Versatile Method for Renormalized Stress-Energy Computation in Black-Hole Spacetimes

We report here on a new method for calculating the renormalized stress-energy tensor (RSET) in black-hole (BH) spacetimes, which should also be applicable to dynamical BHs and to spinning BHs. This new method only requires the spacetime to admit a single symmetry. Thus far, we have developed three variants of the method, aimed for stationary, spherically symmetric, or axially symmetric BHs. We used this method to calculate the RSET of a minimally coupled massless scalar field in Schwarzschild and Reissner-Nordström backgrounds for several quantum states. We present here the results for the RSET in the Schwarzschild case in the Unruh state (the state describing BH evaporation). The RSET is type I at weak field, and becomes type IV at $r\lesssim 2.78M$. Then we use the RSET results to explore violation of the weak and null energy conditions. We find that both conditions are violated all the way from $r\simeq 4.9M$ to the horizon. We also find that the averaged weak energy condition is violated by a class of (unstable) circular timelike geodesics. Most remarkably, the circular null geodesic at $r=3M$ violates the averaged null energy condition.

Figure 1

The various components of the RSET in the Unruh state for a MCMSF in Schwarzschild. The curves are the results obtained in $\theta$ splitting, and the asterisks and plus symbols are results obtained using $t$ and $\phi$ splittings, respectively. Numerical errors start to increase very close to the horizon (say, $r_{*}\lesssim -9$), which results in the tiny wiggles in $T_{\theta }^{\theta }$. Note that energy-momentum conservation guarantees that $r^2{T}_t^r=\text{const}$, yielding the straight horizontal solid line.

Figure 2

The various indicators for WEC and NEC, as a function of $r_{*}$. To the left of the solid vertical line, $r=r_0\simeq 2.78M$, both WEC and NEC are violated as $W<0$. To the right of $r=r_0$, $W$ is positive. NEC then demands $\rho +p_1$, $\rho +p_2\ge 0$, and WEC also requires $\rho \ge 0$. It is clearly seen that $\rho +p_2$ turns negative first (with decreasing $r$); hence, WEC and NEC break together. This happens at a surprisingly large radius $r=r_c\simeq 4.9M$. WEC and NEC are violated all the way from $r_c$ to the horizon.

Figure 3

The projected energy density $T_{\parallel }(r)$ along a circular geodesic at radius $r$. It turns negative at $r\lesssim 3.47M$.