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Paul Tod

Publications and source records attributed to Paul Tod.

At least 19 recordsLinked to original sources
arXiv2026

The massless Boltzmann equation in Minkowski spacetime

We study the spatially homogeneous, massless Boltzmann equation in Minkowski spacetime for a certain range of hard and soft interactions. For hard interactions, we derive a Povzner-type inequality for massless particles and show that solutions exist for all time into the future. For soft interactions, we employ singular weights to control singularities at $ p = 0 $, which arise from the masslessness of particles, to obtain local existence. These results, which are among rather few proofs of existence for the massless Boltzmann equation, are motivated by our earlier work on the massless Einstein--Boltzmann system in certain cosmological settings.

math.AP↗
arXiv2026

Conformal Geodesics Cannot Spiral -- Erratum

Wojciech Kami\'nski has provided a non real-analytic counterexample to our claim in [1] that conformal geodesics cannot spiral. This erratum illustrates how the proof of Lemma 4.6 [1] (on which our claim was based) fails.

math.DG↗
arXiv2024

Rod Structures and Patching Matrices: a review

I review the twistor theory construction of stationary and axisymmetric, Lorentzian signature solutions of the Einstein vacuum equations and the related toric Ricci-flat metrics of Riemannian signature, \cite{W,MW,F,FW}. The construction arises from the Ward construction \cite{W2} of anti-self-dual Yang-Mills fields as holomorphic vector bundles on twistor space, with the observation of Witten \cite{LW} that the Einstein equations for these metrics include the anti-self-dual Yang-Mills equations. The principal datum for a solution is the holomorphic patching matrix $P$ for a holomorphic vector bundle on a reduced twistor space, and $P$ is typically simpler than the corresponding metric to write down. I give a catalogue of examples, building on earlier collections \cite{F,AG}, and consider the inverse problem: how far does the rod structure of such a metric, together with its asymptotics, determine $P$?

math.DG↗
arXiv2024

One-sided type-D metrics with aligned Einstein-Maxwell

We consider four-dimensional, Riemannian metrics for which one or other of the self-dual or anti-self-dual Weyl tensors is type-D and which satisfy the Einstein-Maxwell equations with the corresponding Maxwell field aligned with the type-D Weyl spinor, in the sense of sharing the same Principal Null Directions (or PNDs). Such metrics always have a valence-2 Killing spinor, and therefore a Hermitian structure and at least one Killing vector. We rederive the results of Araneda (\cite{ba}), that these metrics can all be given in terms of a solution of the $SU(\infty)$-Toda field equation, and show that, when there is a second Killing vector commuting with the first, the method of Ward can be applied to show that the metrics can also be given in terms of a pair of axisymmetric solutions of the flat three-dimensional Laplacian. Thus in particular the field equations linearise. Some examples of the constructions are given.

gr-qc↗
arXiv2024

Homogeneous solutions to the Einstein-matter equations with a magnetic field and a conformal gauge singularity

We study massless solutions to the Einstein equations coupled to different matter models with a magnetic field and a conformal gauge singularity assuming spatial homogeneity with three commuting spatial translations. We show that there are no solutions in the case that the matter model is a radiation fluid. If the matter is described via kinetic theory we obtain that there exist unique solutions to the Einstein-Vlasov system and the Einstein-Boltzmann system for a certain range of soft potentials. For both the Vlasov and the Boltzmann case we also obtain asymptotic expansions close to the initial conformal gauge singularity.

gr-qc↗
arXiv2024

Twistor theory of the Chen--Teo gravitational instanton

Toric Ricci--flat metrics in dimension four correspond to certain holomorphic vector bundles over a twistor space. We construct these bundles explicitly, by exhibiting and characterising their patching matrices, for the five--parameter family of Riemannian ALF metrics constructed by Chen and Teo. The Chen--Teo family contains a two--parameter family of asymptotically flat gravitational instantons. The patching matrices for these instantons take a simple rational form.

gr-qc↗
arXiv2023

Conformal Methods in Mathematical Cosmology

When he first introduced the notion of a conformal boundary into the study of asymptotically empty space-times, Penrose noted that that the boundary would be null, space-like or time-like according as the cosmological constant $\Lambda$ was zero, positive or negative. While most applications of the idea of a conformal boundary have been to the zero-$\Lambda$, asymptotically-Minkowskian case, there also has been work on the nonzero cases. Here we review work with a positive $\Lambda$, which is the appropriate case for cosmology of the universe in which we live.

gr-qc↗
arXiv2023

Well-posedness of anisotropic and homogeneous solutions to the Einstein-Boltzmann system with a conformal-gauge singularity

We consider the Einstein-Boltzmann system for massless particles in the Bianchi I space-time with scattering cross-sections in a certain range of soft potentials. We assume that the space-time has an initial conformal gauge singularity and show that the initial value problem is well posed with data given at the singularity. This is understood by considering conformally rescaled equations. The Einstein equations become a system of singular ordinary differential equations, for which we establish an existence theorem which requires several differentiability and eigenvalue conditions on the coefficient functions together with the Fuchsian conditions. The Boltzmann equation is regularized by a suitable choice of time coordinate, but still has singularities in momentum variables. This is resolved by considering singular weights, and the existence is obtained by exploiting singular moment estimates.

gr-qc↗
arXiv2022

Conformal Geodesics Cannot Spiral

We show that conformal geodesics on a Riemannian manifold cannot spiral: there does not exist a conformal geodesic which becomes trapped in every neighbourhood of a point.

math.DG↗
arXiv2022

Some questions about Conformal Cyclic Cosmology

This article is an extended version of a talk given in Oxford in June 2021 as part of an online meeting `Ninety minutes of CCC' to mark the 90th birthday of Sir Roger Penrose. I assemble some questions that I have been asked or have asked myself about CCC.

gr-qc↗
arXiv2021

One-sided type-D Ricci-flat metrics

We consider four-dimensional, Riemannian, Ricci-flat metrics for which one or other of the self-dual or anti-self-dual Weyl tensors is type-D. Such metrics always have a valence-2 Killing spinor, and therefore a Hermitian structure and at least one Killing vector. We rederive the results of Przanowski and collaborators, that these metrics can all be given in terms of a solution of the $SU(\infty)$-Toda field equation, and show that, when there is a second Killing vector commuting with the first, the method of Ward can be applied to show that the metrics can also be given in terms of an axisymmetric solution of the flat three-dimensional Laplacian. Thus in particular the field equations linearise. As a corollary, we show that the same technique linearises the field equations for a four-dimensional Einstein metric with anti-self-dual Weyl tensor and two commuting symmetries. We reduce the Einstein equations with non-zero scalar curvature and one-sided type-D Weyl tensor, excluding the Kähler-Einstein case, to a modified but not integrable Toda equation. Some examples of the constructions are given.

gr-qc↗
arXiv2021

A note on Penrose limits

As a footnote to arXiv:1909.07756, I show that, given a (time-like) umbilic 3-surface $Σ$ in a 4-dimensional space-time $M$, the Penrose limit taken along any null geodesic $Γ$ which lies in $Σ$ is a diagonalisable plane-wave.

gr-qc↗
arXiv2021

The Kijowski--Liu--Yau quasi-local mass of the Kerr black hole horizon

We use the isometric embedding of the spatial horizon of fast rotating Kerr black hole in a hyperbolic space to compute the quasi-local mass of the horizon for any value of the spin parameter $j=J/m^2$. The mass is monotonically decreasing from twice the ADM mass at $j=0$ to $1.76569m$ at $j=\sqrt{3}/2$. It then monotonicaly increases to a maximum around $j=0.99907$, and finally decreases to $2.01966m$ for $j=1$ which corresponds to the extreme Kerr black hole.

gr-qc↗
arXiv2021

Conformal geodesics on gravitational instantons

We study the integrability of the conformal geodesic flow (also known as the conformal circle flow) on the $SO(3)$--invariant gravitational instantons. On a hyper--Kähler four--manifold the conformal geodesic equations reduce to geodesic equations of a charged particle moving in a constant self--dual magnetic field. In the case of the anti--self--dual Taub NUT instanton we integrate these equations completely by separating the Hamilton--Jacobi equations, and finding a commuting set of first integrals. This gives the first example of an integrable conformal geodesic flow on a four--manifold which is not a symmetric space. In the case of the Eguchi--Hanson we find all conformal geodesics which lie on the three--dimensional orbits of the isometry group. In the non--hyper--Kähler case of the Fubini--Study metric on $\CP^2$ we use the first integrals arising from the conformal Killing--Yano tensors to recover the known complete integrability of conformal geodesics.

math.DG↗
arXiv2020

Spacetimes with all Penrose limits diagonalisable

We consider the problem of finding all space-time metrics for which all plane-wave Penrose limits are diagonalisable plane waves. This requirement leads to a conformally invariant differential condition on the Weyl spinor which we analyse for different algebraic types in the Petrov-Pirani-Penrose classification. The only vacuum examples, apart from actual plane waves which are their own Penrose limit, are some of the nonrotating type D metrics, but some nonvacuum solutions are also identified. The condition requires the Weyl spinor, whenever it is nonzero, to be proportional to a valence-4 Killing spinor with a real function of proportionality.

gr-qc↗
arXiv2019

Conformal and isometric embeddings of gravitational instantons

We construct isometric and conformally isometric embeddings of some gravitational instantons in $\mathbb{R}^8$ and $\mathbb{R}^7$. In particular we show that the embedding class of the Einstein--Maxwell instanton due to Burns is equal to $3$. For $\mathbb{CP}^2$, Eguchi--Hanson and anti-self-dual Taub-NUT we obtain upper and lower bounds on the embedding class.

hep-th↗
arXiv2019

Conics, Twistors, and anti-self-dual tri-Kähler metrics

We describe the range of the Radon transform on the space $M$ of irreducible conics in $\CP^2$ in terms of natural differential operators associated to the $SO(3)$-structure on $M=SL(3, \R)/SO(3)$ and its complexification. Following \cite{moraru} we show that for any function $F$ in this range, the zero locus of $F$ is a four-manifold admitting an anti-self-dual conformal structure which contains three different scalar-flat Kähler metrics. The corresponding twistor space ${\mathcal Z}$ admits a holomorphic fibration over $\CP^2$. In the special case where ${\mathcal Z}=\CP^3\setminus\CP^1$ the twistor lines project down to a four-parameter family of conics which form triangular Poncelet pairs with a fixed base conic.

math.DG↗