Modified general relativity

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.