# Curvature Invariants and Lower Dimensional Black Hole Horizons

## TL;DR

This paper explores how curvature invariants can identify black hole horizons in lower-dimensional spacetimes, revealing differences from four-dimensional cases and suggesting horizons are special hypersurfaces.

## Contribution

It analyzes the role of scalar polynomial and Cartan curvature invariants in detecting horizons in (2+1) and (1+1) dimensions, highlighting their distinct applicability.

## Key findings

- Horizon detection methods differ significantly from 4D cases.
- Cartan invariants relate to local extremum of tidal forces.
- Horizon is characterized as a special hypersurface.

## 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.