Gravity-incorporated standard model in a generalized differential geometry

A gravity-incorporated standard model is constructed in a generalized diffe rential geometry (GDG) on R-4 x X-2. Here, R-4 and X-2 are the four-dimensi onal Riemann space and two-point discrete space, respectively. A GDG on R-4 x X-2 is Constructed by adding the basis X-n (n = 1, 2) of the differentia l form on X-2 to the ordinary basis dx(mu) on R-4, and so it is a direct ge neralization of the differential geometry on the continuous manifold. A GDG is a version of non-commutative geometry (NCG). We incorporate gravity by simply replacing the derivative partial derivative (mu) by the covariant de rivative partial derivative (mu) + omega (mu) for a general coordinate tran sformation in the definition of the generalized gauge field on R-4 x X-2, k eeping other parts unchanged. The Yang-Mills-Higgs Lagrangian for the stand ard model is obtained by taking the inner product of two generalized field strengths, whereas the Einstein-Hilbert gravitational Lagrangian is created by the inner product of a generalized field strength and a tensor E-b(a) o n local Lorentz space.