The horizon-entropy increase law for causal and quasi-local horizons and conformal field redefinitions

Abstract

Abstract. We explicitly prove the horizon-entropy increase law for both causal and quasi-locally defined horizons in scalar-tensor and f (R) gravity theories. Contrary to causal event horizons, future outer trapping horizons are not conformally invariant and we provide a modification of trapping horizons to complete the proof, using the idea of generalised entropy. This modification means they are no longer foliated by marginally outer trapped surfaces but fixes the location of the horizon under a conformal transformation. We also discuss the behaviour of horizons in "veiled" general relativity and show, using this new definition, how to locate cosmological horizons in flat Minkowski space with varying units, which is physically identified with a spatially flat FLRW spacetime.

Conformally rescaled spacetimes and Hawking radiation

“…In these cases the proposal in [15] would not locate the horizon at △ = 0, since the Wald entropy would be just one quarter of the area and we would be led to a static, timelike, two-way traversable horizon at θ l = 0, not the null surface △ = 0. This observation in itself is however not enough to show that null horizons should be preferred at all times, since there are many situations where this is questionable too.…”

“…Firstly, the rather simple and rather elegant approach proposed in [13], for dealing with non Einstein gravity theories will not be strictly applicable for vanishing expansion horizons such as the trapping horizon. This drawback though may be partly alleviated by a modified quasi-local horizon definition based on the Wald entropy [15]. This definition is conformally invariant and will allow one to map to the Einstein conformal frame whilst still treating the same physical surface.…”

“…One could imagine a slightly more complicated conformal factor W (v, r) that drops off to one outside some region of interest, far from r = 2M, such that the resulting spacetime is asymptotically Minkowski. Other examples, including cosmological examples, are discussed in [15]. In [14] it was shown that a conformally invariant quasi-local horizon can be defined in terms of the vanishing change of the Wald entropy [28] in the outgoing null direction.…”

“…In dynamical spacetimes quasi-local horizons typically differ from null event horizons and other causal horizons. In recent work [14], [15] the behaviour of quasi-local horizons under conformal transformations of the metric has been investigated. It was shown that the location of the trapping horizon is not conformally invariant.…”

“…It was shown in [15] that the purely classical horizon-entropy increase law for quasilocal horizons could be preserved under a conformal transformation if the standard zero area expansion definition of the horizon is changed. Instead of a trapping horizon surface of zero instantaneous area expansion, one should consider a surface of zero entropy expansion.…”