bibkey: reall2024blackholes
authors: Harvey S. Reall
year: 2024
title: Black Holes
doi: null
url: https://www.damtp.cam.ac.uk/user/hsr1000/black_holes_lectures_2024.pdf
claim: Static Schwarzschild observers have proper acceleration M divided by r squared times the redshift factor in geometric units; local acceleration diverges at the horizon.
strata_touched: []
license: citation-only
triage: anchor

Static Schwarzschild clocks and numerical resource bounds

Source and correspondence

Reall, Black Holes, §9.2, printed p. 116 (PDF p. 117), equations (9.8)–(9.12), derives the acceleration of a stationary Killing orbit and then explicitly specializes its norm to Schwarzschild. The source uses $G=c=1$. In fixed physical units the exterior lapse and the acceleration of a test observer held at fixed areal radius $R>2GM/c^2$ are

$$\alpha =\frac{d\tau }{dt}=\sqrt{1-\frac{2GM}{c^{2}R}},a=\frac{GM}{R^2\alpha }.$$

Here $M>0$ is the source mass and $t$ is normalized at infinity. These are literature-attested background formulas. See also Sean M. Carroll, Lecture Notes on General Relativity, arXiv:gr-qc/9712019, §7, for the Schwarzschild exterior. Setting $dR=d\Omega =0$ in the metric gives the proper-time formula. Scaling $M$ and $R$ by the same positive factor preserves $\alpha$; source mass alone does not specify a clock ratio.

Elementary resource consequence

The following is a repo-derived algebraic application of these classical formulas, rather than a result attributed to the source. Solving for $R$ and substituting gives

$$R=\frac{2GM}{c^2(1-{\alpha }^2)},Ma=\frac{c^4}{4G}\frac{(1-{\alpha }^2{)}^2}{\alpha }.$$

When $0<\alpha \le 1/2$ this implies

$$Ma\ge \frac{9c^4}{64G}{\alpha }^{-1}.$$

Fix numerical units $M_0,a_0>0$. If a family satisfies $M(n)\le M_{0}P(n)$ and $a(n)\le a_{0}Q(n)$, then the two cases $\alpha \le 1/2$ and $\alpha >1/2$ give

$$1\le \alpha (n{)}^{-1}\le \max \left\{2,\frac{64G{M}_0{a}_0}{9c^4}P(n)Q(n)\right\}.$$

For polynomial $P,Q$, the right side has a polynomial majorant. Bounds on the bit lengths of $M$ and $a$ do not supply these numerical bounds. For example, requiring $\alpha =2^{-n}$ for $n\ge 1$ instead forces $Ma\ge 9c^4{2}^n/(64G)$ in this same model. Constant numerical source mass can achieve an arbitrarily small positive lapse only with unbounded suspension acceleration; this is why the two upper bounds are used together.

At fixed $R$, a remote reference interval $T$ corresponds to local waiting $\tau =\alpha T$, between $T/q$ and $T$ when $1/\alpha \le q$. The native protocol application and its charged-operation assumptions are in Native capacity, §20.

Interpretation limits

These are static test-observer statements in a supplied nonrotating vacuum exterior. They do not construct the gravitating source or the suspension, bound total energy expenditure, include backreaction, kinematic time dilation or free fall, or constrain arbitrary spacetimes. They do not specify the proper duration of a CPU primitive. An ideal local CPU and a colocated ideal clock acquire the same gravitational conversion factor relative to $t$; a separate gate implementation is needed to determine the CPU’s local rate. Receiving an external computation result additionally requires a communication and readout protocol.

For a differentiable local Schrödinger evolution, the change of parameter $d\tau =\alpha (t)dt$ gives

$$i\hslash \frac{d\psi }{dt}=\alpha (t)H_{\mathrm{loc}}(\tau (t))\psi .$$

This chain-rule identity compares one specified evolution in two parameters. It does not derive the lapse from information growth or identify an algorithmic copying cost with a universal gravitational redshift.