Abstract
It is known that the event horizon of a black hole can often be identified from the zeroes of some curvature invariants. The situation in lower dimensions has not been thoroughly clarified. In this work we investigate both (2+1)- and (1+1)-dimensional black hole horizons of static, stationary and dynamical black holes, identified with the zeroes of scalar polynomial and Cartan curvature invariants, with the purpose of discriminating the different roles played by the Weyl and Riemann curvature tensors. The situations and applicability of the methods are found to be quite different from that in 4-dimensional spacetime. The suitable Cartan invariants employed for detecting the horizon can be interpreted as a local extremum of the tidal force suggesting that the horizon of a black hole is a genuine special hypersurface within the full manifold, contrary to the usual claim that there is nothing special at the horizon, which is said to be a consequence of the equivalence principle.
Highlights
E, they must be regarded as physical quantities and as mathematical oddities entering the theory equations
The prediction of the energy spectra of the waves emitted by coalescing black holes is almost entirely based on numerical techniques [8,9,10] and analytical methods are still scarce according to the latest review on the experimental and theoretical methods for studying gravitational waves [11]
Note that the notion of the horizon is crucial in these laws: if, eventually, we have the technology to test these laws in the actual Universe, we would need to first correctly identify the horizon of a black hole
Summary
Cornerstone theorems by Hawking and Penrose proved that physical singularities (not to be confused with coordinate singularities which can be removed by a change of the coordinate system) can occur in the theory of classical general relativity [14,15,16,17]. Page 3 of 13 925 horizon, in the third section we will motivate more explicitly our interest in a lower dimensional theory, in section four we will comment on the peculiar properties of (2 + 1)dimensional gravity focusing on the consequences of applying our method, while in the fifth section we will present some explicit applications to the linearly and non-linearly charged Banados–Teitelboim–Zanelli (BTZ) black hole which generally depends on five physical parameters These examples of applications clarify the applicability of the method even for a dynamical configuration currently employed in the description of the formation of a black hole. We will discuss a connection between the Cartan invariants we constructed for locating the horizon and the behavior of tidal forces on its proximity in section seven, and we conclude in section eight summarizing the differences between our analysis and the case of (3 + 1)- and (4 + 1)-dimensional spacetimes
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