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Tevian Dray

Publications and source records attributed to Tevian Dray.

At least 19 recordsLinked to original sources
arXiv2026

Mirror Symmetry and Double Signature Change

The black mirror spacetime proposed by Tzanavaris, Boyle, and Turok [1] connects the two exterior regions of the extended Schwarzschild black hole directly to each other, with no intervening interior region. Using techniques adapted from previous work on signature change, we reexamine the black mirror spacetime as a model of double signature change, and investigate whether there is a surface layer at the horizon, that is, a distributional curvature singularity corresponding to an impulsive gravitational wave. We confirm that the black mirror spacetime does not contain any such singularity, and compare our result with previous claims that the curvature components are analytic. We also discuss the global structure of the black mirror spacetime, and examine what happens to worldlines and curves passing through.

gr-qc↗
arXiv2024

A New Division Algebra Representation of $E_7$

We decompose the Lie algebra $\mathfrak{e}_{8(-24)}$ into representations of $\mathfrak{e}_{7(-25)}\oplus\mathfrak{sl}(2,\mathbb{R})$ using our recent description of $\mathfrak{e}_8$ in terms of (generalized) $3\times3$ matrices over pairs of division algebras. Freudenthal's description of both $\mathfrak{e}_7$ and its minimal representation are therefore realized explicitly within $\mathfrak{e}_8$, with the action given by the (generalized) matrix commutator in $\mathfrak{e}_8$, and with a natural parameterization using division algebras. Along the way, we show how to implement standard operations on the Albert algebra such as trace of the Jordan product, the Freudenthal product, and the determinant, all using commutators in $\mathfrak{e}_8$.

math.GR↗
arXiv2023

A New Division Algebra Representation of $E_6$

We construct the well-known decomposition of the Lie algebra $\mathfrak{e}_8$ into representations of $\mathfrak{e}_6\oplus\mathfrak{su}(3)$ using explicit matrix representations over pairs of division algebras. The minimal representation of $\mathfrak{e}_6$, namely the Albert algebra, is thus realized explicitly within $\mathfrak{e}_8$, with the action given by the matrix commutator in $\mathfrak{e}_8$, and with a natural parameterization using division algebras. Each resulting copy of the Albert algebra consists of anti-Hermitian matrices in $\mathfrak{e}_8$, labeled by imaginary (split) octonions. Our formalism naturally extends from the Lie algebra to the Lie group $E_6\subset E_8$.

math.GR↗
arXiv2022

Octions: An $E_8$ description of the Standard Model

We interpret the elements of the exceptional Lie algebra $\mathfrak{e}_{8(-24)}$ as objects in the Standard Model, including lepton and quark spinors with the usual properties, the Standard Model Lie algebra $\mathfrak{su}(3)+\mathfrak{su}(2)+\mathfrak{u}(1)$, and the Lorentz Lie algebra $\mathfrak{so}(3,1)$. Our construction relies on identifying a complex structure on spinors and then working in the enveloping algebra. The resulting model naturally contains GUTs based on $SO(10)$ (Georgi--Glashow), $SU(5)$ (Georgi), and $SU(4)\times SU(2)\times SU(2)$ (Pati--Salam). We then briefly speculate on the role of the remaining elements of $\mathfrak{e}_8$, and propose a mechanism leading to exactly three generations of particles.

hep-ph↗
arXiv2019

Reflections on the Energy of Black Holes

Inside a black hole, there is no local way to say which side of a sphere is the inside, and which is the outside. One can easily be gulled by this fact into mixing up the sign of the energy. We lead the reader astray with a naïve treatment of the energy of a null shell in black hole spacetimes. We then resolve the confusion, showing that global, rather than local, considerations offer good guidance.

gr-qc↗
arXiv2017

Piecewise Conserved Quantities

We review the treatment of conservation laws in spacetimes that are glued together in various ways, thus adding a boundary term to the usual conservation laws. Several examples of such spacetimes will be described, including the joining of Schwarzschild spacetimes of different masses, and the possibility of joining regions of different signatures. The opportunity will also be taken to explore some of the less obvious properties of Lorentzian vector calculus.

gr-qc↗
arXiv2014

Experts' understanding of partial derivatives using the Partial Derivative Machine

Partial derivatives are used in a variety of different ways within physics. Most notably, thermodynamics uses partial derivatives in ways that students often find confusing. As part of a collaboration with mathematics faculty, we are at the beginning of a study of the teaching of partial derivatives, a goal of better aligning the teaching of multivariable calculus with the needs of students in STEM disciplines. As a part of this project, we have performed a pilot study of expert understanding of partial derivatives across three disciplines: physics, engineering and mathematics. Our interviews made use of the Partial Derivative Machine (PDM), which is a mechanical system featuring four observable and controllable properties, of which any two are independent. Using the PDM, we probed expert understanding of partial derivatives in an experimental context in which there is not a known functional form. Through these three interviews, we found that the mathematicians exhibited a striking difference in their understanding of derivatives relative to the other groups. The physicists and engineers were quick to use measurements to find a numeric approximation for a derivative. In contrast, the mathematicians repeatedly returned to speculation as to the functional form, and although they were comfortable drawing qualitative conclusions about the system from measurements, were reluctant to approximate the derivative through measurement. This pilot study led us to further questions. How do fields differ in their experts' concept image of partial derivatives? What representations of partial derivatives are preferred by experts? We plan to address these questions by means of further interviews with a wider range of disciplinary experts.

physics.ed-ph↗
arXiv2013

Division Algebra Representations of SO(4,2)

Representations of $\text{SO}(4,2)$ are constructed using $4\times4$ and $2\times2$ matrices with elements in $\mathbb{H}'\otimes\mathbb{C}$, and the known isomorphism between the conformal group and $\text{SO}(4,2)$ is written explicitly in terms of the $4\times4$ representation.

math.RA↗
arXiv2013

A Symplectic Representation of $\mathrm{E}_7$

We explicitly construct a particular real form of the Lie algebra $\mathfrak{e}_7$ in terms of symplectic matrices over the octonions, thus justifying the identifications $\mathfrak{e}_7\cong\mathfrak{sp}(6,\mathbb{O})$ and, at the group level, $\mathrm{E}_7\cong\mathrm{Sp}(6,\mathbb{O})$. Along the way, we provide a geometric description of the minimal representation of $\mathfrak{e}_7$ in terms of rank 3 objects called cubies.

math.RA↗
arXiv2013

Toy model studies of tuning and typicality with an eye toward cosmology

We investigate a number of simple toy models to explore interesting relationships between dynamics and typicality. We start with an infinite model that has been proposed as an illustration of how non-ergodic dynamics can produce interesting features that are suggestive for cosmological applications. We consider various attempts to define the infinite model more rigorously as a limit of a finite system. None of our attempts at such rigor were able to preserve the attractive properties. We hope our work will challenge others to find more successful toy models. The difficulty of finding such models suggests that connections between dynamics and typicality we hope for in cosmological theories such as eternal inflation may not be so easy to achieve.

astro-ph.CO↗
arXiv2012

E6, the Group: The structure of SL(3,O)

We present the subalgebra structure of sl(3,O), a particular real form of e6 chosen for its relevance to particle physics and its close relation to generalized Lorentz groups. We use an explicit representation of the Lie group SL(3,O) to construct the multiplication table of the corresponding Lie algebra sl(3,O). Both the multiplication table and the group are then utilized to find various nested chains of subalgebras of sl(3,O), in which the corresponding Cartan subalgebras are also nested where possible. Because our construction involves the Lie group, we simultaneously obtain an explicit representation of the corresponding nested chains of subgroups of SL(3,O).

math.RA↗
arXiv2012

Discovering Real Lie Subalgebras of e6 using Cartan Decompositions

The process of complexification is used to classify a Lie algebra and identify its Cartan subalgebra. However, this method does not distinguish between real forms of a complex Lie algebra, which can differ in signature. In this paper, we show how Cartan decompositions of a complexified Lie algebra can be combined with information from the Killing form to identify real forms of a given Lie algebra. We apply this technique to sl(3,O), a real form of e6 with signature (52,26), thereby identifying chains of real subalgebras and their corresponding Cartan subalgebras within e6. Motivated by an explicit construction of sl(3,O), we then construct an abelian group of order 8 which acts on the real forms of e6, leading to the identification of 8 particular copies of the 5 real forms of e6, which can be distinguished by their relationship to the original copy of sl(3,O).

math.RA↗
arXiv2011

Covariant Derivatives on Null Submanifolds

The degenerate nature of the metric on null hypersurfaces makes it difficult to define a covariant derivative on null submanifolds. Recent approaches using decomposition to define a covariant derivative on null hypersurfaces are investigated, with examples demonstrating the limitations of the methods. Motivated by Geroch's work on asymptotically flat spacetimes, conformal transformations are used to construct a covariant derivative on null hypersurfaces, and a condition on the Ricci tensor is given to determine when this construction can be used. Several examples are given, including the construction of a covariant derivative operator for the class of spherically symmetric hypersurfaces.

gr-qc↗
arXiv2011

Taxicab Angles and Trigonometry

A natural analogue to angles and trigonometry is developed in taxicab geometry. This structure is then analyzed to see which, if any, congruent triangle relations hold. A nice application involving the use of parallax to determine the exact (taxicab) distance to an object is also discussed.

math.MG↗
arXiv2010

Octonionic Cayley Spinors and E6

Attempts to extend our previous work using the octonions to describe fundamental particles lead naturally to the consideration of a particular real, noncompact form of the exceptional Lie group E6, and of its subgroups. We are therefore led to a description of E6 in terms of 3x3 octonionic matrices, generalizing previous results in the 2x2 case. Our treatment naturally includes a description of several important subgroups of E6, notably G2, F4, and (the double cover of) SO(9,1), An interpretation of the actions of these groups on the squares of 3-component "Cayley spinors" is suggested.

math.RA↗
arXiv2010

Octonions, E6, and Particle Physics

In 1934, Jordan et al. gave a necessary algebraic condition, the Jordan identity, for a sensible theory of quantum mechanics. All but one of the algebras that satisfy this condition can be described by Hermitian matrices over the complexes or quaternions. The remaining, exceptional Jordan algebra can be described by 3x3 Hermitian matrices over the octonions. We first review properties of the octonions and the exceptional Jordan algebra, including our previous work on the octonionic Jordan eigenvalue problem. We then examine a particular real, noncompact form of the Lie group E6, which preserves determinants in the exceptional Jordan algebra. Finally, we describe a possible symmetry-breaking scenario within E6: first choose one of the octonionic directions to be special, then choose one of the 2x2 submatrices inside the 3x3 matrices to be special. Making only these two choices, we are able to describe many properties of leptons in a natural way. We further speculate on the ways in which quarks might be similarly encoded.

math.RA↗
arXiv2009

Tensor Generalizations of Affine Symmetry Vectors

A definition is suggested for affine symmetry tensors, which generalize the notion of affine vectors in the same way that (conformal) Killing tensors generalize (conformal) Killing vectors. An identity for these tensors is proved, which gives the second derivative of the tensor in terms of the curvature tensor, generalizing a well-known identity for affine vectors. Additionally, the definition leads to a good definition of homothetic tensors. The inclusion relations between these types of tensors are exhibited. The relationship between affine symmetry tensors and solutions to the equation of geodesic deviation is clarified, again extending known results about Killing tensors.

gr-qc↗