Historical research guide · 0844 · 2017
Quantum tunneling and quasinormal modes in the spacetime of the Alcubierre warp drive
Kimet Jusufi · İzzet Sakallı · Ali Övgün
In one minute
Miguel Alcubierre’s 1994 solution moves a flat bubble of space by expanding spacetime behind it and contracting it in front, and the passenger inside never leaves a locally flat patch. Kimet Jusufi, İzzet Sakallı and Ali Övgün ask what that passenger would actually experience. Rewritten in the flowing-river form the paper uses, the warp bubble behaves like a stream: once its speed passes the speed of light, two horizons appear on either side of the bubble, one acting as a black hole horizon and one as a white hole horizon. Horizons radiate. Treating the radiation as particles tunnelling through the horizon, the authors work out the temperature twice over — once for massive vector particles obeying the Proca equation, once for massive scalars — and get the same expression both times, set by the warp velocity and the steepness of the bubble’s edge. They then strike the bubble with a scalar field and solve for how it rings, finding modes that decay rather than grow.
Connection
This is chapter 4 treating a warp metric the way physicists treat a black hole: not as a line element on a page but as an object with a horizon, a temperature and a ring-down that you could in principle measure from inside. It belongs to chapter 5 too, because the form the authors work in — a Painlevé metric with a shift velocity, space flowing past the observer — is exactly the moving-fluid form that analogue gravity uses, which is why a warp bubble and a sonic horizon in a laboratory fluid share their mathematics. The energy-condition question the paper opens on is treated in full on the site at /library/stm-d1e51c2294.
From the source
Written in Painlevé form, the one-plus-one eternal Alcubierre warp drive has a shift velocity set by the bubble profile — the authors take a bell-shaped profile falling off as the reciprocal of the hyperbolic cosine of the distance from the bubble centre — and two horizons appear once the warp velocity exceeds the speed of light, placed symmetrically about the centre, one with positive surface gravity behaving as a black horizon and one with negative surface gravity behaving as a white horizon.Sect. II, EAWD spacetime, the Painlevé form and the horizon condition
An observer inside the warp drive experiences a thermal flux of Hawking quanta. Solving the Proca equation for massive vector particles in the WKB approximation and reading the tunnelling rate off the Boltzmann relation gives a Hawking temperature equal to the surface gravity over two pi, which for this spacetime is the warp velocity times the slope of the bubble profile at the horizon, divided by two pi.Sect. III, Quantum tunneling of vector particles, the temperature result
The scalar calculation agrees with the vector one. Solving the Klein–Gordon equation by the Hamilton–Jacobi method, with the canonical-invariance correction that closes the tunnelling path from just outside to just inside the horizon, returns exactly the same temperature — so the result does not depend on the spin of the tunnelling particle in this reduction.Sect. IV, Quantum tunneling of scalar particles, in full agreement with the vector result
Adding a generalised uncertainty principle lowers the temperature by a factor of the square root of one minus twice the particle mass squared times the speed of light squared times the GUP parameter, and the original expression is recovered exactly when that parameter is set to zero.Sect. IV, the GUP-corrected temperature
Perturbing the bubble with a massive scalar field reduces the radial Klein–Gordon equation to a hypergeometric equation, which the authors solve analytically for two families of quasinormal frequencies; both families have negative imaginary part — meaning they decay rather than grow — precisely in the range where they also satisfy the wave-identifier condition that distinguishes ingoing from outgoing waves at spatial infinity, and in the highly damped limit the frequencies become independent of the field mass.Sect. V, QNMs of particular EAWD spacetime, the two frequency sets and their stability conditions
The authors are explicit about what is still open: their analytic method captures only the stable modes and is not a proof that no unstable waves exist in this spacetime, which they say should be pursued numerically; the whole calculation is done in the one-plus-one reduction, and the spin dependence of the GUP temperature would only show up in the full three-plus-one warp spacetime; and the greybody factor, absorption cross-section and decay rate are named as the next things to compute.Sect. IV closing paragraph; Sect. V closing paragraph; Sect. VI, Conclusions
The way in
https://doi.org/10.1007/s10714-017-2330-8How to cite it
Kimet Jusufi, İzzet Sakallı, Ali Övgün (2017) Quantum tunneling and quasinormal modes in the spacetime of the Alcubierre warp drive doi:10.1007/s10714-017-2330-8