Intrinsic and extrinsic curvatures in Finsler esque spaces

Abstract

We consider metrics related to each other by functionals of a scalar field $\varphi (x)$ and it’s gradient $\varvec{\nabla }\varphi (x)$, and give transformations of some key geometric quantities associated with such metrics. Our analysis provides useful and elegant geometric insights into the roles of conformal and non-conformal metric deformations in terms of intrinsic and extrinsic geometry of $\varphi$-foliations. As a special case, we compare conformal and disformal transforms to highlight some non-trivial scaling differences. We also study the geometry of 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.

Appendices

Appendix 1: Derivation of Christoffel connection component

We start with the following relations which are straightforward to establish:

$$\begin{aligned} q^m \nabla _m {\overset{\star }{g}}_{bc}&= \left( \nabla _{\varvec{q}} {\Omega ^2}\right) g_{bc} - \epsilon {\mathcal {B}}\left( t_b a_c + t_c a_b \right) - \epsilon \left(\nabla _{\varvec{q}} {\mathcal {B}}\right) t_b t_c \nonumber \\ q^m \nabla _b {\overset{\star }{g}}_{mc}&= - {\mathcal {B}}\nabla _b t_c + \epsilon \left[\nabla _{\varvec{q}} \left({\Omega ^2}-{\mathcal {B}}\right)\right] t_b t_c \end{aligned}$$

(35)

where $\nabla _{\varvec{q}} \equiv q^m \nabla _m$. Using Eq. (35) in Eq. (11), a few steps of algebra give

$$\begin{aligned} {\overset{\star }{\Gamma }}{}^a{}_{bc} T_a&= \sqrt{{\Omega ^2}-{\mathcal {B}}} ~ \Gamma ^a{}_{bc} t_a + \frac{1}{2 \sqrt{{\Omega ^2}-{\mathcal {B}}}} ~ \Delta _{bc} \end{aligned}$$

(36)

where

$$\begin{aligned} \Delta _{bc}&= - \left( \nabla _{\varvec{q}} {\Omega ^2}\right) \; h_{bc} + \epsilon \left[\nabla _{\varvec{q}} \left({\Omega ^2}-{\mathcal {B}}\right)\right] t_b t_c - 2 {\mathcal {B}}K_{(bc)} \nonumber \end{aligned}$$

Appendix 2: Composition law of transformations

As an interesting aside, we note the following composition law for the transformations we are considering in this paper. Using the variables $\left(A={\Omega ^2}, \alpha \right)$, let us consider the following class of metrics, all defined on the same manifold.

$$\begin{aligned} g^{(1)}_{\textit{ab}}&= A_{10} g^{(0)}_{\textit{ab}} - \epsilon \left( A_{10} - \alpha ^{-1}_{10} \right) t^{(0)}_a t^{(0)}_b \nonumber \\ g^{(2)}_{\textit{ab}}&= A_{21} g^{(1)}_{\textit{ab}} - \epsilon \left( A_{21} - \alpha ^{-1}_{21} \right) t^{(1)}_a t^{(1)}_b \end{aligned}$$

(37)

Then, noting that $t^{(1)}_a = t^{(0)}_a/\sqrt{\alpha _{10}}$, it is easy to show that

$$\begin{aligned} g^{(2)}_{\textit{ab}}&= \left( A_{21} A_{10}\right) g^{(0)}_{\textit{ab}} - \epsilon \left[ A_{21} A_{10} - \left( \alpha _{21} \alpha _{10} \right) ^{-1} \right] t^{(0)}_a t^{(0)}_b \end{aligned}$$

(38)

which immediately yields the following simple composition law

$$\begin{aligned} A_{20}&= A_{21} A_{10} \nonumber \\ \alpha _{20}&= \alpha _{21} \alpha _{10} \end{aligned}$$

(39)

or symbolically

(40)