Anisotropic Special Relativity

Abstract

Anisotropic Special Relativity (ASR) is the relativistic theory of nature with a preferred direction in space-time. By relaxing the \textit{full-isotropy} constraint on space-time to the \textit{preference of one direction}, we obtain a perturbative modification of the Minkowski metric as $\mathscr{g}_{\mu\nu}\simeq\eta_{\mu\nu}+2\phi\epsilon_{\mu\nu}$ for a small perturbation parameter $\phi$. The symmetry group of ASR is obtained to have six generators satisfying the full Lorentz group algebra. However, the generators are deformed by the perturbation parameter $\phi$. So, ASR retains the same representations of Special Relativity (SR) but allows for Lorentz-invariant violation at the same time. Any invariant quantity of the theory is the inner product of two contravariant 4-vectors mediated by ${g}_{\mu\nu}$. The mass of a the particle is modified to $m^2=P^{\mu}\mathscr{g}_{\mu\nu}P^{\nu}$ which, in the first approximation level, has the extra term $(2\phi\epsilon_{\mu\nu})P^{\mu}P^{\nu}$ compared to the mass of the particle in SR. So, one application of ASR is, for example, to explain the neutrino flavour oscillation experiments in a natural way without violating the lepton number or adding sterile right-handed neutrinos. The mass of a particle is not the only quantity that is modified in ASR; any scalar quantity such as the Lagrangian of fields are also modified since the anisotropic metric $\mathscr{g}_{\mu\nu}$ is used to contract any pair of covariant-contravariant indices. As a more general consequence of ASR, any Quantum Field Theory (QFT) becomes anisotropic since the Lagrangian must contain the anisotropic metric. So, we provide a procedure to make anisotropic QFTs where the Lorentz-invariant Lagrangians are replaced with their ASR version.