Abstract
We analyze a quantum observer who falls geodesically toward the Cauchy horizon of a (1 + 1)-dimensional eternal black hole spacetime with the global structure of the non-extremal Reissner–Nordström solution. The observer interacts with a massless scalar field, using an Unruh–DeWitt detector coupled linearly to the proper time derivative of the field, and by measuring the local energy density of the field. Taking the field to be initially prepared in the Hartle–Hawking–Israel (HHI) state or the Unruh state, we find that both the detector's transition rate and the local energy density generically diverge on approaching the Cauchy horizon, respectively, proportionally to the inverse and the inverse square of the proper time to the horizon, and in the Unruh state the divergences on approaching one of the branches of the Cauchy horizon are independent of the surface gravities. When the outer and inner horizons have equal surface gravities, the divergences disappear altogether in the HHI state and for one of the Cauchy horizon branches in the Unruh state. We conjecture, on grounds of comparison with the Rindler state in 1 + 1 and 3 + 1 Minkowski spacetimes, that similar properties hold in 3 + 1 dimensions for a detector coupled linearly to the quantum field, but with a logarithmic rather than inverse power-law divergence.
Highlights
It is a great pleasure to dedicate this paper to Roger Penrose, who realized the instability of the Cauchy horizons that occur inside charged and rotating black hole solutions.1 The nature of the instability is a topic of ongoing research, classically and in the presence of quantized fields
Taking the field to be initially prepared in the Hartle–Hawking–Israel (HHI) state or the Unruh state, we find that both the detector’s transition rate and the local energy density generically diverge on approaching the Cauchy horizon, respectively, proportionally to the inverse and the inverse square of the proper time to the horizon, and in the Unruh state the divergences on approaching one of the branches of the Cauchy horizon are independent of the surface gravities
This paper addresses transitions in a time-and-space localized quantum system, coupled to an ambient quantum field, when the system falls geodesically toward a Cauchy horizon
Summary
It is a great pleasure to dedicate this paper to Roger Penrose, who realized the instability of the Cauchy horizons that occur inside charged and rotating black hole solutions. The nature of the instability is a topic of ongoing research, classically and in the presence of quantized fields. Work by Simpson and Penrose led to the strong cosmic censorship conjecture, which states that for generic initial data the spacetime is inextendible beyond the maximal Cauchy development We perform a similar analysis for a geodesic detector approaching the Rindler horizon in (1 þ 1)-dimensional Minkowski spacetime, with the field prepared in the Rindler vacuum, and we contrast the results with a similar analysis in 3 þ 1 dimensions, for a detector coupled linearly to the value (as opposed to the derivative) of the scalar field Based on this comparison, we conjecture that in 3 þ 1 spacetime dimensions, a detector coupled linearly to the value of the scalar field, and approaching a Cauchy horizon, generically has a transition rate that diverges in proper time but only logarithmically.
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