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Quantum mechanics of stationary states of particles in a space–time of classical black holes

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Abstract

We consider interactions of scalar particles, photons, and fermions in Schwarzschild, Reissner–Nordström, Kerr, and Kerr–Newman gravitational and electromagnetic fields with a zero and nonzero cosmological constant. We also consider interactions of scalar particles, photons, and fermions with nonextremal rotating charged black holes in a minimal five-dimensional gauge supergravity. We analyze the behavior of effective potentials in second-order relativistic Schrödinger-type equations. In all cases, we establish the existence of the regime of particles “falling” on event horizons. An alternative can be collapsars with fermions in stationary bound states without a regime of particles “falling.”

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Acknowledgments

The authors thank E. Yu. Popov for the useful discussions and the help in establishing the final form of some analytic expressions. The authors also thank A. L. Novoselova for the essential technical support in preparing the paper.

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  1. Federal State Unitary Enterprise Russian Federal Nuclear Center — All-Russian Research Institute of Experimental Physics, Sarov, Nizhny Novgorod region, Russia

    M. V. Gorbatenko & V. P. Neznamov

  2. National Engineering Physics Institute MEPhI, Moscow, Russia

    V. P. Neznamov

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  1. M. V. Gorbatenko
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A. Effective potential of a Painlevé–Gullstrund field in a Schrödinger-type equation for a scalar particle

We have

$$\begin{aligned} \, &U_{ \mathrm{eff} }^{ \scriptscriptstyle\mathrm{PG} }(\rho)=-\frac{\alpha( \varepsilon ^2-1)}{\rho-2\alpha}- \frac{\alpha\rho \varepsilon ^2}{(\rho-2\alpha)^2}+\frac{1}{2(\rho-2\alpha)^2} \biggl(\frac{\alpha^2}{\rho^2}+\frac{\alpha}{\rho}-1\biggr)+{} \\ &\hphantom{U_{ \mathrm{eff} }^{ \scriptscriptstyle\mathrm{PG} }(\rho){}=}+\frac{1}{2\rho(\rho-2\alpha)} \biggl(1+\frac{\alpha}{\rho}\biggr)-\frac{1}{2\rho(\rho-2\alpha)}l(l+1)+ i\frac{1}{4}\sqrt{\frac{2\alpha}{\rho}}\frac{ \varepsilon }{\rho}, \\ &U_{ \mathrm{eff} }^{ \scriptscriptstyle\mathrm{PG} }|_{\rho\to\infty}=\frac{\alpha}{\rho}(1-2 \varepsilon ^2),\qquad U_{ \mathrm{eff} }^{ \scriptscriptstyle\mathrm{PG} }|_{\rho\to0}=-\frac{1}{8\rho^2}, \\ &U_{ \mathrm{eff} }^{ \scriptscriptstyle\mathrm{PG} }|_{\rho\to 2\alpha}=-\frac{1}{2(\rho-2\alpha)^2} \biggl(\frac{1}{4}+4\alpha^2 \varepsilon ^2\biggr). \end{aligned}$$

B. Effective potentials of gravitational and electromagnetic fields in Schrödinger-type equations for fermions

1. For the Kerr–Newman–(anti-)de Sitter field, in accordance with [6] and Eqs. (113),

$$\begin{aligned} \, U_{ \mathrm{eff} }^{ \scriptscriptstyle\mathrm{KN} }={}&E_{ \mathrm{Schr} }+\frac{3}{8}\frac{1}{B_{ \scriptscriptstyle\mathrm{KN} }^2} \biggl(\frac{dB^{}_{ \scriptscriptstyle\mathrm{KN} }}{dr}\biggr)^2-\frac{1}{4B^{}_{ \scriptscriptstyle\mathrm{KN} }} \frac{d^2B^{}_{ \scriptscriptstyle\mathrm{KN} }}{dr^2}+\frac{1}{4} \frac{d}{dr}(A^{}_{ \scriptscriptstyle\mathrm{KN} }-D^{}_{ \scriptscriptstyle\mathrm{KN} })-{} \nonumber \\ &{}-\frac{1}{4} \frac{(A^{}_{ \scriptscriptstyle\mathrm{KN} }-D^{}_{ \scriptscriptstyle\mathrm{KN} })}{B^{}_{ \scriptscriptstyle\mathrm{KN} }}\frac{dB^{}_{ \scriptscriptstyle\mathrm{KN} }}{dr}+ \frac{1}{8}(A^{}_{ \scriptscriptstyle\mathrm{KN} }-D_{ \scriptscriptstyle\mathrm{KN} })^2+\frac{1}{2}B^{}_{ \scriptscriptstyle\mathrm{KN} }C^{}_{ \scriptscriptstyle\mathrm{KN} }, \end{aligned}$$
(166)
$$\begin{aligned} \, \frac{3}{8}\frac{1}{B_{ \scriptscriptstyle\mathrm{KN} }^2}\biggl(\frac{dB^{}_{ \scriptscriptstyle\mathrm{KN} }}{dr}\biggr)^2={}& \frac{3}{8}\biggl\{\frac{f^{}_{ \scriptscriptstyle\mathrm{KN} }}{\Omega^{}_{ \scriptscriptstyle\mathrm{KN} }+\sqrt{f^{}_{ \scriptscriptstyle\mathrm{KN} }}} \biggl[-\frac{1}{f_{ \scriptscriptstyle\mathrm{KN} }^2}f'^{}_{ \scriptscriptstyle\mathrm{KN} } (\Omega^{}_{ \scriptscriptstyle\mathrm{KN} }+\sqrt{f^{}_{ \scriptscriptstyle\mathrm{KN} }})+{} \nonumber \\ &{}+\frac{1}{f^{}_{ \scriptscriptstyle\mathrm{KN} }}\biggl(\Omega'^{}_{ \scriptscriptstyle\mathrm{KN} }+ \frac{f'^{}_{ \scriptscriptstyle\mathrm{KN} }}{2\,\sqrt{f^{}_{ \scriptscriptstyle\mathrm{KN} }}}\biggr)\biggr]\biggr\}^2, \end{aligned}$$
(167)
$$\begin{aligned} \, -\frac{1}{4}\frac{1}{B^{}_{ \scriptscriptstyle\mathrm{KN} }}\frac{d^2 B^{}_{ \scriptscriptstyle\mathrm{KN} }}{dr^2}={}& -\frac{1}{4}\frac{f^{}_{ \scriptscriptstyle\mathrm{KN} }}{\Omega^{}_{ \scriptscriptstyle\mathrm{KN} }+\sqrt{f^{}_{ \scriptscriptstyle\mathrm{KN} }}} \biggl[\frac{2}{f_{ \scriptscriptstyle\mathrm{KN} }^3}(f'_{ \scriptscriptstyle\mathrm{KN} })^2 (\Omega^{}_{ \scriptscriptstyle\mathrm{KN} }+\sqrt{f^{}_{ \scriptscriptstyle\mathrm{KN} }})-{} \nonumber \\ &{}-\frac{1}{f_{ \scriptscriptstyle\mathrm{KN} }^2}f''^{}_{ \scriptscriptstyle\mathrm{KN} }(\Omega^{}_{ \scriptscriptstyle\mathrm{KN} }+\sqrt{f^{}_{ \scriptscriptstyle\mathrm{KN} }})- \frac{2}{f_{ \scriptscriptstyle\mathrm{KN} }^2}f'^{}_{ \scriptscriptstyle\mathrm{KN} }\biggl(\Omega '^{}_{ \scriptscriptstyle\mathrm{KN} }+ \frac{f'^{}_{ \scriptscriptstyle\mathrm{KN} }}{2\,\sqrt{f^{}_{ \scriptscriptstyle\mathrm{KN} }}}\biggr)+{} \nonumber \\ &{}+\frac{1}{f^{}_{ \scriptscriptstyle\mathrm{KN} }}\biggl(\Omega''^{}_{ \scriptscriptstyle\mathrm{KN} }+ \frac{f''^{}_{ \scriptscriptstyle\mathrm{KN} }}{2\,\sqrt{f^{}_{ \scriptscriptstyle\mathrm{KN} }}} -\frac{(f'_{ \scriptscriptstyle\mathrm{KN} })^2}{4f_{ \scriptscriptstyle\mathrm{KN} }^{3/2}}\biggr)\biggr], \end{aligned}$$
(168)
$$\frac{1}{4}\,\frac{d}{dr}(A-D)=\frac{\lambda}{2} \biggl[\frac{1}{2}\frac{f'^{}_{ \scriptscriptstyle\mathrm{KN} }}{rf_{ \scriptscriptstyle\mathrm{KN} }^{3/3}}+ \frac{1}{r^2f_{ \scriptscriptstyle\mathrm{KN} }^{1/2}}\biggr], $$
(169)
$$-\frac{1}{4}\,\frac{A-D}{B}\,\frac{dB}{dr}=\frac{\lambda}{2rf_{ \scriptscriptstyle\mathrm{KN} }^{1/2}} \biggl(-\frac{f'^{}_{ \scriptscriptstyle\mathrm{KN} }}{f^{}_{ \scriptscriptstyle\mathrm{KN} }}+ \frac{1}{\Omega^{}_{ \scriptscriptstyle\mathrm{KN} }+\sqrt{f^{}_{ \scriptscriptstyle\mathrm{KN} }}}\biggl(\Omega'^{}_{ \scriptscriptstyle\mathrm{KN} }+ \frac{f'^{}_{ \scriptscriptstyle\mathrm{KN} }}{2\,\sqrt{f^{}_{ \scriptscriptstyle\mathrm{KN} }}}\biggr)\biggr), $$
(170)
$$\frac{1}{8}(A-D)^2=\frac{\lambda^2}{2f^{}_{ \scriptscriptstyle\mathrm{KN} }r^2},\qquad \frac{1}{2}BC=-\frac{1}{2f_{ \scriptscriptstyle\mathrm{KN} }^2}(\Omega_{ \scriptscriptstyle\mathrm{KN} }^2-f^{}_{ \scriptscriptstyle\mathrm{KN} }), $$
(171)
where
$$\begin{aligned} \, &f^{}_{ \scriptscriptstyle\mathrm{KN} }=\biggl(1-\frac{\Lambda}{3}r^2\biggr) \biggl(1+\frac{a^2}{r^2}\biggr)-\frac{r_0}{r}+\frac{r_{Q}^2}{r^2}, \\ &f'^{}_{ \scriptscriptstyle\mathrm{KN} }\equiv\frac{df^{}_{ \scriptscriptstyle\mathrm{KN} }}{dr}=-\frac{2\Lambda}{3}r- \frac{2\alpha^2}{3r^3}+\frac{r_0}{r^2}-\frac{2r_Q^2}{r^3}, \\ &f''^{}_{ \scriptscriptstyle\mathrm{KN} }\equiv\frac{d^2f^{}_{ \scriptscriptstyle\mathrm{KN} }}{dr^2}= -\frac{2\Lambda}{3}-\frac{2r_0}{r^3}+\frac{2\alpha^2+6r_Q^2}{r^4}, \\ &\Omega^{}_{ \scriptscriptstyle\mathrm{KN} }=\Xi\biggl[E\biggl(1+\frac{\alpha^2}{r^2}\biggr)- \frac{\alpha m_ \varphi }{r^2}-\frac{qQ}{\Xi r}\biggr], \\ &\Omega'^{}_{ \scriptscriptstyle\mathrm{KN} }\equiv\frac{d\Omega^{}_{ \scriptscriptstyle\mathrm{KN} }}{dr}= \Xi\biggl[-\frac{2E\alpha^2}{r^3}+ \frac{2\alpha m_ \varphi }{r^3}+\frac{qQ}{\Xi r^2}\biggr], \\ &\Omega''^{}_{ \scriptscriptstyle\mathrm{KN} }\equiv\frac{d^2\Omega^{}_{ \scriptscriptstyle\mathrm{KN} }}{dr^2}= \Xi\biggl[\frac{6E\alpha^2}{r^4}-\frac{6\alpha m_ \varphi }{r^4}- \frac{2qQ}{\Xi r^3}\biggr]. \end{aligned}$$

The arithmetic sum of the expressions \(E_{ \mathrm{Schr} }=(E^2-m^2)/2\) and relations (167)–(171) results in an expression for the effective potential \(U_{ \mathrm{eff} }^F\). For the remaining electromagnetic and gravitational fields considered here, the structure of the expressions for the effective potentials is unchanged. Only the expressions for \(f\), \(f'\), \(f''\), \(\Omega\), \(\Omega'\), and \(\Omega''\) change.

2. For the Kerr–(anti-)de Sitter field \((Q=0)\),

$$\begin{aligned} \, &f_{ \scriptscriptstyle\mathrm{K} }=\biggl(1-\frac{\Lambda}{3}r^2\biggr)\biggl(1+ \frac{a^2}{r^2}\biggr)-\frac{r_0}{r},\qquad f'_{ \scriptscriptstyle\mathrm{K} }=-\frac{2\Lambda}{3}r-\frac{2\alpha^2}{3r^{3}}+\frac{r_0}{r^2}, \\ &f''_{ \scriptscriptstyle\mathrm{K} }=-\frac{2\Lambda}{3}-\frac{2r_0}{r^{3}}+\frac{2a^2}{r^4}, \\ &\Omega_{ \scriptscriptstyle\mathrm{K} }=\Xi\biggl[E\biggl(1+\frac{\alpha^2}{r^2}\biggr)- \frac{\alpha m_ \varphi }{r^2}\biggr],\qquad\Omega'_{ \scriptscriptstyle\mathrm{K} }= \Xi\biggl[-\frac{2E\alpha^2}{r^3}+\frac{2\alpha m_ \varphi }{r^3}\biggr], \\ &\Omega''_{ \scriptscriptstyle\mathrm{K} }=\Xi\biggl[\frac{6E\alpha^2}{r^4}- \frac{6\alpha m_ \varphi }{r^4}\biggr]. \end{aligned}$$

3. For the Reissner–Nordström–(anti-)de Sitter field \((a=0)\),

$$\begin{aligned} \, &f_{ \scriptscriptstyle\mathrm{RN} }=1-\frac{\Lambda}{3}r^2-\frac{r_0}{r}+ \frac{r_Q^2}{r^2},\qquad f'_{ \scriptscriptstyle\mathrm{RN} }=-\frac{2\Lambda}{3}r+ \frac{r_0}{r^2}-\frac{2r_Q^2}{r^3}, \\ &f''_{ \scriptscriptstyle\mathrm{RN} }=-\frac{2\Lambda}{3}-\frac{2r_0}{r^3}+\frac{6r_Q^2}{r^4}, \\ &\Omega_{ \scriptscriptstyle\mathrm{RN} }=\Xi E-\frac{qQ}{r},\qquad \Omega'_{ \scriptscriptstyle\mathrm{RN} }=\frac{qQ}{r^2},\qquad \Omega''_{ \scriptscriptstyle\mathrm{RN} }=-\frac{2qQ}{r^3},\qquad\lambda=\kappa, \end{aligned}$$
where \(\kappa\) is the separation constant,
$$\begin{aligned} \, \kappa=\mp1,\mp2,\dots=\begin{cases}-(l-1),&j=l+\dfrac{1}{2}, \\ l,&j=l-\dfrac{1}{2},\end{cases} \end{aligned}$$
and \(j\) and \(l\) are the quantum numbers of the total and orbital momenta of a spin-1/2 particle.

4. For the Schwarzschild–(anti) de Sitter field (\(Q=0\), \(a=0\)),

$$\begin{aligned} \, &f_{ \scriptscriptstyle\mathrm{S} }=1-\frac{\Lambda}{3}r^2 -\frac{r_0}{r},\qquad f'_{ \scriptscriptstyle\mathrm{S} }=-\frac{2\Lambda}{3}r+\frac{r_0}{r^2},\qquad f''_{ \scriptscriptstyle\mathrm{S} }=-\frac{2\Lambda}{3}-\frac{2r_0}{r^{3}}, \\ &\Omega_{ \scriptscriptstyle\mathrm{S} }=\Xi E,\qquad \Omega'_{ \scriptscriptstyle\mathrm{S} }= \Omega''_{ \scriptscriptstyle\mathrm{S} }=0,\qquad\lambda=\kappa. \end{aligned}$$

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Gorbatenko, M.V., Neznamov, V.P. Quantum mechanics of stationary states of particles in a space–time of classical black holes. Theor Math Phys 205, 1492–1526 (2020). https://doi.org/10.1134/S0040577920110070

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  • DOI: https://doi.org/10.1134/S0040577920110070

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