  We study the singularity formation of smooth solutions of the relativistic
Euler equations in $(3+1)$-dimensional spacetime for both finite initial energy
and infinite initial energy. For the finite initial energy case, we prove that
any smooth solution, with compactly supported non-trivial initial data, blows
up in finite time. For the case of infinite initial energy, we first prove the
existence, uniqueness and stability of a smooth solution if the initial data is
in the subluminal region away from the vacuum. By further assuming the initial
data is a smooth compactly supported perturbation around a non-vacuum constant
background, we prove the property of finite propagation speed of such a
perturbation. The smooth solution is shown to blow up in finite time provided
that the radial component of the initial "generalized" momentum is sufficiently
large.
