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Paul Tod

Publications and source records attributed to Paul Tod.

At least 37 records · Page 2Linked to original sources
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↗
arXiv2019

Conformally isometric embeddings and Hawking temperature

We find necessary and sufficient conditions for existence of a locally isometric embedding of a vacuum space-time into a conformally-flat 5-space. We explicitly construct such embeddings for any spherically symmetric Lorentzian metric in $3+1$ dimensions as a hypersurface in $R^{4, 1}$. For the Schwarzschild metric the embedding is global, and extends through the horizon all the way to the $r=0$ singularity. We discuss the asymptotic properties of the embedding in the context of Penrose's theorem on Schwarzschild causality. We finally show that the Hawking temperature of the Schwarzschild metric agrees with the Unruh temperature measured by an observer moving along hyperbolae in $R^{4, 1}$.

gr-qc↗
arXiv2018

Asymptotically $AdS_2\times S^2$ metrics satisfying the Null Energy Condition

We find the most general metric ansatz compatible with the results of Galloway and Graf \cite{GG} constraining asymptotically $AdS_2\times S^2$ space-times (and a differentiability assumption), and then study its curvature subject to a variety of geometrical and physical restrictions. In particular we find explicit examples which are asymptotically $AdS_2\times S^2$ metrics, in the sense of \cite{GG}, and which satisfy the Null Energy Condition but which differ from $AdS_2\times S^2$.

gr-qc↗
arXiv2018

An example of an asymptotically $AdS_2\times S^2$ metric satisfying NEC but which is not exactly $AdS_2\times S^2$

We give an example of an asymptotically $AdS_2\times S^2$ metric, in the sense of \cite{GG}, which satisfies the Null Energy Condition but is not exactly $AdS_2\times S^2$. It is therefore a counterexample to a conjecture of Maldacena mentioned in \cite{GG}, but it does not satisfy field equations. In an appendix we give an example admitting supercovariantly constant spinors as in \cite{t1}, which is asymptotically $AdS_2\times S^2$ on one side only.

gr-qc↗
arXiv2018

Jumps, folds, and singularities of Kodaira moduli spaces

For any integer $k$ we construct an explicit example of a twistor space which contains a one--parameter family of jumping rational curves, where the normal bundle changes from $O(1)+O(1)$ to $O(k)+O(2-k)$. For $k>3$ the resulting anti--self--dual Ricci-flat manifold is a Zariski cone in the space of holomorphic sections of $O(k)$. In the case $k=2$ we recover the canonical example of Hitchin's folded hyper-Kahler manifold, where the jumping lines form a three--parameter family. We show that in this case there exist normalisable solutions to the Schrodinger equation which extend through the fold.

math.DG↗
arXiv2017

The Stützfunktion and the Cut Function

I review some standard theory of convex bodies in $\mathbb{R}^3$ and rephrase it in a formalism of Ted Newman to show the relation between the Stützfunktion of the former theory and the cut function introduced by Ted. This leads to a conjectured inequality for space-like two-spheres in Minkowski space that generalises Minkowski's inequality and is implied by Penrose's cosmic censorship hypothesis.

gr-qc↗
arXiv2016

Non-diagonal four-dimensional cohomogeneity-one Einstein metrics in various signatures

Most known four-dimensional cohomogeneity-one Einstein metrics are diagonal in the basis defined by the left-invariant one-forms, though some essentially non-diagonal ones are known. We consider the problem of explicitly seeking non-diagonal Einstein metrics, and we find solutions which in some cases exhaust the possibilities. In particular we construct new examples of neutral signature non--diagonal Bianchi type VIII Einstein metrics with self--dual Weyl tensor.

gr-qc↗
arXiv2015

A positive Bondi--type mass in asymptotically de Sitter spacetimes

The general structure of the conformal boundary $\mathscr{I}^+$ of asymptotically de Sitter spacetimes is investigated. First we show that Penrose's quasi-local mass, associated with a cut ${\cal S}$ of the conformal boundary, can be zero even in the presence of outgoing gravitational radiation. On the other hand, following a Witten--type spinorial proof, we show that an analogous expression based on the Nester--Witten form is finite only if the Witten spinor field solves the 2-surface twistor equation on ${\cal S}$, and it yields a positive functional on the 2-surface twistor space on ${\cal S}$, provided the matter fields satisfy the dominant energy condition. Moreover, this functional is vanishing if and only if the domain of dependence of the spacelike hypersurface which intersects $\mathscr{I}^+$ in the cut ${\cal S}$ is locally isometric to the de Sitter spacetime. For non-contorted cuts this functional yields an invariant analogous to the Bondi mass.

gr-qc↗
arXiv2015

Some geometry of de Sitter space

In this expository note, I present some basic geometric and twistor theoretic facts about de Sitter space leading up to a discussion of Penrose's quasi-local mass construction for linear gravity theory in the de Sitter background. The corresponding results for Minkowski space and anti--de Sitter space are familiar and can be found in \cite{PR} or \cite{HT} for the former and \cite{KT} for the latter. Some of the formulas given here are also quite familiar, but some are thought to be new and it is convenient to have these facts collected in one place.

gr-qc↗
arXiv2015

Penrose's quasi-local mass for asymptotically anti-de Sitter space-times

Penrose's quasi-local mass construction is carried through for two-surfaces at infinity in asymptotically anti-de Sitter space-times. A modification of the Witten argument is given to prove a positivity property of the resulting conserved quantities. [This work formed part of Ron Kelly's Oxford D.Phil. thesis, and the first person pronoun refers to him. It appeared in hand-written form as `Asymptotically anti-de Sitter space-times' in Twistor Newsletter 20 (1985) pp11-23 (available at http://people.maths.ox.ac.uk/lmason/Tn/TN1-25), but is appearing type-set for the first time here. Footnotes marked ${\bf{PT}}$ have been added for this version by Paul Tod, in the hope of making this work available to a wider audience.]

gr-qc↗
arXiv2015

The equations of CCC

I review the equations of Conformal Cyclic Cosmology given by Penrose. I suggest a slight modification to Penrose's prescription and show how this works out for FRW cosmologies and for Class A Bianchi cosmologies.

gr-qc↗
arXiv2015

Valiente Kroon's obstructions to smoothness at infinity

We conjecture an interpretation in terms of multipole moments of the obstructions to smoothness at infinity found for time-symmetric, conformally-flat initial data by Valiente Kroon (Comm. Math. Phys. 244 (2004), 133-156).

gr-qc↗