The Einstein Equation with Dynamic Curvature

In standard Friedmann–Lemaître–Robertson–Walker (FLRW) cosmology, the spatial curvature is assumed to be constant in time. We investigate the consequences of relaxing this assumption and derive the corresponding Einstein equations without imposing the standard FLRW curvature function. To this end, we formulate a general relativistic framework with a time-dependent curvature parameter X(t). The central concept is geometrical continuity, which unifies positive, flat, and negative spatial curvature within a single analytic representation, replacing the conventional separation into three distinct curvature sectors while preserving finite and differentiable geometrical relations across the transition through k=0.We derive the Christoffel symbols, the Ricci tensor, the Ricci scalar, and the Einstein tensor for the generalized metric. The derivation is verified by recovering the standard FLRW expressions in the limit X=constant and by satisfying the Bianchi identities.The dynamic curvature generates non-vanishing off-diagonal components of the Ricci and Einstein tensors, together with additional diagonal contributions. As a consequence, spatial homogeneity is no longer preserved and the spacetime becomes spatially inhomogeneous.More importantly, isotropy is no longer guaranteed by the Einstein equations alone. Instead, the generalized curvature function S_X (t,χ) must satisfy a nonlinear partial differential equation, which becomes the necessary and sufficient condition for preserving isotropy. In its divergence form, this equation naturally separates into a spatial Wronskian contribution and a term describing the evolution of the dynamic curvature.We further demonstrate that the straightforward generalizationS_X (χ,X(t))=sin⁡(X(t)χ)/X(t) does not satisfy this isotropy condition and therefore cannot describe an isotropic spacetime with dynamic curvature. In the limit X=constant, however, the equation reduces to the standard FLRW solutionS_X (χ)=sin⁡(Xχ)/X.Finally, we show that the Sachs equation is not an independent dynamical field equation in this framework. After substituting D_A=aR_0 S_X, it reduces identically to a trivial identity, implying that the isotropy equation constitutes the fundamental geometrical constraint governing the dynamic curvature.The temporal and radial components of the Einstein equations yield generalized first and second Friedmann equations. In addition, the mixed space-time component produces a new field equation relating the temporal evolution of the geometry to radial energy-momentum transport, while the angular component becomes equivalent to a partial differential equation enforcing isotropy in the case of a perfect fluid. Together, these results reveal that a dynamically varying spatial curvature modifies not only the Friedmann equations but the overall structure of the cosmological Einstein equations.