  We suggest a new perspective on the Cosmological Constant Problem by
scrutinizing its standard formulation. In classical and quantum mechanics
without gravity, there is no definition of the zero point of energy.
Furthermore, the Casimir effect only measures how the vacuum energy {\it
changes} as one varies a geometric modulus. This leads us to propose that the
physical vacuum energy in a Friedman-Lema\^itre-Robertson-Walker expanding
universe only depends on the time variation of the scale factor $a(t)$.
Equivalently, requiring that empty Minkowski space is gravitationally stable is
a principle that fixes the ambiguity in the zero point energy. On the other
hand, if there is a meaningful bare cosmological constant, this prescription
should be viewed as a fine-tuning.
  We describe two different choices of vacuum, one of which is consistent with
the current universe consisting only of matter and vacuum energy. The resulting
vacuum energy density $\rhovac$ is constant in time and approximately $k_c^2
H_0^2$, where $k_c$ is a momentum cut-off and $H_0$ is the current Hubble
constant; for a cut-off close to the Planck scale, values of $\rhovac$ in
agreement with astrophysical measurements are obtained. Another choice of
vacuum is more relevant to the early universe consisting of only radiation and
vacuum energy, and we suggest it as a possible model of inflation.
