Renormalized stress-energy tensor for stationary black holes

We continue the presentation of the pragmatic mode-sum regularization (PMR) method for computing the renormalized stress-energy tensor (RSET). We show in detail how to employ the $t$-splitting variant of the method, which was first presented for ${⟨{\varphi }^2⟩}_{\text{ren}}$, to compute the RSET in a stationary, asymptotically flat background. This variant of the PMR method was recently used to compute the RSET for an evaporating spinning black hole. As an example for regularization, we demonstrate here the computation of the RSET for a minimally coupled, massless scalar field on Schwarzschild background in all three vacuum states. We discuss future work and possible improvements of the regularization schemes in the PMR method.

Figure 1

(a) The solid line is the numerically calculated $F_{rr}$ ($\omega$, $r=6$); it is proportional to ${\omega }_3$ for large $\omega$ and clearly divergent. The dashed line is the analytically computed singular part, $F_{rr}^{\text{Sing}}$ ($\omega$, $r=6$), which captures the nonoscillatory divergent part of $F_{rr}$ ($\omega$, $r=6$). (b) The result of subtracting the singular part $F_{rr}^{\text{Sing}}$ ($\omega$, $r=6$) from $F_{rr}$ ($\omega$, $r=6$), which is the definition of $F_{rr}^{\text{Reg}}$ ($\omega$, $r=6$) in Eq. (3.8). It is dominated by oscillations with an amplitude proportional to ${\omega }^{5/2}$.

Figure 2

(a) The integral function $H_{rr}$ ($\omega$, $r=6$) vs $\omega$. The amplitude of the oscillations is proportional to ${\omega }^{5/2}$. (b) $H_{rr}^{*}$ ($\omega$, $r=6$) vs $\omega$, obtained by applying self-cancellation on $H_{rr}$ ($\omega$, $r=6$). Notice the quick convergence and also the numerical deviations at $\omega \gtrsim 1.5$. The red cross denotes the point chosen by an algorithm we designed to pick the optimal $\omega$ value to estimate the $\omega \to \infty$ limit.

Figure 3

The solid curves represent the RSET in the Boulware state, calculated using the $t$-splitting variant. The plus symbols are results by Paul Anderson, obtained using the traditional regularization method [9]. The RSET in the Boulware state is divergent at the horizon, and decays at infinity like $r^{-4}$. To clarify the picture we multiplied the components by factors of $f(r)\equiv 1-2M/r$ and $r$.

Figure 4

The RSET in the Unruh state. Here the RSET is finite on the horizon and we multiply by $f(r)\equiv 1–2M/r$ to treat the coordinate singularity. Note that $-r^2{T}_t^r$ is a conserved quantity, representing the outgoing Hawking radiation to infinity.

Figure 5

The RSET in the Hartle-Hawking state. Here the RSET is finite at the horizon, and the multiplication by $f(r)\equiv 1–2M/r$ is for scaling purposes only.