# Potential energy

File:Irrotational vortex.gif
Potential energy is rest mass.<ref name="RestMass">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."</ref> 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.<ref>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."</ref><ref>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."</ref>

• Actual energy (E = hf) is nonzero‑frequency angular momentum.<ref>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."</ref>
• Potential energy (rest mass<ref name="RestMass"/>) is zero‑frequency angular momentum.

The potential fields are irrotationally radial ("electric") fluxes of the vacuum<ref name="Ziegler">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."</ref><ref name="Volovik">Volovik, G. E. The Universe in a Helium Droplet. OUP, 2003, p. 60. "The non-viscous flow of the vacuum should be potential (irrotational)."</ref> and divide into two classes:

• The gravitoelectric fields;<ref>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."</ref>
• The electric fields.<ref>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)."</ref>

The potential energy is negative.<ref name = "PotEnIsNeg">Why is the Potential Energy Negative? HyperPhysics</ref> 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, and, in accordance with the second law of thermodynamics, tends to be radiated away, so that the object acquires a net negative potential energy, also known as the object's binding energy.<ref>Young, Hugh D.; Freedman, ‎Roger A. University Physics. Addison-Wesley, 2000, p. 736. "A crystal of table salt is made of ions of sodium (Na+) and chlorine (Cl) and has a net negative potential energy. To dissolve salt in water, energy must be added to separate the ions."</ref><ref>Shu, Frank. The Physical Universe: An Introduction to Astronomy. University Science Books, 1982, p. 66. "Remember, more binding energy means the system is more bound—has greater negative energy."</ref>

In accordance with the minimum total potential energy principle, the universe's matter flows towards ever more negative 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.

## Potential energy in cosmology

The minimum total potential energy principle dictates that the universe begins its 13.8-billion-year-long gravitational life cycle in a state of maximal (i.e., zero) potential energy, and minimal (i.e., zero) actual energy.

The gravitational potential fields of uniformly distrubuted matter waves cancel each other, so that the total gravitational potential energy is maximal (i.e., zero), while the total actual energy of matter waves is minimal (i.e., zero). Therefore, at the outset of the universe's 13.8-billion-year-long gravitational life cycle, the Compton wavelength or radius of each proton is extremely big, so that protons fill the entire volume of the universe and there are no empty spaces between them. Also, the primordial protons are uniformly distributed, so that the universe's hierarchicity factor (G) is zero:

Accordingly the primordial state of things which I picture is an even distribution of protons and electrons, extremely diffuse and filling all (spherical) space, remaining nearly balanced for an exceedingly long time until its inherent instability prevails. We shall see later that the density of this distribution can be calculated; it was about one proton and electron per litre. There is no hurry for anything to begin to happen. But at last small irregular tendencies accumulate, and evolution gets under way. The first stage is the formation of condensations ultimately to become the galaxies; this, as we have seen, started off an expansion, which then automatically increased in speed until it is now manifested to us in the recession of the spiral nebulae.

Eddington, Arthur. The Expanding Universe CUP, 1933, pp. 56–57

The gravitational potential energy of a spherical universe is:

${\displaystyle U=-\left({\frac {3}{5}}\right){\frac {GM^{2}}{R}}}$

where G is the universe's hierarchicity factor, M is the mass of the universe, and R is the radius of the universe.<ref name=Kutner158>Kutner, Marc L. Astronomy: A Physical Perspective. CUP, 2003, p. 158</ref>

In accordance with the minimum total potential energy principle, the universe's gravitational potential energy (U) becomes ever more negative, which is achieved by the tick-tock principle: a gradual shrinkage of the universe's radius (R) is followed by a sudden increase in the universe's hierarchicity factor (G).<ref name="HierarchFactor">Moskvitch, Katia. Strength of gravity shifts—and this time it's serious. New Scientist, 11 September 2013</ref>

## Gravitational potential energy

### Self‑gravitating sphere

The gravitational potential energy of a massive spherical cloud is proportional to its radius<ref name="Mihos">Mihos, Chris. Gravitational Energy. Case Western Reserve University</ref> and causes the sphere to fall towards its own centre.

Initially, the cold and rarefied cloud is transparent to its own blackbody radiation, so the collapse is isothermal (because any increase in temperature is immediately radiated away) and continues on a free-fall timescale tff,<ref>Benacquista, Matthew. An Introduction to the Evolution of Single and Binary Stars. Springer, 2012, p. 136</ref> which 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,<ref name="Doig">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."</ref> 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).

Once the density of the gas is high enough that the cloud becomes opaque and can no longer freely radiate away the energy gained through gravitation, then the collapse will become stepwise—an adiabatic compression to a higher temperature will be followed by a period of radiative cooling, ending with another adiabatic compression to a still higher temperature:

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
1. Lane's law dictates that the temperature of a self‑gravitating ideal-gas sphere is inversely proportional to its radius: rT(r) = constant.<ref>Reid, Neill I. Hawley, Suzanne L. New Light on Dark Stars: Red Dwarfs, Low-Mass Stars, Brown Dwarfs. Springer, 2013, p. 84</ref> So, when the sphere's radius (r) decreases tenfold, the sphere's temperature (T) increases tenfold.<ref name="Doig"/>
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.

Stage of collapse Description Increase in speed of collapse per tenfold decrease in radius
1
A cold and rarefied cloud is transparent to its own blackbody radiation, so the collapse is isothermal (because any increase in temperature is immediately radiated away) 31.6228-fold
2
The condensed cloud has become opaque to its own blackbody radiation and can no longer freely radiate away the energy gained through gravitation, so that its collapse has become stepwise—an adiabatic compression to a higher temperature is followed by a period of radiative cooling, ending with another adiabatic compression to a still higher temperature 100-fold

So, 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). It is known that a system in the ground state (the lowest potential energy state,<ref>Atkins, Peter; Paula, Julio de; Friedman, Ronald. Quanta, Matter, and Change. OUP, 2009, p. 107. "It is sometimes awkward to keep in mind what ‘higher’ and ‘lower’ potential energy means. When attractions are dominant, the potential energy is negative and a lower (more negative) potential energy is ‘more stable’ and a higher (less negative) potential energy is ‘less stable’."</ref> or, which is the same, the state of a zero rest mass<ref name="RestMass"/>) has a zero temperature.<ref>Koechner, Walter. Solid-State Laser Engineering. Springer, 2013, p. 4. "At absolute zero temperature, Boltzmann's statistics predicts that all atoms will be in the ground state."</ref> Therefore, since the proton has a nonzero rest mass, it is not in the ground state and must have a nonzero temperature. 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,<ref>Willis, Bill. Collisions to melt the vacuum. New Scientist, 3 October 1983, p. 10</ref> which is the approximate temperature of the proton. The Stefan–Boltzmann law dictates that because of its enormously high temperature, the proton must radiate its energy away at a feverish rate, and consequently shrink to ever higher temperatures in accordance with Lane's law.

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.<ref>Hirsch, Arthur. Our universe is expanding faster than scientists predicted, study suggests. Hub, 3 June 2016</ref> 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

File:Grande-dixence.jpg
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