arXiv subjects

Xavier Lachaume

Publications and source records attributed to Xavier Lachaume.

4 recordsLinked to original sources

arXiv2019

On the number of terms in the Lovelock products

In this short note we wonder about the explicit expression of the expanding of the $p$-th Lovelock product. We use the 1990's works of S. A. Fulling et al. on the symmetries of the Riemann tensor, and we show that the number of independent scalars appearing in this expanding is equal to the number of Young diagrams with all row lengths even in the decomposition of the $p$-th plethysm of the Young diagram representing the symmetries of the Riemann tensor.

gr-qc↗

arXiv2018

The constraint equations of Lovelock gravity theories: a new $σ_k$-Yamabe problem

This paper is devoted to the study of the constraint equations of the Lovelock gravity theories. In the case of an empty, compact, conformally flat, time-symmetric, and space-like manifold, we show that the Hamiltonian constraint equation becomes a generalisation of the $σ_k$-Yamabe problem. That is to say, the prescription of a linear combination of the $σ_k$-curvatures of the manifold. We search solutions in a conformal class for a compact manifold. Using the existing results on the $σ_k$-Yamabe problem, we describe some cases in which they can be extended to this new problem. This requires to study the concavity of some polynomial. We do it in two ways: regarding the concavity of an entire root of this polynomial, which is connected to algebraic properties of the polynomial; and seeking analytically a concavifying function. This gives several cases in which a conformal solution exists. At last we show an implicit function theorem in the case of a manifold with negative scalar curvature, and find a conformal solution when the Lovelock theories are close to General Relativity.

math-ph↗

arXiv2018

$n+1$ formalism of $f($Lovelock$)$ gravity

In this note we perform the $n+1$ decomposition, or Arnowitt Deser Misner (ADM) formulation of $f($Lovelock$)$ gravity theory. The hamiltonian form of Lovelock gravity was known since the work of C. Teitelboim and J. Zanelli in 1987, but this result had not yet been extended to $f($Lovelock$)$ gravity. Besides, field equations of $f($Lovelock$)$ have been recently be computed by P. Bueno et al., though without ADM decomposition. We focus on the non-degenerate case, ie. when the Hessian of $f$ is invertible. Using the same Legendre transform as for $f(\mathrm{R})$ theories, we can identify the partial derivatives of $f$ as scalar fields, and consider the theory as a generalised scalar-tensor theory. We then derive the field equations, and project them along a $n+1$ decomposition. We obtain an original system of constraint equations for $f($Lovelock$)$ gravity, as well as dynamical equations. We give explicit formulas for the $f(\mathrm{R},$ Gauss-Bonnet$)$ case.

gr-qc↗

arXiv2017

On the concavity of a sum of elementary symmetric polynomials

We introduce a new problem on the elementary symmetric polynomials $σ_k$, stemming from the constraint equations of some modified gravity theory. For which coefficients is a linear combination of $σ_k$ $1/p$-concave, with $0 \leq k \leq p$? We establish connections between the $1/p$-concavity and the real-rootedness of some polynomials built on the coefficients. We conjecture that if the restriction of the linear combination to the positive diagonal is a real-rooted polynomial, then the linear combination is $1/p$-concave. Using the theory of hyperbolic polynomials, we show that this would be implied by a short algebraic statement: if the polynomials $P$ and $Q$ of degree $n$ are real-rooted, then $\sum_{k=0}^n P^{(k)}Q^{(n-k)}$ is real-rooted as well. This is not proven yet. We conjecture more generally that the global $1/p$-concavity is equivalent to the $1/p$-concavity on the positive diagonal. We prove all our guessings for $p=2$. The way is open for further developments.

math.CA↗