1. Nicolas Franco, Michał Eckstein
Causality in noncommutative two-sheeted space-times
, vol. xxx, pp. xxx-xxx (2015).
[abstract] [preprint]

Abstract:
We investigate the causal structure of two-sheeted space-times using the tools of Lorentzian spectral triples. We show that the noncommutative geometry of these spaces allows for causal relations between the two sheets. The computation is given in details when the sheet is a 2- or 4-dimensional globally hyperbolic spin manifold. The conclusions are then generalised to a point-dependent distance between the two sheets resulting from the fluctuations of the Dirac operator.

2. Nicolas Franco, Michał Eckstein
Exploring the Causal Structures of Almost Commutative Geometries
SIGMA, vol. 10, 010 (2014).
[abstract] [preprint] [journal] [download]

Abstract:
We investigate the causal relations in the space of states of almost commutative Lorentzian geometries. We fully describe the causal structure of a simple model based on the algebra $\mathcal{S}(\mathbb{R}^{1,1}) \otimes M_2(\mathbb{C})$, which has a non-trivial space of internal degrees of freedom. It turns out that the causality condition imposes restrictions on the motion in the internal space. Moreover, we show that the requirement of causality favours a unitary evolution in the internal space.

3. Nicolas Franco, Michał Eckstein
An algebraic formulation of causality for noncommutative geometry
Class. Quantum Grav., vol. 30, p. 135007 (2013).
[abstract] [preprint] [journal]

Abstract:
We propose an algebraic formulation of the notion of causality for spectral triples corresponding to globally hyperbolic manifolds with a well defined noncommutative generalization. The causality is given by a specific cone of Hermitian elements respecting an algebraic condition based on the Dirac operator and a fundamental symmetry. We prove that in the commutative case the usual notion of causality is recovered. We show that, when the dimension of the manifold is even, the result can be extended in order to have an algebraic constraint suitable for a Lorentzian distance formula.

4. Piotr T. Chruściel, Michał Eckstein, Luc Nguyen and Sebastian J. Szybka
Existence of singularities in two-Kerr black holes
Class. Quantum Grav., vol. 28, p. 245017 (2011).
[abstract] [preprint] [journal]

Abstract:
We show that the angular momentum—area inequality 8π|J| ≤ A for weakly stable minimal surfaces would apply to I+-regular many-Kerr solutions, if any existed. Hence, we remove the undesirable hypothesis in the Hennig–Neugebauer proof of non-existence of well-behaved two-component solutions.
*supported by the grant from The John Templeton Foundation

5. Piotr T. Chruściel, Michał Eckstein, Sebastian J. Szybka
On smoothness of Black Saturns
Journal of High Energy Physics, vol. 2010, pp. 1-39 (2010).
[abstract] [preprint] [journal]

Abstract:
We prove smoothness of the domain of outer communications (d.o.c.) of the Black Saturn solutions of Elvang and Figueras. We show that the metric on the d.o.c. extends smoothly across two disjoint event horizons with topology R x S^3 and R x S^1 x S^2. We establish stable causality of the d.o.c. when the Komar angular momentum of the spherical component of the horizon vanishes, and present numerical evidence for stable causality in general.