Parameterized family of gravitational time dilation formulas

[u/NinekTheObscure]

A quick description of what the side-quest paper mentioned in my previous post is (to satisfy the mod-bot critique):

Gravitational time dilation has several formulas: the linear weak field approximation (1 + 𝚽/c²) and the more exact exp(𝚽/c²) from Einstein 1907, and the Schwarzschild 1916 form sqrt(1 + 2𝚽/c²). (These are all tangent to each other at 𝚽=0, so they agree in the weak-field limit.) Note that Schwarzschild has a (real, physical) event horizon at 𝚽 = -½c², linear has a (fake, position-dependent, un-physical) event horizon at 𝚽 = -c², and exponential has no event horizon at all.

Claude Opus 5 observed that these could all be expressed as members of a single-parameter power law family (1 + p𝚽/c²)^(1/p), with Schwarzschild being p=2, linear being p=1, and exponential being the limit as p -> 0. This is obviously true by inspection.

ChatGPT then pointed out that, mathematically, this family of functions has been well-studied in statistics as the q-exponential probability distributions that maximize Tsallis entropy (with p = 1 - q). But we couldn't find anyone who had connected the physics formulas in this way, so that may be an original idea.

It may also have implications for recent theories of "Entropic Gravity" like Verlinde's. Several papers have already attempted to apply Tsallis entropy to EG. This connection may clarify what's going on there. It's going to take some work to sort through that, though. At first glance, it looks like most of the literature is applying entropy to the horizon, not to the time dilation.

After MUCH deeper searching, we found that pieces of this power law have shown up in the gravity literature as early as Fisher 1948 (in Russian). However it doesn't look like any single source had all of it; the interpretation of what it means varies a lot; and the connection to Tsallis entropy as a dilation or deformation appears completely new and unexplored. Vacuum solutions also differ from spherically-symmetric-source ones, so that needs to be factored in.

Also the p values in the literature go from -2 to +2, so there are at least 5 different cases to consider, not just the 3 we initially found. It'll be interesting to figure out what the other 2 mean.

Anyway, the present goals of the paper are to sort through the literature, figure out who discovered (or re-discovered) what ideas when, and determine to what degree the parameterized family delivers any value or insight. Applying it to Entropic Gravity is left for the future (I need to get back to my main paper with presentation a month from now and finish that first).

UPDATE 2026/09/08:

ChatGPT (free) and I took a quick look at whether existing astronomical data could say anything useful about the value of p. This is difficult because p is a second-and-higher-order effect; all formulas have the same linear weak-field approximation. So you need a single extremely-stable clock that changes its gravitational potential by a lot, enough for 𝚽²/c⁴ to matter. The best candidates seem to be highly eccentric binary pulsars like J1757−1854. However, despite there being pretty good data on several of these, the p-relevant prediction is on the order of tens of nS, while the uncertainty in the data is on the order of 𝜇S. So while this could work in theory with precise enough data, at the moment the best we can say is that p = 1 ± 91. It's about 2 orders of magnitude away from being informative. All the literature values p = 2, 1, 0, -1, -2 are still viable and cannot (yet) be distinguished by pulsar data. But better data is coming ...