Intrinsic and Extrinsic curvatures in Finsler-esque spaces

Abstract

We consider metrics related to each other by functionals of a scalar field $\phi (x)$ and it's gradient $\nabla \phi (x)$, and give transformations of some key geometric quantities associated with such metrics. Our analysis provides useful and elegant geometric insights into the roles of {\it conformal} and {\it non-conformal} metric deformations in terms of intrinsic and extrinsic geometry of $\phi$-foliations. As a special case, we compare {\it conformal} and {\it disformal} transforms to highlight some non-trivial scaling differences. We also study the geometry of {\it equi-geodesic} surfaces formed by points $p$ at constant geodesic distance $\sigma (p,P)$ from a fixed point $P$, and apply our results to a specific disformal geometry based on $\sigma (p,P)$ which was recently shown to arise in the context of spacetime with a minimal length.

arXiv comment8 pages, 1 figure; Sec IV now includes a more refined expression for Ricci scalar completely in terms of conformal (3-)geometry, and discusses a couple of applications; to appear in Gen. Rel. Grav._

References in corpus (6)

• Relativistic gravitation theory for the MOND paradigm
• The motion of point particles in curved spacetime
• The Relation between Physical and Gravitational Geometry
• Resembling dark energy and modified gravity with Finsler-Randers cosmology
• Minimal Length and Small Scale Structure of Spacetime
• Relativistic Finsler geometry

Cited by in corpus (18)

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• Raychaudhuri equation with zero point length
• Quantum metric for null separated events and spacetime atoms
• Action and Observer dependence in Euclidean quantum gravity
• Euclidean Action and the Einstein tensor
• A Local First Law of Gravity
• Minimum-length Ricci scalar for null separated events
• Information content and minimum-length metric: A drop of light
• Effective null Raychaudhuri equation
• Normal coordinates based on curved tangent space
• Zero-point gravitational field equations
• Spacetime atoms and extrinsic curvature of equi-geodesic surfaces
• Disformal transformations and the motion of a particle in semi-classical gravity
• Expectation values of minimum-length Ricci scalar
• Quantum states for a minimum-length spacetime