### 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
for a free particle and assume 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
where is the Newtonian gravitational potential . Applying Hamilton’s equations to solve for
the acceleration yields
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 . Of course, if the 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 [140, 138, 256].