  For relativistic quantum field theories, we consider Lorentz breaking,
spatially homogeneous field configurations or states that evolve in time along
a symmetry direction. We dub this situation "spontaneous symmetry probing"
(SSP). We mainly focus on internal symmetries, i.e. on symmetries that commute
with the Poincare group. We prove that the fluctuations around SSP states have
a Lagrangian that is explicitly time independent, and we provide the field
space parameterization that makes this manifest. We show that there is always a
gapless Goldstone excitation that perturbs the system in the direction of
motion in field space. Perhaps more interestingly, we show that if such a
direction is part of a non-Abelian group of symmetries, the Goldstone bosons
associated with spontaneously broken generators that do not commute with the
SSP one acquire a gap, proportional to the SSP state's "speed". We outline
possible applications of this formalism to inflationary cosmology.
