative loss of the concomitantly generated collapse‑impeding actual energy is taken into account, the free-fall timescale tff is inversely proportional to the sphere's density ρ, and the above‑quoted equation 13.8 takes the following form:

${\displaystyle t_{\mathrm {ff} }={\frac {3\pi }{32G\rho }}.}$

This equation dictates that the densest self‑gravitating sphere shrinks fastest, and, in accordance with Lane's law, has the highest temperature. The densest object in the universe is the proton (5.96 × 1014 g/cm3). In 1983, numerical calculations on large computers predicted that as the temperature is raised the colour‑repelling physical vacuum should flip into the simple vacuum, of which protons consist, at a temperature of 2 × 1012 K,[13] which is the approximate temperature of the proton. Since material standards of length are themselves made of shrinking protons, the exponentially accelerating self‑gravitational shrinkage of the proton can only be inferred from the relative expansion of intergalactic spaces, which are less dense and thus self‑gravitationally shrink progressively slower than the proton:

All change is relative. The universe is expanding relatively to our common material standards; our material standards are shrinking relatively to the size of the universe. The theory of the "expanding universe" might also be called the theory of the "shrinking atom". <...>

Let us then take the whole universe as our standard of constancy, and adopt the view of a cosmic being whose body is composed of intergalactic spaces and swells as they swell. Or rather we must now say it keeps the same size, for he will not admit that it is he who has changed. Watching us for a few thousand million years, he sees us shrinking; atoms, animals, planets, even the galaxies, all shrink alike; only the intergalactic spaces remain the same. The earth spirals round the sun in an ever‑decreasing orbit. It would be absurd to treat its changing revolution as a constant unit of time. The cosmic being will naturally relate his units of length and time so that the velocity of light remains constant. Our years will then decrease in geometrical progression in the cosmic scale of time. On that scale man's life is becoming briefer; his threescore years and ten are an ever‑decreasing allowance. Owing to the property of geometrical progressions an infinite number of our years will add up to a finite cosmic time; so that what we should call the end of eternity is an ordinary finite date in the cosmic calendar. But on that date the universe has expanded to infinity in our reckoning, and we have shrunk to nothing in the reckoning of the cosmic being.

We walk the stage of life, performers of a drama for the benefit of the cosmic spectator. As the scenes proceed he notices that the actors are growing smaller and the action quicker. When the last act opens the curtain rises on midget actors rushing through their parts at frantic speed. Smaller and smaller. Faster and faster. One last microscopic blurr of intense agitation. And then nothing.

—Eddington, Arthur. The Expanding Universe CUP, 1933, pp. 90–92

In 1998, Adam Riess and his team discovered that the apparent expansion of intergalactic spaces is accelerating. On 5 April 2016, Adam Riess et al. announced that the rate of the acceleration is itself increasing—over the three years since 21 March 2013, when the Planck space observatory published the local Hubble constant value, the apparent expansion of intergalactic spaces had accelerated by nine percent more than expected.[14] It means that over the three years from 2013 to 2016, the speed of the proton's exponentially accelerating self‑gravitational shrinkage relative to the intergalactic spaces had increased by nine percent more than expected. The 13.8‑billion‑year‑long "drama for the benefit of the cosmic spectator" has come to its finale.

### Earth

Hyrdroelectric power plants use the gravitational potential energy of water (in the form of a difference in height) to produce electricity.

If an object is lifted a certain distance from the surface from the Earth, the force experienced is caused by weight and height. Work is defined as force over a distance, and work is another word for energy. This means gravitational potential energy is equal to

${\displaystyle U=F\Delta h}$
where
${\displaystyle F}$ is the force of gravity
${\displaystyle \Delta h}$ is the change in height

or

${\displaystyle U=mgh}$

Total work done by gravitational potential energy in a moving object from position 1 to position 2 can be found by:

${\displaystyle \Delta W=U_{1}-U_{2}}$

or

${\displaystyle \Delta W=mgh_{1}-mgh_{2}}$
where
${\displaystyle m}$ is the mass of the object
${\displaystyle g}$ is the acceleration caused by gravity (constant)
${\displaystyle h_{1}}$ is the first position
${\displaystyle h_{2}}$ is the second position

## Electric potential energy

Electric potential energy is experienced by charges both different and alike, as they repel or attract each other. Charges can either be positive (+) or negative (-), where opposite charges attract and similar charges repel. If two charges were placed a certain distance away from each other, the potential energy stored between the charges can be calculated by:

${\displaystyle U={\frac {kQq}{r}}}$
where
${\displaystyle k}$ is 1/4πє (for air or vacuum it is ${\displaystyle 9x10^{9}Nm^{2}/C^{2}}$)
${\displaystyle Q}$ is the first charge
${\displaystyle q}$ is the second charge
${\displaystyle r}$ is the distance apart

## Elastic potential energy

Elastic potential energy is experienced when a rubbery material is pulled away or pushed together. The amount of potential energy the material has depends on the distance pulled or pushed. The longer the distance pushed, the greater the elastic potential energy the material has. If a material is pulled or pushed, the potential energy can be calculated by:

${\displaystyle U={\frac {1}{2}}kx^{2}}$
where
${\displaystyle k}$ is the spring force constant (how well the material stretches or compresses)
${\displaystyle x}$ is the distance the material moved from its original position

## References

1. Heighway, Jack. Einstein, the Aether and Variable Rest Mass. HeighwayPubs, 2011, p. 36. "Understanding why rest masses are reduced in a gravitational field only requires a simple insight: when an object is raised in a gravitational field, the gravitational potential energy increase is real, and exists as an increase, usually tiny, in the rest mass of the object."
2. Morrison, Faith A. An Introduction to Fluid Mechanics. CUP, 2013, p. 442. "Energy may be stored in the state of a system—for example, as kinetic energy stored in the speed of the system, as potential energy stored in the position of the system in a potential field, or as internal energy stored in the chemical state of a system."
3. Best, Myron G. Igneous and Metamorphic Petrology. John Wiley & Sons, 2013, p. 21. "Potential energy can be equated with the amount of work required to move a body from one position to another in a potential field, in this instance, the gravitational field of the Earth."
4. Biedenharn, L. C.; Louck, J. D. Angular Momentum in Quantum Physics. Addison-Wesley Pub. Co., Advanced Book Program, 1981. "The Planck quantum of action, h, has precisely the dimensions of an angular momentum, and, moreover, the Bohr quantization hypothesis specified the unit of (orbital) angular momentum to be ħ = h/2π. Angular momentum theory and quantum physics are thus clearly linked."
5. Ziegler, Franz. Mechanics of Solids and Fluids. Springer, 1995, p. 167. "Force in such a potential field is a flux in the sense of a mechanical driving agent."
6. Volovik, G. E. The Universe in a Helium Droplet. OUP, 2003, p. 60. "The non-viscous flow of the vacuum should be potential (irrotational)."
7. Grøn, Øyvind; Hervik, Sigbjørn. Einstein's General Theory of Relativity with Modern Applications in Cosmology. Springer, 2007, pp. 201, 203. "φ is the Newtonian or 'gravitoelectric' potential: φ = −Gm/r. ... In the Newtonian theory there will not be any gravitomagnetic effects; the Newtonian potential is the same irrespective of whether or not the body is rotating. Hence the gravitomagnetic field is a purely relativistic effect. The gravitoelectric field is the Newtonian part of the gravitational field, while the gravitomagnetic field is the non-Newtonian part."
8. Soviet Physics, Uspekhi. Vol. 40, issues 1–6, American Institute of Physics, 1997, p. 39. "From Maxwell equations (6.20) it follows that the electric field is potential: E(r) = −gradφ(r)."
9. Why is the Potential Energy Negative? HyperPhysics
10. Mihos, Chris. Gravitational Energy. Case Western Reserve University
11. Doig, Peter. An Outline of Stellar Astronomy. Hutchinson, 1947, p. 76. "Lane reached the apparently paradoxical result that a star by losing heat and contracting actually grew hotter. A star shrinking under gravitation to half its linear size and remaining built on the same model, or "homologous" (i.e., the densities at two corresponding points at any two stages remaining the same fraction of the mean density) would be eight times as dense, and the internal pressures would be sixteen times as great as the overlying material is attracted four times as strongly and its weight is held up on only a quarter of the area. From the formula connecting temperature with pressure and density, given earlier in the chapter, it will be seen that the temperature in this example would be twice as great. By such reasoning, Lane concluded that as stars get smaller they grow hotter to withstand gravitation and resist collapse."
12. Reid, Neill I. Hawley, Suzanne L. New Light on Dark Stars: Red Dwarfs, Low-Mass Stars, Brown Dwarfs. Springer, 2013, p. 84
13. Willis, Bill. Collisions to melt the vacuum. New Scientist, 3 October 1983, p. 10
14. Hirsch, Arthur. Our universe is expanding faster than scientists predicted, study suggests. Hub, 3 June 2016