Abstract
The E = 4πc² of Paper 12 has the form of the surface area of a sphere of radius c. Paper 60 showed that this 4π is a mass, a choice of unit (a convention), and Paper 61 that, once a unit of time is fixed so that t = 1, the area of the light sphere really is 4πc². Energy is not an area. Is there, then, a reading in which the area of the light sphere enters the bookkeeping of the energy? Verlinde (2011) set up a bookkeeping in which a sphere’s surface is divided into cells about one Planck area in size and each cell receives an energy kT/2, half of kT, so that the total is Mc². Following Section 4.1 of that paper, the temperature T is set from the gravitational acceleration on the skin in the Unruh form. This paper does not adopt Verlinde’s claim that gravity is an entropic force; it takes only the bookkeeping and applies it to the light sphere in Yamagishi’s units.
In Yamagishi’s units (Għ = 1 with G = 4π and ħ = 1/(4π), m = 4π, k = 1, and t = 1 so that the light sphere has radius c), the light sphere has 4πc⁵ cells (four times the entropy πc⁵ of Paper 65), each at a temperature of 2/c³ and carrying an energy of 1/c³, and the total is exactly 4πc². The coefficient 1 is ½ × 4π ÷ 2π: for any mass M the total is Mc², and the 4π of the skin’s area cancels against the 2π of the Unruh form (Paper 142) and the 1/2 of equipartition. With M = 4π, the 4π left in E = 4πc² is the 4π of M. In this bookkeeping the total is not the area itself but the number of cells, which is proportional to the area, times the energy per cell.
The bookkeeping is built so that it returns Mc² once Newtonian gravity is put in; the total coming out as Mc² is not in itself a new coincidence. What this paper counts is the condition under which the total does not depend on the size of the skin. On a skin of radius R the number of cells grows as R² and the temperature falls as 1/R²; on the light sphere at time t the factors t² and 1/t² cancel in the same way, and the total stays at 4πc². The light sphere is not a special skin. Compared within fields of Yukawa form, the total falls by the factor (1 + κR)exp(−κR). At κ = 1 and R = 0.5, 1, 2 and 4, these values times 4π equal the point-source fluxes 11.432834, 9.245819, 5.102020 and 1.150806 in the table of Paper 1107 (the two are the same function, so this is a check of the numbers). The total is the flux through the skin times c²/(4πG), and its independence of the skin’s size is the same condition as Paper 1107’s "4π stays on the skin only when the field is massless".
In SI units, counting a mass of 1 kg on the light sphere at t = 1 s gives 4.32346×10⁸⁷ cells at a temperature of 3.01132×10⁻⁴⁸ K, and the total is Mc² = 8.98755178737×10¹⁶ J. Applying S = N/4 of Paper 73 to a skin gives E = 2TS; with the skin at the Schwarzschild radius, the temperature of the bookkeeping equals the Hawking temperature and N/4 equals the black-hole entropy, and this becomes Smarr’s relation itself, in the form Mc² = 2TS, for a black hole. On the skin of the light sphere (t = 1), S = N/4 is an assumption carried over. The separator is whether the total of the energy shared out over the cells moves when the size of the skin is moved.
Keywords: equipartition, Unruh temperature, light sphere, emergent gravity, Yukawa field, Gauss’s law, 4π, separator.
Scope (what is not claimed)
This paper claims no new theorem or law. Verlinde’s bookkeeping (Verlinde 2011: the number of cells N = Ac³/(Għ), equipartition, E = Mc² and the Gauss-law form), the Unruh temperature (Unruh 1976), the Yukawa form of a field, Smarr’s relation (Smarr 1973) and the CODATA 2022 values (NIST 2024) are all known. Verlinde’s equations were checked against the original on arXiv (1001.0785). In Section 3.2 Verlinde fixes the temperature by equipartition; in Section 4.1 he sets it from the acceleration (equation (4.17)) and obtains Gauss’s law. This paper follows Section 4.1. What is added is applying this bookkeeping to the light sphere in Yamagishi’s units and counting 4πc⁵ cells, a temperature of 2/c³ per cell and a total of 4πc²; separating out that the 4π of the skin’s area cancels against the 2π of the temperature and the 1/2 of equipartition, while the 4π of the mass remains, as seen by moving M; showing that the total is the same on a skin of any radius and on the light sphere at any time, that this is the same condition as a massless field, and that in a Yukawa field the total falls in the same way as the flux of Paper 1107; and the SI numbers. That combining S = N/4 of Paper 73 with Verlinde’s equipartition gives E = 2TS, and that on the Schwarzschild horizon this becomes the same equation as Smarr’s relation, is not claimed as a new relation (whether this form is already in the literature has not been checked here). Paper 12 read E = 4πc² as the form of a sphere’s surface area. Paper 60 wrote that its 4π is a mass and a convention, Paper 61 described the light sphere, and Paper 65 gave the light sphere’s entropy πc⁵. Paper 73 wrote how the 1/4 of the entropy is assembled and the equipartition reading, and placed the light sphere (the surface along which light travels) as a local causal horizon in Jacobson’s sense. The skin in this paper is the sphere of radius ct at time t, taken as an equipotential surface; it is not a black-hole horizon. Paper 74 placed the system within the family of emergent gravity, and Paper 142 showed that the 2π of the Unruh temperature is a smoothness condition. Paper 1107 wrote that 4π stays on the skin only when the field is massless, and Paper 1114 read m = 4π as the mass of a skin of density 1. This paper sits among them. This paper does not say that gravity is an entropic force (that is Verlinde’s claim, and it is still debated). It does not say that energy is an area, nor that the light sphere is a special skin. It does not derive G = 4π or S = A/4 from the system’s own structure (the open problem of Paper 74). It does not say that space arises from a skin or from light. Strong gravity in general relativity is not treated.
1. Is energy an area?
Paper 12 read $E=4\pi c^2$, obtained by putting $m=4\pi$ into $E=mc^2$, as having the form of the surface area of a sphere of radius $c$. Paper 60 showed that this $4\pi$ is not a solid angle but a mass, a choice of unit (a convention). Paper 61 wrote that, once a unit of time is fixed so that $t=1$, the area of the sphere over which light spreads (the light sphere) really is $4\pi c^2$. Even so, energy is measured in joules and area in square metres. Energy is not an area.
Is there, then, a reading in which the area of the light sphere enters the bookkeeping of the energy? There is one in the work that reads gravity in the language of heat. Verlinde (2011) divided the surface of a sphere around a body into cells about one Planck area in size, gave each cell an energy $k_BT/2$, set up the bookkeeping so that the total is $Mc^2$, and derived Newtonian gravity from it. This paper does not adopt Verlinde’s claim that gravity is an entropic force. Verlinde himself called the arguments heuristic (Section 6.1 of Verlinde 2011), and Paper 74 wrote that emergent gravity is still debated. This paper uses only the bookkeeping: it applies it to the light sphere in Yamagishi’s units and counts what remains and what cancels.
2. The bookkeeping of the cells
Verlinde’s bookkeeping is made of four equations. With the area of the sphere’s surface $A=4\pi R^2$, the number of cells is
$$N=\frac{Ac^3}{G\hbar}$$(equation (3.10) of Verlinde 2011: the area divided by the Planck area $G\hbar/c^3$). The total energy is shared out equally over the cells, $E=Nk_BT/2$ (equation (3.11): equipartition), and $E=Mc^2$ (equation (3.12)). The temperature is set in one of two ways. In Section 3.2 it is fixed by equipartition and $E=Mc^2$, and the Unruh formula (3.8) is said not to be needed. In Section 4.1 the temperature is set, after (3.8), from the acceleration on the skin as $k_BT=\hbar a/(2\pi c)$ (equation (4.17); the form of the Unruh temperature, Unruh 1976), equipartition is written as an integral over the skin (equation (4.19)), and Gauss’s law follows. The bookkeeping in this paper follows Section 4.1 and sets the temperature from the acceleration. Putting in the Newtonian acceleration $a=GM/R^2$ produced by the mass $M$ inside gives
$$\frac{1}{2}Nk_BT=\frac{1}{2}\cdot\frac{4\pi R^2c^3}{G\hbar}\cdot\frac{\hbar GM}{2\pi cR^2}=Mc^2,$$and the skin’s radius $R$ drops out. Verlinde’s bookkeeping is built so that it returns $Mc^2$ once Newtonian gravity is put in. The total coming out as $Mc^2$ is not in itself a new coincidence.
The coefficient 1 is made of $1/2\times4\pi\div2\pi$. The $4\pi$ is the full set of directions of the skin’s area $4\pi R^2$ (the solid angle) and is in the area for any $M$. The $2\pi$ is that of the Unruh temperature. The $1/2$ is that of equipartition. That the coefficient comes out exactly 1 is due to the formula for the number of cells carrying no numerical factor: Verlinde wrote that equation (3.10) may simply be regarded as the definition of $G$, so the cells are counted in such a way that Newton’s $G$ comes out right. So the cancellation of the area’s $4\pi$ against the 1/2 and the $2\pi$ comes from the way the bookkeeping is set up. Verlinde also wrote, in Section 3.1, that he put a $2\pi$ into the entropy change (3.6) so that it cancels the $2\pi$ of the Unruh temperature and Newton’s second law $F=ma$ comes out exactly. Paper 142 showed that the $2\pi$ of the Unruh temperature is not a solid angle but the period of a circle in imaginary time (the smoothness condition that leaves no conical point). Paper 73 wrote that the 1/4 of black-hole entropy is made of the same $2\pi$, the $4\pi$ of the solid angle and a rational factor from the surface gravity.
The total can also be written with the flux through the skin. Since $4\pi R^2a$ is the gravitational flux through the skin,
$$\frac{1}{2}Nk_BT=\frac{c^2}{4\pi G}\,(4\pi R^2a).$$Verlinde extended this form to general skins that are equipotential surfaces, obtained Gauss’s law $M=(1/4\pi G)\int\nabla\Phi\cdot d\mathbf A$ (equation (4.20)), and wrote that any equipotential surface could have been chosen (Section 4.2).
3. Counting the light sphere in Yamagishi’s units
In the units of Paper 44 and Paper 65, $G=4\pi$ and $\hbar=1/(4\pi)$, so $G\hbar=1$ and the Planck area is $1/c^3$. Add $m=4\pi$ (Paper 02 and Paper 12) and $k_B=1$, and, following Paper 61, fix a unit of time so that $t=1$: the light sphere then has radius $c$ and area $4\pi c^2$. Here $c$ is the number for the speed of light in those units, not 299792458. Taking 1 s as the unit of time, with units of length and mass chosen so that $G=4\pi$ and $\hbar=1/(4\pi)$, gives $c=2.0292\times10^{17}$, and $4\pi c^5$ is then the same as the SI number of cells, $4.32346\times10^{87}$, in Section 6.
| skin | radius | area | cells $N$ | temperature per cell $k_BT$ | sum $Nk_BT/2$ |
|---|---|---|---|---|---|
| light sphere ($t=1$) | $c$ | $4\pi c^2$ | $4\pi c^5$ | $2/c^3$ | $4\pi c^2$ |
| light sphere (time $t$) | $ct$ | $4\pi c^2t^2$ | $4\pi c^5t^2$ | $2/(c^3t^2)$ | $4\pi c^2$ |
| skin of radius 1 | $1$ | $4\pi$ | $4\pi c^3$ | $2/c$ | $4\pi c^2$ |
| skin of radius $R$ | $R$ | $4\pi R^2$ | $4\pi R^2c^3$ | $2/(R^2c)$ | $4\pi c^2$ |
The light sphere has $N=4\pi c^5$ cells, four times the entropy $S=N/4=\pi c^5$ counted in Paper 65. The acceleration produced on the light sphere by the mass $4\pi$ inside is $a=GM/c^2=16\pi^2/c^2$, so the temperature of one cell is $k_BT=2/c^3$ and the energy of one cell is $1/c^3$. The total is
$$\frac{1}{2}Nk_BT=4\pi c^5\cdot\frac{1}{c^3}=4\pi c^2,$$the $E=4\pi c^2$ of Paper 12.
Which $4\pi$ remains? On the face of this product, the $4\pi$ of $M$ is cancelled inside the energy per cell by the 1/2 and the $2\pi$, and the $4\pi$ of the area shows on the side of the number of cells. The product alone does not decide which $4\pi$ remains; moving $M$ does. The total moves as $Mc^2$ when $M$ is moved, and the coefficient 1 ($1/2\times4\pi\div2\pi$) does not depend on $M$. The area’s $4\pi$ cancels inside the bookkeeping for any $M$, so the $4\pi$ left in $E=4\pi c^2$ when $M=4\pi$ is the $4\pi$ of $M$, the $4\pi$ of the mass (the third of Paper 60, the convention). The first (the solid angle) is used up inside the bookkeeping. In this bookkeeping the total is not the area itself but the number of cells, which is proportional to the area, times the energy per cell.
4. On any skin, at any time
As the table shows, the total does not depend on the skin’s radius. On a skin of radius $R$ the number of cells is $4\pi R^2c^3$, growing as $R^2$, and the temperature per cell is $2/(R^2c)$, falling as $1/R^2$, so the total stays at $4\pi c^2$. A skin of radius 1 and mass $4\pi$, as in Paper 1114 (counted here in Yamagishi’s units with $G=4\pi$, using the acceleration just outside the skin), also gives $4\pi c^2$, with $4\pi c^3$ cells at temperature $2/c$.
On the light sphere at time $t$ (radius $ct$), the number of cells grows as $t^2$ and the temperature per cell thins out as $1/t^2$. Paper 61 read the thinning of the intensity of light on the light sphere as $1/r^2$; the same $1/r^2$ form appears here as the thinning of the temperature per cell. As the light sphere spreads, the total does not move from $4\pi c^2$.
So the light sphere is not a special skin in the bookkeeping. Every skin around the mass inside, of whatever size, gives the same total. The total is not a property of one skin but a single number shared by every skin.
5. When the field has a mass
The total does not depend on the size of the skin because gravity falls off as $1/r^2$ all the way out. Compare within fields of Yukawa form. Give the field a mass and make the potential energy $-GMe^{-\kappa r}/r$; the acceleration on the skin becomes $GM(1+\kappa R)e^{-\kappa R}/R^2$, and the same bookkeeping gives
$$\frac{Nk_BT/2}{Mc^2}=(1+\kappa R)\,e^{-\kappa R},$$which falls as the skin grows. The total is independent of the skin’s size only at $\kappa=0$. At $\kappa=1$ the acceleration was obtained from a numerical derivative of the potential energy.
| $R$ | sum $\div Mc^2$ | times $4\pi$ | flux in Paper 1107 |
|---|---|---|---|
| 0.5 | 0.9097959896 | 11.432834 | 11.432834 |
| 1 | 0.7357588823 | 9.245819 | 9.245819 |
| 2 | 0.4060058497 | 5.102020 | 5.102020 |
| 4 | 0.09157819444 | 1.150806 | 1.150806 |
The values times $4\pi$ equal the table in which Paper 1107 counted the point-source flux $4\pi(1+\kappa R)e^{-\kappa R}$. The two are the same function, so this is a check of the numbers. The total of the bookkeeping is the flux through the skin times $c^2/(4\pi G)$; in terms of the values in the table of Paper 1107 (the flux for a source of strength 1), it is those values times $Mc^2/(4\pi)$. Paper 1107 kept apart that, in a Yukawa field, one number of pattern 0 still comes out while the flux falls with the radius and no longer represents that number directly. The total of the bookkeeping follows the flux. Paper 1107 counted that $4\pi$ stays on the skin only when the field is massless; the independence of the total from the skin’s size is the same condition.
6. SI numbers
A mass of 1 kg is counted on the light sphere at $t=1$ s and at $t=1$ year (CODATA 2022; NIST 2024). The Planck area $G\hbar/c^3$ is $2.61228\times10^{-70}$ m$^2$.
| quantity | $t=1$ s | $t=1$ year (365.25 days) |
|---|---|---|
| radius of the light sphere (m) | 299792458 | $9.460730473\times10^{15}$ |
| area (m$^2$) | $1.12941\times10^{18}$ | $1.12476\times10^{33}$ |
| cells $N$ | $4.32346\times10^{87}$ | $4.30566\times10^{102}$ |
| acceleration $GM/R^2$ (m/s$^2$) | $7.42616\times10^{-28}$ | $7.45687\times10^{-43}$ |
| temperature $T$ (K) | $3.01132\times10^{-48}$ | $3.02377\times10^{-63}$ |
| energy per cell $k_BT/2$ (J) | $2.07879\times10^{-71}$ | $2.08738\times10^{-86}$ |
| sum $Nk_BT/2$ (J) | $8.98755178737\times10^{16}$ | $8.98755178737\times10^{16}$ |
Over a year the number of cells and the temperature move by a factor of about $10^{15}$ in opposite directions, but in both cases the total is $Mc^2=8.98755178737\times10^{16}$ J. The relative difference is $1.97\times10^{-31}$ at 1 s and 0 at 1 year, the size of the rounding in a 30-digit calculation. The total equals $Mc^2$ identically, so this is a check of the numbers, not a new coincidence. The energy per cell is of order $10^{-71}$ J, far from anything measurable. This reading cannot be tested directly with these numbers.
7. Equipartition and entropy
Paper 73 wrote, as the equipartition reading (Padmanabhan), the entropy of a horizon as $S=N/4$ (in the words of Paper 73, one bit per four Planck areas). Applying the same $S=N/4$ to a skin gives, on that skin,
$$E=\frac{1}{2}Nk_BT=2T\cdot\frac{N}{4}=2TS$$($k_B=1$).
With the skin at the Schwarzschild radius $R=2GM/c^2$, the Newtonian acceleration $GM/R^2$ equals the Schwarzschild surface gravity $c^4/(4GM)$. The temperature of the bookkeeping equals the Hawking temperature $\hbar c^3/(8\pi GMk_B)$, $N/4$ equals the black-hole entropy $4\pi GM^2/(\hbar c)$, and $E=2TS$ becomes Smarr’s relation itself for a black hole that neither rotates nor carries charge (Smarr 1973; written with the Hawking temperature and entropy, $Mc^2=2TS$). On the horizon the bookkeeping and Smarr’s relation become the same equation. In the Schwarzschild spacetime, the acceleration of a static observer times the redshift factor is exactly $GM/r^2$ at every radius and equals the surface gravity on the horizon; this paper does not enter general relativity and only records this fact.
On the skin of the light sphere ($t=1$), $T=2/c^3$ and $S=\pi c^5$, so $2TS=4\pi c^2$. This skin, however, is not a black-hole horizon, and applying $S=N/4$ to it is an assumption. Paper 73 placed the light sphere (the surface along which light travels) as a local causal horizon in Jacobson’s sense. The skin in this paper is the sphere of radius $ct$ at time $t$, taken as an equipotential surface; it is not a black-hole horizon. $E=2TS$ is the form obtained by combining $S=N/4$ of Paper 73 with Verlinde’s equipartition; whether it is already in the literature has not been checked here, and it is not claimed as a new relation.
8. Same root or distinct roots
(1) The $4\pi$ of $E=4\pi c^2$ and the $4\pi$ of the light sphere’s area $4\pi c^2$: the first is a mass (the third of Paper 60), the second a solid angle (the first), and their roots are distinct. Inside the bookkeeping the area’s $4\pi$ cancels against the $2\pi$ of the temperature and the 1/2 of equipartition, and the $4\pi$ of the mass, which moves when $M$ is moved, remains.
(2) The independence of the total from the skin’s size, and Paper 1107’s “$4\pi$ stays on the skin”: both are the flux of Gauss’s law and come from the field being massless. They share a root.
(3) The 1/2 of equipartition and the 1/4 of entropy: the energy per cell and the amount of information per cell, with distinct roots. Together they give the 2 of $E=2TS$.
(4) $E=2TS$ and Smarr’s relation $Mc^2=2TS$: on the Schwarzschild horizon they become the same equation. The reason lies on the side of general relativity (the redshifted acceleration of static observers being $GM/r^2$), and since this paper does not enter general relativity, it does not decide whether they share a root. On the skin of the light sphere $S=N/4$ is an assumption, and $E=2TS$ there is a form borrowed from the horizon.
9. The separator
The separator is whether the total of the energy shared out over the cells moves when the size of the skin is moved. Compared within fields of Yukawa form, for a massless field the number of cells grows in proportion to the area, the temperature per cell falls in inverse proportion to it, and the total does not move from $Mc^2$ ($4\pi c^2$ in Yamagishi’s units). The same holds as the light sphere spreads. For a massive field the total falls by the factor $(1+\kappa R)e^{-\kappa R}$.
Closing
E = 4πc² reads, in Verlinde’s bookkeeping, as the total of the energy shared out equally over the cells of the light sphere. In Yamagishi’s units (G = 4π, ħ = 1/(4π), m = 4π, t = 1) there are 4πc⁵ cells, each at a temperature of 2/c³, and the total is 4πc². The 4π of the skin’s area cancels against the 2π of the Unruh-form temperature and the 1/2 of equipartition, and the 4π that moves when M is moved, the 4π of the mass, remains. In this bookkeeping the total is not the area itself but the number of cells, which is proportional to the area, times the energy per cell. The total is the same on a skin of any size and on the light sphere however far it has spread, and, compared within fields of Yukawa form, this is the same condition as the field being massless. With a massive field the total falls in the same way as the flux of Paper 1107. This paper does not say that gravity is an entropic force; it uses only the bookkeeping.
References
- National Institute of Standards and Technology (2024). CODATA Recommended Values of the Fundamental Physical Constants: 2022. NIST.
- Smarr, L. (1973). Mass formula for Kerr black holes. Physical Review Letters.
- Unruh, W. G. (1976). Notes on black-hole evaporation. Physical Review D.
- Verlinde, E. (2011). On the origin of gravity and the laws of Newton. Journal of High Energy Physics.
- Yamagishi, Y. (2026). The $4\pi$ Mass Cancellation Theorem [Paper 02]. YOI: https://www.yamagishiyuuki.com/yoi/2; DOI: 10.5281/zenodo.21403931
- Yamagishi, Y. (2026). From Boundary Energy to Volume Energy [Paper 12]. YOI: https://www.yamagishiyuuki.com/yoi/12; DOI: 10.5281/zenodo.21404703
- Yamagishi, Y. (2026). The Power of $4\pi$ Is the Mass Dimension [Paper 44]. YOI: https://www.yamagishiyuuki.com/yoi/44; DOI: 10.5281/zenodo.21602231
- Yamagishi, Y. (2026). $c$ as the Ball of Velocity Space [Paper 60]. YOI: https://www.yamagishiyuuki.com/yoi/60; DOI: 10.5281/zenodo.21598012
- Yamagishi, Y. (2026). $c$ as the Light Sphere [Paper 61]. YOI: https://www.yamagishiyuuki.com/yoi/61; DOI: 10.5281/zenodo.21598127
- Yamagishi, Y. (2026). The Bekenstein--Hawking Entropy of the Causal Sphere and $c^5$ [Paper 65]. YOI: https://www.yamagishiyuuki.com/yoi/65; DOI: 10.5281/zenodo.21616466
- Yamagishi, Y. (2026). The “1/4” in Black-Hole Entropy [Paper 73]. YOI: https://www.yamagishiyuuki.com/yoi/73; DOI: 10.5281/zenodo.21666086
- Yamagishi, Y. (2026). The Yamagishi Patch within Emergent Gravity [Paper 74]. YOI: https://www.yamagishiyuuki.com/yoi/74; DOI: 10.5281/zenodo.21670102
- Yamagishi, Y. (2026). Move It, and Only the $2\pi$ Survives [Paper 142]. YOI: https://www.yamagishiyuuki.com/yoi/142; DOI: 10.5281/zenodo.22082281
- Yamagishi, Y. (2026). When the Field Equation Is of Second Order, the Interior Can Be Read from Outside a Closed Skin Only as One Number per Angular Pattern [Paper 1107]. YOI: https://www.yamagishiyuuki.com/yoi/1107
- Yamagishi, Y. (2026). m = 4π Is the Mass of a Skin of Radius 1 at Density 1, and the 1/r² Ball Is That Skin Puffed Out Evenly [Paper 1114]. YOI: https://www.yamagishiyuuki.com/yoi/1114
Reference
Yamagishi, Y. (2026). E = 4πc² Reads as Energy Shared Out over the Cells of the Light Sphere, and Its Total Is the Same on Every Skin Only for a Massless Field [Paper 1115]. YOI: https://www.yamagishiyuuki.com/yoi/1115
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 1115. This paper is a preprint and has not been peer reviewed.