Compactness and Curvature Scales

Compactness, acceleration, and curvature answer different physical questions. For a spherical system of mass $M$ and areal radius $R$, compactness measures the depth of the gravitational potential; acceleration refers to a chosen worldline; tidal scales compare neighboring free-fall trajectories; and curvature invariants characterize the spacetime geometry. This card restores $c$ throughout and defines

$$r_g\equiv\frac{GM}{c^2}.$$

Compactness and weak-field scales

The dimensionless compactness is

$$\mathcal C\equiv\frac{GM}{Rc^2}=\frac{r_g}{R}.$$

In the Newtonian regime,

$$\Phi(R)=-\frac{GM}{R}=-\mathcal Cc^2, \qquad \frac{v_{\rm esc}^2}{c^2}=2\mathcal C.$$

Thus $\mathcal C\ll1$ is a dimensionless weak-potential condition. The escape-speed relation is Newtonian: extrapolating it to $v_{\rm esc}=c$ suggests the Schwarzschild scale but does not derive an event horizon. The Schwarzschild metric, its large-radius Newtonian limit, and the interpretation of its mass parameter are given in Reall 2013, § 12.1, printed pp. 131–133, Eqs. (12.2)–(12.5), PDF and Carroll 1997, § 7, printed pp. 170–172, Eqs. (7.28)–(7.29), PDF.

Two dimensional scales that are often confused with compactness are

$$a_N\equiv\frac{GM}{R^2}=\frac{c^2\mathcal C}{R}, \qquad \bar\rho\equiv\frac{3M}{4\pi R^3}.$$

$a_N$ is the Newtonian surface-acceleration scale, while $\bar\rho$ is a mean matter-density or enclosed-mass scale. Neither alone determines whether the field is relativistically strong. For a static observer outside a Schwarzschild black hole, the invariant magnitude of proper acceleration is instead

$$a_{\rm stat}(r) =\frac{GM}{r^2\sqrt{1-2r_g/r}}.$$

It diverges as $r\to2r_g$ because remaining static there requires unbounded thrust. A freely falling observer has zero proper acceleration while still measuring tidal curvature, and an accelerated observer in flat spacetime can have nonzero proper acceleration with no curvature.

Tidal, invariant, and orbital scales

For Schwarzschild geometry, neighboring freely falling trajectories have the relative-acceleration scale

$$\frac{\Delta a}{L}\sim\frac{GM}{r^3}, \qquad \mathcal R_{\rm tid}\sim\frac{GM}{c^2r^3}=\frac{r_g}{r^3},$$

where $L$ is their small physical separation and $\mathcal R_{\rm tid}$ has curvature dimension $L^{-2}$. This is the operational content of geodesic deviation; see Reall 2013, §§ 6.5–6.6, printed pp. 61–63, especially Eqs. (6.19)–(6.23), PDF.

The Schwarzschild Kretschmann scalar and its associated curvature length are

$$\begin{aligned} K&\equiv R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\\ &=\frac{48G^2M^2}{c^4r^6} =\frac{48r_g^2}{r^6},\\ L_K&\equiv K^{-1/4} =\frac{r^{3/2}}{48^{1/4}\sqrt{r_g}}. \end{aligned}$$

Direct contraction fixes the coefficient $48$. Carroll uses the same invariant diagnostic in Carroll 1997, § 7, printed pp. 171–172, Eq. (7.30), PDF, but that equation prints $12$ for the metric in Eq. (7.29), a factor-of-four typo; its conclusion that $r=2r_g$ is curvature-regular is unchanged.

The Keplerian dynamical scale is

$$\Omega_{\rm orb}\sim\sqrt{\frac{GM}{r^3}}.$$

For Schwarzschild circular geodesics this is the exact angular frequency with respect to asymptotic Schwarzschild time wherever such circular timelike geodesics exist; the effective-potential derivation and its strong-field domain are in Carroll 1997, § 7, printed pp. 174–179, Eqs. (7.47)–(7.54), PDF. The same combination sets the inverse tidal timescale, but it is not a local scalar curvature invariant.

Horizon and mass-scaling checks

At the Schwarzschild horizon $r_H=2r_g$,

$$\mathcal C_H=\frac{r_g}{r_H}=\frac12, \qquad K_H=\frac{48r_g^2}{(2r_g)^6}=\frac{3}{4r_g^4}.$$

Now rescale $M\mapsto\lambda M$ while holding $x=r/r_g$ fixed and scaling any compared surface radius $R$ in the same way. Then

$$\begin{aligned} \mathcal C&\mapsto\mathcal C, &a_N&\mapsto\lambda^{-1}a_N,\\ \bar\rho&\mapsto\lambda^{-2}\bar\rho, &\mathcal R_{\rm tid}&\mapsto\lambda^{-2}\mathcal R_{\rm tid}. \end{aligned}$$ $$\begin{aligned} K&\mapsto\lambda^{-4}K, &L_K&\mapsto\lambda L_K,\\ \Omega_{\rm orb}&\mapsto\lambda^{-1}\Omega_{\rm orb}. \end{aligned}$$

Consequently every Schwarzschild black hole has the same horizon compactness, while a more massive hole has a smaller horizon acceleration scale $a_N(r_H)$, tidal curvature, Kretschmann scalar, and dynamical frequency. The static-observer proper acceleration still diverges at the horizon. “Strong field” must therefore name the diagnostic being compared.

Finally, scalar invariants are valuable but not complete. Schwarzschild vacuum has $R_{\mu\nu}=0$ and $R=0$ while its Weyl curvature and $K$ are nonzero. Conversely, $K=0$ alone does not imply flatness in Lorentzian signature, and some nonflat wave spacetimes have vanishing scalar polynomial curvature invariants. Use frame tidal components or Cartan data when scalar contractions do not decide the question. For material models, radii, and density limits, continue to Relativistic Matter and Compact Objects.

References

• Carroll, Sean M. “Lecture Notes on General Relativity.” NSF-ITP/97-147, arXiv:gr-qc/9712019v1 (1997), § 7, printed pp. 170–180. PDF.
• Reall, Harvey S. General Relativity. Cambridge Part III lecture notes (2013), §§ 6.5–6.6, printed pp. 61–63, and § 12.1, printed pp. 131–133. PDF.