arXiv subjects

Amir Babak Aazami

Publications and source records attributed to Amir Babak Aazami.

25 records · Page 2Linked to original sources

arXiv2019

On the principal Ricci curvatures of a Riemannian 3-manifold

We study global obstructions to the eigenvalues of the Ricci tensor on a Riemannian 3-manifold. As a topological obstruction, we first show that if the 3-manifold is closed, then certain choices of the eigenvalues are prohibited: in particular, there is no Riemannian metric whose corresponding Ricci eigenvalues take the form $(-μ,f,f)$, where $μ$ is a positive constant and $f$ is a smooth positive function. We then concentrate on the case when one of the eigenvalues is zero. Here we show that if the manifold is complete and its Ricci eigenvalues take the form $(0,λ,λ)$, where $λ$ is a positive constant, then its universal cover must split isometrically. If the manifold is closed, scalar-flat, and its zero eigenspace contains a unit length vector field that is geodesic and divergence-free, then the manifold must be flat. Our techniques also apply to the study of Ricci solitons in dimension three.

math.DG↗

arXiv2016

Symplectic 4-manifolds via Lorentzian geometry

We observe that, in dimension four, symplectic forms may be obtained via Lorentzian geometry; in particular, null vector fields can give rise to exact symplectic forms. That a null vector field is nowhere vanishing yet orthogonal to itself is essential to this construction. Specifically, we show that on a Lorentzian 4-manifold $(M,g)$, if ${\boldsymbol k}$ is a complete null vector field with geodesic flow along which $\text{Ric}({\boldsymbol k},{\boldsymbol k}) > 0$, and if $f$ is any smooth function on $M$ with ${\boldsymbol k}(f)$ nowhere vanishing, then $dg(e^f{\boldsymbol k},\cdot)$ is a symplectic form and ${\boldsymbol k}/{\boldsymbol k}(f)$ is a Liouville vector field; any null surface to which ${\boldsymbol k}$ is tangent is then a Lagrangian submanifold. Even if the Ricci curvature condition is not satisfied, one can still construct such symplectic forms with additional information from ${\boldsymbol k}$; we give an example of this, with ${\boldsymbol k}$ a complete Liouville vector field, on the maximally extended "rapidly rotating" Kerr spacetime.

math.DG↗

arXiv2016

Penrose's singularity theorem in a Finsler spacetime

We translate Penrose's singularity theorem to a Finsler spacetime. To that end, causal concepts in Lorentzian geometry are extended, including definitions and properties of focal points and trapped surfaces, with careful attention paid to the differences that arise in the Finslerian setting.

math.DG↗

arXiv2016

The geometry of gravitational lensing magnification

We present a definition of unsigned magnification in gravitational lensing valid on arbitrary convex normal neighborhoods of time oriented Lorentzian manifolds. This definition is a function defined at any two points along a null geodesic that lie in a convex normal neighborhood, and foregoes the usual notions of lens and source planes in gravitational lensing. Rather, it makes essential use of the van Vleck determinant, which we present via the exponential map, and Etherington's definition of luminosity distance for arbitrary spacetimes. We then specialize our definition to spacetimes, like Schwarzschild's, in which the lens is compact and isolated, and show that our magnification function is monotonically increasing along any geodesic contained within a convex normal neighborhood.

gr-qc↗

arXiv2015

The Newman-Penrose Formalism for Riemannian 3-manifolds

We adapt the Newman-Penrose formalism in general relativity to the setting of three-dimensional Riemannian geometry, and prove the following results. Given a Riemannian 3-manifold without boundary and a smooth unit vector field ${\boldsymbol k}$ with geodesic flow, if an integral curve of ${\boldsymbol k}$ is hypersurface-orthogonal at a point, then it is so at every point along that curve. Furthermore, if ${\boldsymbol k}$ is complete, hypersurface-orthogonal, and satisfies $\text{Ric}({\boldsymbol k},{\boldsymbol k}) \geq 0$, then its divergence must be nonnegative. As an application, we show that if the Riemannian 3-manifold is closed and a unit length ${\boldsymbol k}$ with geodesic flow satisfies $\text{Ric}({\boldsymbol k},{\boldsymbol k}) > 0$, then ${\boldsymbol k}$ cannot be hypersurface-orthogonal, thus recovering a recent result. Turning next to scalar curvature, we derive an evolution equation for the scalar curvature in terms of unit vector fields ${\boldsymbol k}$ that satisfy the condition $R({\boldsymbol k},\cdot,\cdot,\cdot) = 0$. When the scalar curvature is a nonzero constant, we show that a hypersurface-orthogonal unit vector field ${\boldsymbol k}$ satisfies $R({\boldsymbol k},\cdot,\cdot,\cdot) = 0$ if and only if it is a Killing vector field.

math.DG↗

arXiv2014

Blowup solutions of Jang's equation near a spacetime singularity

We study Jang's equation on a one-parameter family of asymptotically flat, spherically symmetric Cauchy hypersurfaces in the maximally extended Schwarzschild spacetime. The hypersurfaces contain apparent horizons and are parametrized by their proximity to the singularity at $r = 0$. We show that on those hypersurfaces sufficiently close to the singularity, every radial solution to Jang's equation blows up. The proof depends only on the geometry in an arbitrarily small neighborhood of the singularity, suggesting that Jang's equation is in fact detecting the singularity. We comment on possible applications to the weak cosmic censorship conjecture.

gr-qc↗

arXiv2006

Substructure and the Cusp and Fold Relations

Gravitational lensing of a background source by a foreground galaxy lens occasionally produces four images of the source. The cusp and the fold relations impose conditions on the ratios of magnifications of these four-image lenses. In this theoretical investigation, we explore the sensitivity of these relations to the presence of substructure in the lens. Starting with a smooth lens potential, we add varying amounts of substructure, while keeping the source position fixed, and find that the fold relation is a more robust indicator of substructure than the cusp relation for the images. This robustness is independent of the detailed spatial distribution of the substructure, as well as of the ellipticity of the lensing potential and the presence of external shear.

astro-ph↗