Stress-Energy Tensor for a Massless Spin1/2Field in Static Black Hole Spacetimes

Abstract

The stress-energy tensor for the massless spin 1/2 field is numerically computed outside and on the event horizons of both charged and uncharged static nonrotating black holes, corresponding to the Schwarzschild, Reissner-Nordström, and extreme Reissner-Nordström solutions of Einstein's equations. The field is assumed to be in a thermal state at the black hole temperature. Comparison is made between the numerical results and previous analytic approximations for the stress-energy tensor in these spacetimes. For the Schwarzschild (charge zero) solution, it is shown that the stress energy differs even in sign from the analytic approximation. For the Reissner-Nordström and extreme Reissner-Nordström solutions, divergences predicted by the analytic approximations are shown not to exist.

Macroscopic effects of the quantum trace anomaly

“…The electrically charged Reisner-Nordstrom black hole for which f (r) = 1 − 2M/r + e 2 /r 2 is an example in which shows that it is possible to obtain a regular stress-energy on the horizon with (3.36) even in the extreme Reisner-Nordstrom case with appropriate choice of auxiliary field integration constants, in agreement with the direct numerical results of [41]. Thus the anomaly induced effective action can reproduce the correct behaviors of the quantum stress tensor also in this case, in contrast to the approximation of [35,36,37] which predicts a divergent stress-energy on the horizon of an extreme…”

“…conditions (5.14), which is a four parameter restriction of the eight parameters in the auxiliary fields of the form ( [40] and [41] respectively. In the spin 1 case the dashed curve represents the numerical results of ref.…”

Approximate stress-energy tensor of the massless spin-1/2field in Schwarzschild spacetime

“…Unfortunately, as has been pointed out in Ref. [2], T µ ν analytic for the spin-1/2 field gives a wrong sign of the energy density at the event horizon, invalidating thus any prospect applications. On the other hand, however, the expectation value of the stress-energy tensor of the conformally invariant massless fields in the Hartle-Hawking state in the Schwarzschild geometry is known to possess some general features [10].…”

“…Now we can compare the approximate tensors (9-11) to (12)(13)(14)(15) and to the numerical results presented in Ref. [2]. By construction the tensors are exact at infinity and very close to the exact value at the event horizon.…”

“…Further, a piece of the numerical data, such as the exact value of one of the components of the stress-energy tensor, say T θ θ ren , on the event horizon may be used in the final determination of the model. It should be noted that calculations of the horizon value of the stress-energy tensor could be regarded as a relatively simple task since it is sufficient to retain only the two smallest frequencies in the mode sums [2,11]. In this sense this information could be regarded as independent of the numerical calculations of the stressenergy tensor for r > 2M.…”