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Spontaneous Emission of Light Quanta from the Vacuum

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Negative Frequency at the Horizon

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Abstract

In the previous chapter of this dissertation we used the classical theory of Physics that rules the dynamics of the Universe on large scales—General Relativity—to study the behaviour of spacetime around spherical bodies. We introduced the idea of black holes, regions of spacetime bounded by their event horizon from which nothing can escape. In this section, we will try to tie General Relativity with Thermodynamics—broadly speaking, the theory that rules the organization of the Universe. For this purpose, we will follow the arguments which scientists of the early 1970s had to contend with, and see how they found that these theories can be united at the event horizon of black holes. This will eventually lead us to call upon Quantum Physics to explain how black holes can be in a state of thermal equilibrium—thus introducing the concept of spontaneous emission of light quanta from the vacuum.

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Notes

  1. 1.

    The nuclear mass loss associated with fusion results in energy release that heats up the star.

  2. 2.

    In that sense, the black hole represents the state of maximal entropy, that is the equilibrium end state of gravitational collapse.

  3. 3.

    The future of a set is the collection of all spacetime points that can be reached by future-going timelike or null curves from that set.

  4. 4.

    The past of future null infinity of \(\mathcal {S}\), \(j^-(\mathcal {F}^+)\), physically represents the set of all events from which an observer could escape to the asymptotic region.

  5. 5.

    This is a closed, spacelike, 2-surface whose ingoing and outgoing null normal geodesics are both converging. For example, a sphere at constant r and v in Eddington-Finkelstein coordinates is a trapped surface if it lies inside the horizon.

  6. 6.

    A partial Cauchy surface is a hypersurface which is intersected by any causal curve at most once.

  7. 7.

    A spacetime in which certain observers can never escape to the asymptotic region, i.e., for which the past of future null infinity is not the entire spacetime, is a spacetime that has an event horizon. It is said to possess a black hole.

  8. 8.

    \(\delta A\) is the change in surface area of the event horizon of the black hole.

  9. 9.

    Note that it was Hawking who discovered that black hole horizons must grow if there is only positive energy that falls in, and Bekenstein who later established the link between this observation and entropy.

  10. 10.

    The acceleration of the particle arbitrarily close to the horizon goes to infinity, but from afar this is multiplied by the redshit factor, which also tends to infinity in this case, yielding a finite constant.

  11. 11.

    The first law of black hole mechanics states that \(S\leftrightarrow A\).

  12. 12.

    The calculation (3.38)–(3.43) was historically performed by means of \(\Gamma \)-functions [3, 26].

  13. 13.

    Particles will be present because \(\bar{\left| 0\right\rangle }\) will not be annihilated by \(a_\omega \):

    figure a

    .

  14. 14.

    Black hole explosion refers to the fact that the emission rate goes as \(1/M^2\) so that for small holes this becomes very large, and the lifetime (which goes as \(M^3\)) becomes very small.

  15. 15.

    Although a step-like profile models an infinite slope at the horizon, which would correspond to an infinite surface gravity and temperature, the calculations show a totally different result. As we will see, the spectral densities we calculate are finite. I think this is because, ultimately, the amplitude of waves is limited by dispersion.

  16. 16.

    In the experiment only smooth profiles can be realised. Calculations with an infinitely steep profile only have a suggestive role in understanding the experiment.

  17. 17.

    The model does not account for the dispersion changes due to the finiteness of the intersites distance.

  18. 18.

    Note that in this section, the partial derivative with respect to a variable is denoted by \(\partial _t\equiv \frac{\partial }{\partial t}\). We do not use the relativistic-covariant formulation.

  19. 19.

    Note that the lowest branch is approximately a massless polariton: it can be fitted with a dispersion relation of the form \(|\omega |=c|k|\) for low wavenumbers (close to \(k=0\)).

  20. 20.

    Note that (2.78) is an approximate version of this dispersion relation where we have assumed that \(\omega <|\Omega |\) for a medium with only one resonant frequency.

  21. 21.

    Note that, by replacing the conjugate momenta of the electromagnetic and polarisation fields by their expression in terms of derivatives of the fields (Eqs. 3.53 and 3.54), one obtains the usual form of the pseudo norm—as in Eq. (1.12), with \(\phi \) a field. Because of dispersion, this expression would of course be slightly more complicated, although as readily computable.

  22. 22.

    An alternative proof follows from the observation that, given \(\partial _\tau \rho =0\) and \(\left\langle V_1,V_2\right\rangle =\alpha \left\langle V_1,V_1\right\rangle +\sum _{i=1}^3\bar{\alpha }_i\left\langle V_i,V_i^\dagger \right\rangle \), being the second term of the latter equation zero, the assessment of time conservation consists in calculating \(\partial _\tau \int \alpha \left\langle V_1,V_1\right\rangle \mathrm {d}\zeta +\partial _\tau \left\langle V_1,V_1\right\rangle =\int \partial _\tau \alpha \left\langle V_1,V_1\right\rangle \mathrm {d}\zeta \). \(\partial _\tau \alpha \left\langle V_1,V_1\right\rangle =0\), and thus \(\partial _\tau \alpha \left\langle V_1,V_2\right\rangle \)=0.

  23. 23.

    Remark that the change in the refractive index described by (3.71) is frequency-dependent.

  24. 24.

    The magnitude of the refractive index change giving rise to the various mode configurations depends on the medium properties. For the sake of the argument presented in this section it suffices to identify three categories of refractive index change: small, medium, and large—exact numbers will be provided by the numerical analysis carried in Sect. (4.3).

  25. 25.

    The commutation of the out modes on the in modes gives zero and all the mixed terms go to zero.

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  1. Faculty of Physics, University of Vienna, Vienna, Austria

    Maxime J. Jacquet

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Jacquet, M.J. (2018). Spontaneous Emission of Light Quanta from the Vacuum. In: Negative Frequency at the Horizon. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-91071-0_3

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