arXiv research

Unified view on big bang singularities from initial data.

Localized big bang singularities found without background solutions.

We show that the Big Bang singularity of the Friedmann-Lemaitre-Robertson-Walker model does not raise major problems to General Relativity. We prove a theorem showing that the Einstein equation can be written in a non-singular form, which allows the extension of the spacetime before the Big Bang. The physical interpret…

Einstein's equation, in its standard form, breaks down at the Big Bang singularity. A new version, equivalent to Einstein's whenever the latter is defined, but applicable in wider situations, is proposed. The new equation remains smooth at the Big Bang singularity of the Friedmann-Lemaitre-Robertson-Walker model. It is…

The paper proves conditions for curvature blow-up in quiescent big bang singularities.

The paper proves that certain FLRW spacetimes cannot be extended past the big bang.

The Weyl curvature hypothesis of Penrose attempts to explain the high homogeneity and isotropy, and the very low entropy of the early universe, by conjecturing the vanishing of the Weyl tensor at the Big-Bang singularity. In previous papers it has been proposed an equivalent form of Einstein's equation, which extends i…

We consider branes $N=I\times\so$, where $\so$ is an $n$\ndash dimensional space form, not necessarily compact, in a Schwarzschild-AdS_{(n+2)} bulk $\mc N$. The branes have a big crunch singularity. If a brane is an ARW space, then, under certain conditions, there exists a smooth natural transition flow through the sin…

We consider spacetimes $N$ satisfying some structural conditions, which are still fairly general, and prove convergence results for the leaves of an inverse mean curvature flow. Moreover, we define a new spacetime $\hat N$ by switching the light cone and using reflection to define a new time function, such that the two…

Study proves stability of big bang singularity in complex system.

Develops geometric framework for analyzing big bang singularities without symmetry assumptions.

Proves initial data on big bang singularities for Einstein-nonlinear scalar field equations lead to unique solutions.

We consider branes $N$ in a Schwarzschild-$\text{AdS}_{(n+2)}$ bulk, where the stress energy tensor is dominated by the energy density of a scalar fields map $\f:N\ra \mc S$ with potential $V$, where $\mc S$ is a semi-Riemannian moduli space. By transforming the field equation appropriately, we get an equivalent field …

New obstruction prevents certain spacetimes with both big bang and big crunch.

Einstein's equation is rewritten in an equivalent form, which remains valid at the singularities in some major cases. These cases include the Schwarzschild singularity, the Friedmann-Lemaître-Robertson-Walker Big Bang singularity, isotropic singularities, and a class of warped product singularities. This equation is co…

We consider the inverse mean curvature flow in Robertson-Walker spacetimes that satisfy the Einstein equations and have a big crunch singularity and prove that under natural conditions the rescaled inverse mean curvature flow provides a smooth transition from big crunch to big bang. We also construct an example showing…

Study shows intrinsic timed Hausdorff convergence leads to Gromov-Hausdorff and big bang convergence.

New proof of past stability for Kasner solutions in $(3+1)$-dimensional Einstein vacuum spacetime.

New null distance bounds confirm Big Bang singularity in cosmological models.

We study the asymptotic behavior of the solution curves of the dynamics of spacetimes of the topological type $Σ_{p}\times \mathbb{R}$, $p>1$, where $Σ_{p}$ is a closed Riemann surface of genus $p$, in the regime of $2+1$ dimensional classical general relativity. The configuration space of the gauge fixed dynamics is i…

We consider the so-called inverse $F$-curvature flow (IFCF) $\dot x = -F^{-1}ν$ in ARW spaces, i.e. in Lorentzian manifolds with a special future singularity. Here, $F$ denotes a curvature function of class $(K^*)$, which is homogenous of degree one, e.g. the $n$-th root of the Gaussian curvature, and $ν$ the past dire…

In General Relativity the metric can be recovered from the structure of the lightcones and a measure giving the volume element. Since the causal structure seems to be simpler than the Lorentzian manifold structure, this suggests that it is more fundamental. But there are cases when seemingly healthy causal structure an…

In this article the degenerate warped products of singular semi-Riemannian manifolds are studied. They were used recently by the author to handle singularities occurring in General Relativity, in black holes and at the big-bang. One main result presented here is that a degenerate warped product of semi-regular semi-Rie…

This work presents the foundations of Singular Semi-Riemannian Geometry and Singular General Relativity, based on the author's research. An extension of differential geometry and of Einstein's equation to singularities is reported. Singularities of the form studied here allow a smooth extension of the Einstein field eq…

This paper is a continuation of the work by the same authors on the Cartan group equipped with the sub-Finsler $\ell_\infty$ norm. We start by giving a detailed presentation of the structure of bang-bang extremal trajectories. Then we prove upper bounds on the number of switchings on bang-bang minimizers. We prove that…

We point out that algebraically special Einstein fields with twisting rays exhibit the basic properties of conformal Universes considered recently by Roger Penrose.

Seminar held at JINR, Dubna, May 15, 2012. In General Relativity, spacetime singularities raise a number of problems, both mathematical and physical. One can identify a class of singularities - with smooth but degenerate metric - which, under a set of conditions, allow us to define proper geometric invariants, and to w…

The paper examines gravitational singularities in spacetimes and proves inextendibility.

We consider spacetimes consisting of a manifold with Lorentzian metric and a weight function or scalar field. These spacetimes admit a Bakry-Émery-Ricci tensor which is a natural generalization of the Ricci tensor. We impose an energy condition on the Bakry-Émery-Ricci tensor and obtain singularity theorems of a cosmol…

Milne-like spacetimes are a class of FLRW models which admit $C^0$ spacetime extensions through the big bang. The boundary of a Milne-like spacetime can be identified with a null cone in the extension. We find that the comoving observers all emanate from a single point in the extension. This suggests that something phy…

In this paper we study a sub-Finsler geometric problem on the free-nilpotent group of rank 2 and step 3. Such a group is also called Cartan group and has a natural structure of Carnot group, which we metrize considering the $\ell_\infty$ norm on its first layer. We adopt the point of view of time-optimal control theory…

A series of old and recent theoretical observations suggests that the quantization of gravity would be feasible, and some problems of Quantum Field Theory would go away if, somehow, the spacetime would undergo a dimensional reduction at high energy scales. But an identification of the deep mechanism causing this dimens…

In this paper, we extend the persona-based sequence-to-sequence (Seq2Seq) neural network conversation model to a multi-turn dialogue scenario by modifying the state-of-the-art hredGAN architecture to simultaneously capture utterance attributes such as speaker identity, dialogue topic, speaker sentiments and so on. The …

A large collection of financial contracts offering guaranteed minimum benefits are often posed as control problems, in which at any point in the solution domain, a control is able to take any one of an uncountable number of values from the admissible set. Often, such contracts specify that the holder exert control at a…

Given a time function $τ$ on a spacetime $M$, we define a `null distance function', $\hat{d}_τ$, built from and closely related to the causal structure of $M$. In basic models with timelike $\nabla τ$, we show that 1) $\hat{d}_τ$ is a definite distance function, which induces the manifold topology, 2) the causal struct…

Under the optimal withdrawal strategy of a policyholder, the pricing of variable annuities with Guaranteed Minimum Withdrawal Benefit (GMWB) is an optimal stochastic control problem. The surrender feature available in marketed products allows termination of the contract before maturity, making it also an optimal stoppi…

Consider a smooth manifold $M$ with a smooth cometric $g^{\ast}$ which changes the bilineal type by transverse way, on a hypersurface $D^{\infty}$. Suppose that the radical annihilator hyperplane is tangent to $D^{\infty}$. We examine the geometry of the ($g^{\ast}$-dual) covariant metric $g$ on $M-$ $D^{\infty}$, prov…

The two-dimensional renormalization group acting as the Ricci flow $Λ\frac{\partial}{\partialΛ} g_{μν} = R_{μν}$ produces a specific 1+3 dimensional space-time metric which describes an expanding universe that starts with a big bang $a \sim t^{\scriptscriptstyle 1/\sqrt3}$ then decelerates until $z=0.2$ then accelerate…

Uniform volume estimate for Kähler metrics in big cohomology classes.

Let $X$ be a compact Kähler manifold and $\{θ\}$ be a big cohomology class. We prove several results about the singularity type of full mass currents, answering a number of open questions in the field. First, we show that the Lelong numbers and multiplier ideal sheaves of $θ$-plurisubharmonic functions with full mass a…

New findings show cosmological constant as initial condition for non-isotropic spacetimes.

The Miyaoka-Yau inequality is proven for certain singular varieties with big canonical or anticanonical divisors.

Gravitational wave memory increases faster than Brownian motion in early universe and astrophysical sources.

We show that normalized currents of integration along the common zeros of random $m$-tuples of sections of powers of $m$ singular Hermitian big line bundles on a compact Kähler manifold distribute asymptotically to the wedge product of the curvature currents of the metrics. If the Hermitian metrics are Hölder with sing…

We use integrable systems techniques to study the singularities of timelike non-minimal constant mean curvature (CMC) surfaces in the Lorentz-Minkowski 3-space. The singularities arise at the boundary of the Birkhoff big cell of the loop group involved. We examine the behaviour of the surfaces at the big cell boundary,…

J. Nitsche proved that an isolated singularity of a conformal hyperbolic metric is either a conical singularity or a cusp one. We prove by developing map that there exists a complex coordinate $z$ centered at the singularity where the metric has the expression of either $\displaystyle{\frac{4α^2\vert z \vert^{2α-2}}{(1…

We introduce a combinatorial argument to study closed minimal hypersurfaces of bounded area and high Morse index. Let $(M^{n+1},g)$ be a closed Riemannian manifold and $Σ\subset M$ be a closed embedded minimal hypersurface with area at most $A>0$ and with a singular set of Hausdorff dimension at most $n-7$. We show the…

Let $(B_t)_{0\leq t\leq T}$ be either a Bernoulli random walk or a Brownian motion with drift, and let $M_t:=\max\{B_s: 0\leq s\leq t\}$, $0\leq t\leq T$. This paper solves the general optimal prediction problem \sup_{0\leqτ\leq T}\sE[f(M_T-B_τ)], where the supremum is over all stopping times $τ$ adapted to the natural…