arXiv subjects

Blake Temple

Publications and source records attributed to Blake Temple.

At least 19 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

The essential regularity of singular connections in geometry

This paper, a culmination of the authors' theory of the RT-equations, accomplishes the following: (i) We discover there is a true (geometric) regularity associated with every affine connection, its ``essential regularity'', the highest possible regularity achievable by coordinate transformation, a geometric property independent of starting atlas. (ii) We give a checkable necessary and sufficient condition for determining whether or not a connection is at its essential regularity in a given atlas, based on the relative regularity of the connection and its Riemann curvature. (iii) We introduce a computable procedure based on the RT-equations for lifting any $L^p$ affine connection given in a starting atlas, to a new atlas in which the connection exhibits its essential regularity. This resolves the long-standing problem of determining whether or not a singularity in an affine connection is removable or essential, applicable to any connection with components locally in $L^p$, $p>n$, general enough to include GR shock wave and cusp singularities in General Relativity. Since a manifold by itself does not carry an intrinsic level of regularity, the authors propose that the essential regularity of a connection marks the point at which an intrinsic level of regularity enters the subject of geometry.

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↗

arXiv2024

Time-periodic solutions of the compressible Euler equations and the Nonlinear Theory of Sound

We prove the existence of ``pure tone'' nonlinear sound waves of all frequencies. These are smooth, time periodic, oscillatory solutions of the $3\times3$ compressible Euler equations satisfying periodic or acoustic boundary conditions in one space dimension. This resolves a centuries old problem in the theory of Acoustics, by establishing that the pure modes of the linearized equations are the small amplitude limits of solutions of the nonlinear equations. Riemann's celebrated 1860 proof that compressions always form shocks is known to hold for isentropic and barotropic flows, but our proof shows that for generic entropy profiles, shock-free periodic solutions containing nontrivial compressions and rarefactions exist for every wavenumber $k$.

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↗

arXiv2023

On Weak Solutions to the Geodesic Equation in the Presence of Curvature Bounds

We show that taking account of bounded curvature reduces the threshold regularity of connection coefficients required for existence and uniqueness of solutions to the geodesic equation, to $L^p_\text{loc}$, one derivative below the regularity $W^{1,p}_\text{loc}$ required if one does not take account of curvature, ($L_{loc}^p$ for existence, $W_{loc}^{1,p}$ for uniqueness). Our argument is based on authors' theory of the RT-equations for regularizing connections to optimal regularity by coordinate transformation. The incoming regularity is too low to formulate a weak version of the geodesic equation based on the standard method of multiplying by smooth test functions and integrating by parts, so alternatively, we define weak solutions by coordinate transformation and we give an explicit procedure for mollifying the original connection such that the correct weak solution is indeed a limit of smooth solutions of the mollified equations in the original coordinates. This is an example where limits under suitable mollification are more fundamental than a weak formulation, indicative of more complicated PDE's in which the standard weak formulation of the equations does not adequately rule out unphysical solutions. Our results apply to general second order ODE's in which the lack of regularity can be isolated in the connection coefficients. The results apply to General Relativity.

math.AP↗

arXiv2023

The Nonlinear Theory of Sound

We prove the existence of ``pure tone'' nonlinear sound waves of all frequencies. These are smooth, space and time periodic, oscillatory solutions of the $3\times3$ compressible Euler equations in one space dimension. Being perturbations of solutions of a linear wave equation, they provide a rigorous justification for the centuries old theory of Acoustics. In particular, Riemann's celebrated 1860 proof that compressions always form shocks holds for isentropic and barotropic flows, but for generic entropy profiles, shock-free periodic solutions containing nontrivial compressions and rarefactions exist for every wavenumber $k$.

math.AP↗

arXiv2022

Optimal Regularity and Uhlenbeck Compactness for General Relativity and Yang-Mills Theory

We announce the extension of optimal regularity and Uhlenbeck compactness to the general setting of connections on vector bundles with non-compact gauge groups over non-Riemannian manifolds, including the Lorentzian metric connections of General Relativity. Compactness is the essential tool of mathematical analysis for establishing validity of approximation schemes. Our proofs are based on the theory of the RT-equations for connections with $L^p$ curvature. Solutions of the RT-equations furnish coordinate and gauge transformations which give a non-optimal connection a gain of one derivative over its Riemann curvature, (i.e., to optimal regularity). The RT-equations are elliptic regardless of metric signature, and regularize singularities in solutions of the hyperbolic Einstein equations. As an application, singularities at GR shock waves are removable, implying geodesic curves, locally inertial coordinates and the Newtonian limit all exist. By the extra derivative we extend Uhlenbeck compactness from Uhlenbeck's setting of vector bundles with compact gauge groups over Riemannian manifolds, to the case of compact and non-compact gauge groups over non-Riemannian manifolds. Our version of Uhlenbeck compactness can also be viewed as a "geometric" improvement of the Div-Curl Lemma, improving weak continuity of wedge products to strong convergence.

gr-qc↗

arXiv2021

On the Optimal Regularity Implied by the Assumptions of Geometry II: Connections on Vector Bundles

We extend authors' prior results on optimal regularity and Uhlenbeck compactness for affine connections to general connections on vector bundles. This is accomplished by deriving a vector bundle version of the RT-equations, and establishing a new existence theory for these equations. These new RT-equations, non-invariant elliptic equations, provide the gauge transformations which transform the fibre component of a non-optimal connection to optimal regularity, i.e., the connection is one derivative more regular than its curvature in $L^p$. The existence theory handles curvature regularity all the way down to, but not including, $L^1$. Taken together with the affine case, our results extend optimal regularity of Kazden-DeTurck and the compactness theorem of Uhlenbeck, applicable to Riemannian geometry and compact gauge groups, to general connections on vector bundles over non-Riemannian manifolds, allowing for both compact and non-compact gauge groups. In particular, this extends optimal regularity and Uhlenbeck compactness to Yang-Mills connections on vector bundles over Lorentzian manifolds as base space, the setting of General Relativity.

math.DG↗

arXiv2020

Decay and Subluminality of Modes of all Wave Numbers in the Relativistic Dynamics of Viscous and Heat Conductive Fluids

To further confirm the causality and stability of a second-order hyperbolic system of partial differential equations that models the relativistic dynamics of barotropic fluids with viscosity and heat conduction (H. Freist\"uhler and B. Temple, J. Math. Phys. 59 (2018)), this paper studies the Fourier-Laplace modes of this system and shows that all such modes, relative to arbitrary Lorentz frames, (a) decay with increasing time and (b) travel at subluminal speeds. Stability is also shown for the related model of non-barotropic fluids (H. Freist\"uhler and B. Temple. Proc. R. Soc. A 470 (2014) and Proc. R. Soc. A 473 (2017)). Even though these properties had been known for a while in the sense of numerical evidence, the fully analytical proofs for the subluminality of modes of arbitrary wave numbers in arbitrary frames given here appear to be the first regarding any five-field formulation of dissipative relativistic fluid dynamics.

physics.flu-dyn↗

arXiv2019

On the Optimal Regularity Implied by the Assumptions of Geometry I: Connections on Tangent Bundles

We resolve the problem of optimal regularity and Uhlenbeck compactness for affine connections in General Relativity and Mathematical Physics. First, we prove that any affine connection $\Gamma$, with components $\Gamma \in L^{2p}$ and components of its Riemann curvature ${\rm Riem}(\Gamma)$ in $L^p$, in some coordinate system, can be smoothed by coordinate transformation to optimal regularity, $\Gamma \in W^{1,p}$ (one derivative smoother than the curvature), $p> \max\{n/2,2\}$, dimension $n\geq 2$. For Lorentzian metrics in General Relativity this implies that shock wave solutions of the Einstein-Euler equations are non-singular -- geodesic curves, locally inertial coordinates and the Newtonian limit, all exist in a classical sense, and the Einstein equations hold in the strong sense. The proof is based on an $L^p$ existence theory for the Regularity Transformation (RT) equations, a system of elliptic partial differential equations (introduced by the authors) which determine the Jacobians of the regularizing coordinate transformations. Secondly, this existence theory gives the first extension of Uhlenbeck compactness from Riemannian metrics, to general affine connections bounded in $L^\infty$, with curvature in $L^{p}$, $p>n$, including semi-Riemannian metrics, and Lorentzian metric connections of relativistic Physics. We interpret this as a "geometric" improvement of the generalized Div-Curl Lemma. Our theory shows that Uhlenbeck compactness and optimal regularity are pure logical consequences of the rule which defines how connections transform from one coordinate system to another -- what one could take to be the "starting assumption of geometry".

math-ph↗

arXiv2018

How to smooth a crinkled map of spacetime: Uhlenbeck compactness for $L^\infty$ connections and optimal regularity for general relativistic shock waves by the Reintjes-Temple-equations

We present authors' new theory of the RT-equations, nonlinear elliptic partial differential equations which determine the coordinate transformations which smooth connections $\Gamma$ to optimal regularity, one derivative smoother than the Riemann curvature tensor ${\rm Riem}(\Gamma)$. As one application we extend Uhlenbeck compactness from Riemannian to Lorentzian geometry; and as another application we establish that regularity singularities at GR shock waves can always be removed by coordinate transformation. This is based on establishing a general multi-dimensional existence theory for the RT-equations, by application of elliptic regularity theory in $L^p$ spaces. The theory and results announced in this paper apply to arbitrary $L^\infty$ connections on the tangent bundle $T\mathcal{M}$ of arbitrary manifolds $\mathcal{M}$, including Lorentzian manifolds of General Relativity.

gr-qc↗

arXiv2018

Inversion of a Non-uniform Difference Operator

The problem of applying Nash-Moser Newton methods to obtain periodic solutions of the compressible Euler equations has led authors to identify the main obstacle, namely, how to invert operators which impose periodicity when they are based on non-uniform shift operators. Here we begin a theory for finding the inverses of such operators by proving that a scalar non-uniform difference operator does in fact have a bounded inverse on its range. We argue that this is the simplest example which demonstrates the need to use direct rather than Fourier methods to analyze inverses of linear operators involving nonuniform shifts.

math.AP↗

arXiv2018

Optimal metric regularity in General Relativity follows from the RT-equations by elliptic regularity theory in $L^p$-spaces

Shock wave solutions of the Einstein equations have been constructed in coordinate systems in which the gravitational metric is only Lipschitz continuous, but the connection $\Gamma$ and curvature $Riem(\Gamma)$ are both in $L^{\infty}$. At this low level of regularity, the physical meaning of such gravitational metrics remains problematic. Here we address the mathematical problem as to whether the condition that $Riem(\Gamma)$ has the same regularity as $\Gamma$, is sufficient for the existence of a coordinate transformation which perfectly cancels out the jumps in the leading order derivatives of $\delta\Gamma$, thereby raising the regularity of the connection and the metric by one order--a subtle problem. We have now discovered, in a framework much more general than GR shock waves, that the regularization of non-optimal connections is determined by a nonlinear system of elliptic equations with matrix valued differential forms as unknowns, the Regularity Transformation equations, or RT-equations. In this paper we establish the first existence theory for the nonlinear RT-equations in the general case when $\Gamma, {\rm Riem}(\Gamma)\in W^{m,p}$, $m\geq1$, $n<p< \infty$, where $\Gamma$ is any affine connection on an $n$-dimensional manifold. From this we conclude that for any such connection $\Gamma(x) \in W^{m,p}$ with ${\rm Riem}(\Gamma) \in W^{m,p}$, $m\geq1$, $n<p< \infty$, given in $x$-coordinates, there always exists a coordinate transformation $x\to y$ such that $\Gamma(y) \in W^{m+1,p}$. That is, $\Gamma$ exhibits optimal regularity in $y$-coordinates. The problem of optimal regularity for the hyperbolic Einstein equations is thus resolved by elliptic regularity theory in $L^p$-spaces applied to the RT-equations.

gr-qc↗

arXiv2018

The Regularity Transformation Equations: An elliptic mechanism for smoothing gravitational metrics in General Relativity

A central question in General Relativity (GR) is how to determine whether singularities are geometrical properties of spacetime, or simply anomalies of a coordinate system used to parameterize the spacetime. In particular, it is an open problem whether there always exist coordinate transformations which smooth a gravitational metric to optimal regularity, two full derivatives above the curvature tensor, or whether regularity singularities exist. We resolve this open problem above a threshold level of smoothness by proving in this paper that the existence of such coordinate transformations is equivalent to solving a system of nonlinear elliptic equations in the unknown Jacobian and transformed connection, both viewed as matrix valued differential forms. We name these the Regularity Transformation equations, or RT-equations. In a companion paper we prove existence of solutions to the RT-equations for connections $\Gamma\in W^{m,p},$ curvature ${\rm Riem}(\Gamma) \in W^{m,p}$, assuming $m\geq1$, $p>n$. Taken together, these results imply that there always exist coordinate transformations which smooth arbitrary connections to optimal regularity, (one derivative more regular than the curvature), and there are no regularity singularities, above the threshold $m\geq1$, $p>n$. Authors are currently working on extending these methods to the case of GR shock waves, when gravitational metrics are only Lipschitz continuous, ($m=0$, $p=\infty$), and optimal regularity is required to recover basic properties of spacetime.

gr-qc↗

arXiv2016

Shock Wave Interactions and the Riemann-flat Condition: The Geometry behind Metric Smoothing and the Existence of Locally Inertial Frames in General Relativity

We prove that the essential smoothness of the gravitational metric at shock waves in GR, a PDE regularity issue for weak solutions of the Einstein equations, is determined by a geometrical condition which we introduce and name the {\it Riemann-flat condition}. The Riemann-flat condition determines whether or not the essential smoothness of the gravitational metric is two full derivatives more regular than the Riemann curvature tensor. This provides a geometric framework for the open problem as to whether {\it regularity singularities} (points where the curvature is in $L^\infty$ but the essential smoothness of the gravitational metric is only Lipschitz continuous) can be created by shock wave interaction in GR, or whether metrics Lipschitz at shocks can always be smoothed one level to $C^{1,1}$ by coordinate transformation. As a corollary of the ideas we give a proof that locally inertial frames always exist in a natural sense for shock wave metrics in spherically symmetric spacetimes, independent of whether the metric itself can be smoothed to $C^{1,1}$ locally. This latter result yields an explicit procedure (analogous to Riemann Normal Coordinates in smooth spacetimes) for constructing locally inertial coordinates for Lipschitz metrics, and is a new regularity result for GR solutions constructed by the Glimm scheme.

gr-qc↗

arXiv2015

"Regularity Singularities" and the Scattering of Gravity Waves in Approximate Locally Inertial Frames

It is an open question whether solutions of the Einstein-Euler equations are smooth enough to admit locally inertial coordinates at points of shock wave interaction, or whether "regularity singularities" can exist at such points. The term {\it regularity singularity} was proposed by the authors as a point in spacetime where the gravitational metric tensor is Lipschitz continuous ($C^{0,1}$), but no smoother, in any coordinate system of the $C^{1,1}$ atlas. An existence theory for shock wave solutions in $C^{0,1}$ admitting arbitrary interactions has been proven for the Einstein-Euler equations in spherically symmetric spacetimes, but $C^{1,1}$ is the requisite smoothness required for space-time to be locally flat. Thus the open problem of regularity singularities is the problem as to whether locally inertial coordinate systems exist at shock waves within the larger $C^{1,1}$ atlas. To clarify this open problem, we identify new "Coriolis type" effects in the geometry of $C^{0,1}$ shock wave metrics and prove they are essential in the sense that they can never be made to vanish within the atlas of {\it smooth} coordinate transformations, the atlas usually assumed in classical differential geometry. Thus the problem of existence of regularity singularities is equivalent to the question as to whether or not these Coriolis type effects are essentially non-removable and `real', or merely coordinate effects that can be removed, (in analogy to classical Coriolis forces), by going to the less regular atlas of $C^{1,1}$ transformations. If essentially non-removable, it would argue strongly for a `real' new physical effect for General Relativity, providing a physical context to the open problem of regularity singularities.

gr-qc↗

arXiv2015

No Regularity Singularities Exist at Points of General Relativistic Shock Wave Interaction between Shocks from Different Characteristic Families

We give a constructive proof that coordinate transformations exist which raise the regularity of the gravitational metric tensor from $C^{0,1}$ to $C^{1,1}$ in a neighborhood of points of shock wave collision in General Relativity. The proof applies to collisions between shock waves coming from different characteristic families, in spherically symmetric spacetimes. Our result here implies that spacetime is locally inertial and corrects an error in our earlier RSPA-publication, which led us to the false conclusion that such coordinate transformations, which smooth the metric to $C^{1,1}$, cannot exist. Thus, our result implies that regularity singularities, (a type of mild singularity introduced in our RSPA-paper), do not exist at points of interacting shock waves from different families in spherically symmetric spacetimes. Our result generalizes Israel's celebrated 1966 paper to the case of such shock wave interactions but our proof strategy differs fundamentally from that used by Israel and is an extension of the strategy outlined in our original RSPA-publication. Whether regularity singularities exist in more complicated shock wave solutions of the Einstein Euler equations remains open.

gr-qc↗