arXiv subjects

Christopher Alexander

Publications and source records attributed to Christopher Alexander.

6 recordsLinked to original sources

arXiv2025

The Instability of the Critical Friedmann Spacetime at the Big Bang as an Alternative to Dark Energy

We characterize the local instability of pressureless Friedmann spacetimes to radial perturbation at the Big Bang. The analysis is based on a formulation of the Einstein-Euler equations in self-similar variables $(t,\xi)$, with $\xi=r/t$, conceived to realize the critical ($k=0$) Friedmann spacetime as a stationary solution whose character as an unstable saddle rest point $SM$ is determined via an expansion of smooth solutions in even powers of $\xi$. The eigenvalues of $SM$ imply the $k\neq0$ Friedmann spacetimes are unstable solutions within the unstable manifold of $SM$. We prove that all solutions smooth at the center of symmetry agree with a Friedmann spacetime at leading order in $\xi$, and with an eye toward Cosmology, we focus on $\mathcal{F}$, the set of solutions which agree with a $k<0$ Friedmann spacetime at leading order, providing the maximal family into which generic underdense radial perturbations of the unstable critical Friedmann spacetime will evolve. We prove solutions in $\mathcal{F}$ generically accelerate away from Friedmann spacetimes at intermediate times but decay back to the same leading order Friedmann spacetime asymptotically as $t\to\infty$. Thus instabilities inherent in the Einstein-Euler equations provide a natural mechanism for an accelerated expansion without recourse to a cosmological constant or dark energy.

gr-qc↗

arXiv2024

Cosmic Accelerations Characterize the Instability of the Critical Friedmann Spacetime

We give a definitive characterization of the instability of the pressureless ($p=0$) critical ($k=0$) Friedmann spacetime to smooth radial perturbations. We use this to characterize the global accelerations away from $k\leq0$ Friedmann spacetimes induced by the instability in the underdense case. The analysis begins by incorporating the Friedmann spacetimes into a mathematical analysis of smooth spherically symmetric solutions of the Einstein field equations expressed in self-similar coordinates $(t,\xi)$ with $\xi=\frac{r}{t}<1$, conceived to realize the critical Friedmann spacetime as an unstable saddle rest point $SM$. We identify a new maximal asymptotically stable family $\mathcal{F}$ of smooth outwardly expanding solutions which globally characterize the evolution of underdense perturbations. Solutions in $\mathcal{F}$ align with a $k<0$ Friedmann spacetime at early times, generically introduce accelerations away from $k<0$ Friedmann spacetimes at intermediate times and then decay back to the same $k<0$ Friedmann spacetime as $t\to\infty$ uniformly at each fixed radius $r>0$. We propose $\mathcal{F}$ as the maximal asymptotically stable family of solutions into which generic underdense perturbations of the unstable critical Friedmann spacetime will evolve and naturally admit accelerations away from Friedmann spacetimes within the dynamics of solutions of Einstein's original field equations, that is, without recourse to a cosmological constant or dark energy.

gr-qc↗

arXiv2023

Supersonic Gravitational Collapse for Non-Isentropic Gaseous Stars

We show the existence of a new class of initially smooth spherically symmetric self-similar solutions to the non-isentropic Euler-Poisson system. These solutions exhibit supersonic gravitational implosion in the sense that the density blows-up in finite time while the fluid velocity remains supersonic. In particular, they occupy a portion of the phase space that is far from the recently constructed isentropic self-similar implosion. At the heart of our proof is the presence of a two-parameter scaling invariance and the reduction of the problem to a non-autonomous system of ordinary differential equations. We use the requirement of smoothness of the flow as a selection principle that constrains the choice of scaling indices. An important consequence of our analysis is that for all the solutions we construct, the polytropic index $\gamma$ is strictly bigger than $\frac{4}{3}$, which is in sharp contrast to the known results in the isentropic case.

math.AP↗

arXiv2023

Shock-Wave Refinement of the Friedmann-Robertson-Walker Metric

The mathematics of general relativistic shock waves is introduced and considered in a cosmological context. In particular, an expanding Friedmann-Roberson-Walker metric is matched to a Tolman-Oppenheimer-Volkoff metric across a spherical shock surface. This is the general relativistic analogue of a shock-wave explosion within a static singular isothermal fluid sphere and may be regarded as a model for the Big Bang. These shock waves are constructed both within and beyond the Hubble radius, which corresponds to a universe outside and inside its Schwarzschild radius respectively. Certain self-similar perturbations of the FRW metric lead to an accelerated expansion, even without a cosmological constant, and thus it is conjectured that such a mechanism may account for the anomalous acceleration observed today without recourse to dark energy.

gr-qc↗

arXiv2022

Automating UAV Flight Readiness Approval using Goal-Directed Answer Set Programming

We present a novel application of Goal-Directed Answer Set Programming that digitizes the model aircraft operator's compliance verification against the Academy of Model Aircrafts (AMA) safety code. The AMA safety code regulates how AMA flyers operate Unmanned Aerial Vehicles (UAVs) for limited recreational purposes. Flying drones and their operators are subject to various rules before and after the operation of the aircraft to ensure safe flights. In this paper, we leverage Answer Set Programming to encode the AMA safety code and automate compliance checks. To check compliance, we use the s(CASP) which is a goal-directed ASP engine. By using s(CASP) the operators can easily check for violations and obtain a justification tree explaining the cause of the violations in human-readable natural language. Further, we implement an algorithm to help the operators obtain the minimal set of conditions that need to be satisfied in order to pass the compliance check. We develop a front-end questionnaire interface that accepts various conditions and use the backend s(CASP) engine to evaluate whether the conditions adhere to the regulations. We also leverage s(CASP) implemented in SWI-Prolog, where SWI-Prolog exposes the reasoning capabilities of s(CASP) as a REST service. To the best of our knowledge, this is the first application of ASP in the AMA and Avionics Compliance and Certification space.

cs.AI↗

arXiv2021

General Relativistic Shock Waves that Exhibit an Accelerated Expansion

This paper concerns the construction and analysis of a new family of exact general relativistic shock waves. The construction resolves the long-standing open problem of determining the expanding waves created behind a shock-wave explosion within a static isothermal sphere, akin to stellar ignition. In particular, these shock-wave spacetimes are formed through matching a family of self-similar asymptotically Friedmann spacetimes to the family of Tolman-Oppenheimer-Volkoff spacetimes. The matching is accomplished in Schwarzschild coordinates where the shock waves appear one derivative less regular than they actually are. Separately, both families contain singularities, but as matched shock-wave spacetimes, they are singularity free. It was previously unknown whether the matching of the two families could be achieved within a region where both families are nonsingular. Indeed, for a pure radiation equation of state, the matching occurs near the sonic point of the interior expanding wave and this makes the analysis quite delicate, both numerically and formally. It is for this reason the construction is accompanied by a rigorous existence proof in the pure radiation case, which forms the main result of this paper. The analysis extends to consider the matching of any self-similar expanding wave with a Tolman-Oppenheimer-Volkoff spacetime, with potentially distinct equations of state each side of the shock. The asymptotically Friedmann shock-waves exhibit an accelerated expansion, analogous to the acceleration modelled by the cosmological constant in the Standard Model of Cosmology. However, unlike in the Standard Model, these shock-wave spacetimes solve the Einstein field equations in the absence of a cosmological constant, opening up the question of whether a purely mathematical mechanism could account for the accelerated expansion observed today, rather than dark energy.

gr-qc↗