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2.5 Lorentz violation and the equivalence principle

Lorentz violation implies a violation of the equivalence principle. Intuitively this is clear: In order for there to be Lorentz violation particles must travel on world-lines that are species dependent (and not fully determined by the mass). In various papers dealing with Lorentz violating dispersion relations one will sometimes see the equivalence principle being cited as a motivation for keeping the Lorentz violating terms equal for all particle species. We now give a pedagogical example to show that the equivalence principle is violated even in this case. Consider a dispersion modification of the form
f(4) E2 = m2 + p2 + ---|p|4 (9) EPl
for a free particle and assume f(4) is independent of particle species. If we assume Hamiltonian dynamics at low energy and use the energy as the Hamiltonian, then for a non-relativistic particle in a weak gravitational field we have
2 4 H = m + p--+ f(4)--p----+ V(x), (10) 2m 2mE2Pl
where V(x) is the Newtonian gravitational potential mP(x). Applying Hamilton’s equations to solve for the acceleration yields
( ) i @P-- (4)m2xi--.xi ¨x = - @xi 1 + 6f E (11) Pl
to lowest order in the Lorentz violating term. From this expression it is obvious that the acceleration is mass dependent and the equivalence principle is violated (albeit slightly) for particles of different masses with the same f(4). Of course, if the f (n) terms are different, as is natural with some Lorentz violating models [110], then it is also obviously violated. As a consequence one cannot preserve the equivalence principle with Lorentz violation unless one also modifies Hamiltonian dynamics. Equivalence principle tests are therefore able to also look for Lorentz violation and vice versa (for an explicit example see [13]). Other examples of the relationship between equivalence principle violation and Lorentz violation can be found in [140138256Jump To The Next Citation Point].
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