Abstract
In this study, we located and compared different types of horizons in the spherically symmetric Vaidya solution. The horizons we found were trapping horizons, which can be null, timelike, or spacelike, null surfaces with constant area change and also conformal Killing horizons. The conformal Killing horizons only exist for certain choices of the mass function. Under a conformal transformation, the conformal Killing horizons can be mapped into true Killing horizons. This allows conclusions drawn in the dynamical Vaidya spacetime to be related to known properties of static spacetimes. We found the conformal factor that performs this transformation and wrote the new metric in explicitly static coordinates. Using this construction we found that the tunneling argument for Hawking radiation does not umabiguously support Hawking radiation being associated with the trapping horizon. We also used this transformation to derive the form of the surface gravity for a class of physical observers in Vaidya spacetimes.
Highlights
That quantum effects are relevant at the horizons of black holes is shown by the Hawking effect.While the energy flux due to Hawking radiation is widely expected to be small, a longstanding debate has existed about the nature of quantum fields in the vicinity of black hole horizons and the true nature of the expectation value of the renormalized stress energy tensor
Conformal invariance of the relevant physics suggests that Hawking radiation should be associated with the conformal Killing horizon, not the trapping horizon even in the Vaidya spacetime, which is a full solution of the Einstein equations with reasonable matter content formulated in the canonical conformal frame
While this result has been obtained using the standard techniques valid in static spacetimes, its immediate utility is that it can be transformed via the conformal transformation to the surface gravity that would be measured by observers following the same trajectory in the Vaidya spacetime
Summary
That quantum effects are relevant at the horizons of black holes is shown by the Hawking effect. A key difference between the causal horizons and those based on marginally trapped surfaces is that the former are always by definition null surfaces generated by null vector fields, while the latter can be null and spacelike and timelike Another key difference between these two types of horizons is their behavior under a conformal transformation of the spacetime metric. A natural physical phenomenon to associate with the boundary of a black hole is Hawking radiation This is relevant in the context of entropy microstates and information retention but may have relevance to renormalized stress-energy and the quantum vacuum. The second section defines conformally static coordinates for the Vaidya solution and uses these to study again the Hamilton-Jacobi tunneling argument.
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