  In a Lorentzian spacetime there exists a smooth regular line element field
$(\bm{X},-\bm{X}) $ and a unit vector $ \bm{u} $ collinear with one of the pair
of vectors in the line element field. An orthogonal decomposition of symmetric
tensors can be constructed in terms of the Lie derivative along $ \bm{X} $ of
the metric and a product of the unit vectors; and a linear sum of
divergenceless symmetric tensors. A modified Einstein equation of general
relativity is then obtained by using the principle of least action, the
decomposition and a fundamental postulate of general relativity. The
decomposition introduces a new symmetric tensor $ \varPhi_{\alpha\beta} $ which
describes the energy-momentum of the gravitational field. It completes
Einstein's equation and addresses the energy localization problem. Variation of
the action with respect to $ X^{\mu} $ restricts $u_{\mu}$ to a particular
value, which defines the possible Lorentzian metrics. $ \Phi $, the trace of $
\varPhi_{\alpha\beta} $, describes dark energy. The cosmological constant is
dynamically replaced by $ \Phi $. A cyclic universe that developed after the
Big Bang is described. The dark energy density provides a natural explanation
of why the vacuum energy density is so small, and why it dominates the present
epoch. Assuming dark matter does not exist, a solution to the modified Einstein
equation introduces two additional terms into the Newtonian radial force
equation, from which the baryonic Tully-Fisher relation is obtained.
