Surface gravity of a Killing horizon

Question

I have two questions about this:

1. Surface gravity is defined on the Killing horizon by $\xi^\mu \nabla_\nu \xi^\nu = \kappa \xi^\nu$ for the Killing vector $\xi$. Why can we interpret this as the force required at infinity to hold a unit mass at rest on the horizon?

2. Is there an obvious reason why $\nabla^\mu \xi^2$ is orthogonal to the horizon? Clearly, I can expand this and get $\nabla^\mu \xi^2 = - 2 \xi^\rho \nabla_\rho \xi^\mu =-2 \kappa \xi^\mu$ which is orthogonal since $\xi$ is orthogonal and $\kappa$ is just a constant. But, I want to know if there is an obvious, intuitive way to see this without doing any calculations (the author says it's "obvious") or if the person that wrote the notes was expecting me to do the above calculation to see this?

Answer 1

Number (2) follows from the fact that for any scalar function $\Phi(x^{\nu})$, the gradient $\partial_{\mu} \Phi$ is orthogonal to the level set $\Phi(x^{\nu})=const$.

Intuitively, this should hold since $\Phi$ doesn't change in directions tangent to the hypersurface. A proof follows from the chain rule. Let $\gamma^\mu (s)$ be a curve which lies entirely in the hypersurface $H$ defined by $\Phi(x^{\nu})=c$. Then since $\Phi(\gamma^{\mu}(t))=c$ , $$\frac{d\Phi(\gamma^{\mu}(t))}{dt} = \frac{\partial \Phi}{\partial \gamma^\mu} \frac{d\gamma^{\mu}}{dt} = \partial_\mu \Phi \; \frac{d\gamma^{\mu}}{dt} = 0 .$$ The derivative $\frac{d\gamma^{\mu}}{dt}$ of an arbitrary curve in $H$ gives an arbitrary tangent vector to $H$, so $\partial_\mu \Phi$ is normal to $H$. This is a property of differential geometry, and has nothing to do with null vectors or horizons.

In particular, then, the magnitude $\xi^2$ of the Killing vector field is a scalar function on spacetime, with the constant value $\xi^2=0$ on the horizon. So the gradient $\nabla_\mu \xi^2=\partial_\mu \xi^2$ must be normal to the horizon.

As for (1), take a look at Wald eqn 12.5.18 and problem 6.4.

First time giving an answer... hope it helps!

Answer 2

1) Going to Rindler coordinates, in the near horizon limit of the schwarzschild black hole, you get that the surface gravity is precisely the constant acceleration of the Rindler observer. Indeed this is the equivalence principle at work: gravitation "=" acceleration.

For a static observer you can do an explicit calculation by defining the four-acceleration as $a^{\mu}=U^{\sigma} \nabla_{\sigma} U^{\mu}$, where $U^{\mu}$ is the velocity, and taking the time-translation Killing field $ξ^{\mu}=\sqrt{-ξ^2} U^{\mu}$. See Carroll's book pag. 246 for reference.

2)By definition the Killing vector is null on the horizon, so $ξ^2=0$ on the horizon. Moreover this means that is normal to himself and to the horizon.