  In this thesis, a non-standard geometric framework, the "quasi-metric"
framework (QMF), is used to define relativistic space-time. The QMF consists of
a 4-dimensional space-time manifold equipped with two one-parameter families of
Lorentzian 4-metrics ${\bf g}_t$ and ${\bf {\bar g}}_t$ parametrized by a
(unique) global time function $t$. The global time function represents one
extra degenerate time dimension and it defines a "distinguished" foliation of
quasi-metric space-time into spatial hypersurfaces. The metric family ${\bf
{\bar g}}_t$ is found as solutions of field equations, whereas the metric
family ${\bf g}_t$ is found via a local transformation ${\bf {\bar
g}}_t{\rightarrow}{\bf g}_t$ and is used in the equations of motion. The role
of the degenerate dimension is to describe global scale changes between
gravitational and non-gravitational systems. In particular, this yields an
alternative description of the expansion of the Universe. In this thesis, a
quasi-metric theory of gravity is constructed. Like General Relativity, the
field equations have two independent propagating dynamical degrees of freedom.
However, a number of non-standard features makes the field equations unsuitable
for a standard PPN-analysis. This implies that the experimental status of the
theory is not completely clear at this point in time. But the non-metric part
of the theory may be tested rather independently. That is, the theory predicts
that gravitational fields in vacuum and gravitationally bound bodies made of
ideal gas expand like the expansion of the Universe. Several observations
suggest this; e.g., the "Pioneer effect", the mean acceleration of the Moon,
the spin-down of the Earth, palaeo-tidal records, etc. Thus quasi-metric
relativity has experimental support where metric theories fail.
