Double Foliation Research Articles

Spherically symmetric, static black holes with scalar hair, and naked singularities in nonminimally coupled k -essence

We apply a recently developed 2+1+1 decomposition of spacetime, based on a nonorthogonal double foliation for the study of spherically symmetric, static black hole solutions of Horndeski scalar-tensor theory. Our discussion proceeds in an effective field theory (EFT) of modified gravity approach, with the action depending on metric and embedding scalars adapted to the nonorthogonal 2+1+1 decomposition. We prove that the most generic class of Horndeski Lagrangians compatible with observations can be expressed in this EFT form. By studying the first order perturbation of the EFT action we derive three equations of motion, which reduce to those derived earlier in an orthogonal 2+1+1 decomposition, and a fourth equation for the metric parameter N related to the nonorthogonality of the foliation. For the Horndeski class of theories with vanishing $G_3$ and $G_5$, but generic functions $G_2(\phi,X)$ (k-essence) and $G_4(\phi)$ (nonminimal coupling to the metric) we prove the unicity theorem that no action beyond Einstein--Hilbert allows for the Schwarzschild solution. Next we integrate the EFT field equations for the case with only one independent metric function obtaining new solutions characterized by a parameter interpreted as either mass or tidal charge, the cosmological constant and a third parameter. These solutions represent naked singularities, black holes with scalar hair or have the double horizon structure of the Schwarzschild--de Sitter spacetime. Solutions with homogeneous Kantowski--Sachs type regions also emerge. Finally, one of the solutions obtained for the function $G_4$ linear in the curvature coordinate, in certain parameter range exhibits an intriguing logarithmic singularity lying outside the horizon. The newly derived hairy black hole solutions evade previously known unicity theorems by being asymptotically nonflat, even in the absence of the cosmological constant.

Gravitational dynamics in a 2+1+1 decomposed spacetime along nonorthogonal double foliations: Hamiltonian evolution and gauge fixing

Motivated by situations with temporal evolution and spatial symmetries both singled out, we develop a new 2+1+1 decomposition of spacetime, based on a nonorthogonal double foliation. Time evolution proceeds along the leaves of the spatial foliation. We identify the gravitational variables in the velocity phase-space as the 2-metric (induced on the intersection $\Sigma_{t\chi }$ of the hypersurfaces of the foliations), the 2+1 components of the spatial shift vector, together with the extrinsic curvature, normal fundamental form and normal fundamental scalar of $\Sigma _{t\chi }$, all constructed with the normal to the temporal foliation. This work generalizes a previous decomposition based on orthogonal foliations, a formalism lacking one metric variable, now reintroduced. The new metric variable is related to (i) the angle of a Lorentz-rotation between the nonorthogonal bases adapted to the foliations, and (ii) to the vorticity of these basis vectors. As a first application of the formalism, we work out the Hamiltonian dynamics of general relativity in terms of the variables identified as canonical, generalizing previous work. As a second application we present the unambiguous gauge-fixing suitable to discuss the even sector scalar-type perturbations of spherically symmetric and static spacetimes in generic scalar-tensor gravitational theories, which has been obstructed in the formalism of orthogonal double foliation.

Hamiltonian Dynamics of Doubly-Foliable Space-Times

The 2+1+1 decomposition of space-time is useful in monitoring the temporal evolution of gravitational perturbations/waves in space-times with a spatial direction singled-out by symmetries. Such an approach based on a perpendicular double foliation has been employed in the framework of dark matter and dark energy motivated scalar-tensor gravitational theories for the discussion of the odd sector perturbations of spherically symmetric gravity. For the even sector however the perpendicularity has to be suppressed in order to allow for suitable gauge freedom, recovering the 10th metric variable. The 2+1+1 decomposition of the Einstein-Hilbert action leads to the identification of the canonical pairs, the Hamiltonian and momentum constraints. Hamiltonian dynamics is then derived via Poisson brackets.

Covariant double-null dynamics: 2 + 2-splitting of the Einstein equations

We develop a (2 + 2)-imbedding formalism adapted to a double foliation of spacetime by a net of two intersecting families of lightlike hypersurfaces. The intersections define a set of 2-spaces with hypersurface-orthogonal lightlike normals. The formalism is two-dimensionally covariant, and leads to simple, geometrically transparent and tractable expressions for the Einstein field equations and the Einstein - Hilbert action, and it should find a variety of applications. It is applied here to elucidate the structure of the characteristic initial-value problem of general relativity.

Chaos with Few Degrees of Freedom

As a mathematician, astronomer, and physicist, Poincaré was the first to study the phenomenon of chaos in classical mechanics, in particular the three-body-problem of the Moon orbiting around the Earth under the perturbation of the Sun. His work helps us to understand atomic and molecular physics, because our intuition in these areas is always based on a classical picture for the motion of the nuclei and electrons. The Moon's motion was found to be multi-periodic by the early observers; Newton explained the main frequencies and calculated the lowest terms in the corresponding Fourier expansion. But inspite of ingenious efforts, like the work of Hill involving as a starting point a periodic orbit of the full problem, Poincaré showed that perturbation theory is unable to guarantee the convergence of the Fourier expansion. In this traditional representation, the lunar trajectory would be restricted to a four-dimensional invariant torus although the available phase-space has eight dimensions. The invariant tori are destroyed because of increasing phase-lock due to perturbations, whenever the system is close to a resonance. The KAM-theorem gives a mathematical criterion for this loss, which is best seen in a surface of section for such simple systems as the Anisotropic and the Diamagnetic Kepler Problem (AKP and DKP). Instead of invariant tori, the flow in phase space involves a double foliation where each trajectory is the intersection of one leaf from the unstable foliation with a leaf from the stable foliation. Neighboring trajectories drift exponentially away from each other in the future along the unstable leaf, and approach each another in the stable leaf coming from the past. This situation, called “hard chaos”, also allows for a symbolic description of each trajectory in terms of a simple code like binary sequences in the AKP. Einstein showed that the usual connection with quantum mechanics is valid only for the case of “regular behavior”, i.e. the presence of invariant tori. For classically chaotic systems, VanVleck's expression for the propagator seems to be the best starting point, provided it is summed over all the classical trajectories that go from the source to the defector. Its trace becomes a sum over periodic orbits, which lends additional weight to Poincaré's emphasis on their importance in classical mechanics. In trying to understand the trace of an operator as a function of time, or the scattering phase-shift as a function of momentum, one runs into almost-periodic functions. They are able to mimic locally any arbitrary smooth function provided the spectrum of frequencies, although discrete, is rich enough. This unexpected, but smooth behavior can be seen as a symptom of quantum chaos, in contrast to the fractal nature of classical chaos.