Surface Gravity of a Stationary Black Hole

1. From Event Horizon to Killing Horizon

We begin by establishing the geometric arena where surface gravity is defined. Our setting is a stationary spacetime containing a black hole. As you know, stationarity implies the existence of a timelike Killing vector field $\xi$, which outside the black hole corresponds to time translations (in adapted coordinates, $\xi ={\partial }_t$).

The event horizon, $H$, is a null hypersurface. Because the spacetime is stationary, the flow generated by $\xi$ must map the horizon to itself. A remarkable result known as the Rigidity Theorem (first proved by Hawking) makes this connection much more precise. It states that, under reasonable physical assumptions (e.g., for analytic solutions to the Einstein-Maxwell equations), the event horizon of a stationary black hole is a Killing horizon.

A Killing horizon is a null hypersurface whose null geodesic generators are the orbits of a specific Killing vector field, which we will call $\chi$. This special vector $\chi$ is constructed from the symmetries of the spacetime:

• For a static (non-rotating) black hole like Schwarzschild, $\chi$ is simply the stationary Killing vector $\xi$. On the horizon, $\xi$ becomes null.
• For a stationary, axisymmetric (rotating) black hole like Kerr, $\chi$ is a specific linear combination of the stationary Killing vector $\xi ={\partial }_t$ and the axial Killing vector $\eta ={\partial }_{\varphi }$: $$\chi =\xi +{\Omega }_H\eta$$ Here, ${\Omega }_H$ is a constant called the angular velocity of the horizon. It is the angular velocity with which the horizon "drags" spacetime around. $\chi$ is the unique combination of these two Killing vectors that becomes null on the horizon.

The concept of the event horizon as a Killing horizon is the essential geometric foundation for defining surface gravity.

2. The Formal Definition of Surface Gravity

The null geodesic generators of the horizon are the integral curves of the Killing vector $\chi$. You'll recall from our study of geodesics that a curve $x^{\mu }(\lambda )$ with tangent vector $V^{\mu }=d{x}^{\mu }/d\lambda$ is a geodesic if it is parallel-transported, i.e., ${\nabla }_{V}V=0$. If this holds, $\lambda$ is called an affine parameter.

However, the integral curves of a Killing vector are not, in general, affinely parameterized geodesics. The extent to which they fail to be is measured by the "acceleration" vector ${\nabla }_{\chi }\chi$. On a Killing horizon, a remarkable thing happens: this acceleration vector turns out to be proportional to the tangent vector $\chi$ itself. The constant of proportionality is the surface gravity, $\kappa$.

Formally, the surface gravity $\kappa$ of a Killing horizon is defined by the relation:

This equation holds at every point on the event horizon $H$. Because $\kappa$ measures the failure of the parameter along the Killing vector's orbits to be an affine one, $\kappa$ is also known as the non-affinity coefficient of the horizon's generators.

For our primary examples, the surface gravity is:

• Schwarzschild: $\kappa =\frac{1}{4M}$
• Kerr: $\kappa =\frac{\sqrt{M^2-a^2}}{2M(M+\sqrt{M^2-a^2})}=\frac{r_{+}-r_{-}}{2(r_{+}^2+a^2)}$, where $a=J/M$.

Notice that for the extremal Kerr black hole ($a=M$), the surface gravity is zero.

An alternative, but equivalent, definition that is often useful for calculations is given in terms of the gradient of the norm of the Killing field $\chi$:

where $|\chi |=\sqrt{-{\chi }_{\mu }{\chi }^{\mu }}$ is the redshift factor, which approaches zero at the horizon.

3. Physical Interpretations of Surface Gravity

The formal definition ${\nabla }_{\chi }\chi =\kappa \chi$ is geometrically elegant but physically abstract. Fortunately, $\kappa$ has several powerful and intuitive physical interpretations.

Interpretation 1: The Redshifted Force at the Horizon

Imagine an observer trying to hover at a fixed radius $r$ outside a black hole. To counteract gravity, they must continuously fire their rockets, experiencing a proper acceleration $a$. As this observer gets closer and closer to the event horizon, the required acceleration $a$ diverges to infinity.

At the same time, any signal the observer sends out is subject to gravitational redshift. The redshift factor for a signal received by a distant observer is given by $V=\sqrt{-{\chi }_{\mu }{\chi }^{\mu }}$. As the observer approaches the horizon, this factor goes to zero.

The surface gravity is the finite, invariant quantity that emerges from the product of these two effects in the limit as the observer reaches the horizon:

In essence, surface gravity is the acceleration required to remain just at the horizon, as measured by an observer at infinity.

For a static black hole, this has an even simpler interpretation: $\kappa$ is the force that must be applied at infinity (e.g., via a massless string) to hold a particle of unit mass stationary just above the event horizon.

Interpretation 2: A Measure of Temperature

The most profound interpretation of $\kappa$ comes from its analogy with temperature. This can be seen in a few ways. First, as shown by a clever thought experiment involving a Carnot-like heat engine operating between two black holes, one can define a thermodynamic temperature $T_{\text{BH}}$ that is directly proportional to $\kappa$.

The definitive connection was made by Stephen Hawking. By applying quantum field theory in the curved spacetime of a black hole, he showed that black holes are not truly black but radiate particles as if they were a blackbody with a physical temperature given by:

In geometric units ($\hslash =c=k_B=G=1$), this is simply $T_H=\kappa /2\pi$. This stunning result links the geometry of the horizon ($\kappa$), quantum mechanics ($\hslash$), and thermodynamics ($T$).

4. The Zeroth Law of Black Hole Mechanics

The analogy with temperature is solidified by a fundamental property of $\kappa$. For a stationary black hole, one might think that $\kappa$ could vary from point to point on the horizon (e.g., it might be different at the poles than at the equator of a Kerr black hole). However, a key theorem proves this is not the case.

The Zeroth Law of Black Hole Mechanics: For any stationary black hole solution to the Einstein equations (satisfying certain energy conditions), the surface gravity $\kappa$ is constant over the entire event horizon.

This is immediately analogous to the Zeroth Law of Thermodynamics, which states that a body in thermal equilibrium has a uniform temperature throughout. The fact that the formula for $\kappa$ for a Kerr black hole depends only on $M$ and $J$ (or $a$) and not on the polar angle $\theta$ is a concrete manifestation of this powerful law.

Conclusion

In this lesson, we have moved from the geometry of spacetime to the physics of the event horizon. We have given a rigorous definition of surface gravity and uncovered its deep physical meaning.

Key Takeaways:

• The event horizon of a stationary black hole is a Killing horizon, a null surface generated by a special Killing vector $\chi$.
• Surface gravity, $\kappa$, is formally defined as the non-affinity of the horizon's null generators, via the equation ${\nabla }_{\chi }\chi =\kappa \chi$.
• Physically, $\kappa$ can be interpreted as the redshifted force required to hold a particle at the horizon. Counter-intuitively, $\kappa$ is smaller for more massive black holes.
• The Zeroth Law of Black Hole Mechanics states that $\kappa$ is constant over the entire event horizon of a stationary black hole.
• This constancy, combined with other arguments, establishes $\kappa$ as the direct analogue of temperature in thermodynamics, a connection made precise by Hawking's formula for black hole temperature, $T_H\propto \kappa$.

Preview of Next Lesson:

The Zeroth Law is just the beginning. The analogy between black hole mechanics and thermodynamics is remarkably complete. In our next lesson, we will state and interpret the other three laws of black hole mechanics. We will define black hole entropy in terms of the horizon area and see how these laws form a perfect parallel to the familiar laws of thermodynamics, further cementing the revolutionary idea that black holes are not just geometric objects, but are profound thermodynamic systems.