Stress-Energy Tensor for a Massless Spin $1/2$ Field in Static Black Hole Spacetimes

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.

Figure 1

The analytic (dashed line) and numerical (solid line) values for the quantity $⟨T_t^t⟩$ as a function of the distance from the event horizon $r_{+}$ in Schwarzschild spacetime.

Figure 2

The numerical values for the quantities $⟨T_{\theta }^{\theta }⟩$ and $⟨T_r^r⟩$ from top to bottom at the event horizon $r_{+}$ in Schwarzschild spacetime.

Figure 3

The numerical values for the quantity $⟨T_t^t⟩$ for $|Q|/M=1$, 0.99, 0.95, 0.8, 0 from top to bottom at the event horizon $r_{+}$ for the case $\mu M=1$.

Figure 4

The numerical values for the quantity $(⟨T_r^r⟩-⟨T_t^t⟩)/f$ for $|Q|/M=1$, 0.9999, 0.999, 0.99 from top to bottom at the event horizon $r_{+}$. For these plots, the value $\mu M=1$ has been chosen.