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Modeling Transverse Relative Locality

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Abstract

We investigate some aspects of relativistic classical theories with “relative locality”, in which pairs of events established to be coincident by nearby observers may be described as non-coincident by distant observers. While previous studies focused mainly on the case of longitudinal relative locality, where the effect occurs along the direction connecting the distant observer to the events, we here focus on transverse relative locality, in which instead the effect is found in a direction orthogonal to the one connecting the distant observer to the events. Our findings suggest that, at least for theories of free particles, transverse relative locality is as significant as longitudinal relative locality both conceptually and quantitatively. And we observe that “dual gravity lensing” can be viewed as one of two components of transverse relative locality. We also speculate about a type of spacetime noncommutativity for which transverse relative locality could be particularly significant.

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Notes

  1. One can qualify this sort of effects as “dual-gravity lensing” in light of the thesis put forward in Refs. [1, 2] which characterizes relative locality as a manifestation of the, possibly curved, geometry of momentum space. The standard gravitational lensing is caused by spacetime curvature, and this relative-locality-induced “lensing” can be attributed, in light of Refs. [1, 2], to the “dual gravity” of momentum space.

  2. We work in units such that the speed-of-light scale c is set to 1.

  3. The coefficients of the terms ℓΩ 3 and \(\ell\varOmega\vec{P}^{2}\) in C ℓ were arranged in Ref. [3] just so that this speed law for massless particles, 1−ℓ|p|, would be produced. This is how from the more general two-parameter case \(\mathcal{C}_{\ell}= \varOmega^{2} - \vec{P}^{2} +\ell( \gamma' \varOmega^{3} + \gamma'' \varOmega\vec{P}^{2})\) one arrives at the one-parameter case considered here and in Ref. [3]: \(\mathcal{C}_{\ell}= \varOmega^{2} - \vec{P}^{2} +\ell( 2\gamma\varOmega^{3} + (1-2 \gamma) \varOmega\vec{P}^{2})\).

  4. Our choice of considering two soft massless particles and one hard massless particle is somewhat redundant but helps us keep the presentation clearer. With only one soft particle, plus the hard particle, one could already infer all the properties of the transverse relative locality which we are going to discuss (in fact in Fig. 1 only one soft and one hard particle are shown). By contemplating two soft particles we have the luxury of seeing explicitly that coincidences of emission events of soft particles still behave with absolute locality, and this then renders more evident how the event of emission of a hard particle behaves anomalously (with relative locality). Moreover, there is a “relativist tradition” of viewing an event as a crossing of two worldlines, and from that perspective our 3 simultaneous emission events can be viewed as two independent crossing events: the crossing of two soft worldlines and the crossing of the hard worldline with one of the soft worldlines (it is of course irrelevant which one of the soft worldlines is taken into account for this). From this traditional relativist perspective one would describe the relativity of locality as the fact that for Alice the two crossings coincide whereas according to the coordinates of distantly boosted Camilla they do not coincide.

  5. Note that just like a 2+1D κ-Minkowski noncommutativity is linked to the algebra hom(2) (Euclidean-homotheties algebra) our ρ-Minkowski noncommutativity is linked to the algebra e(2) (Euclidean algebra).

  6. In particular, it is easy to verify that instead the standard translations would not satisfy the Jacobi identities with ρ-Minkowski coordinates. In particular, if {P i ,x j }=−δ ij ,{P i ,t}=0, then from (26) it follows that

    $$ \bigl\{t,\{x_i,P_j\}\bigr\}+\bigl\{x_i, \{P_j,t\}\bigr\}+\bigl\{P_j,\{t,x_i\}\bigr \}=\rho\, \epsilon_{ij} $$
    (27)

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Acknowledgements

We are grateful to Michele Arzano for valuable feedback on a first draft of this manuscript. Our work also benefited from discussions of various aspects of relative locality with Laurent Freidel, Giulia Gubitosi, Jerzy Kowalski-Glikman, Flavio Mercati, Giacomo Rosati and Lee Smolin.

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Authors and Affiliations

  1. Dipartimento di Fisica, Università di Roma “La Sapienza”, P.le A. Moro 2, 00185, Roma, Italy

    Giovanni Amelino-Camelia, Leonardo Barcaroli & Niccoló Loret

  2. INFN, Sez. Roma1, P.le A. Moro 2, 00185, Roma, Italy

    Giovanni Amelino-Camelia & Niccoló Loret

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  1. Giovanni Amelino-Camelia
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  2. Leonardo Barcaroli
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  3. Niccoló Loret
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Correspondence to Giovanni Amelino-Camelia.

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Amelino-Camelia, G., Barcaroli, L. & Loret, N. Modeling Transverse Relative Locality. Int J Theor Phys 51, 3359–3375 (2012). https://doi.org/10.1007/s10773-012-1216-5

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