A generalized relativistic billiard is the following dynamical system: a particle moves under the influence of some force fields in the interior of a domain with pseudo-Riemannian metric, and as the particle hits the boundary of the domain, its velocity is transformed as if the particle underwent an elastic collision with a moving wall, considered in the framework of the special theory of relativity. We study a periodic and a ‘‘monotone’’ action of the boundary. We prove that in both cases under some general conditions the invariant manifold in the velocity phase space of the generalized relativistic billiard, where the point velocity equals the velocity of light, is an exponential attractor or contains one, and for an open set of initial conditions the particle energy tends to infinity.