Abstract
In units with ħ, c, the Boltzmann constant and G set to 1, the entropy of a black hole is S = A/4, where A is the area of the horizon. In three-dimensional space A = 4πr², so S is the same number as πr², the area of the horizon’s cross-sectional disk. This paper examines with numbers where this match comes from and how it moves with the dimension of space and with the way the gravitational constant is fixed, and shows that the shadow disk of a black hole seen in the photographs is a different disk. No new theorem or law is claimed.
First, the 1/4 cancels Archimedes’ 4. The area of a sphere is 4 times the disk of its great circle, and this ratio is 4 only in three dimensions (Paper 1081: 4.712388980 in four dimensions, 5.333333333 in five). For the black hole in d dimensions (the Tangherlini solution), the coefficient of the entropy is the same number as the horizon’s cross-section (a ball of one dimension less) only in three dimensions, among d = 3 to 10. The three formulas for mass, temperature and entropy satisfy the first law to within 2.845e−11.
Second, and this is the core: the disk reading moves with the way the gravitational constant is fixed. From the first law, the entropy is 2π times “the mass added up after dividing by the surface gravity”, and the 2π is the 2π of the temperature formula. The remaining factor changes with the way G is fixed. When G is fixed by Newton’s force, the factor of the area of the sphere disappears and the coefficient becomes 2π×(d−2)/(d−1). It is the same number as the cross-section in only two dimensions, three (π) and four (4π/3, the volume of the three-dimensional ball cut at the equator), and the match holds shell by shell (a ring of the disk in three dimensions, a shell of the ball in four). In five dimensions the two separate into 4.712388980 and 4.934802201. There are only two because there are only two balls without π, the point and the segment. With 8πG set to 1 (the units of Paper 1092), the entropy is 2π times the area in every dimension and no disk appears.
Third, the shadow in the photographs is a different disk. The edge of the shadow is the image of the circle where light can orbit (Paper 1046), and in three dimensions it lies at $\mathbf{2.598076211}$ times the radius of the horizon. The area of the shadow disk is 27/4 = 6.75 times the horizon’s cross-section, exactly 8 times in four dimensions and $\mathbf{9.425061169}$ times in five, growing with the dimension. With G and the mass set to 1, the entropy disk is 4π and the shadow disk 27π (Paper 1069), so the shadow is larger even than the area 16π of the horizon sphere.
To be candid, the content of the entropy (what it counts) is not entered. No claim is made that information sits on the cross-sectional disk, or that the holographic principle is true. Because the match with the disk depends on the way the gravitational constant is fixed, the same number is not called the same root (Paper 300). The photographed shadow does not measure the entropy. What has been found is that the entropy disk of three dimensions is a form of number that appears under one way of fixing the gravitational constant, and that it is a different disk from the photographed shadow.
Keywords: black-hole entropy, horizon, black-hole shadow, photon sphere, Archimedes, gravitational constant, dimension of space.
Scope (what is not claimed)
No new theorem or law is claimed. That the entropy of a black hole is one quarter of the area of its horizon (Bekenstein 1973; Hawking 1975), the Schwarzschild solution in d-dimensional space (Tangherlini 1963) and the way its mass, temperature and entropy are written (Myers and Perry 1986), the circle where light can orbit and the impact parameter of the edge of the shadow, and that the area of a sphere is 4 times the disk of its great circle (Archimedes) are all known. That the entropy is 2π times the area when 8πG is set to 1 is also a rewriting of the formula. The microscopic content of the entropy is not treated. What is counted to give one quarter is not entered. No claim is made that information sits on the cross-sectional disk. The match with the disk is treated as a match of numbers that appears under one way of fixing the gravitational constant. No claim is made that the holographic principle is true. No claim is made that the photographed shadow measured the entropy. The shadow is the image of the circle where light can orbit, a different disk from the horizon’s cross-section. Rotation and charge are not included. The shadow with rotation was counted in Paper 1069. Relation to earlier papers: Paper 73 anchored the one quarter as a composite of the spatial solid angle 4π, the period 2π of Euclidean time in the temperature, and the factor of the surface gravity. Here the same one quarter is also, in three dimensions, the number that turns the sphere into the disk of its great circle, and it is counted how this moves with the way the gravitational constant is fixed. Paper 65 wrote the entropy of the causal sphere as πc⁵. That π, too, is the π left after this cancellation. Paper 1081 counted how the ratio of the area of a sphere to the disk of its great circle moves from dimension to dimension, and that it is 4 only in three dimensions. Here that ratio is set beside the one quarter of the entropy. Paper 1085 counted the trace-reversal factor (d−1)/(d−2), and Paper 1092 measured Newton’s G from the acceleration of the black-hole solution and used units with 8πG set to 1. Here the same measurement is used to write the coefficient of the entropy in three ways of fixing the constant. Paper 1046 showed that the edge of the shadow is not the horizon but the image of the circle where light can orbit, and Paper 1069 counted that the area of the shadow is 27π and stays a circle even with rotation. Here that shadow disk is compared with the entropy disk, and the ratio is counted as the dimension changes. Paper 35 and Paper 60 treated placing 4π in the gravitational constant or in a mass as a choice of units, and Paper 300 set up the criterion that the same number is not called the same root. The G = 1, Newton’s G and 8πG = 1 here are also treated as ways of fixing the constant. What is added: that the entropy in three dimensions is the same number as the horizon’s cross-sectional disk, with its origin (Archimedes’ 4), that with Newton’s G the match holds in three and four dimensions, shell by shell, with the reason (the two balls without π), that with 8πG = 1 the entropy is 2π times the area in every dimension, and that the photographed shadow disk is 27/4 times the cross-section and grows with the dimension.
1. The question
The entropy of a black hole is one quarter of the area $A$ of its horizon (Bekenstein 1973; Hawking 1975). Written with the Planck length $\ell_P=\sqrt{\hbar G/c^{3}}$, it is
$$\frac{S}{k_B}=\frac{A}{4\ell_P^{2}}.$$For a Schwarzschild black hole in three-dimensional space $A=4\pi r_s^{2}$, so
$$\frac{S}{k_B}=\frac{\pi r_s^{2}}{\ell_P^{2}},$$and the entropy equals the area of the horizon’s cross-sectional disk (the disk of the great circle of radius $r_s$) measured in units of the Planck length squared. Paper 73 anchored the one quarter as a composite of the spatial solid angle $4\pi$, the period $2\pi$ of Euclidean time in the temperature, and the factor of the surface gravity. This paper reads the same one quarter as the number that turns the sphere into the disk of its great circle, and counts how that reading moves with the dimension of space and with the way the gravitational constant is fixed.
There is another disk. The photograph of the centre of M87 released by the Event Horizon Telescope in 2019 shows a dark disk (Event Horizon Telescope Collaboration 2019). Paper 1046 showed that its edge is not the horizon but the image of the circle where light can orbit. This paper compares the two disks with numbers.
2. Three formulas, and the 2π of the first law
From here on $\hbar=c=k_B=1$. The black hole of $d$-dimensional space is the Tangherlini solution (Tangherlini 1963); with $f=1-\mu/r^{d-2}$ and horizon radius $r$,
$$M=\frac{(d-1)S_{d-1}}{16\pi G}\,r^{d-2},\qquad T=\frac{\kappa}{2\pi}=\frac{d-2}{4\pi r},\qquad S=\frac{S_{d-1}r^{d-1}}{4G}$$(written with the $G$ of the action, as in Myers and Perry 1986). Here $\kappa=f'(r)/2=(d-2)/(2r)$ is the surface gravity, $S_{d-1}$ the area of the sphere of radius 1, and $V_k$ the volume of the $k$-dimensional ball of radius 1, with $V_d\times d=S_{d-1}$. That the three satisfy the first law $dM=T dS$ was checked for $d=3$ to 10 at three radii. The largest deviation was $2.845\times10^{-11}$. In three dimensions with $G=1$, $M=r/2$ and $S=\pi r^{2}=4\pi M^{2}$, which is $12.566370614$ for $M=1$.
Rewriting the first law with $T=\kappa/(2\pi)$ gives
$$S=2\pi\int\frac{dM}{\kappa},$$and the $2\pi$ is the $2\pi$ of the temperature formula. The remaining factor (the mass added up after dividing by the surface gravity) changes with the way the gravitational constant is fixed. The integral was done numerically for $d=3$ to 8 in three ways of fixing the constant and compared with the formulas (largest deviation $3.234\times10^{-12}$).
| way of fixing the gravitational constant | $\int dM/\kappa$ | entropy |
|---|---|---|
| $G$ of the action set to 1 | $A/(8\pi)$ | $A/4$ |
| Newton’s $G_N$ set to 1 | $(d-2) r^{d-1}/(d-1)$ | $2\pi(d-2) r^{d-1}/(d-1)$ |
| $8\pi G$ set to 1 | $A$ | $2\pi A$ |
The next three sections take one row each.
3. The G of the action: the same number only in three dimensions
The cross-section of the horizon is the ball of one dimension less obtained by cutting the horizon at its equator, with volume $V_{d-1}r^{d-1}$. It was set beside the coefficient of the entropy, $S\times G/r^{d-1}=S_{d-1}/4$.
| $d$ | coefficient of entropy $S_{d-1}/4$ | cross-section $V_{d-1}$ | sphere ÷ cross-section |
|---|---|---|---|
| 3 | $3.141592654$ | $3.141592654$ | $4.000000000$ |
| 4 | $4.934802201$ | $4.188790205$ | $4.712388980$ |
| 5 | $6.579736267$ | $4.934802201$ | $5.333333333$ |
| 6 | $7.751569170$ | $5.263789014$ | $5.890486225$ |
Among $d=3$ to 10, they are the same number only in three dimensions. The reason is that the ratio of the area of the sphere to its cross-section is 4 only in three dimensions. The 4 of three dimensions is Archimedes’ ratio, the area of a sphere being 4 times the disk of its great circle; Paper 1081 counted that this ratio moves as $\pi$, 4, $4.712388980$, $5.333333333$ and $5.890486225$ from two to six dimensions. The one quarter of the entropy cancels this 4 of Archimedes exactly only in three dimensions. The $\pi$ in the entropy $\pi c^{5}$ of the causal sphere in Paper 65 is also the $\pi$ left after this cancellation.
4. Newton’s G: three and four dimensions, and shell by shell
The gravitational constant can also be fixed by Newton’s force $F=G_N Mm/r^{d-1}$. As in Paper 1092, it was measured, without copying a formula, from the gravitational acceleration $\tfrac12 f'(r)$ of the solution. $G_N/G$ is $1.000000000$ in three dimensions, $0.848826363$ in four and $0.716197244$ in five, agreeing with $8\pi(d-2)/((d-1)S_{d-1})$ to within $1.5\times10^{-9}$. This ratio comes from the trace-reversal factor $(d-1)/(d-2)$ counted in Paper 1085.
Written with Newton’s $G_N$, the entropy becomes
$$S=\frac{2\pi(d-2)}{d-1}\,\frac{r^{d-1}}{G_N},$$and the factor $S_{d-1}$ of the area of the sphere disappears. The coefficient and the cross-section are set side by side.
| $d$ | coefficient of entropy $2\pi(d-2)/(d-1)$ | cross-section $V_{d-1}$ |
|---|---|---|
| 3 | $3.141592654$ | $3.141592654$ |
| 4 | $4.188790205$ | $4.188790205$ |
| 5 | $4.712388980$ | $4.934802201$ |
| 6 | $5.026548246$ | $5.263789014$ |
| 7 | $5.235987756$ | $5.167712780$ |
| 8 | $5.385587406$ | $4.724765970$ |
Among $d=3$ to 10, they are the same number in only two dimensions, three and four. In three dimensions it is $\pi$ (the disk of the great circle), in four $4\pi/3$ (the volume of the three-dimensional ball obtained by cutting the three-dimensional sphere of the horizon at its equator).
The match holds not only for the whole but shell by shell. The entropy added with each piece of mass, $dS=2\pi dM/\kappa$, was compared with the outermost shell of the cross-sectional ball, $S_{d-2}r^{d-2}dr$ (at $r=1.7$). $dS/dr$ divided by $r^{d-2}$ is $2\pi(d-2)$: $6.283185307$, $12.566370614$, $18.849555922$, $25.132741229$, $31.415926536$ and $37.699111843$ for three to eight dimensions, while the shell of the cross-section is $6.283185307$, $12.566370614$, $19.739208802$, $26.318945070$, $31.006276680$ and $33.073361792$; they are the same only in three and four dimensions. In three dimensions $M=r/2$ and $\kappa=1/(2r)$, so $dM/\kappa=r dr$ and the entropy grows by $2\pi r dr$. This is exactly how the area of a disk of radius $r$ grows when a ring of width $dr$ is added. In four dimensions it grows by $4\pi r^{2}dr$, which is how the volume of a three-dimensional ball grows when a shell is added.
The reason there are only two lies in the recurrence for the volume of the ball, $V_{d-1}=2\pi V_{d-3}/(d-1)$ (checked for $d=3$ to 10). With it, the match between coefficient and cross-section becomes
$$d-2=V_{d-3}.$$The $2\pi$ and the $d-1$ are common to both sides; what remains is a comparison between the dimension of space minus 2 and the volume of the ball two dimensions lower. $V_k=k+1$ only for $k=0$ (a point, $V_0=1$) and $k=1$ (a segment of length 2, $V_1=2$); from $k=2$ on the balls carry $\pi$ ($V_2-3=0.141592654$). There are only two balls without $\pi$, the point and the segment, so there are only two matches.
In three dimensions $G_N=G$, so the reading with the $G$ of the action (the $\pi$ of the disk of the great circle) and the reading with Newton’s $G$ (the $\pi$ obtained by multiplying the $2\pi$ of the temperature by $(d-2)/(d-1)=1/2$) point to the same number.
5. 8πG = 1: no disk appears
In units with $8\pi G$ set to 1 (the units used in Paper 1092), the first law becomes
$$dM=\kappa\,dA$$(largest deviation $7.068\times10^{-11}$ for $d=3$ to 10 at three radii). The entropy is therefore, in every dimension,
$$S=2\pi A$$(largest deviation $2.220\times10^{-16}$), which is $8\pi^{2}=78.956835209$ in three dimensions at radius 1. The $2\pi$ that remains is the $2\pi$ of the temperature formula, the $2\pi$ that Paper 73 called the period of Euclidean time, and the $\pi$ of a disk appears nowhere. The disk reading is a form of number that appears only in three dimensions (and also in four with Newton’s $G$) when the $G$ of the action or Newton’s $G$ is set to 1. Just as Paper 35 and Paper 60 treated placing $4\pi$ in the gravitational constant or in a mass as a choice of units, which way of fixing the constant to take is a choice; this paper only sets the three side by side and does not call any of them correct.
6. The shadow in the photographs is a different disk
The circle where light can orbit is at the radius where $f/r^{2}$ is greatest, $r^{d-2}=(d/2) r_h^{d-2}$. The impact parameter $b$ of the edge of the shadow is the smallest value of $r/\sqrt{f}$. Both were searched for numerically and compared with the formulas: the orbiting circle agreed to 7 decimal places and the edge of the shadow to 9. They are listed with the radius of the horizon set to 1.
| $d$ | orbiting circle | edge of shadow $b$ | shadow cross-section ÷ horizon cross-section |
|---|---|---|---|
| 3 | $1.500000000$ | $2.598076211$ | $6.750000000$ |
| 4 | $1.414213562$ | $2.000000000$ | $8.000000000$ |
| 5 | $1.357208808$ | $1.752149037$ | $9.425061169$ |
| 6 | $1.316074013$ | $1.611854898$ | $10.880020560$ |
| 7 | $1.284735157$ | $1.520119138$ | $12.338596449$ |
| 8 | $1.259921050$ | $1.454831515$ | $13.793958065$ |
In three dimensions the orbiting circle is at 1.5 times the horizon and $f$ there is $1/3$, so $b=1.5\times\sqrt3=2.598076211$ (the $5.196152423$ of Paper 1046 is in units where the mass converted to a length is 1, so that the horizon radius is 2). The area of the shadow disk is $b^{2}=27/4=6.75$ times the cross-section of the horizon. In four dimensions the orbiting circle is at $\sqrt2$ times and $f$ there is $1/2$, so $b$ is exactly 2 and the area ratio is $2^{3}=8$. The ratio grows with the dimension.
In three dimensions with $G$ and the mass set to 1, the horizon radius is 2; the area of the entropy disk is $4\pi=12.566370614$, the shadow disk $27\pi=84.823001647$ (the shadow area of Paper 1069), and the area of the horizon sphere $16\pi=50.265482457$. The shadow disk is $6.750000000$ times the entropy disk and $1.687500000$ times the horizon sphere, larger than the area of the horizon itself.
7. Two disks
The entropy disk is the cross-section obtained by cutting the horizon itself at its equator. Its radius is the radius of the horizon, and its $\pi$ is the $\pi$ of the disk of the great circle when written with the $G$ of the action, half the $2\pi$ of the temperature when written with Newton’s $G$, and it disappears when $8\pi G$ is set to 1. It is not seen in a photograph; it is a form of number that appears under one way of fixing the gravitational constant.
The shadow disk is the image of the circle where light can orbit. Its radius is $2.598076211$ times the horizon, and the 27 comes from the square of the 1.5 of the orbiting circle times the $1/f=3$ there, taken 4 times ($27/4=1.5^{2}\times3$). This one is seen in the photographs.
The two have the same form $\pi R^{2}$, but their radii differ and their $\pi$ come from different places. By the criterion of Paper 300 they cannot be called the same root. Calling the dark disk of the photographs “the entropy disk” confuses the two disks.
8. What was watched in the check
When Newton’s $G_N$ was measured from the acceleration of the solution, the loss of digits met in Paper 1092 (subtracting numbers close to 1) was avoided by taking the difference of $\mu/r^{d-2}$ itself. The orbiting circle was found by a golden-section search for the smallest $b=r/\sqrt f$ and compared with the formula. In a draft of the check, a condition that always passes (comparing $2\pi\times0.5$ with $\pi$) was written into a test; it was replaced by a comparison of the coefficient itself.
Closing
Closing: the entropy S = A/4 of a black hole in three dimensions is the same number as the area πr² of the horizon’s cross-sectional disk, and the one quarter cancels Archimedes’ 4 between the sphere and its great circle. From the first law, the entropy is the 2π of the temperature formula times the mass added up after dividing by the surface gravity, and the latter factor changes with the way the gravitational constant is fixed. With the G of the action the match with the disk holds only in three dimensions; with Newton’s G it holds in three and four, shell by shell, because there are only two balls without π, the point and the segment. With 8πG set to 1 the entropy is 2π times the area in every dimension and no disk appears. The shadow disk in the photographs is the image of the circle where light can orbit, 27/4 times the horizon’s cross-section, 8 times in four dimensions and $\mathbf{9.425061169}$ times in five. To be candid, neither the content of the entropy nor where information sits has been entered. What has been found is that the entropy disk of three dimensions is a form of number that appears under one way of fixing the gravitational constant, and a different disk from the photographed shadow. The separator is the dimension of space and the way the gravitational constant is fixed.
References
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Reference
Yamagishi, Y. (2026). In Three Dimensions the Entropy of a Black Hole Is the Same Number as the Area of the Horizon’s Cross-Sectional Disk, and the Shadow Disk in the Photographs Is 27/4 Times Larger [Paper 1093]. YOI: https://www.yamagishiyuuki.com/yoi/1093
ORCID: 0009-0006-8587-3450
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The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 1093. This paper is a preprint and has not been peer reviewed.