This paper proposes a new method called "Einstein Tensor Cycle Transformation (ETC transformation)" for a unified understanding of gravity and electromagnetism, two fundamental forces in nature. This approach applies the "exponential" mathematical structure that has been successful in electromagnetic theory to gravitational theory as well. In electromagnetic theory, the phase factor $e^{i\theta(x)}$ plays a crucial role; by extending this structure to gravity and using the Einstein tensor $G_{\mu\nu}$ cyclically as a new metric, we can treat both forces within a common framework. In cosmological expansion models, we demonstrate the existence of fixed points $G'_{\mu\nu} = G_{\mu\nu}$ for ETC transformation when the scale factor is $a(t) = a_0 e^{H_0 t}$ (flat universe) or $a(t) \sim \sinh(H_0 t)$, $\cosh(H_0 t)$ (curved universe). This discovery provides a new perspective on quantum gravity theory with potential for verification through future cosmological observations.