Stationary Horizons, Surface Gravity, and Redshift

For a nonextremal stationary horizon, surface gravity is the conversion rate between Killing time and a regular affine parameter on the horizon. The same rate controls the near-horizon redshift and the dimensionless thermal ratio $\omega/\kappa$. Its numerical value is meaningful only after the horizon generator has been normalized.

Required background. Horizon taxonomy identifies when a Killing horizon exists, and Tolman redshift and local temperature supplies the stationary redshift law.

Helpful background. Accelerated detectors gives the Rindler comparison, and WKB and turning-point methods supports later mode propagation.

Surface gravity and regular null coordinates

For a static spherical metric

$$ds^2=f(r)dt^2-\frac{dr^2}{f(r)}-r^2d\Omega_2^2,$$

assume a simple zero $f(r_h)=0$, $f'(r_h)\ne0$, and normalize $\chi=\partial_t$ in the stated exterior. Then

$$\kappa=\frac12 f'(r_h)$$

for the future outer horizon orientation. Define the tortoise coordinate and null coordinates by

$$\frac{dr_*}{dr}=\frac1{f(r)}, \qquad u=t-r_*, \qquad v=t+r_*.$$

Near the horizon,

$$r_* = \frac{1}{2\kappa}\log|r-r_h|+O(1).$$

The outgoing Killing coordinate $u$ is singular at the future horizon. A regular Kruskal coordinate is

$$U=-\frac1\kappa e^{-\kappa u},$$

up to a positive affine rescaling and shift. Along the horizon $U$ is affine, whereas translation $u\mapsto u+\Delta u$ rescales $U$ exponentially. Equivalently,

$$\chi=-\kappa U\partial_U+\kappa V\partial_V.$$

This exponential is the local kinematic input to both the collapse Bogoliubov transform and horizon KMS analyticity. It requires a simple, nonextremal zero; an extremal double zero gives a different, typically power-law relation.

The relation between a bifurcate Killing horizon, its normalized generator, and the corresponding thermal state is conditional on global state existence, not a consequence of the coordinate change alone (Kay and Wald 1991, §§ 5–7).

A static observer has four-velocity $u^\mu=\chi^\mu/\sqrt{\chi^2}$ and measures

$$\omega_{\rm loc}=\frac{\omega_\chi}{\sqrt{\chi^2}}, \qquad T_{\rm loc}=\frac{T_\chi}{\sqrt{\chi^2}}.$$

The divergent local temperature near the horizon describes the acceleration of static worldlines; a freely falling detector in a horizon-regular state need not measure the same response.

The structure map places $\kappa$ in the first box because its normalization propagates into state frequencies, the KMS or mode relation, and the scattering flux.

Surface gravity as the scale connecting geometry to near-horizon QFT. The diagram is schematic and not to scale; scattering and asymptotic flux remain separate later steps.

The failure map tests a pure convention error: rescaling the generator while leaving $\kappa$, frequency, or temperature unchanged produces a spurious physical difference.

Normalization failure for stationary horizons. This schematic, not-to-scale map licenses only ratios and observables formed with one consistently normalized time flow.

Application: the exponential affine relation

For Schwarzschild, $f=1-2M/r$, $r_h=2M$, and asymptotic normalization gives $\kappa=1/(4M)$. On a fixed-$t$ slice near the horizon,

$$r-2M\propto e^{2\kappa r_*} \propto e^{-2\kappa u},$$

where the second proportionality holds because $t$ is fixed, while the regular affine label itself obeys $U\propto-e^{-\kappa u}$. A positive-frequency horizon mode $e^{-i\Omega U}$ therefore becomes a non-monochromatic function of $u$. Conversely, a late-time Killing mode behaves near the horizon as

$$e^{-i\omega u}=(-\kappa U)^{i\omega/\kappa},$$

whose analytic continuation across $U=0$ produces the relative factor $e^{-\pi\omega/\kappa}$ underlying the Planck ratio.

Now rescale $\chi'=a\chi$ with $a>0$. Then

$$\kappa'=a\kappa, \qquad \omega_{\chi'}=a\omega_\chi, \qquad T_{\chi'}=aT_\chi,$$

so $\omega/\kappa$ and $\omega/T$ are invariant. In an asymptotically flat spacetime, imposing $\chi^2\to1$ fixes $a$. In de Sitter or a finite static region there may be no such infinity, so the normalization convention must be reported with any temperature.

Domain and failure conditions

See the chapter domain and failure-conditions table. The result assumes a nondegenerate Killing horizon, one time orientation, and a stated normalization of $\chi$. It licenses the exponential affine relation and the scale $\kappa/(2\pi)$ with respect to that flow. The decisive checks are a simple zero of the lapse and invariance of $\omega/\kappa$ under generator rescaling. A dynamical horizon or an extremal double zero requires a different map; inconsistent rescaling downgrades any numerical temperature to a convention artifact.

Exercise

Derive $r_*\sim(2\kappa)^{-1}\log|r-r_h|$ from a simple zero of $f$.

Solution

Expand $f(r)=f'(r_h)(r-r_h)+O((r-r_h)^2)=2\kappa(r-r_h)+\cdots$. Then

$$r_*=\int^r\frac{dr'}{f(r')} =\frac{1}{2\kappa}\log|r-r_h|+O(1).$$

Exponentiating $u=t-r_*$ gives the regular coordinate $U\propto-e^{-\kappa u}$.

Handoff

Gravitational collapse supplies a physical state preparation and a ray-tracing map whose late-time form contains this exponential relation.

References

• Hawking, Stephen W. “Particle Creation by Black Holes.” Communications in Mathematical Physics 43 (1975): 199–220. doi:10.1007/BF02345020.
• Kay, Bernard S., and Robert M. Wald. “Theorems on the Uniqueness and Thermal Properties of Stationary, Nonsingular, Quasifree States on Spacetimes with a Bifurcate Killing Horizon.” Physics Reports 207 (1991): 49–136. doi:10.1016/0370-1573(91)90015-E.
• Wald, Robert M. Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics. Chicago: University of Chicago Press, 1994. Publisher record.