<p>The curved spacetime counterparts γ^<i>a</i> of the Dirac γ matrices presented in the curved spacetime Dirac equation is a g-valued vector field on the spacetime manifold. It determines the spacetime metric <i>g</i>_<i>ab</i> and spacetime geometry, and can be used as the dynamical variable of the gravitational field and the spacetime geometry. The Riemann curvature tensor of the metric connection, the Lagrangian density of the gravitational field and the Einstein-Hilbert action can be expressed in terms of the γ^<i>a</i> vector field. The dynamical equation of the gravitational field in terms of γ^<i>a</i> has been derived by means of the principle of least action, which is equivalent to Einstein field equation. The stress-energy–momentum tensors of the matter fields are expressed in terms of the derivative of the Lagrangian density of the matter field with respect to the γ^<i>a</i> field. The γ^<i>a</i> field has enormous gauge freedom, three kinds of constraints including the strong, weak and weakest constraints can be imposed on the γ^<i>a</i> field. A detailed examination was conducted of the gauge invariance of the gravitational field γ^<i>a</i> and the unitary equivalence of the gravitational field. The availability of the gauges fixed by the constraints has been systematically studied, and the necessary and sufficient conditions for the existence of the constraints-fixed gauges have been provided.</p>