Mode-sum regularization of $⟨{\varphi }^2⟩$ in the angular-splitting method

The computation of the renormalized stress-energy tensor or ${⟨{\varphi }^2⟩}_{\text{ren}}$ in curved spacetime is a challenging task, at both the conceptual and technical levels. Recently we developed a new approach to compute such renormalized quantities in asymptotically flat curved spacetimes, based on the point-splitting procedure. Our approach requires the spacetime to admit some symmetry. We already implemented this approach to compute ${⟨{\varphi }^2⟩}_{\text{ren}}$ in a stationary spacetime using $t$ splitting, namely splitting in the time-translation direction. Here we present the angular-splitting version of this approach, aimed for computing renormalized quantities in a general (possibly dynamical) spherically symmetric spacetime. To illustrate how the angular-splitting method works, we use it here to compute ${⟨{\varphi }^2⟩}_{\text{ren}}$ for a quantum massless scalar field in Schwarzschild background, in various quantum states (Boulware, Unruh, and Hartle-Hawking states). We find excellent agreement with the results obtained from the $t$-splitting variant and also with other methods. Our main goal in pursuing this new mode-sum approach was to enable the computation of the renormalized stress-energy tensor in a dynamical spherically symmetric background, e.g. an evaporating black hole. The angular-splitting variant presented here is most suitable to this purpose.

Figure 1

(a) Solid curve: The numerically calculated $E_{\omega ,l=1}$, as defined in Eq. (3.28), evaluated at $r=6M$. Its large-$\omega$ asymptotic behavior $\propto {\omega }^{-1}$ is evident from the dashed line, which is $\omega E_{\omega ,l=1}$. (b) Solid curve: The subtraction $E_{\omega ,l=1}-E_{\omega ,l=0}$ at $r=6M$. It decays as ${\omega }^{-3}$, as seen from the dashed line, which is the same subtraction multiplied by $100{\omega }^3$.

Figure 2

(a) The solid curve represents $F(l,r=6M)$, calculated according to Eq. (3.13). The dashed curve is the analytic function $F_{\mathrm{sing}}(l,r=6M)$, given in Eq. (4.2). (b) The graph displays $F_{\text{reg}}(l,r=6M)$, which is simply the difference between the two curves in (a). It rapidly approaches a constant, the blind spot discussed in Sec. 3a3.

Figure 3

(a) The sequence $H(l,r)$ constructed according to Eq. (3.25). The fast convergence is evident. (b) A close-up on the plateau in (a). In this enlargement, one can also notice the growth of numerical error at large $l$ (say $l>16$). The red cross indicates the estimated optimal $l$ value [for numerically evaluating the large-$l$ limit of $H(l,r)$; $l=12$ in the present case]. This estimated optimal $l$ is automatically selected by an algorithm that assesses where the numerical error starts to grow.

Figure 4

(a) The solid line represents the results for ${⟨{\varphi }^2⟩}_{\text{ren}}$ in the Boulware state, calculated using the new angular-splitting variant. There is excellent agreement with both the results obtained using the $t$-splitting variant (the crosses) and previous results by Anderson (the asterisks). (b) A near-horizon close-up of $(1-2M/r){⟨{\varphi }^2⟩}_{\text{ren}}$, showing that it is clearly finite. Here we also give the analytic horizon value calculated by Candelas (the dotted line).

Figure 5

(a) The solid curve displays ${⟨{\varphi }^2⟩}_{\text{ren}}$ in the Unruh state calculated using angular splitting. The results obtained using the $t$-splitting variant are marked by crosses. The dashed curve is aimed to demonstrate the $\propto (r^{-2})$ large-$r$ asymptotic behavior of ${⟨{\varphi }^2⟩}_{\text{ren}}$, by multiplying the latter by $r^2/10$. (b) A near-horizon enlargement of ${⟨{\varphi }^2⟩}_{\text{ren}}$ in the Unruh state, demonstrating its regularity at $r\to 2M$.

Figure 6

(a) The solid curve displays ${⟨{\varphi }^2⟩}_{\text{ren}}$ in the Hartle-Hawking state calculated using the angular-splitting method. The results obtained from the $t$-splitting variant are marked by crosses. The analytical asymptotic values calculated by Candelas are represented by the two dotted horizontal lines (top line, horizon; bottom line, infinity). (b) A near-horizon enlargement of ${⟨{\varphi }^2⟩}_{\text{ren}}$ in the Hartle-Hawking state. One can see the agreement with the analytical value at the horizon calculated by Candelas (the dotted horizontal line).