Quantum vacuum energy near a black hole: the Maxwell field

The author considers a quantised Maxwell field propagating in the gravitational field of a Schwarzschild black hole. The vector Hartle-Hawking propagator is defined on the Riemannian section of the analytically continued space-time and expanded in terms of four-dimensional vector spherical harmonics. The equations for the radial functions appearing in this expansion are derived for both odd and even parity. Using the expansion of the vector Hartle-Hawking propagator the author then derives the point-separated expectation value of the Maxwellian energy-momentum tensor in the Hartle-Hawking vacuum. The renormalised values of radial pressure, tangential pressure and energy density are obtained near the horizon of the black hole. The author thus extends previous work of various authors on vacuum polarisation of a massless scalar field near a black hole to the more realistic case of the Maxwell field. In contrast to the scalar field, the Maxwell field exhibits a positive energy density near the horizon in the Hartle-Hawking vacuum state.