I'm trying to understand the force required to hold a particle near the event horizon of a black hole. In particular I'm trying to fill in some details of Carroll's text around equations 6.15 to 6.17.

I'm working in a spacetime that has a time like Killing vector $K_\mu$ that is normalised such that $K^\mu K_\mu = 1$ at infinity (spacetime is asymptotically flat) and the Killing horizon for $K$ is a hyper surface $\Sigma$.

It's straightforward to show that if we have a static particle (i.e. four velocity $U_a$ is proportional to $K_a$ i.e. $K^a = VU^a$ where in order to have a proper time parametrisation we should have $V = \sqrt{-K^aK_a}$) then the acceleration of this observer is $A^a = \nabla^a \log(V)$ and so, assuming this particle has unit mass, the local "force" required to hold it there is $F = A^aA_a$.

Now I'm trying to understand how this force is "red-shifted" at infinity. If we imagine a string connecting this particle to an observer at infinity then what is the local force that the observer at infinity needs to exert on the string to hold the particle at a fixed radius (not necessarily precisely at the Killing horizon)?

I have been trying to think about energy conservation and the fact that $E = K^a V_a$ is conserved along geodesics, so perhaps we could start the particle at infinity with four velocity proportional to $K^a$, let it fall on a time-like geodesic to a point near the Killing horizon, and then somehow using a conservation of energy argument? I haven't been able to make much progress with this so far!

The reason for being interested in this is that I'm trying to understand how we can interpret the surface gravity as the limit of this force as the particle approaches the Killing horizon.


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