A Quantum Peek Inside the Black Hole Event Horizon

Abstract

We solve the Klein-Gordon equation for a scalar field, in the background geometry of a dust cloud collapsing to form a black hole in the (1+1) spacetime. This has been performed both inside and outside the event horizon and arbitrarily close to the curvature singularity. This allows us to determine the expectation value of regularized stress tensor, everywhere in the appropriate quantum state (viz., the Unruh vacuum) of the field. Using this expectation value we have studied the behaviour of energy density and flux measured by both static and radially freely falling observers at any given event. Outside the black hole, energy density and flux lead to the standard results expected from the Hawking radiation. While inside the collapsing dust ball, the energy densities of both matter and scalar field diverge near the singularity in both (1+1) and (1+3) spacetime dimensions. Intriguingly the energy density of the field dominates over that of classical matter. These results suggest that the back-reaction effects are significant in the region close to the singularity.

Appendices

Appendix A

In this Appendix, we present the steps for deriving various results mentioned in Chap. 9.

A.1 Stress-Energy Tensor: Explicit Derivation

1.1 A.1.1 Exterior Region

The various derivatives of the conformal factor in the exterior region have the following expressions:

$$\begin{aligned} \partial _{+}C=\frac{r-1}{2r^{3}}\frac{dB/dU}{dA/dU};\partial _{+}^{2}C=\frac{(r-1)(3-2r)}{4r^{5}}\frac{dB/dU}{dA/dU} \end{aligned}$$

(9.60)

$$\begin{aligned} \partial _{-}C=\frac{r-1}{r}\partial _{-}\left( \frac{dB/dU}{dA/dU}\right) -\frac{r-1}{2r^{3}} \left( \frac{dB/dU}{dA/dU} \right) ^{2} \end{aligned}$$

(9.61)

$$\begin{aligned} \partial _{-}^{2}C&=-\frac{3}{2}\frac{r-1}{r^{3}}\frac{dB/dU}{dA/dU}\partial _{-} \left( \frac{dB/dU}{dA/dU} \right) +\frac{r-1}{r}\partial _{-}^{2}\left( \frac{dB/dU}{dA/dU}\right) \nonumber \\&+\frac{(3-2r)(r-1)}{4r^{5}}\left( \frac{dB/dU}{dA/dU}\right) ^{3} \end{aligned}$$

(9.62)

$$\begin{aligned} \partial _{-}\partial _{+}C=\frac{r-1}{2r^{3}}\partial _{-}\left( \frac{dB/dU}{dA/dU}\right) -\left( \frac{dB/dU}{dA/dU}\right) ^{2}\frac{(3-2r)(r-1)}{4r^{5}} \end{aligned}$$

(9.63)

With the following expressions for energy momentum tensor:

$$\begin{aligned} \langle T_{++}\rangle =\frac{\kappa ^{2}}{48\pi }\left( \frac{3}{r^{4}}-\frac{4}{r^{3}}\right) ;\langle T_{+-}\rangle =\frac{\kappa ^{2}}{12\pi }\frac{r-1}{r^{4}}\frac{dB/dU}{dA/dU} \end{aligned}$$

(9.64)

$$\begin{aligned} \langle T_{--}\rangle&= \frac{\kappa ^{2}}{48\pi }\Bigg [\left( \frac{dB/dU}{dA/dU}\right) ^{2} \left( \frac{3}{r^{4}}-\frac{4}{r^{3}}\right) \nonumber \\&+16\left\{ \frac{1}{2}\frac{\partial _{-}^{2}\left( \frac{dB/dU}{dA/dU}\right) }{\frac{dB/dU}{dA/dU}}-\frac{3}{4}\left( \frac{\partial _{-}\left( \frac{dB/dU}{dA/dU}\right) }{\frac{dB/dU}{dA/dU}} \right) ^{2} \right\} \Bigg ] \nonumber \\&=\frac{\kappa ^{2}}{48\pi }\left( \frac{dB/dU}{dA/dU}\right) ^{2} \Bigg [\left( \frac{3}{r^{4}}-\frac{4}{r^{3}}\right) +\frac{16}{\left( \frac{dB}{dU}\right) ^{2}} \nonumber \\&\times \Big [\left\{ \frac{1}{2}\frac{\partial _{U}^{2}\left( dB/dU\right) }{dB/dU} -\frac{3}{4}\left( \frac{\partial _{U}\left( dB/dU\right) }{dB/dU} \right) ^{2} \right\} \nonumber \\&-\left\{ \frac{1}{2}\frac{\partial _{U}^{2}\left( dA/dU\right) }{dA/dU} -\frac{3}{4}\left( \frac{\partial _{U}\left( dA/dU\right) }{dA/dU} \right) ^{2} \right\} \Big ]\Bigg ] \end{aligned}$$

(9.65)

these relations can be simplified to arrive at,

$$\begin{aligned} \langle T_{+-}\rangle =\frac{\kappa ^{2}}{12\pi }\frac{r-1}{r^{4}} \left( \frac{\cot \chi _{0}+\tan \left( \frac{U-\chi _{0}}{2}\right) }{\cot \chi _{0}-\tan \left( \frac{U+\chi _{0}}{2}\right) }\right) \frac{\cos ^{2}\left( \frac{U+\chi _{0}}{2}\right) }{\cos ^{2}\left( \frac{U-\chi _{0}}{2}\right) } \end{aligned}$$

(9.66)

$$\begin{aligned} \langle T_{--}\rangle&=\frac{\kappa ^{2}}{48\pi } \left( \left( \frac{\cot \chi _{0}+\tan \left( \frac{U+\chi _{0}}{2}\right) }{\cot \chi _{0}-\tan \left( \frac{U+\chi _{0}}{2}\right) }\right) \frac{\cos ^{2}\left( \frac{U+\chi _{0}}{2}\right) }{\cos ^{2}\left( \frac{U-\chi _{0}}{2}\right) } \right) ^2 \left[ \left( \frac{3}{r^{4}}-\frac{4}{r^{3}}\right) \right. \nonumber \\&+\left( \frac{a_{max}\cos ^{2}\left( \frac{U+\chi _{0}}{2}\right) }{\sin \chi _{0}\left( \cot \chi _{0}-\tan \left( \frac{U+\chi _{0}}{2}\right) \right) }\right) ^{-2} \nonumber \\&\times \left( -15\left[ \tan ^{2}\left( \frac{U+\chi _{0}}{2}\right) -\tan ^{2}\left( \frac{U-\chi _{0}}{2}\right) \right] \right. \nonumber \\&-6\cot \chi _{0}\left[ \tan \left( \frac{U+\chi _{0}}{2}\right) +\tan \left( \frac{U-\chi _{0}}{2}\right) \right] \nonumber \\&+\frac{4\cot \chi _{0}}{\sin ^{2}\chi _{0}}\left[ \frac{1}{\cot \chi _{0}-\tan \left( \frac{U+\chi _{0}}{2}\right) }\right. \nonumber \\&\left. -\frac{1}{\cot \chi _{0}+\tan \left( \frac{U-\chi _{0}}{2}\right) }\right] +\frac{a_{max}}{\sin \chi _{0}}\left[ \frac{1}{\left( \cot \chi _{0} -\tan \left( \frac{U+\chi _{0}}{2}\right) \right) ^{2}}\right. \nonumber \\&-\left. \left. \left. \frac{1}{\left( \cot \chi _{0}+\tan \left( \frac{U-\chi _{0}}{2}\right) \right) ^{2}} \right] \right) \right] \end{aligned}$$

(9.67)

In arriving at the above relations we have used the following expressions for the various derivatives dA / dU and dB / dU are respectively:

$$\begin{aligned} \frac{dA}{dU}=\frac{a_{max}\cos ^{2}\left( \frac{U-\chi _{0}}{2}\right) }{\sin \chi _{0}\left( \cot \chi _{0}+\tan \left( \frac{U-\chi _{0}}{2}\right) \right) };\frac{dB}{dU}=\frac{a_{max}\cos ^{2}\left( \frac{U+\chi _{0}}{2}\right) }{\sin \chi _{0}\left( \cot \chi _{0}-\tan \left( \frac{U+\chi _{0}}{2}\right) \right) } \end{aligned}$$

(9.68)

$$\begin{aligned} \frac{d^{2}A}{dU^{2}}=-\frac{1}{2}\frac{\left[ 1+2\sin ^{2}\left( \frac{U-\chi _{0}}{2}\right) +\cot \chi _{0}\sin \left( U-\chi _{0}\right) \right] }{\sin ^{4}\chi _{0}\left( \cot \chi _{0}+\tan \left( \frac{U-\chi _{0}}{2}\right) \right) ^{2}} \end{aligned}$$

(9.69)

$$\begin{aligned} \frac{d^{2}B}{dU^{2}}=\frac{1}{2}\frac{\left[ 1+2\sin ^{2}\left( \frac{U+\chi _{0}}{2}\right) -\cot \chi _{0}\sin \left( U+\chi _{0}\right) \right] }{\sin ^{4}\chi _{0}\left( \cot \chi _{0}-\tan \left( \frac{U+\chi _{0}}{2}\right) \right) ^{2}} \end{aligned}$$

(9.70)

as well as the following derivatives:

$$\begin{aligned} \frac{1}{2}&\frac{\partial _{U}^{2}\left( dA/dU\right) }{dA/dU} -\frac{3}{4}\left( \frac{\partial _{U}\left( dA/dU\right) }{dA/dU} \right) ^{2} = -\frac{1}{16}\Big [15\tan ^{4}\left( \frac{U-\chi _{0}}{2}\right) \nonumber \\&+24\cot \chi _{0} \tan ^{3}\left( \frac{U-\chi _{0}}{2}\right) +10\tan ^{2} \left( \frac{U-\chi _{0}}{2} \right) \nonumber \\&+4\cot ^{2}\chi _{0}\left\{ 1+2\tan ^{2}\left( \frac{U-\chi _{0}}{2}\right) \right\} \nonumber \\&+16\cot \chi _{0}\tan \left( \frac{U-\chi _{0}}{2}\right) -1\Big ] \left[ \cot \chi _{0}+\tan \left( \frac{U-\chi _{0}}{2}\right) \right] ^{-2} \nonumber \\&=-\frac{1}{16}\Big [15\tan ^{2}\left( \frac{U-\chi _{0}}{2}\right) -6\cot \chi _{0}\tan \left( \frac{U-\chi _{0}}{2}\right) +5\left( 1+\sin ^{-2}\chi _{0}\right) \nonumber \\&-\frac{4\sin ^{-2}\chi _{0}\cot \chi _{0}}{\cot \chi _{0}+\tan \left( \frac{U-\chi _{0}}{2}\right) } -\frac{a_{max}}{\sin \chi _{0}\left( \cot \chi _{0} +\tan \left( \frac{U-\chi _{0}}{2}\right) \right) ^{2}}\Big ] \end{aligned}$$

(9.71)

and

$$\begin{aligned} \frac{1}{2}&\frac{\partial _{U}^{2}\left( dB/dU\right) }{dB/dU} -\frac{3}{4}\left( \frac{\partial _{U}\left( dB/dU\right) }{dB/dU} \right) ^{2} = -\frac{1}{16}\Big [15\tan ^{4}\left( \frac{U+\chi _{0}}{2}\right) \nonumber \\&-24\cot \chi _{0} \tan ^{3}\left( \frac{U+\chi _{0}}{2}\right) -10\tan ^{2}\left( \frac{U+\chi _{0}}{2} \right) \nonumber \\&+4\cot ^{2}\chi _{0}\left\{ 1+2\tan ^{2}\left( \frac{U+\chi _{0}}{2}\right) \right\} \nonumber \\&-16\cot \chi _{0}\tan \left( \frac{U+\chi _{0}}{2}\right) -1\Big ] \left[ \cot \chi _{0}-\tan \left( \frac{U+\chi _{0}}{2}\right) \right] ^{-2} \nonumber \\&=-\frac{1}{16}\Big [15\tan ^{2}\left( \frac{U+\chi _{0}}{2}\right) +6\cot \chi _{0}\tan \left( \frac{U+\chi _{0}}{2}\right) +5\left( 1+\sin ^{-2}\chi _{0}\right) \nonumber \\&-\frac{4\sin ^{-2}\chi _{0}\cot \chi _{0}}{\cot \chi _{0}-\tan \left( \frac{U+\chi _{0}}{2}\right) } -\frac{a_{max}}{\sin \chi _{0}\left( \cot \chi _{0} -\tan \left( \frac{U+\chi _{0}}{2}\right) \right) ^{2}}\Big ] \end{aligned}$$

(9.72)

1.2 A.1.2 Interior Region

For the interior region the various derivatives of the conformal factors are:

$$\begin{aligned} \frac{1}{C}\partial _{+}C&=\frac{1}{\left( dA/dV\right) }\left[ \frac{1}{a^{2}}\frac{da^{2}}{dV}- \frac{d^{2}A/dV^{2}}{dA/dV}\right] \nonumber \\ \frac{1}{C}\partial _{-}C&=\frac{1}{\left( dA/dU\right) }\left[ \frac{1}{a^{2}}\frac{da^{2}}{dU}- \frac{d^{2}A/dU^{2}}{dA/dU}\right] \end{aligned}$$

(9.73)

$$\begin{aligned} \frac{1}{C}\partial _{+}^{2}C&=\frac{1}{\left( dA/dV\right) ^{2}} \Bigg [\frac{1}{a^{2}}\frac{d^{2}a^{2}}{dV^{2}} -\frac{3}{a^{2}}\frac{da^{2}}{dV}\frac{1}{dA/dV}\frac{d^{2}A}{dV^{2}} \nonumber \\&-\frac{1}{dA/dV}\frac{d^{3}A}{dV^{3}} +3\left( \frac{1}{dA/dV}\frac{d^{2}A}{dV^{2}}\right) ^{2} \Bigg ] \end{aligned}$$

(9.74)

$$\begin{aligned} \frac{1}{C}\partial _{-}^{2}C&=\frac{1}{\left( dA/dU\right) ^{2}} \Bigg [\frac{1}{a^{2}}\frac{d^{2}a^{2}}{dU^{2}} -\frac{3}{a^{2}}\frac{da^{2}}{dU}\frac{1}{dA/dU}\frac{d^{2}A}{dU^{2}} -\frac{1}{dA/dU}\frac{d^{3}A}{dU^{3}} \nonumber \\&+3\left( \frac{1}{dA/dU}\frac{d^{2}A}{dU^{2}}\right) ^{2} \Bigg ] \end{aligned}$$

(9.75)

$$\begin{aligned} \frac{1}{C}\partial _{-}\partial _{+}C&=\frac{1}{\frac{dA}{dV}\frac{dA}{dU}} \left[ \frac{1}{a^{2}}\frac{d^{2}a^{2}}{dUdV} -\frac{1}{a^{2}}\frac{da^{2}}{dU}\frac{1}{dA/dV}\frac{d^{2}A}{dV^{2}} -\frac{1}{a^{2}}\frac{da^{2}}{dV}\frac{1}{dA/dU}\frac{d^{2}A}{dU^{2}}\right. \nonumber \\&\left. +\frac{1}{dA/dV}\frac{d^{2}A}{dV^{2}}\frac{1}{dA/dU}\frac{d^{2}A}{dU^{2}}\right] \end{aligned}$$

(9.76)

Then the components of the stress energy tensor in the inside region are:

$$\begin{aligned} \langle T_{++}\rangle&=\frac{1}{12\pi }\frac{1}{\left( dA/dV\right) ^{2}} \left[ \left\{ \frac{1}{2a^{2}}\frac{d^{2}a^{2}}{dV^{2}} -\frac{3}{4}\left( \frac{1}{a^{2}}\frac{da^{2}}{dV}\right) ^{2} \right\} -\left\{ \frac{1}{2}\frac{1}{dA/dV}\frac{d^{3}A}{dV^{3}}\right. \right. \nonumber \\&\left. \left. -\frac{3}{4}\frac{1}{\left( dA/dV\right) ^{2}}\left( \frac{d^{2}A}{dV^{2}}\right) ^{2} \right\} \right] \nonumber \\&=\frac{1}{12\pi }\frac{1}{\left( dA/dV\right) ^{2}}\Bigg [-\frac{1}{8} \left( \frac{3a_{max}\sin \chi _{0}}{r\left( \frac{U+V}{2}\right) }-2\right) \nonumber \\&-\left\{ \frac{1}{2}\frac{\partial _{V}^{2}\left( dA/dV\right) }{\left( dA/dV\right) } -\frac{3}{4}\left( \frac{\partial _{V}\left( dA/dV\right) }{\left( dA/dV\right) }\right) ^{2} \right\} \Bigg ] \end{aligned}$$

(9.77)

$$\begin{aligned} \langle T_{--}\rangle&=\frac{1}{12\pi }\frac{1}{\left( dA/dU\right) ^{2}} \left[ \left\{ \frac{1}{2a^{2}}\frac{d^{2}a^{2}}{dU^{2}} -\frac{3}{4}\left( \frac{1}{a^{2}}\frac{da^{2}}{dU}\right) ^{2} \right\} -\left\{ \frac{1}{2}\frac{1}{dA/dU}\frac{d^{3}A}{dU^{3}}\right. \right. \nonumber \\&\left. \left. -\frac{3}{4}\frac{1}{\left( dA/dU\right) ^{2}}\left( \frac{d^{2}A}{dU^{2}}\right) ^{2} \right\} \right] \nonumber \\&=\frac{1}{12\pi }\frac{1}{\left( dA/dU\right) ^{2}} \left[ -\frac{1}{8} \left( \frac{3a_{max}\sin \chi _{0}}{r\left( \frac{U+V}{2}\right) }-2\right) -\left\{ \frac{1}{2}\frac{\partial _{U}^{2}\left( dA/dU\right) }{\left( dA/dU\right) }\right. \right. \nonumber \\&\left. \left. -\frac{3}{4}\left( \frac{\partial _{U}\left( dA/dU\right) }{\left( dA/dU\right) }\right) ^{2} \right\} \right] \end{aligned}$$

(9.78)

$$\begin{aligned} \langle T_{+-}\rangle&=\frac{1}{24\pi }\frac{1}{dA/dV}\frac{1}{dA/dU} \left[ \frac{1}{a^{2}}\frac{d^{2}a^{2}}{dUdV} -\frac{1}{a^{2}}\frac{da^{2}}{dU}\frac{1}{a^{2}}\frac{da^{2}}{dV}\right] \nonumber \\&=-\frac{1}{48\pi }\frac{1}{dA/dV}\frac{1}{dA/dU}\frac{1}{1+\cos \left( \frac{U+V}{2} \right) } \end{aligned}$$

(9.79)

where various derivatives of the quantity A are given in Appendix F.1.1.

A.2 Energy Density and Flux For Various Observers

1.1 A.2.1 Static Observer

Below we provide the full expressions for energy density and flux calculated for static observer:

$$\begin{aligned} \mathcal {U}&=\langle T_{++}\rangle \left( \dot{V}^{+}\right) ^{2} +\langle T_{--}\rangle \left( \dot{V}^{-}\right) ^{2} +2\langle T_{+-}\rangle \dot{V}^{+}\dot{V}^{-} \nonumber \\&=\frac{\kappa ^{2}}{48\pi }\left( \frac{r}{r-1}\right) \Big [\left( -\frac{2}{r^{4}}\right) +\frac{\sin ^{2}\chi _{0}}{a_{max}^{2}\cos ^{4} \left( \frac{U+\chi _{0}}{2}\right) }\left( \cot \chi _{0}-\tan \left( \frac{U+\chi _{0}}{2}\right) \right) ^{2} \nonumber \\&\times \Big \lbrace -15\left[ \tan ^{2}\left( \frac{U+\chi _{0}}{2}\right) -\tan ^{2}\left( \frac{U-\chi _{0}}{2}\right) \right] \nonumber \\&-6\cot \chi _{0}\left[ \tan \left( \frac{U+\chi _{0}}{2}\right) +\tan \left( \frac{U-\chi _{0}}{2}\right) \right] \nonumber \\&+\frac{4\cot \chi _{0}}{\sin ^{2}\chi _{0}}\left[ \frac{1}{\cot \chi _{0}-\tan \left( \frac{U+\chi _{0}}{2}\right) }-\frac{1}{\cot \chi _{0}+\tan \left( \frac{U-\chi _{0}}{2}\right) }\right] \Big \rbrace \nonumber \\&+\frac{\sin \chi _{0}}{a_{max}\cos ^{4}\left( \frac{U+\chi _{0}}{2}\right) }\left[ 1 -\frac{\left( \cot \chi _{0}-\tan \left( \frac{U+\chi _{0}}{2}\right) \right) ^{2}}{\left( \cot \chi _{0}+\tan \left( \frac{U-\chi _{0}}{2}\right) \right) ^{2}} \right] \Big ] \end{aligned}$$

(9.80)

and the expression for flux turns out to be:

$$\begin{aligned} \mathcal {F}&=-\langle T_{ab}\rangle u^{a}n^{b}=-\langle T_{++}\rangle \left( \dot{V}^{+}\right) ^{2} +\langle T_{--}\rangle \left( \dot{V}^{-}\right) ^{2} \nonumber \\&=\frac{\kappa ^{2}}{48\pi }\left( \frac{r}{r-1}\right) \Big [\frac{\sin ^{2}\chi _{0}}{a_{max}^{2}\cos ^{4} \left( \frac{U+\chi _{0}}{2}\right) }\left( \cot \chi _{0}-\tan \left( \frac{U+\chi _{0}}{2}\right) \right) ^{2} \nonumber \\&\times \Big \lbrace -15\left[ \tan ^{2}\left( \frac{U+\chi _{0}}{2}\right) -\tan ^{2}\left( \frac{U-\chi _{0}}{2}\right) \right] \nonumber \\&-6\cot \chi _{0}\left[ \tan \left( \frac{U+\chi _{0}}{2}\right) +\tan \left( \frac{U-\chi _{0}}{2}\right) \right] \nonumber \\&+\frac{4\cot \chi _{0}}{\sin ^{2}\chi _{0}}\left[ \frac{1}{\cot \chi _{0}-\tan \left( \frac{U+\chi _{0}}{2}\right) }-\frac{1}{\cot \chi _{0}+\tan \left( \frac{U-\chi _{0}}{2}\right) }\right] \Big \rbrace \nonumber \\&+\frac{\sin \chi _{0}}{a_{max}\cos ^{4}\left( \frac{U+\chi _{0}}{2}\right) }\left[ 1 -\frac{\left( \cot \chi _{0}-\tan \left( \frac{U+\chi _{0}}{2}\right) \right) ^{2}}{\left( \cot \chi _{0}+\tan \left( \frac{U-\chi _{0}}{2}\right) \right) ^{2}} \right] \Big ] \end{aligned}$$

(9.81)

1.2 A.2.2 Radially In-Falling Observers: Inside

The energy density for radially in-falling observer has the following expression:

$$\begin{aligned} \mathcal {U}&=\frac{\kappa ^{2}}{48\pi }\frac{1}{a^{2}\left( \eta \right) }\Big [-8 \sec ^{2}\frac{\eta }{2} +4+\frac{1}{2}\Big \lbrace 15\tan ^{2}\left( \frac{\eta -\chi _{0}-\tilde{\chi }}{2}\right) \nonumber \\&-6\cot \chi _{0}\tan \left( \frac{\eta -\chi _{0}-\tilde{\chi }}{2}\right) +5\left( 1+\sin ^{-2}\chi _{0}\right) \nonumber \\&-\frac{4\sin ^{-2}\chi _{0}\cot \chi _{0}}{\cot \chi _{0} +\tan \left( \frac{\eta -\chi _{0}-\tilde{\chi }}{2}\right) } -\frac{a_{max}}{\sin \chi _{0}\left( \cot \chi _{0} +\tan \left( \frac{\eta -\chi _{0}-\tilde{\chi }}{2}\right) \right) ^{2}} \nonumber \\&+15\tan ^{2}\left( \frac{\eta -\chi _{0}+\tilde{\chi }}{2}\right) -6\cot \chi _{0}\tan \left( \frac{\eta -\chi _{0}+\tilde{\chi }}{2}\right) +5\left( 1+\sin ^{-2}\chi _{0}\right) \nonumber \\&-\frac{4\sin ^{-2}\chi _{0}\cot \chi _{0}}{\cot \chi _{0} +\tan \left( \frac{\eta -\chi _{0}+\tilde{\chi }}{2}\right) } -\frac{a_{max}}{\sin \chi _{0}\left( \cot \chi _{0} +\tan \left( \frac{\eta -\chi _{0}+\tilde{\chi }}{2}\right) \right) ^{2}} \Big \rbrace \Big ] \end{aligned}$$

(9.82)

while the flux has the following expression:

$$\begin{aligned} \mathcal {F}&=\frac{\kappa ^{2}}{48\pi }\frac{1}{a^{2}\left( \eta \right) }\frac{1}{2} \Big \lbrace 15\tan ^{2}\left( \frac{\eta -\chi _{0}-\tilde{\chi }}{2}\right) -6\cot \chi _{0}\tan \left( \frac{\eta -\chi _{0}-\tilde{\chi }}{2}\right) \nonumber \\&+5\left( 1+\sin ^{-2}\chi _{0}\right) -\frac{4\sin ^{-2}\chi _{0}\cot \chi _{0}}{\cot \chi _{0} +\tan \left( \frac{\eta -\chi _{0}-\tilde{\chi }}{2}\right) } \nonumber \\&-\frac{a_{max}}{\sin \chi _{0}\left( \cot \chi _{0} +\tan \left( \frac{\eta -\chi _{0}-\tilde{\chi }}{2}\right) \right) ^{2}} -15\tan ^{2}\left( \frac{\eta -\chi _{0}+\tilde{\chi }}{2}\right) \nonumber \\&+6\cot \chi _{0}\tan \left( \frac{\eta -\chi _{0}+\tilde{\chi }}{2}\right) -5\left( 1+\sin ^{-2}\chi _{0}\right) \nonumber \\&+\frac{4\sin ^{-2}\chi _{0}\cot \chi _{0}}{\cot \chi _{0} -\tan \left( \frac{\eta -\chi _{0}+\tilde{\chi }}{2}\right) } +\frac{a_{max}}{\sin \chi _{0}\left( \cot \chi _{0} +\tan \left( \frac{\eta -\chi _{0}+\tilde{\chi }}{2}\right) \right) ^{2}} \Big \rbrace \end{aligned}$$

(9.83)

1.3 A.2.3 Radially In-falling Observers: Outside

For radially in-falling observer outside the dust ball has the following expression for energy density:

$$\begin{aligned} \mathcal {U}&=\langle T_{++}\rangle \left( \dot{V}^{+}\right) ^{2} +\langle T_{--}\rangle \left( \dot{V}^{-}\right) ^{2} +2\langle T_{+-}\rangle \dot{V}^{+}\dot{V}^{-} \nonumber \\&=\frac{\kappa ^{2}}{48\pi } 4E^{2}\left( \frac{r}{r-1}\right) ^{2} \left( \frac{3}{r^{4}}-\frac{4}{r^{3}}\right) +\frac{\kappa ^{2}}{24\pi }\left( \frac{r}{r-1}\right) \left( -\frac{7}{r^{4}}+\frac{8}{r^{3}}\right) \nonumber \\&+\frac{\kappa ^{2}}{48\pi }\left( \frac{r}{r-1}\right) ^{2} \left( E+\sqrt{E^{2}-\frac{r-1}{r}}\right) ^{2} \nonumber \\&\times \Big [\frac{\sin ^{2}\chi _{0}}{a_{max}^{2}\cos ^{4} \left( \frac{\eta +\chi _{0}-\tilde{\chi }}{2}\right) }\left( \cot \chi _{0}-\tan \left( \frac{\eta +\chi _{0}-\tilde{\chi }}{2}\right) \right) ^{2} \nonumber \\&\times \Big \lbrace -15\left[ \tan ^{2}\left( \frac{\eta +\chi _{0}-\tilde{\chi }}{2}\right) -\tan ^{2}\left( \frac{\eta -\chi _{0}-\tilde{\chi }}{2}\right) \right] \nonumber \\&-6\cot \chi _{0}\left[ \tan \left( \frac{\eta +\chi _{0}-\tilde{\chi }}{2}\right) +\tan \left( \frac{\eta -\chi _{0}-\tilde{\chi }}{2}\right) \right] \nonumber \\&+\frac{4\cot \chi _{0}}{\sin ^{2}\chi _{0}}\left[ \frac{1}{\cot \chi _{0}-\tan \left( \frac{\eta +\chi _{0}-\tilde{\chi }}{2}\right) }-\frac{1}{\cot \chi _{0}+\tan \left( \frac{\eta -\chi _{0}-\tilde{\chi }}{2}\right) }\right] \Big \rbrace \nonumber \\&+\frac{\sin \chi _{0}}{a_{max}\cos ^{4}\left( \frac{\eta +\chi _{0}-\tilde{\chi }}{2}\right) }\left[ 1 -\frac{\left( \cot \chi _{0}-\tan \left( \frac{\eta +\chi _{0}-\tilde{\chi }}{2}\right) \right) ^{2}}{\left( \cot \chi _{0}+\tan \left( \frac{\eta -\chi _{0}-\tilde{\chi }}{2}\right) \right) ^{2}} \right] \Big ] \end{aligned}$$

(9.84)

and the flux has the following expression:

$$\begin{aligned} \mathcal {F}&=-\langle T_{ab}\rangle u^{a}n^{b}=-\langle T_{++}\rangle \left( \dot{V}^{+}\right) ^{2} +\langle T_{--}\rangle \left( \dot{V}^{-}\right) ^{2} \nonumber \\&=\frac{\kappa ^{2}}{48 \pi }4E\sqrt{E^{2}-\frac{r-1}{r}} \left( \frac{r}{r-1}\right) ^{2} \left( \frac{3}{r^{4}}-\frac{4}{r^{3}}\right) +\frac{\kappa ^{2}}{48\pi }\left( \frac{r}{r-1}\right) ^{2} \nonumber \\&\times \left( E+\sqrt{E^{2}-\frac{r-1}{r}}\right) ^{2}\Big [\frac{\sin ^{2}\chi _{0}}{a_{max}^{2}\cos ^{4} \left( \frac{\eta +\chi _{0}-\tilde{\chi }}{2}\right) }\left( \cot \chi _{0}-\tan \left( \frac{\eta +\chi _{0}-\tilde{\chi }}{2}\right) \right) ^{2} \nonumber \\&\times \Big \lbrace -15\left[ \tan ^{2}\left( \frac{\eta +\chi _{0}-\tilde{\chi }}{2}\right) -\tan ^{2}\left( \frac{\eta -\chi _{0}-\tilde{\chi }}{2}\right) \right] \nonumber \\&-6\cot \chi _{0}\left[ \tan \left( \frac{\eta +\chi _{0}-\tilde{\chi }}{2}\right) +\tan \left( \frac{\eta -\chi _{0}-\tilde{\chi }}{2}\right) \right] \nonumber \\&+\frac{4\cot \chi _{0}}{\sin ^{2}\chi _{0}}\left[ \frac{1}{\cot \chi _{0}-\tan \left( \frac{\eta +\chi _{0}-\tilde{\chi }}{2}\right) }-\frac{1}{\cot \chi _{0}+\tan \left( \frac{\eta -\chi _{0}-\tilde{\chi }}{2}\right) }\right] \Big \rbrace \nonumber \\&+\frac{\sin \chi _{0}}{a_{max}\cos ^{4}\left( \frac{\eta +\chi _{0}-\tilde{\chi }}{2}\right) }\left[ 1 -\frac{\left( \cot \chi _{0}-\tan \left( \frac{\eta +\chi _{0}-\tilde{\chi }}{2}\right) \right) ^{2}}{\left( \cot \chi _{0}+\tan \left( \frac{\eta -\chi _{0}-\tilde{\chi }}{2}\right) \right) ^{2}} \right] \Big ] \end{aligned}$$

(9.85)

A.3 Effective temperature for various observers

In this section we present the full expression for effective temperature measured by detectors in various trajectories. For static observer the effective temperature turns out to be:

$$\begin{aligned} T_{-}&=\frac{1}{2\pi } \left| \frac{\ddot{V}^{-}}{\dot{V}^{-}}\right| \nonumber \\&=\frac{1}{2\pi }\left| \dot{u}~\left( \frac{\frac{d^{2}A}{dU^{2}}}{\frac{dA}{dU}\frac{dB}{dU}} -\frac{\frac{d^{2}B}{dU^{2}}}{\left( \frac{dB}{dU}\right) ^{2}} \right) \right| \nonumber \\&=\frac{1}{4\pi }\sqrt{\frac{r}{r-1}}\sin ^{4}\chi _{0} \nonumber \\&\times \Big \lbrace \frac{\left[ 1+2\sin ^{2}\left( \frac{U-\chi _{0}}{2}\right) +\cot \chi _{0}\sin \left( U-\chi _{0}\right) \right] \left( \cot \chi _{0}-\tan \left( \frac{U+\chi _{0}}{2}\right) \right) }{\left( \cot \chi _{0}+\tan \left( \frac{U-\chi _{0}}{2}\right) \right) \cos ^{2}\left( \frac{U+\chi _{0}}{2}\right) \cos ^{2}\left( \frac{U-\chi _{0}}{2}\right) } \nonumber \\&+\frac{\left[ 1+2\sin ^{2}\left( \frac{U+\chi _{0}}{2}\right) -\cot \chi _{0}\sin \left( U+\chi _{0}\right) \right] }{\cos ^{4}\left( \frac{U+\chi _{0}}{2}\right) } \Big \rbrace \end{aligned}$$

(9.86)

For radially in-falling observes inside the dust sphere the effective temperature turns out to be:

$$\begin{aligned} T_{-}=\frac{1}{4\pi }\left| \frac{1 +2\sin ^{2}\left( \frac{\eta -\chi _{0}-\tilde{\chi }}{2}\right) +\cot \chi _{0}\sin \left( \frac{\eta -\chi _{0}-\tilde{\chi }}{2}\right) }{\cos ^{2}\left( \frac{\eta -\chi _{0}-\tilde{\chi }}{2}\right) \left\{ \cot \chi _{0}+\tan \left( \frac{\eta -\chi _{0}-\tilde{\chi }}{2}\right) \right\} } \right| \end{aligned}$$

(9.87)

Finally, for radially in-falling observers in the Schwarzschild spacetime effective temperature takes the following expression:

$$\begin{aligned} T_{-}&=\frac{1}{2\pi }\Bigg \vert \frac{\ddot{u}}{\dot{u}} -\dot{u}\frac{\sin ^{4}\chi _{0}}{2} \nonumber \\&\Big \lbrace \frac{\left[ 1+2\sin ^{2}\left( \frac{\eta -\chi _{0}-\tilde{\chi }}{2}\right) +\cot \chi _{0}\sin \left( \eta -\chi _{0}-\tilde{\chi }\right) \right] \left( \cot \chi _{0}-\tan \left( \frac{\eta +\chi _{0}-\tilde{\chi }}{2}\right) \right) }{\left( \cot \chi _{0}+\tan \left( \frac{\eta -\chi _{0}-\tilde{\chi }}{2}\right) \right) \cos ^{2}\left( \frac{\eta +\chi _{0}-\tilde{\chi }}{2}\right) \cos ^{2}\left( \frac{\eta -\chi _{0}-\tilde{\chi }}{2}\right) } \nonumber \\&+\frac{\left[ 1+2\sin ^{2}\left( \frac{\eta +\chi _{0}-\tilde{\chi }}{2}\right) -\cot \chi _{0}\sin \left( \eta +\chi _{0}-\tilde{\chi }\right) \right] }{\cos ^{4}\left( \frac{\eta +\chi _{0}-\tilde{\chi }}{2}\right) } \Big \rbrace \Bigg \vert \end{aligned}$$

(9.88)