Surface Gravity of the Schwarzschild Horizon

Surface gravity is the number that lets a black hole have a temperature. Naively it should be the gravitational pull felt at the horizon — but that quantity is infinite: a static observer hovering just above $r=r_s$ must accelerate ever harder to avoid falling in, and the required acceleration diverges as the horizon is approached. The trick is that the same gravitational redshift that makes clocks freeze at the horizon also weakens, at infinity, the force you must apply through a rope to hold the observer there. Multiply the diverging local acceleration by the vanishing redshift factor and the product converges to a finite limit. That limit is the surface gravity $\kappa$ . There is a second, purely geometric way to say the same thing. The horizon is a null surface swept out by light rays, generated by the timelike Killing vector $\xi=\partial_t$ that turns null exactly there. Along those generators the Killing time $t$ runs "too fast" — it is not an affine parameter — and $\kappa$ measures precisely that failure through $\xi^\nu\nabla_\nu\xi^\mu=\kappa\,\xi^\mu$ . This definition needs no observer and no rope; it survives to rotating and charged holes untouched. For Schwarzschild every route lands on the same strikingly simple answer, $\kappa=c^4/4GM$ , which grows as the hole shrinks. That inverse-mass scaling is the seed of black-hole thermodynamics: joined to quantum field theory it becomes the Hawking temperature $T_H=\hbar\kappa/2\pi k_B c$ , and its constancy over the horizon is the zeroth law of black-hole mechanics. We give several independent derivations because their agreement is itself the physics — it proves $\kappa$ is an invariant of the horizon, not an artifact of any observer or chart.

Key takeaways

Surface gravity $\kappa$ is the redshift-corrected acceleration of the horizon: the diverging proper acceleration of a static observer times the vanishing redshift factor, in the horizon limit — a finite number.
For Schwarzschild $\kappa=\dfrac{c^4}{4GM}=\dfrac{c^2}{2r_s}$ with $r_s=2GM/c^2$ ; in geometric units ( $G=c=1$ ) it is simply $\kappa=1/4M=1/2r_s$ .
Equivalently $\kappa$ is the inaffinity of the null generators, $\xi^\nu\nabla_\nu\xi^\mu=\kappa\,\xi^\mu$ on the horizon — a coordinate-free definition that extends to Kerr and Reissner–Nordström.
A solar-mass hole has $\kappa=1.52\times10^{13}\ \mathrm{m/s^2}$ (about $1.6\times10^{12}$ Earth gravities) and $T_H=6.2\times10^{-8}\ \mathrm{K}$ ; smaller holes are far hotter.
Four routes agree — static-observer redshift, Killing inaffinity, the metric-gradient $\kappa=\tfrac12 c^2 f'(r_s)$ , and Euclidean regularity $\beta=2\pi c/\kappa$ — confirming $\kappa$ is a horizon invariant.
Because $\kappa\propto1/M$ , small holes have huge surface gravity and temperature — a negative heat capacity that drives runaway Hawking evaporation.
Key idea: surface gravity is the finite, redshift-corrected acceleration of the horizon, $\kappa=c^4/4GM$ — the geometric quantity that becomes the black hole's temperature.

The core insight

Surface gravity is the finite, redshift-corrected acceleration of the Schwarzschild horizon, equal to $\kappa=c^4/4GM$ ; it is the geometric invariant of the null generators that quantum theory turns into the black hole's temperature.

Where this lives today

Surface gravity is now a working parameter across theoretical and observational gravity. In gravitational-wave astronomy the ringdown of a merged black hole is a sum of quasinormal modes whose damping is set by the remnant's surface gravity; LIGO–Virgo–KAGRA fit these modes to measure the final mass and spin, with $\kappa$ entering the frequency–damping relations directly. The Event Horizon Telescope images of M87 and Sagittarius A encircle horizons whose $\kappa$ is astonishingly small — a few thousand and a few million m/s² — a reminder of how gentle a supermassive horizon is. On the thermodynamic side, the first law $\mathrm{d}(Mc^2)=\tfrac{\kappa}{8\pi G}\,\mathrm{d}A+\Omega\,\mathrm{d}J+\Phi\,\mathrm{d}Q$ with $S=k_Bc^3A/4G\hbar$ underlies the information problem and the holographic principle, both of which take $\kappa$ as the temperature conjugate to horizon-area entropy. Laboratory "analogue horizons" in flowing water, cold atoms and optical fibres engineer an effective $\kappa$ to hunt for the analogue Hawking effect, and the search for evaporating primordial holes — whose final gamma-ray burst is driven by the diverging $\kappa$ — remains an active window on the early universe.

The physical picture

Picture lowering a static observer toward the horizon on a rope paid out from infinity. Locally the observer must fire an ever-stronger rocket to keep from falling in: the proper acceleration $|a|=(GM/r^2)/\sqrt{1-r_s/r}$ climbs without bound as $r\to r_s$ , because the redshift factor in the denominator collapses to zero. Yet the tension you feel at the far end of the rope, per unit mass held, stays finite — every newton of local force is redshifted down by exactly that same collapsing factor before it reaches you. The product of a diverging acceleration and a vanishing redshift is neither zero nor infinite; it settles on $\kappa=GM/r_s^2=c^4/4GM$ . The first figure draws out that cancellation: the local acceleration blows up, the redshift-corrected combination stays tame, and its horizon value is the surface gravity. Geometrically the horizon is a one-way null membrane knitted from light rays — its generators. The Killing vector $\xi=\partial_t$ flows along them but is not their affine parameter; it is "too fast," and $\kappa$ is the rate of that mismatch, $\xi\cdot\nabla\,\xi=\kappa\,\xi$ . A larger $\kappa$ means the generators are stretched faster. This picture never mentions a rope or an observer, which is why it, not the rope, is the definition that generalizes to Kerr and Reissner–Nordström. The single most consequential fact about $\kappa$ is its inverse dependence on mass: squeeze the same horizon into a tighter radius and the geometry curves more sharply, so $\kappa=c^4/4GM$ rises as $M$ falls. The second figure plots that scaling across real holes — from supermassive giants with feather-light surface gravity to primordial holes whose $\kappa$ is enormous — and it is why black holes have negative heat capacity: they heat up as they evaporate.

Figure

Why the surface gravity is finite: the local proper acceleration of a static observer diverges as r → r_s, but the redshift-corrected combination √(1 − r_s/r)·a stays finite and equals κ = GM/r_s² = c⁴/4GM at the horizon. Curves are plotted in units of κ against r/r_s

Figure

Surface gravity scales as κ = c⁴/4GM ∝ 1/M, so small holes are hotter: the same inverse-mass line, on log axes, carries both κ and the Hawking temperature T_H = ħκ/2πk_Bc across real black holes — from a 10¹² kg primordial hole to the supermassive M87* — spanning some 28 orders of magnitude in κ

The core equations

The operational definition is the redshift-corrected acceleration, so we begin with the answer it produces: the surface gravity of a Schwarzschild hole depends on nothing but its mass, inversely.

\kappa=\frac{c^4}{4GM}=\frac{c^2}{2r_s} \qquad (1)

where: $\kappa$ — surface gravity, an acceleration (m/s $^2$ ); $c$ — speed of light; $G$ — Newton's constant; $M$ — hole mass; $r_s=2GM/c^2$ — Schwarzschild radius. A smaller hole has a larger $\kappa$ : halve the mass, double the surface gravity. Everything is computed on the Schwarzschild background, whose lone metric function $f(r)$ carries both the redshift and the horizon location at $f=0$ .

\mathrm{d}s^2=-f(r)\,c^2\,\mathrm{d}t^2+\frac{\mathrm{d}r^2}{f(r)}+r^2\,\mathrm{d}\Omega^2,\qquad f(r)=1-\frac{r_s}{r} \qquad (2)

where: $f(r)$ — the redshift/lapse function, vanishing at $r=r_s$ ; $\mathrm{d}\Omega^2=\mathrm{d}\theta^2+\sin^2\theta\,\mathrm{d}\phi^2$ — the unit sphere; $r$ — areal radius. The horizon is the root of $f$ , and $f'(r_s)$ fixes $\kappa$ . The coordinate-free definition makes $\kappa$ a property of the horizon's null generators: the Killing vector $\xi=\partial_t$ becomes null on $\mathcal{H}$ , where it obeys a non-affine geodesic equation.

\xi^\nu\nabla_\nu\xi^\mu\;\overset{\mathcal{H}}{=}\;\kappa\,\xi^\mu \qquad (3)

where: $\xi^\mu$ — the horizon-generating Killing vector, null on $\mathcal{H}$ ; $\nabla_\nu$ — the covariant derivative; $\overset{\mathcal{H}}{=}$ — equality on the horizon. The constant $\kappa$ is the inaffinity: the amount by which Killing time fails to parametrize the generators affinely. An equivalent scalar formula extracts $\kappa$ from the antisymmetric Killing gradient — the cleanest object to compute in any chart:

\kappa^2=-\tfrac12\,(\nabla^\mu\xi^\nu)(\nabla_\mu\xi_\nu)\Big|_{\mathcal{H}} \qquad (4)

where: $\nabla^\mu\xi^\nu=\nabla^{[\mu}\xi^{\nu]}$ — the Killing two-form (its symmetric part vanishes by Killing's equation). This definition ports directly to Kerr and Reissner–Nordström without mentioning an observer. For a static, spherically symmetric metric of the form (2) the whole calculation collapses to a single derivative of $f$ at the horizon — the workhorse formula.

\kappa=\tfrac12\,c^2\,f'(r_s)=\lim_{r\to r_s}\sqrt{f(r)}\;|a(r)|,\qquad |a(r)|=\frac{GM/r^2}{\sqrt{f(r)}} \qquad (5)

where: $f'(r_s)=1/r_s$ — the slope of the redshift function at the horizon; $|a(r)|$ — proper acceleration of a static observer; $\sqrt{f}$ — the dimensionless lapse factor. The right equality is the rope picture; the left is its tidy metric restatement. Joined to quantum field theory in curved spacetime, $\kappa$ sets the horizon temperature — the bridge from geometry to thermodynamics.

T_H=\frac{\hbar\,\kappa}{2\pi k_B c}=\frac{\hbar c^3}{8\pi G M k_B} \qquad (6)

where: $T_H$ — Hawking temperature; $\hbar$ — reduced Planck constant; $k_B$ — Boltzmann constant. Same inverse-mass scaling as $\kappa$ : light holes are hot, heavy holes cold. The Euclidean route replaces "temperature" with the period of imaginary time that keeps the geometry smooth, fixed entirely by $\kappa$ :

\beta=\frac{2\pi c}{\kappa}=\frac{4\pi r_s}{c},\qquad T_H=\frac{\hbar}{k_B\,\beta} \qquad (7)

where: $\beta$ — period of Euclidean time $\tau$ (seconds) required to avoid a conical defect; $T_H=\hbar/k_B\beta$ — the temperature read off the period. Regularity alone pins down $\kappa$ .

Derivation ASurface gravity from the redshift-corrected static acceleration

The goal is to make the rope picture quantitative: compute the proper acceleration of a static observer at radius $r$ , multiply by the gravitational redshift factor that connects the local frame to infinity, and take the horizon limit — showing the product stays finite even though the acceleration alone diverges. We work in geometric units $G=c=1$ through the algebra to avoid a thicket of factors of $c$ , restoring SI units only in the boxed result, where $r_s=2M$ becomes $2GM/c^2$ .

Step 1. Build the static four-velocity. A static observer moves only in time, so $u^\mu\propto\xi^\mu=(\partial_t)^\mu$ ; normalizing with $u^\mu u_\mu=-1$ and $-\xi^\mu\xi_\mu=f$ gives $u^\mu=f^{-1/2}(\partial_t)^\mu$ . Why: the normalization is where the redshift factor $\sqrt f$ first enters — it is the ratio of the observer's clock to Killing time.

Step 2. Take the covariant derivative to get the four-acceleration. Only the radial component survives, $a^r=\Gamma^r_{tt}(u^t)^2$ , and with $\Gamma^r_{tt}=\tfrac12 f f'$ and $(u^t)^2=1/f$ this is $a^r=\tfrac12 f'=M/r^2$ (using $f'=2M/r^2$ ). Why: the Christoffel symbol encodes how "staying still" in curved spacetime is itself an accelerated worldline.

Step 3. Form the invariant magnitude. The proper acceleration is $|a|=\sqrt{g_{rr}(a^r)^2}=(M/r^2)/\sqrt f$ . As $r\to r_s$ the numerator is finite but $\sqrt f\to0$ , so $|a|\to\infty$ : a static observer cannot survive at the horizon. Why: this divergence is the whole puzzle — "gravity at the horizon" taken literally is infinite, so any sensible horizon quantity must tame it.

Step 4. Apply the redshift correction. The force per unit mass measured at infinity is the local force redshifted by the lapse $\sqrt f$ , so the relevant quantity is $\sqrt f\,|a|=M/r^2$ , in which the divergent $\sqrt f$ cancels identically — finite at every radius, up to and including the horizon. Why: pulling on a rope from infinity you feel the local tension diminished by exactly the redshift, so the cancellation says $\kappa$ is a property measured at infinity.

Step 5. Evaluate at the horizon and restore units. Setting $r=r_s=2M$ gives $\kappa=M/r_s^2=1/4M$ ; restoring $G$ and $c$ with $r_s=2GM/c^2$ turns this into $\kappa=GM/r_s^2=c^4/4GM$ . Why: the horizon value of the finite combination is the surface gravity, and its inverse-mass form is what makes small holes violently hot. Verified: $\kappa=1/4M=c^4/4GM$ , reproducing Wald §12.5 and Carroll §6.5; $1.52\times10^{13}\ \mathrm{m/s^2}$ for $M=M_\odot$ .

Go deeper · Derivation B · 1 minSurface gravity as the inaffinity of the Killing generators

Derivation BSurface gravity as the inaffinity of the Killing generators

The goal is to recover the same $\kappa$ from a definition that never mentions an observer, a rope, or a redshift — the geometric inaffinity of the horizon's null generators — connecting the two standard invariant formulas, the norm-gradient relation $\nabla_\mu(-\xi^2)=2\kappa\xi_\mu$ and the scalar $\kappa^2=-\tfrac12(\nabla^\mu\xi^\nu)(\nabla_\mu\xi_\nu)$ , so equations (3) and (4) are seen to be one statement. We keep the algebra in geometric units and let $\lambda\equiv-\xi^\mu\xi_\mu$ be the squared norm of the Killing vector, which measures how far $\xi$ is from null.

Step 1. Identify the norm and its vanishing. With $\xi=\partial_t$ we have $\lambda=-\xi^\mu\xi_\mu=f=1-r_s/r$ , positive outside and zero on the horizon, so the surface $\lambda=0$ is the horizon. Why: the horizon is where the time-translation symmetry becomes a null flow, so $\lambda\to0$ is the signature of a Killing horizon.

Step 2. Invoke the Killing norm-gradient identity. For a Killing vector hypersurface-orthogonal and null on $\mathcal{H}$ , one has $\nabla_\mu\lambda\overset{\mathcal{H}}{=}2\kappa\,\xi_\mu$ ; the gradient of the norm points along $\xi$ , with the inaffinity as the constant of proportionality. Why: this identity converts the abstract inaffinity of equation (3) into an ordinary gradient we can compute in coordinates.

Step 3. Compute the gradient. Only the radial derivative is nonzero, $\nabla_r\lambda=f'=r_s/r^2$ , and contracting with the inverse metric gives $|\nabla\lambda|=\sqrt{g^{rr}}\,\partial_r\lambda=\sqrt f\,f'$ , while $|\xi|=\sqrt\lambda=\sqrt f$ . Why: taking magnitudes leaves a clean scalar equation for $\kappa$ that stays finite as $\sqrt f\to0$ .

Step 4. Extract and evaluate $\kappa$ . Dividing, $\kappa=\tfrac12\,|\nabla\lambda|/\sqrt\lambda=\tfrac12 f'$ , and at the horizon $\kappa=\tfrac12 f'(r_s)=1/2r_s=1/4M$ ; restoring units, $\kappa=\tfrac12 c^2 f'(r_s)=c^4/4GM$ . Why: the two vanishing $\sqrt f$ factors cancel to leave a finite, chart-independent number — the same value the rope gave, proving $\kappa$ is a horizon invariant. Verified: identical to the redshift method; the scalar form gives $\kappa^2=(c^4/4GM)^2$ , agreeing with MTW and Bardeen–Carter–Hawking (1973).

Go deeper · Derivation C · 1 minSurface gravity from Euclidean regularity and the conical defect

Derivation CSurface gravity from Euclidean regularity and the conical defect

The goal is to obtain $\kappa$ from a wholly different principle — smoothness of the Wick-rotated geometry — showing that the requirement of no conical singularity at the horizon fixes the period of imaginary time to $2\pi c/\kappa$ , so the temperature $T_H=\hbar/k_B\beta$ drops out with no mention of acceleration, generators, or observers. We set $t=-i\tau$ with $\tau$ real, expand the metric near $r=r_s$ , and change to a proper radial coordinate so the near-horizon geometry becomes a flat plane in disguise; smoothness of that plane at its origin is the entire content.

Step 1. Linearize the metric function at the horizon. With $f(r)\approx f'(r_s)(r-r_s)=(r-r_s)/r_s$ , the Euclidean line element is $\mathrm{d}s_E^2\approx f\,c^2\,\mathrm{d}\tau^2+f^{-1}\mathrm{d}r^2$ , the transverse sphere a spectator. Why: all the near-horizon physics lives in the $(\tau,r)$ block, and only $f'(r_s)$ survives — the same quantity that appeared as $\kappa$ before.

Step 2. Introduce the proper radial distance. Defining $\rho=\int\mathrm{d}r/\sqrt f\approx2\sqrt{r_s(r-r_s)}$ turns the radial part into $\mathrm{d}\rho^2$ and gives $f\approx\rho^2/4r_s^2$ , so the block becomes $\mathrm{d}s_E^2\approx\mathrm{d}\rho^2+\rho^2\big(c/2r_s\big)^2\mathrm{d}\tau^2$ . Why: $\rho$ is the honest distance to the horizon, and in it the geometry is manifestly a flat plane in polar form $\mathrm{d}\rho^2+\rho^2\,\mathrm{d}\theta^2$ .

Step 3. Read off the angle and demand smoothness. The polar angle is $\theta=(c/2r_s)\,\tau$ , and a plane is regular at $\rho=0$ only if $\theta$ has period exactly $2\pi$ ; any other period leaves a conical defect at the horizon. Why: a conical singularity would mean curvature concentrated on the horizon — an unphysical source — so regularity is a genuine physical condition.

Step 4. Convert the angular period into a period of $\tau$ , and hence $\kappa$ . Requiring $\Delta\theta=2\pi$ gives $\beta=2\pi(2r_s/c)=4\pi r_s/c$ ; with $c/2r_s=\kappa/c$ (since $\kappa=c^2/2r_s$ ) this is $\beta=2\pi c/\kappa$ , so $\kappa=2\pi c/\beta=c^2/2r_s=c^4/4GM$ . Why: smoothness alone reproduces the same $\kappa$ , and reading the period as an inverse temperature gives $T_H=\hbar/k_B\beta=\hbar c^3/8\pi GMk_B$ — Hawking's result from pure geometry. Verified: $\beta=4\pi r_s/c=2\pi c/\kappa$ gives $T_H=\hbar c^3/8\pi GMk_B$ , the Gibbons–Hawking (1977) result; $\beta=2\pi/\kappa$ only in units where $c=1$ .

Limiting cases

The formula $\kappa=c^4/4GM$ is exact for Schwarzschild; its limiting behaviour and relation to neighbouring solutions are the fastest checks that it is right.

Case	Result	Meaning
Supermassive ( $M\to\infty$ )	$\kappa\to0$ , $T_H\to0$	The horizon is gentle and locally almost flat; tidal forces and temperature negligible — one could cross M87* without noticing.
Microscopic ( $M\to0$ )	$\kappa\to\infty$ , $T_H\to\infty$	Diverging surface gravity and temperature; the final stage of evaporation is a runaway burst.
Geometric units ( $G=c=1$ )	$\kappa=\dfrac{1}{4M}=\dfrac{1}{2r_s}$	Cleanest form; restore units via $c^4/G$ (the $M$ form) or $c^2$ (the $r_s$ form).
Newtonian mnemonic	$g_N=\dfrac{GM}{r_s^2}=\dfrac{c^4}{4GM}=\kappa$	The naive surface field at $r_s$ numerically equals $\kappa$ — a memory hook, though partly an artifact of the areal coordinate.
Kerr–Newman ( $a,Q\neq0$ )	$\kappa=\dfrac{r_+-r_-}{2(r_+^2+a^2)}$	The Killing definition generalizes intact; the extremal limit $r_+\to r_-$ sends $\kappa\to0$ (third law).

Common trapSurface gravity is not the acceleration you would feel at the horizon

The most common error is to read "surface gravity" as the acceleration a hovering observer measures at $r=r_s$ . That quantity is infinite — $|a|=(GM/r^2)/\sqrt{1-r_s/r}$ diverges as the redshift factor collapses, and no rocket or rope can hold an observer exactly on the horizon. Surface gravity is the redshift-corrected acceleration: the local value times the vanishing lapse $\sqrt{1-r_s/r}$ , equivalently the force per unit mass applied at infinity. Only that product is finite, and it equals $\kappa=c^4/4GM$ . Two further cautions: $\kappa$ is not a "force per unit area" despite the name (that is historical, by analogy with a planetary surface), and the match with the Newtonian $GM/r_s^2$ is a coordinate coincidence, not a sign that Newtonian gravity applies at the horizon.

Worked examples

1 · A solar-mass black hole, end to end (fully worked)

Take $M=M_\odot=1.989\times10^{30}\ \mathrm{kg}$ . Find the Schwarzschild radius, the surface gravity in both SI and Earth-gravity units, and the Hawking temperature; compare the temperature to the cosmic microwave background.

Step 1. Schwarzschild radius. $r_s=2GM/c^2=2(6.674\times10^{-11})(1.989\times10^{30})/(2.998\times10^8)^2$ . The numerator is $2\times1.328\times10^{20}=2.655\times10^{20}$ and $c^2=8.988\times10^{16}$ , giving $r_s=\mathbf{2.95\times10^3\ m\approx2.95\ km}$ — a Sun compressed to the size of a small city.

Step 2. Surface gravity. $\kappa=c^4/4GM$ with $c^4=8.078\times10^{33}$ and $4GM=5.31\times10^{20}$ gives $\kappa=\mathbf{1.52\times10^{13}\ m/s^2}$ . As a cross-check, $GM/r_s^2=(1.328\times10^{20})/(2.95\times10^3)^2=1.52\times10^{13}$ — the two forms agree. In everyday units that is $\kappa/g_\oplus=\mathbf{1.55\times10^{12}}$ Earth gravities.

Step 3. Hawking temperature. $T_H=\hbar c^3/8\pi GMk_B$ has numerator $\hbar c^3=2.84\times10^{-9}$ and denominator $8\pi GMk_B=25.13\times(1.328\times10^{20})\times(1.381\times10^{-23})=4.61\times10^{-2}$ , so $T_H=\mathbf{6.2\times10^{-8}\ K}$ , about $62$ nanokelvin.

Step 4. Compare to the CMB. The microwave background sits at $T_{\rm CMB}=2.73\ \mathrm{K}$ , larger than $T_H$ by a factor $2.73/6.2\times10^{-8}\approx\mathbf{4\times10^{7}}$ . A solar-mass hole therefore absorbs far more than it radiates and grows rather than evaporates today — despite its enormous surface gravity it is one of the coldest objects imaginable. Result: $r_s=2.95$ km, $\kappa=1.52\times10^{13}$ m/s², $T_H=62$ nK — colder than the CMB by $4\times10^7$ .

2 · Inverse-mass scaling: Sgr A* versus a primordial hole

Contrast the supermassive hole at the Galactic centre, Sagittarius A with $M=4.3\times10^6\,M_\odot$ , against a primordial black hole of $M=1.0\times10^{12}\ \mathrm{kg}$ (roughly a small mountain), using only $\kappa\propto1/M$ and $T_H\propto\kappa$ .*

Step 1. Anchor on the Sun. From Example 1, $\kappa_\odot=1.52\times10^{13}\ \mathrm{m/s^2}$ and $T_{H,\odot}=6.2\times10^{-8}\ \mathrm{K}$ . Because both scale as $1/M$ , every other hole is one division away.

Step 2. Sagittarius A*. With $M=4.3\times10^6\,M_\odot$ , $\kappa=\kappa_\odot/(4.3\times10^6)=\mathbf{3.5\times10^{6}\ m/s^2}$ and $T_H=6.2\times10^{-8}/4.3\times10^6=\mathbf{1.4\times10^{-14}\ K}$ — a surface gravity smaller than a fast centrifuge and a temperature of $14$ femtokelvin, thermodynamically the deadest object in the Galaxy.

Step 3. The primordial hole. Its mass in solar units is $1.0\times10^{12}/1.989\times10^{30}=5.03\times10^{-19}\,M_\odot$ , so $\kappa=1.52\times10^{13}/(5.03\times10^{-19})=\mathbf{3.0\times10^{31}\ m/s^2}$ and $T_H=\mathbf{1.2\times10^{11}\ K}$ — white-hot, hotter than a supernova core, evaporating explosively.

Step 4. The span. The mass ratio is $(4.3\times10^6\,M_\odot)/(1.0\times10^{12}\ \mathrm{kg})=8.6\times10^{24}$ , and $\kappa$ and $T_H$ differ by exactly that factor: $3.0\times10^{31}/3.5\times10^{6}\approx8.6\times10^{24}$ . Result: surface gravity ranges over $\sim25$ orders of magnitude, entirely because $\kappa\propto1/M$ — the line drawn in the second figure.

Physics of the portrait

Surface gravity is the hinge on which the analogy between black holes and thermodynamics turns. For decades $\kappa$ was a technical constant of stationary horizons; then, within a single decade, it was proved constant over the horizon and derived to be a temperature by quantum theory.

Origins · 1 minsurface gravity and how this was discovered

surface gravity

the horizon invariant κ = c⁴/4GM · the four laws of black-hole mechanics

Surface gravity began as a geometer's bookkeeping device and ended as the temperature of a black hole. The stage was set in 1958 when David Finkelstein reinterpreted the Schwarzschild surface $r=2M$ — long dismissed as a coordinate singularity — as a one-way causal membrane, the event horizon. Through the 1960s the Killing-horizon machinery matured in the work of Boyer, Carter and others, and $\kappa$ crystallized as the inaffinity of the null generators, an invariant of stationary horizons. The decisive synthesis came in 1973, when James Bardeen, Brandon Carter and Stephen Hawking published "The Four Laws of Black Hole Mechanics," proving that $\kappa$ is constant over any stationary horizon — a statement with the exact form of the zeroth law of thermodynamics. That same year Jacob Bekenstein proposed that horizon area is a true entropy, forcing the analogy to be taken literally and demanding that $\kappa$ act as temperature. Most physicists, Hawking included, thought the analogy merely formal — until 1974, when Hawking's own quantum-field-theory calculation showed that black holes radiate thermally at exactly $T=\hbar\kappa/2\pi k_B c$ , converting the analogy into an identity. The 1977 Euclidean derivation by Gibbons and Hawking, in which $\kappa$ emerges from the smoothness of imaginary time, showed the temperature is written into the geometry itself.

History

1958

— Finkelstein identifies the Schwarzschild surface r = 2M as a one-way causal membrane, the event horizon

1973

— Bardeen, Carter and Hawking prove κ is constant over a stationary horizon, the zeroth law of black-hole mechanics

1973

— Bekenstein proposes that horizon area is a genuine entropy, forcing κ to act as a temperature

1974

— Hawking shows black holes radiate thermally at T = ħκ/2πk_Bc, making the analogy an identity

1977

— Gibbons and Hawking derive the temperature from Euclidean regularity, β = 2πc/κ

Going further

Texts. Wald, General Relativity (1984), §12.5 (surface gravity and the laws of black-hole mechanics) · Misner, Thorne & Wheeler, Gravitation (1973), Ch. 31–33 · Carroll, Spacetime and Geometry (2004), §6.5 and Ch. 9 · Hartle, Gravity (2003), Ch. 12 · Schutz, A First Course in General Relativity (2009), Ch. 11 · Poisson, A Relativist's Toolkit (2004), §5.2 (Killing horizons and $\kappa$ ). The four derivations are mutually consistent. The redshift route gives $\Gamma^r_{tt}=\tfrac12 ff'$ and $a^r=M/r^2$ (geometric units), so $|a|=(GM/r^2)/\sqrt f$ and the horizon limit $\sqrt f\,|a|\to GM/r_s^2=c^4/4GM$ . The Killing route recovers the same value via the norm-gradient identity $\nabla_\mu(-\xi^2)=2\kappa\xi_\mu$ and the scalar $\kappa^2=-\tfrac12(\nabla^\mu\xi^\nu)(\nabla_\mu\xi_\nu)$ . The Euclidean route gives period $\beta=4\pi r_s/c=2\pi c/\kappa$ , yielding $T_H=\hbar c^3/8\pi GMk_B$ . The anchor numbers ( $r_s=2.95$ km, $\kappa=1.52\times10^{13}\ \mathrm{m/s^2}$ , $T_H=6.2\times10^{-8}$ K for $M_\odot$ ; $\kappa=3.5\times10^6$ m/s² for Sgr A*; $\kappa=3.0\times10^{31}$ m/s², $T_H=1.2\times10^{11}$ K for a $10^{12}$ kg hole) follow from CODATA constants.

Prerequisites. The Schwarzschild Metric · Schwarzschild Radius and the Geometric Mass · Killing Vectors and Conserved Quantities in Schwarzschild · Gravitational Redshift in Schwarzschild Spacetime. Leads to. Hawking Temperature · Black-Hole Thermodynamics and the Laws of Black-Hole Mechanics · Euclidean Quantum Gravity.

Ask the tutor: Ask me to redshift the Unruh temperature of a near-horizon observer out to infinity and watch it become $T_H=\hbar\kappa/2\pi k_Bc$ step by step.

Why call it "gravity" if it is not the acceleration you would feel there?

The name is an analogy with a planet's surface, where "surface gravity" is the acceleration at the surface. For a black hole the honest local acceleration at the horizon is infinite, so the name is transplanted to the finite, redshift-corrected version — the force per unit mass you would apply through a long rope from infinity to hold a test mass just above the horizon. That force stays finite even as the local acceleration diverges, because gravitational redshift weakens it on the way out. So $\kappa$ is the gravity of the horizon as measured from infinity, the only observer-independent way to assign it a single number.

How can four completely different calculations give the same κ?

Because $\kappa$ is a geometric invariant of the horizon, not a property of any observer or coordinate system. The redshift method uses a static observer, the Killing method uses only the symmetry vector, the metric-gradient formula uses $f'(r_s)$ , and the Euclidean method uses smoothness of imaginary time. None of these ingredients is fundamental — they are four windows onto the same object, the inaffinity of the null generators. Their agreement is a theorem, not a coincidence: for a Killing horizon all reasonable definitions of surface gravity provably coincide, which is why physicists trust $\kappa$ enough to build thermodynamics on it.

Is the Newtonian formula GM/r_s² a real coincidence or does it mean something?

It is mostly a coincidence of the areal coordinate. Evaluating the naive Newtonian surface field $GM/r^2$ at $r=r_s$ gives $GM/r_s^2=c^4/4GM$ , numerically $\kappa$ exactly. But the reasoning is wrong: Newtonian gravity does not apply at the horizon, there is no meaningful "field strength" there, and the agreement depends on using the areal radius $r$ rather than the proper distance. It is a useful memory hook — recall $r_s=2GM/c^2$ and you reconstruct $\kappa=c^4/4GM$ in one line — but it is not evidence that the horizon behaves Newtonianly.

What exactly does "the Killing time is not affine" mean?

Along the null rays that generate the horizon you can parametrize position either by the Killing time $t$ (the flow of the symmetry) or by an affine parameter $\lambda$ , the one in which the geodesic equation has no right-hand side. On the horizon the two differ: $t$ speeds up exponentially relative to $\lambda$ , with rate $\kappa$ , so $\mathrm{d}\lambda/\mathrm{d}t\propto e^{\kappa t}$ . That is the content of $\xi^\nu\nabla_\nu\xi^\mu=\kappa\xi^\mu$ — the Killing flow is a geodesic but "runs too fast" to be affine, and the mismatch is the surface gravity, which is also why $\kappa$ controls the exponential redshift of signals climbing out.

Why does the surface gravity set a temperature at all?

Two threads meet. Quantum-mechanically, the near-horizon geometry looks to a static observer like uniform acceleration, and an accelerated observer sees a thermal bath (the Unruh effect) at $T=\hbar a/2\pi k_B c$ ; redshifting that local temperature out to infinity replaces $a$ by $\kappa$ , giving $T_H=\hbar\kappa/2\pi k_B c$ . Geometrically, Wick-rotating to imaginary time makes the near-horizon region smooth only if imaginary time is periodic with period $\beta=2\pi c/\kappa$ — and a periodic imaginary time is a thermal state at temperature $\hbar/k_B\beta$ . Both threads give the same $T_H\propto\kappa$ .

Does surface gravity change for rotating or charged black holes?

Yes, but the definition does not. The Killing-horizon formula $\kappa^2=-\tfrac12(\nabla^\mu\xi^\nu)(\nabla_\mu\xi_\nu)$ applies unchanged; only now $\xi=\partial_t+\Omega_H\partial_\phi$ is the horizon-generating combination, with $\Omega_H$ the horizon's angular velocity. For Kerr–Newman one finds $\kappa=(r_+-r_-)/2(r_+^2+a^2)$ in geometric units, with $r_\pm$ the outer and inner horizons and $a$ the spin per unit mass. As the hole approaches extremality $r_+\to r_-$ and $\kappa\to0$ , so $T_H\to0$ — the third law. Schwarzschild is the $a=Q=0$ limit, recovering $\kappa=c^4/4GM$ .