A spacetime satisfies the non-timelike boundary version of the Penrose property if the timelike future of any point on I− contains the whole of I+ . This property was first discussed for asymptotically flat spacetimes by Penrose, along with an equivalent definition (the finite version). In this paper we consider the Penrose property in greater generality. In particular we consider spacetimes with a non-zero cosmological constant and we note that the two versions of the property are no longer equivalent. In asymptotically AdS spacetimes it is necessary to re-state the property in a way which is more suited to spacetimes with a timelike boundary. We arrive at a property previously considered by Gao and Wald. Curiously, this property was shown to fail in spacetimes which focus null geodesics. This is in contrast to our findings in asymptotically flat and asymptotically de Sitter spacetimes. We then move on to consider some further example spacetimes (with zero cosmological constant) which highlight features of the Penrose property not previously considered. We discuss spacetimes which are the product of a Lorentzian and a compact Riemannian manifold. Perhaps surprisingly, we find that both versions of the Penrose property are satisfied in this product spacetime if and only if they are satisfied in the Lorentzian spacetime only. We also discuss the Ellis–Bronnikov wormhole (an example of a spacetime with more than one asymptotically flat end) and the Hayward metric (an example of a non-singular black hole spacetime).

中文翻译：

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具有宇宙学常数的彭罗斯性质

一个时空满足彭罗斯性质的非类时边界版本，如果 我− 包含全部 我+ . Penrose 首先针对渐近平坦时空讨论了此属性，以及等效定义（有限版本）。在本文中，我们更普遍地考虑彭罗斯性质。特别是我们考虑具有非零宇宙学常数的时空，我们注意到该属性的两个版本不再等价。在渐近 AdS 时空中，有必要以更适合具有类时边界的时空的方式重新陈述该属性。我们到达了 Gao 和 Wald 之前考虑过的一处房产。奇怪的是，这个属性在聚焦零测地线的时空中被证明是失败的。这与我们在渐近平坦和渐近德西特时空中的发现形成对比。然后，我们继续考虑一些进一步的时空示例（宇宙学常数为零），这些时空突出了之前未考虑过的彭罗斯属性的特征。我们讨论时空，它是洛伦兹流形和紧黎曼流形的乘积。也许令人惊讶的是，我们发现当且仅当它们仅在洛伦兹时空中得到满足时，彭罗斯性质的两个版本在这个乘积时空中都得到满足。我们还讨论了 Ellis–Bronnikov 虫洞（具有多个渐近平端的时空示例）和 Hayward 度量（非奇异黑洞时空的示例）。我们发现当且仅当它们仅在洛伦兹时空中满足时，彭罗斯性质的两个版本在这个乘积时空中都得到满足。我们还讨论了 Ellis–Bronnikov 虫洞（具有多个渐近平端的时空示例）和 Hayward 度量（非奇异黑洞时空的示例）。我们发现当且仅当它们仅在洛伦兹时空中满足时，彭罗斯性质的两个版本在这个乘积时空中都得到满足。我们还讨论了 Ellis–Bronnikov 虫洞（具有多个渐近平端的时空示例）和 Hayward 度量（非奇异黑洞时空的示例）。