Move it, and only the 2 pi survives – a census of Unruh and Hawking.
Paper 52 stated that the eight pi of the Hawking temperature is four times two pi, that this two pi is a periodicity rather than a solid angle, and that it shares its root with the two pi of the Unruh effect. This paper proves that and then tests the decomposition by moving the black hole. NO NEW MATHEMATICAL THEOREM AND NO NEW PHYSICAL LAW IS CLAIMED. Unruh (1976), Hawking (1975), the Gibbons-Hawking Euclidean method and the relation between surface gravity and temperature are all standard. This is a paper that confirms a shared root rather than separating roots, and for the corpus that confirmation is part of the same roll-call.
First, the two pi is a smoothness condition. The Euclidean Rindler metric is, near the origin, a plane written in polar coordinates, with the imaginary time multiplied by the acceleration playing the role of the angle. The condition for having no conical singularity is that this angle have period exactly two pi: a period of pi leaves a deficit of pi and a period of four pi a deficit of minus two pi, and both are cones (checked here). A period of two pi in the angle means a period of two pi over the acceleration in imaginary time, and since the period of imaginary time is the inverse temperature, the temperature is the acceleration divided by two pi. The address of this two pi is therefore smoothness, and not the total solid angle of a sphere (Papers 35, 45).
Second, the four lives in the surface gravity. Taking the Schwarzschild function to be one minus two M over r, the surface gravity at the horizon is half its derivative there, namely one over four M (computed here symbolically). The temperature is therefore the surface gravity divided by two pi, namely one over eight pi M. So the four in eight pi equals two pi times four comes from the relation between surface gravity and mass, and not from any periodicity. Paper 52’s decomposition is confirmed concretely here.
Third, and this is the core. Move the black hole and the four moves while the two pi does not. For Reissner-Nordstrom the surface gravity is the difference of the outer and inner horizons divided by twice the square of the outer horizon. The temperature normalised by the mass is 0.039789 at zero charge, exactly one over eight pi; 0.039584 at a charge-to-mass ratio of one half; 0.033648 at nine tenths; 0.017243 at 0.99; and zero at extremality. Kerr behaves the same way: 0.036932 at an angular-momentum ratio of one half, 0.024157 at nine tenths, 0.004731 at 0.998, and zero at extremality. So eight pi is not universal: charge it or spin it and the coefficient moves, vanishing at extremality, while the division by two pi never once changes. What is universal is therefore the relation that the temperature is the surface gravity divided by two pi, and not the form one over eight pi M. Paper 52’s decomposition has survived the test: on moving things, the moving part and the fixed part came apart. Papers 117 and 132 said that the separator is what has to be moved for a number to move, and here the separators are the charge and the angular momentum, and the same test pulls the four apart from the two pi.
Fourth, near the horizon the local temperature is the Unruh temperature itself. By Tolman’s law the local temperature is the temperature at infinity divided by the square root of the metric function, and the proper acceleration of a static observer is the mass divided by the square of the radius times that square root. Their ratio, as the radius approaches the horizon, runs 0.16235396, 0.15918678, 0.15915526 and 0.15915495, agreeing to eight digits with one over two pi, namely 0.15915494. The Hawking temperature measured locally near the horizon is thus exactly the Unruh temperature of the local acceleration, and the two two pi are one and the same. Paper 52’s claim is confirmed numerically as well.
Fifth, they are nevertheless not the same phenomenon. The Unruh effect needs no gravity, no horizon of finite area and no evaporation; it occurs in flat spacetime and brings with it no area law for entropy and no information problem. That the two pi share a root does not therefore make the two the same phenomenon. What Paper 17 asked for was an address, and two things can share an address while being different buildings. And one last point. The corpus usually says that roots differ, but confirming that they agree belongs to the same discipline: both come from the single habit of not asserting what has not been checked, and saying that two things share a root also required moving them.
IMPORTANT (scope): a census of established standard material (Layer 2). Every work cited is standard. What is not claimed: any new mathematical theorem (there is none) / any new physical law (there is none) / that the Hawking or Unruh temperatures are in error (they are not) / that the Unruh and Hawking effects are the same phenomenon (they are not; only the two pi share a root). What is added: demonstrating by a conical-deficit computation the shared root asserted in Paper 52, testing the decomposition by charging and spinning the hole, and confirming that the near-horizon ratio converges to one over two pi to eight digits. What remains: on moving things, only the two pi was left standing. What changed was the surface gravity, and the face called eight pi was one that Schwarzschild happened to be wearing. There is only one way to tell what is universal: move it and see.
On the making of this work: The ideas and content of this work stem from the author’s own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author’s. Feedback and corrections are sincerely appreciated.
Reference
Yamagishi, Y. (2026). Move It, and Only the $2\pi$ Survives [Paper 142]. Zenodo.
DOI: 10.5281/zenodo.22082281
ORCID: 0009-0006-8587-3450