# Tom LawrenceRonin Institute · Department of Physics

Tom Lawrence

PhD, University of Southampton; MPhys University of Exeter

## About

17

Publications

2,210

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4

Citations

Introduction

I research geometry and group theory aspects of unification physics. I look at the nonlinear realisations of both internal and spacetime symmetries of classical fields, including the constraints and potentials that induce spontaneous symmetry breaking.
I juggle this with work on localism and public sector finance, particularly the funding of English local authorities.

Additional affiliations

September 2001 - July 2002

Position

- Problems class assistant and script marker

Description

- I helped to run problems classes in association with a lecture course given by Prof D J Dunstan, and presented two lectures when he was unable to attend. I also marked students' solutions to problems associated with this course and one other.

October 1997 - February 2002

Education

October 1993 - June 1997

## Publications

Publications (17)

This presentation (slides with audio and video commentary) explains my programme of research to extend the overall framework of general relativity to include the other fundamental interactions of nature. I call it "covariant compactification".
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The presentation is in Microsoft PowerPoint format. Due to formatting incompatibilities, the equation...

The core principle of General Relativity (GR) is that curved spacetime is manifested as gravity. Kaluza-Klein theories use this principle to interpret non-gravitational interactions. Most require spacetime to be extended in a complex way. If, instead, spacetime is extended in the most natural way, orthogonal gauge symmetries result. These represent...

This paper considers the relationship between geometry, symmetry and fundamental interactions — gravity and those mediated by gauge fields. We explore product spacetimes which (a) have the necessary symmetries for gauge interactions and four-dimensional gravity and (b) reduce to an [Formula: see text]-dimensional isotropic universe in their flat sp...

In these notes, we consider classical field theory in a universe with one spatial dimension which is finite in extent. We consider a single scalar field which is continuous in the spatial coordinate and in time. From this starting point, we find that features of quantum mechanics arise more naturally that they do when starting from classical partic...

My presentation "Do the symmetries of product spaces hold the key to unification?" to Symmetry 2021 on Friday 13 August 2021.
This shows how a careful study of product spaces reveals that they naturally contain gauge fields. This gives rise to a new type of Kaluza-Klein theory, in which all spatial dimensions are identical in the decompactificati...

My Lightning Talk to Ronin Institute on 25 May 2021, summarising my research for a general audience in 10 minutes. You can listen to me giving the talk at https://www.youtube.com/watch?v=cfdL13zaHts

Product manifolds play a key role in Kaluza-Klein theories. In existing theories, the internal symmetries generally act directly on the compact space. We take an alternative approach, in which the compact space has real extra spatial dimensions, so that in the decompactification limit, all of the spatial dimensions are identical. We study the geome...

A presentation I gave to the Geometric Foundations of Gravity conference on Friday 2 July 2021. It explains the coset formulation of gravity which I have developed. This improves on the tetrad formulation, which has deficiencies which have come to light during research on teleparallel theories. This formulation draws on i) coset space methods which...

This paper looks at how changes of coordinates on a pseudo-Riemannian manifold induce homogeneous linear transformations on its tangent spaces. We see that a pseudo-orthonormal frame in a given tangent space is the basis for a set of Riemann normal coordinates. A Lorentz subgroup of the general linear transformations preserves this pseudo-orthonorm...

This paper looks at how changes of coordinates on a pseudo-Riemannian manifold induce homogeneous linear transformations on its tangent spaces. We see that a pseudo-orthonormal frame in a given tangent space is the basis for a set of Riemann normal coordinates. A Lorentz subgroup of the general linear transformations preserves this pseudo-orthonorm...

This paper looks at connections on a pseudo-Riemannian manifold and the symmetries of its tangent spaces. In particular, it looks at a coset decomposition of the general linear group of Jacobian matrices, and the relationship between this, the Levi-Civita connection and the Weitzenböck connection. It then addresses various issues which have been th...

This paper looks at connections on a pseudo-Riemannian manifold and the symmetries of its tangent spaces. In particular, it looks at a coset decomposition of the general linear group of Jacobian matrices, and the relationship between this, the Levi-Civita connection and the Weitzenböck connection. It then addresses the role of translations in gener...

We present a geometric field theory in which the Lagrangian has full general covariance in a higher-dimensional spacetime. Covariant constraints on a vector field cause the extra dimensions to compactify spontaneously. Changes of coordinates induce transformations under which the values of the covariant derivative of the vector form orbits. Constra...

This paper analyses the geometry of the Lie algebra of SO(6) by
making use of its homomorphism with SU(4). We study the vector
space of 4×4 traceless, Hermitian matrices from four
different viewpoints and examine the connections between them.
We review the strata of this space under group transformations
using established techniques for su(N) alge...

Non-linear realisations of the groups SU(2) and SO(1,4) are analysed, described by the coset spaces SU(2)/U(1) and SO(1,4)/SO(1,3). The analysis consists of determining the transformation properties of the Goldstone bosons, constructing the most general possible Lagrangian for the realisations and finding the metric of the coset space. The Lie alge...

## Questions

Question (1)

By a Lorentzian manifold, I mean any pseudo-Riemannian manifold with signature +,-,-,-,... and any number of spacelike dimensions. If the metric is reducible, (can be written in block diagonal form across any chart of the manifold), is it necessarily true that the manifold is the product of geodesically complete factor spaces? If there's anyone who can answer this, I'd be grateful if you could tell me how this can be shown, or provide a reference if it's a known theorem.

## Projects

Projects (4)

Investigating how quantum phenomena and relations could emerge from classical field theory

Providing a template/worked example for research into spontaneous symmetry breaking and non-linear realisations