1 ) 1 f R ω l 1 n Y l m e i ω t . (18)

In the limit $E\to 0$ , the proper response rate of the charge can be written 

$\frac{{R}_{0lm}}{\sqrt{f\left({r}_{0}\right)}}=4\text{π}\underset{E\to 0}{lim}\frac{{|{\mathcal{T}}_{Elm}^{\to }|}^{2}}{\sqrt{f\left({r}_{0}\right)}\beta E}$ (19)

where ${\mathcal{T}}_{Elm}^{\to }$ is the index $n=\to$ in transition amplitude ${\mathcal{T}}_{\omega lm}^{n}$ which has the form

${\mathcal{T}}_{\omega lm}^{n}\equiv \frac{1}{2\text{π}\delta \left(\omega -E\right)}\int \text{ }{\text{d}}^{p+2}x\sqrt{-g}{j}^{\mu }〈1n;\omega lm|{\stackrel{^}{A}}_{\mu }|0〉$ (20)

Now, we return to calculate ${\mathcal{T}}_{Elm}^{\to }$ . The Equation (8) can be written

$\frac{1}{{r}^{2}}\frac{\text{d}}{\text{d}r}\left[f{r}^{2-p}\frac{\text{d}}{\text{d}r}\left({r}^{p}{R}_{\omega l}^{\left(1\to \right)}\left(r\right)\right)\right]-\frac{l\left(l+p-1\right)}{{r}^{2}}{R}_{\omega l}^{\left(1\to \right)}\left(r\right)+\frac{{\omega }^{2}}{f}{R}_{\omega l}^{\left(1\to \right)}\left(r\right)=0.$ (21)

After introducing the Wheeler tortoise coordinate and a function $\phi$

${R}_{\omega l}^{\left(1\to \right)}\left(r\right)\equiv \frac{\sqrt{l\left(l+p-1\right)}}{\omega }{r}^{-\frac{p}{2}-1}{\phi }_{\omega l}^{\left(1\to \right)}\left(r\right),$ (22)

the Equation (21) changes into

$\left({\omega }^{2}+\frac{{\text{d}}^{2}}{\text{d}{r}^{*2}}-{V}_{1}\left({r}^{*}\right)\right){\phi }_{\omega l}^{\left(1\to \right)}\left(r\right)=0$ (23)

with

${V}_{1}\left[{r}^{*}\left(r\right)\right]=f\frac{l\left(l+p-1\right)}{{r}^{2}}+{f}^{2}\frac{p\left(p-2\right)}{4{r}^{2}}-{f}^{\prime }f\frac{\left(p-2\right)}{2r}$ (24)

For the small $\omega$ and the condition $\left(r-{r}_{h}\ll {\omega }^{2}{r}_{h}^{3},|\omega {r}^{*}|\ll 1\right)$ , the wave coming from the past horizon ${H}^{-}$ is almost completely reflected back by the potential toward the horizon

${\phi }_{\omega l}^{\left(1\to \right)}\approx -2\omega {r}^{*}+\text{const}$ (25)

Generally, it is hard to find the analytic expression for the Wheeler tortoise coordinate ${r}^{*}$ . Fortunately, what we need is just the behavior of ${r}^{*}$ near horizon. Substituting the expression of $f\left(r\right)$ , the leading term of the Wheeler tortoise coordinate can be written

${r}^{*}\approx \frac{2}{p-3}{\left(\frac{2M}{\alpha }\right)}^{\frac{1}{p-3}}ln\left(z-1\right)$ (26)

by using the transition

$z=\frac{2}{B}{r}^{\frac{p-3}{2}}-1.$ (27)

The boundary condition of ${R}_{\omega l}^{\left(1\to \right)}$ reads

${R}_{\omega l}^{\left(1\to \right)}\approx -\frac{\sqrt{l\left(l+p-1\right)}}{p-3}{2}^{\frac{3p-4}{p-3}}{B}^{-\frac{p}{p-3}}\mathrm{ln}\left(z-1\right),\left(r-{r}_{H}\ll {\omega }^{2}{r}_{H}^{3},|\omega {r}^{*}|\ll 1\right)$ (28)

In terms of variable z, Equation (21) can be written

$\begin{array}{l}\frac{1}{4}\left({z}^{2}-1\right)\frac{{\text{d}}^{2}q\left(z\right)}{\text{d}{z}^{2}}+\frac{1}{4}\left[\frac{3p-1}{p-3}\left(z-1\right)+2\right]\frac{\text{d}q\left(z\right)}{\text{d}z}\\ +\frac{1}{{\left(p-3\right)}^{2}}\left[-l\left(l+p-1\right)+\frac{p\left(p-5\right)}{z+1}+p+{\omega }^{2}\frac{z+1}{z-1}{\left(\frac{B}{2}\left(z+1\right)\right)}^{\frac{4}{p-3}}\right]q\left(z\right)=0\end{array}$ (29)

and this equation can be solved explicitly for small $\omega$ limit. Combining the asymptotic behavior ${R}_{\omega l}^{\left(1\to \right)}\to 0$ as $z\to +\infty$ , the solution of Equation (29) is

$\begin{array}{c}{R}_{\omega l}^{\left(1\to \right)}=\frac{\sqrt{l\left(l+p-1\right)}}{p-3}{2}^{\frac{3p-4}{p-3}}{B}^{-\frac{p}{p-3}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}×\text{F}\left(\left[\frac{2l}{p-3},\frac{-2l-2p+2}{p-3}\right],\left[\frac{-2p+2}{p-3}\right],\frac{z+1}{2}\right){\left(z+1\right)}^{-\frac{2p}{p-3}}\end{array}$ (30)

where the coefficient has been appropriately chosen to agreement with the boundary condition Equation (28).

Thus, the expression of the proper response rate of the charge is

$\frac{{R}_{0lm}}{\sqrt{f\left({r}_{0}\right)}}=\frac{\left(p-3\right){q}^{2}{\left({z}_{0}+1\right)}^{-\frac{2p+2}{p-3}}{B}^{\frac{2-2p}{p-3}}{2}^{\frac{6}{p-3}}}{\text{π}l\left(l+p-1\right)\sqrt{f\left({r}_{0}\right)}}{\left(\frac{{z}_{0}-1}{{z}_{0}+1}\right)}^{2}{\left[\frac{\text{d}}{\text{d}{z}_{0}}{\text{F}}_{l}\left({z}_{0}\right)\right]}^{2}{|{Y}_{lm}|}^{2}$ (31)

ubstituting the all parameters for d-dimensional GB black hole

${z}_{0}=\frac{2}{B}{r}_{0}^{\frac{p-3}{2}}-1$ (32)

$f\left({r}_{0}\right)=1-\frac{B}{{r}_{0}^{\frac{d-5}{2}}}$ (33)

${\text{F}}_{l}\left({z}_{0}\right)=\text{F}\left(\left[\frac{2l}{p-3},\frac{-2l-2p+2}{p-3}\right],\left[\frac{-2p+2}{p-3}\right],\frac{z+1}{2}\right)$ (34)

the total transition probability per proper time of the charge is given by

$\begin{array}{c}{R}^{\text{tot}}=\underset{l}{\overset{+\infty }{\sum }}\underset{m=-1}{\overset{l}{\sum }}\frac{{R}_{0lm}}{\sqrt{f\left({r}_{0}\right)}}\\ =\underset{l}{\overset{+\infty }{\sum }}\frac{\left(p-3\right){q}^{2}{\left({z}_{0}+1\right)}^{-\frac{2p+2}{p-3}}{B}^{\frac{2-2p}{p-3}}{2}^{\frac{6}{p-3}}}{\text{π}l\left(l+p-1\right)\sqrt{f\left({r}_{0}\right)}}{\left(\frac{{z}_{0}-1}{{z}_{0}+1}\right)}^{2}{\left[\frac{\text{d}}{\text{d}{z}_{0}}{\text{F}}_{l}\left({z}_{0}\right)\right]}^{2}\frac{G\left(l\right)}{{\Omega }_{p}}\end{array}$ (35)

We can give the main contribution of the result of the Schwarzschild black hole and GB black hole,

${R}^{\text{tot}}~{\left(\frac{{r}_{h}}{{r}_{0}}\right)}^{p-3},\left(\text{Schwarzschild}\right)$ (36)

${R}^{\text{tot}}~{\left(\frac{{r}_{h}}{{r}_{0}}\right)}^{\frac{3p-1}{2}}.\left(\text{GB}\right)$ (37)

Assuming that there is a charge outside an unknown black hole, the radius of the horizon ( ${r}_{h}$ ), the mass of the black hole (M) and the location of the charge ( ${r}_{0}$ ) are known. Meanwhile, we can measure the response rate of the charge outside this unknown black hole. Compare the measured value with Equation (36) and Equation (37), then we can tell the black hole is the GB type or not.

4. Conclusions

In this paper, we quantized the free electrodynamics in static spherically symmetric spacetime of arbitrary dimensions in a modified Feynman gauge. Then we examined the Gupta-Bleuler quantization in this modified Feynman gauge. The results obtained were applied to compute the total response rate of a static charge outside the d-dimensional GB black hole in the Unruh vacuum.

For the Einstein-Gauss-Bonnet gravity, one can follow the same procedure. The Wheeler tortoise coordinate ${r}^{*}$ has a asymptotic behavior ${r}^{*}\approx {f}^{\prime }{\left({r}_{h}\right)}^{-1}ln\left(r-{r}_{h}\right)$ as near horizon. The boundary condition Equation (25) changes into ${\phi }_{\omega l}^{\left(1\to \right)}\approx -2\omega /{f}^{\prime }\left({r}_{h}\right)ln\left(r-{r}_{h}\right)$ . One problem is that we do not find a new variable to simplify the Equation (21) and its analytic solution. An applicable way is to find a series solution of Equation (21), the results can not be expressed in terms of familiar special functions and we neglect it here.

Such an outcome is not only a simple promotion work for what Crispino et al.  have done. Having the specific form of the free quantum electrodynamics in static spherically symmetric spacetime of arbitrary dimensions, It may provide us a chance for further investigating quantum field theory in high-dimensional curved spacetime. For instance, some authors (    ) have researched whether or not a quantum version of the equivalence principle could be formulated and show some equivalence for low-frequency quantum phenomena in flat and curved spacetime. The same problem could be reconsidered in high-dimensional spacetime and discuss the dimension dependence of the results.

Acknowledgements

This work has been supported by the Scientific Research Program Funded by Shaanxi Provincial Education Department under Program No.16JK1394.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

Conflicts of Interest

The authors declare no conflicts of interest.

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