Near the Horizon

Near the horizon the exterior of the black hole may be described by the Rindler metric

      dτ = ρ² dω² − dρ² − dX² − dY ²

It is useful to replace ρ by a tortoise-like coordinate which again goes to −∞ at the horizon. We define

u = logρ     (1.1)

and the metric near the horizon becomes

 (1.2)

The scalar field action becomes

 (1.3)

where ∂_⊥χ = (∂_X, ∂_Y ).Instead of using spherical waves, near the horizon we can decompose χ into transverse plane waves with transverse wave vector

 (1.4)

the action for a given wave number k is

 (1.5)

Thus the potential is

V (k, u) = k² e^2u       (1.6)

The correspondence between the momentum vector k and the angular momentum ℓ is given by the usual connection between momentum and angular momentum. If the horizon has circumference 2π(2MG), then a wave with wave vector k_⊥ will correspond to an angular momentum |ℓ| = |k| r = 2MG|k|. Thus the potential in equation 1.6 is seen to be proportional to ℓ². For very low angular momentum the approximation is not accurate, but qualitatively is correct for ℓ > 0.I n approximating a sum over ℓ and m by an integral over k, the integral should be infrared cut off at |k| ∼ 1/MG. From the action in equation 1.5 we obtain the equation of motion

 (1.7)

A solution which behaves like exp(iνt) in Schwarzschild time has the form

 (1.8)

The time independent form of the equation of motion is

 (1.9)

Once again we see that unless k = 0, there is a potential confining quanta to the region near the horizon. Qualitatively, the behavior of a quantum field in a black hole background differs from the Rindler space approximation in that for the black hole, the potential barrier is cut off when ρ = e^u is greater than MG. By contrast, in the Rindler case V increases as e^u without bound.

      We have not thus far paid attention to the boundary conditions at the horizon where u → −∞.Since in this region the field χ(u) behaves like a free massless field, the boundary condition would be expected to be that the field is in the usual quantum ground state. In the next section we will see that this is not so.