Kasner eons in Lovelock black holes

In the vicinity of spacelike singularities, general relativity predicts that the metric behaves, at each point, as a Kasner space which undergoes a series of “Kasner epochs” and “eras” characterized by certain transition rules. The period during which this process takes place defines a “Kasner eon,” which comes to an end when higher-curvature or quantum effects become relevant. When higher-curvature densities are included in the action, spacetime can undergo transitions into additional Kasner eons. During each eon, the metric behaves locally as a Kasner solution to the higher-curvature density controlling the dynamics. In this paper we identify the presence of Kasner eons in the interior of static and spherically symmetric Lovelock gravity black holes. We determine the conditions under which eons occur and study the Kasner metrics which characterize them, as well as the transitions between them. We show that the null energy condition implies a monotonicity property for the effective Kasner exponent at the end of the Einsteinian eon. We also characterize the Kasner solutions of more general higher-curvature theories of gravity. In particular, we observe that the Einstein gravity condition that the sum of the Kasner exponents adds up to 1, <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msubsup><a:mo>∑</a:mo><a:mrow><a:mi>i</a:mi><a:mo>=</a:mo><a:mn>1</a:mn></a:mrow><a:mrow><a:mi>D</a:mi><a:mo>−</a:mo><a:mn>1</a:mn></a:mrow></a:msubsup><a:msub><a:mi>p</a:mi><a:mi>i</a:mi></a:msub><a:mo>=</a:mo><a:mn>1</a:mn></a:math>, admits a universal generalization in the form of a family of Kasner metrics satisfying <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:msubsup><c:mo>∑</c:mo><c:mrow><c:mi>i</c:mi><c:mo>=</c:mo><c:mn>1</c:mn></c:mrow><c:mrow><c:mi>D</c:mi><c:mo>−</c:mo><c:mn>1</c:mn></c:mrow></c:msubsup><c:msub><c:mi>p</c:mi><c:mi>i</c:mi></c:msub><c:mo>=</c:mo><c:mn>2</c:mn><c:mi>n</c:mi><c:mo>−</c:mo><c:mn>1</c:mn></c:math> which exists for any order-<e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>n</e:mi></e:math> higher-curvature density and in general dimensions. Published by the American Physical Society 2024