arXiv subjects
Publications and source records attributed to Moritz Reintjes.
A limiting case is worked out in which the causal action principle for causal fermion systems describing Minkowski space gives rise to the linear Fock space dynamics of perturbative quantum field theory including non-abelian gauge fields and Dirac fields.
We prove a low-regularity version of Hawking's singularity theorem for Lorentzian metrics in $W^{1,p}$ with Riemann curvature in $L^p$, where $p>2n$ and $n$ the dimension of spacetime. This extends previous results beyond the Lipschitz regime. Under suitable lower Ricci bounds and upper mean curvature assumptions, expressed in terms of temporal functions, we establish both the globally hyperbolic version of Hawking's theorem, in the form of an upper bound on the time separation from a spacelike Cauchy hypersurface, and the version with a compact achronal spacelike hypersurface, yielding timelike RT-geodesic incompleteness. The proof combines regularisations, based on the elliptic RT-equations, to raise the regularity of the metric by one derivative, with a refinement of the previously used manifold convolution. We introduce a new smeared-out notion of mean curvature adapted to the low metric regularity before, and the $W^{2,p}$-hypersurfaces arising after regularisation. As further consequences, we show that $W^{1,p}$-Lorentzian metrics with $L^p$-bounded curvature are causally plain, and we prove a corresponding low-regularity version of Myers's theorem in the Riemannian setting.
We give the first general construction of solutions of the static spherically symmetric Einstein-Euler equations, the Tolman-Oppenheimer-Volkoff (TOV-)equation, with prescribed density functions allowed to be discontinuous and non-uniform; these solutions describe stellar phase transitions in General Relativity. Boundedness of the resulting pressure functions solving the TOV-equations, from the boundary down to the stellar center, is obtained by identifying a novel condition on the prescribed density, in generalization of the classical Buchdahl limit. Moreover, we introduce a new necessary condition for the existence of such bounded pressure functions, which in the special case of a uniform density state reduces to the classical Buchdahl limit on the stellar mass-radius relationship. We present various examples to study the stellar mass-radius relationships resulting from our new conditions.
Zero viscosity limits are central to the study of classical shock waves. By identifying the correct physical (Lax admissible) shocks, they are a cornerstone in the design of analytical and numerical schemes. For relativistic fluid flow, however, the underlying dissipation mechanism, based on the Euclidean Laplace operator (so-called ``artificial viscosity''), violates Lorentz invariance, the fundamental principle of Special Relativity ensuring the speed of light bound. In this paper we show that replacing the Laplacian on conserved quantities by the wave operator on the fluid four-velocity alone, (not involving the density), provides a simplest Lorentz invariant description of dissipative relativistic fluid flow. We prove the resulting equations are causal and well-posed in one spatial dimension, and we establish their dissipativity by proving decay of Fourier Laplace modes near steady states. Moreover, we prove shock waves have profiles (a unique viscous travelling wave approximation in $L^2$) if and only if the shock wave is Lax admissible, and we prove that entropy production of travelling wave solutions is positive if and only if they obey the speed of light bound. This establishes the dissipative relativistic Euler equations introduced in this paper as an efficient model for the study of relativistic shock waves in the zero viscosity limit, both in analytical and numerical approaches, consistent with the laws of Relativity.
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.
A limiting case is considered in which the causal action principle for causal fermion systems describing Minkowski space gives rise to the linear Fock space dynamics of quantum electrodynamics. The quantum nature of the bosonic field is a consequence of the stochastic description of a multitude of fluctuating fields coupled to non-commuting operators, taking into account dephasing effects. The scaling of all error terms is specified. Our analysis leads to the concept of holographic mixing, which is introduced and explained in detail.
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.
In this paper we propose a weaker version of Penrose's much heeded Strong Cosmic Censorship (SCC) conjecture, asserting inextentability of maximal Cauchy developments by manifolds with Lipschitz continuous Lorentzian metrics and Riemann curvature bounded in $L^p$. Lipschitz continuity is the threshold regularity for causal structures, and curvature bounds rule out infinite tidal accelerations, arguing for physical significance of this weaker SCC conjecture. The main result of this paper, under the assumption that no extensions exist with higher connection regularity $W^{1,p}_\text{loc}$, proves in the affirmative this SCC conjecture with bounded curvature for $p$ sufficiently large, ($p>4$ with uniform bounds, $p>2$ without uniform bounds).
This paper is dedicated to a detailed analysis and computation of quantum states of causal fermion systems. The mathematical core is to compute integrals over the unitary group asymptotically for a large dimension of the group, for various integrands with a specific scaling behavior in this dimension. It is shown that, in a well-defined limiting case, the localized refined pre-state is positive and allows for the description of general entangled states.
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.
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.
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. Freistühler 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. Freistühler 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.
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 $Γ\in W^{m,p},$ curvature ${\rm Riem}(Γ) \in W^{m,p}$, 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.
We present authors' new theory of the RT-equations, nonlinear elliptic partial differential equations which determine the coordinate transformations which smooth connections $Γ$ to optimal regularity, one derivative smoother than the Riemann curvature tensor ${\rm Riem}(Γ)$. 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.
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 $Γ$ and curvature $Riem(Γ)$ 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 $Riem(Γ)$ has the same regularity as $Γ$, is sufficient for the existence of a coordinate transformation which perfectly cancels out the jumps in the leading order derivatives of $δΓ$, 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 $Γ, {\rm Riem}(Γ)\in W^{m,p}$, $m\geq1$, $n<p< \infty$, where $Γ$ is any affine connection on an $n$-dimensional manifold. From this we conclude that for any such connection $Γ(x) \in W^{m,p}$ with ${\rm Riem}(Γ) \in W^{m,p}$, $m\geq1$, $n<p< \infty$, given in $x$-coordinates, there always exists a coordinate transformation $x\to y$ such that $Γ(y) \in W^{m+1,p}$. That is, $Γ$ 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.
The previous functional analytic construction of the fermionic projector on globally hyperbolic Lorentzian manifolds is extended to space-times of infinite lifetime. The construction is based on an analysis of families of solutions of the Dirac equation with a varying mass parameter. It makes use of the so-called mass oscillation property which implies that integrating over the mass parameter generates decay of the Dirac wave functions at infinity. We obtain a canonical decomposition of the solution space of the massive Dirac equation into two subspaces, independent of observers or the choice of coordinates. The constructions are illustrated in the examples of ultrastatic space-times and de Sitter space-time.
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".
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.