Abstract
An Ermakov–Pinney-like equation associated with the scalar wave equation in curved space-time is here studied. The example of Schwarzschild space-time considered in the present work shows that this equation can be viewed more as a “model equation,” with interesting applications in black hole physics. Other applications studied involve cosmological space-times (de Sitter) and pulse of plane gravitational waves: in all these cases the evolution of the Ermakov–Pinney field seems to be consistent with a rapid blow-up, unlike the Schwarzschild case where spatially damped oscillations are allowed. Eventually, the phase function is also evaluated in many of the above space-time models.
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Notes
Note that a harmonic time dependence leads to a term linear in y and hence corresponds to the \(p \not =0\) case.
Generalizations with \(\alpha \) a complex scalar field are also possible.
We follow the convention according to which Greek indices from the beginning of the alphabet are Lie-algebra indices. When necessary, this index specification is explicitly repeated in the text to avoid confusion.
In general, all our formulae for currents are particular cases of the general expressions
$$\begin{aligned} J=J^{\mu }{\partial \over \partial x^{\mu }}, \; J_{\nu }=J^{\mu }g_{\mu \nu }, \; J^{\flat }=J_{\nu }dx^{\nu }. \end{aligned}$$Note that, as discussed above in the Schwarzschild case, the second term here is obtained from the first by replacing \(f_c\rightarrow -f_c\).
References
Cohen, J.M., Kegeles, L.S.: Electromagnetic fields in curved spaces: a constructive procedure. Phys. Rev. D 10, 1070 (1974)
Treves, F.: Introduction to Pseudodifferential and Fourier Integral Operators. Fourier Integral Operators, vol. 2. Plenum Press, New York (1980)
Esposito, G., Battista, E., Di Grezia, E.: Bicharacteristics and Fourier integral operators in Kasner spacetime. Int. J. Geom. Methods Mod. Phys. 12, 1550060 (2015)
Esposito, G., Minucci, M.: A new perspective on the Ermakov–Pinney and scalar wave equations. Lett. High Energy Phys. LHEP 3, 5 (2019). arXiv:1905.09382 [math.GM]
Ermakov, V.P.: Second-order differential equations: conditions of complete integrability. Univ. Izv. Kiev Ser. III 9, 1 (1880)
Pinney, E.: The non-linear differential equation \(y^{\prime \prime }+p(x)y+cy^{-3}=0\). Proc. Am. Math. Soc. 1, 681 (1950)
Detweiler, S.L., Messaritaki, E., Whiting, B.F.: Selfforce of a scalar field for circular orbits about a Schwarzschild black hole. Phys. Rev. D 67, 104016 (2003). https://doi.org/10.1103/PhysRevD.67.104016. [arXiv:gr-qc/0205079 [gr-qc]]
Ronveaux, A.: Heun’s Differential Equations. Oxford University Press, Oxford (1995)
Stephani, H., Kramer, D., MacCallum, M.A.H., Hoenselaers, C., Herlt, E.: Exact Solutions of Einstein’s Field Equations. Cambridge University Press, Cambridge (2003). https://doi.org/10.1017/CBO9780511535185
Schmidt, B.G., Stewart, J.M.: The scalar wave equation in a Schwarzschild spacetime. Proc. R. Soc. Lond. A 367, 503 (1979)
Casadio, R., Luzzi, M.: The method of comparison equations for Schwarzschild black holes. Phys. Rev. D 74, 085005 (2006)
Tsamparlis, M., Paliathanasis, A.: Generalizing the autonomous Kepler–Ermakov system in a Riemannian space. J. Phys. A 45, 275002 (2012)
Acknowledgements
G. Esposito is grateful to the Dipartimento di Fisica “Ettore Pancini” of Federico II University for hospitality and support.
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Bini, D., Esposito, G. New solutions of the Ermakov–Pinney equation in curved space-time. Gen Relativ Gravit 52, 60 (2020). https://doi.org/10.1007/s10714-020-02713-y
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DOI: https://doi.org/10.1007/s10714-020-02713-y