# Rainbow metric formalism and Relative Locality

N. LORET∗, L. BARCAROLI† Dipartimento di Fisica, Università di Roma ”La
Sapienza”,
and INFN, Sez. Roma1,
P.le A. Moro 2, 00185 Roma, EU.
†E-mail<EMAIL_ADDRESS>∗E-mail:
<EMAIL_ADDRESS>http://www.phys.uniroma1.it

###### Abstract

This proceeding is based on a talk prepared for the XIII Marcell Grossmann
meeting. We summarise some results of work in progress in collaboration with
Giovanni Amelino-Camelia about momentum dependent (Rainbow) metrics in a
Relative Locality framework and we show that this formalism is equivalent to
the Hamiltonian formalization of Relative Locality obtained in
arXiv:1102.4637.

Relative Locality; Rainbow metrics; $\kappa$-Poincaré; Curved momentum-space.

##  I Introduction

In literature Planck length111We adopt units such that the speed-of-light
scale is $1$ ($c=1$) $L_{P}=\sqrt{\hbar G}$ is considered a natural guess as
quantum gravity characteristic length scale. However, $L_{P}$ is non zero only
in a regime in which $G$ and $\hbar$ are non zero too, so this hypotesis
requires a full fledged quantum gravity theory to elaborate it. We here focus
on a ”classical non-gravitational” regime bob ; principle ; kbob  of quantum
gravity. In this regime $\hbar$ and $G$ are both neglected while their ratio
is fixed: $M_{P}=\sqrt{\hbar/G}$. In this paper from now on we will formalize
the features of obstruction of measurability and momentum-space deformation
with $\kappa$-Poincaré Hopf algebra majruegg ; lukruegg  in 1+1D:

$\displaystyle\\{{\cal N},p_{0}\\}=p_{i}\;,\;\\{{\cal
N},p_{1}\\}=\left(\frac{1-e^{-2\ell
p_{0}}}{2\ell}-\frac{\ell}{2}p^{2}\right)\;,\;\\{p_{1},p_{0}\\}=0\;,$ (1)

where ${\cal N}$ is the boost operator, $P_{\alpha}$ are the translation
operators and where the Casimir operator $\mathcal{C}$ is:

${\cal C}=\left(\frac{2}{\ell}\sinh\left(\frac{\ell
p_{0}}{2}\right)\right)^{2}-p_{1}^{2}e^{\ell p_{0}}.$ (2)

The algebric sector of $\kappa$-Poincaré can be interpreted as an example of
DSR theorygacIJMPD  with $\ell\sim 1/M_{P}$ the scale of deformation. It is,
therefore, always possible to rely on the existence of a ”classical” limit by
turning off this deformation ($\ell\rightarrow 0$). In this context moreover
it is easier to give a simple interpretation of ”Relative Locality” bob ;
principle  phenomena on physical observables and thus implement a
phenomenology kbob ; transverse . It is well known in literature (see exempli
gratia dawn ; synchrotron ) that this kind of phenomenology is rather elusive
to theoretical investigation. It would be of paramount importance, then, to
formalize some kind of rainbow metrics in relative locality, to try to study
Planckian effects in curved spacetimes models. In order to do so we need to
define the infinitesimal translation of coordinates between two observers
labeled by parameters $\epsilon^{\alpha}$:

$\delta x^{\beta}=\epsilon^{\alpha}\\{p_{\alpha},x^{\beta}\\},$ (3)

using Poisson brackets to express the action of translation operators on
coordinates, as defined in principle ; bob ; kbob .

##  II Some issues with Rainbow Metrics

Smolin and Magueijo SmolinRainbow  proposed that in a framework where free
field theories have plane wave solutions, even though the 4-momentum they
carry satisfies deformed dispersion relations of the form

$p_{0}^{2}f^{2}\left(p_{0}/E_{P}\right)-(p\cdotp
p)g^{2}\left(p_{0}/E_{P}\right)=m^{2},$ (4)

spacetime metric should be modified according to the energy of the particle we
use to probe it. In fact relation (4) can be realized by the action of a
nonlinear map from momentum space to itself, denoted as
$U\,:\,\mathcal{P}\rightarrow\mathcal{P}$, given by

$U\cdotp(p_{0},p_{i})=(U_{0},U_{i})=\left(f\left(p_{0}/E_{P}\right)p_{0},g\left(p_{0}/E_{P}\right)p_{i}\right),$

which implies that momentum space has a nonlinear norm, given by
$|p|^{2}\,=\,\eta^{ab}U_{a}(p)U_{b}(p)$. If one still wants to have at his
disposal a plane wave solution for free fields, since momentum transforms
nonlinearly, the contraction between position and momentum must remain linear.
Smolin and Magueijo suggested that in case momentum transforms nonlinearly,
this can be obtained imposing that
$\zeta^{\alpha\gamma}\tilde{g}_{\gamma\beta}=\delta_{\beta}^{\alpha}$, where
$\zeta^{\alpha\gamma}$ is the metric of momentum space and
$\tilde{g}_{\gamma\beta}$ is the so called ”Rainbow metric”, such as the
spacetime interval

$ds^{2}=\tilde{g}_{\gamma\beta}dx^{\gamma}dx^{\beta}=(dx^{0})^{2}/f^{2}-(dx^{i})^{2}/g^{2}\,,$
(5)

is explicitely energy-dependent. This approach, althought extremely valuable
from the phenomenologic point of view, is problematic in case one is
interested in avoiding to break the line-element invariance. Let’s consider,
for example the first order expansion of the dispersion relation we can obtain
from casimir (2):

$m^{2}=p_{0}^{2}-p_{1}^{2}-\ell p_{1}^{2}p_{0}\,.$ (6)

The mass of (6) is clearly invariant under a boost generator of the form

${\cal N}=x^{0}p_{1}+x^{1}\left(p_{0}-\ell
p_{0}^{2}-\frac{\ell}{2}p_{1}^{2}\right)\,,$ (7)

obtained imposing $\\{{\cal N},{\cal C}\\}=0$, assuming
$\\{p_{\alpha},x^{\beta}\\}=\delta_{\alpha}^{\beta}$. On the other hand, from
equation (6) we can set

$f^{2}(p_{0})=1\;,\;\;\;g^{2}(p_{0})=1+\ell p_{0}\,,$ (8)

and then may identify the element line in the flat spacetime case to be

$ds^{2}=(dx^{0})^{2}-(1-\ell p_{0})(dx^{1})^{2}\,,$ (9)

which is not invariant under the $\ell$-deformed boost (7).

##  III Metric formalism in Relative Locality

We will show that is possible to obtain a similar scenario for the
$\kappa$-Minkowski spacetime framework in which coordinates satisfy the
relation $\\{\chi^{1},\chi^{0}\\}=\ell\chi^{1}$, and its deformed symmetry
generators algebra $\kappa$-Poincar time-to-the-right basis defined in Eq.
(1). Those relations and operators agree with a deformed symplectic sector:

$\begin{array}[]{ll}\\{p_{0},\chi^{0}\\}=1\,,&\,\\{p_{1},\chi^{0}\\}=-\ell
p_{i}\,,\\\ \\{p_{0},\chi^{1}\\}=0\,,&\,\\{p_{1},\chi^{1}\\}=1\,.\end{array}$
(10)

We can easily obtain the metric formalism as generalization of Pitagora’s
theorem, expressing the action of translations with Poisson brakets as in
Eq.(3):

$ds^{2}\equiv(\epsilon^{\alpha}\pi_{\alpha}\rhd\xi^{\gamma})(\epsilon^{\beta}\pi_{\beta}\rhd\xi_{\gamma})=\\{p_{\kappa},\chi^{\mu}\\}\\{p_{\lambda},\chi^{\nu}\\}\frac{\partial\pi_{\alpha}}{\partial
p_{\kappa}}\frac{\partial\xi^{\gamma}}{\partial\chi^{\mu}}\frac{\partial\pi_{\beta}}{\partial
p_{\lambda}}\frac{\partial\xi_{\gamma}}{\partial\chi^{\nu}}d\chi^{\alpha}d\chi^{\beta}\,,$
(11)

in which $(\xi,\pi)$ is the set of coordinates used by an observer at rest
with respect with the center of mass of a certain process, while $(\chi,p)$ is
the set used by a generic one. In the $\ell\rightarrow 0$ limit, in which
$\\{p_{\alpha},\chi^{\mu}\\}\equiv\\{p_{\alpha},x^{\mu}\\}=\delta_{\alpha}^{\mu}$,
the (11) reduces to

$ds^{2}=\eta_{\alpha\beta}e^{\gamma}_{\alpha}e^{\delta}_{\beta}dx^{\alpha}dx^{\beta}$
(12)

where the tetrads are defined as
$e^{\gamma}_{\alpha}(x)\equiv\partial\xi^{\gamma}/\partial x^{\alpha}$. In our
case, however we have to take into account the curvature of momentum-space
contribution, then
$e^{\gamma}_{\alpha}(\chi,p)\equiv\frac{\partial\xi^{\gamma}}{\partial\chi^{\mu}}\frac{\partial\pi_{\alpha}}{\partial
p_{\mu}}$. Using (11) it is also possible to find a metric implementing the
inferences of momentum-space curvature (also known as Relative Locality
effects) on the particle localization process. In fact Eqs. (10) and (11)
define a momentum dipendent (rainbow) metric, at all orders in $\ell$, of the
form:

$\tilde{g}_{\alpha\beta}(\chi,p)=\left(\begin{array}[]{cc}e^{0}_{0}(\chi,p)^{2}&-\ell
p_{1}e^{0}_{0}(\chi,p)^{2}\\\ -\ell
p_{1}e^{0}_{0}(\chi,p)^{2}&-(e^{1}_{1}(\chi,p)^{2}-\ell^{2}p_{1}^{2}e^{0}_{0}(\chi,p)^{2})\end{array}\right),$

which generates an invariant line-element exploring . The minkowskian limit
$\tilde{\eta}_{\alpha\beta}(p)$ of (III) is the result of the inference of a
deSitter-like curvature of momentum-space on flat spacetime, formalized with
the algebra of simmetries we showed in (1). In this regime, using the geodesic
equation for a photon
$\tilde{\eta}_{\alpha\beta}\dot{x}^{\alpha}\dot{x}^{\beta}=0$, one can finally
find the relation

$1-2\ell
p_{1}\frac{d\chi^{1}}{d\chi^{0}}-(e^{1}_{1}(p)^{2}-\ell^{2}(p_{1})^{2})\left(\frac{d\chi^{1}}{d\chi^{0}}\right)^{2}=0$
(13)

This equation, using the dispersion relation we can obtain from Eq. (2), gives
the expression of a wordline (in terms of commutative coordinates
$x^{\alpha}$) for a massless particle:

$x^{1}-\bar{x}^{1}=-e^{\ell p_{0}}(x^{0}-\bar{x}^{0}),$ (14)

which is the same result defined in Ref. kbob ; lateshift  for momentum-
dependant massless wordlines. Rainbow metrics formalism is then equivalent to
the Hamiltonian one in the spacetime Minkowskian limit of Relative Locality,
but it can be even more useful in phenomenology since it naturally implements
spacetime curvature. Further analyses should be dedicated to the comparison
between this approach and the promising investigation of Relative Locality in
curved spacetimes described in lateshift ; jackcurvo ; finsler .

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