Higher order equations of motion and the dynamics of cosmic expansion

Empirical observations confirm that the universe’s components are diverging at an accelerating rate. This acceleration suggests that the motion of celestial bodies over vast distances is differentiable to a higher order with respect to time. Hubble’s law captures this time-dependent expansion, derived from the exponential time-displacement relation. To describe these large-scale dynamics, while maintaining translational symmetry and energy conservation, higher-order Lagrangian equations are employed to derive equations of motion across cosmic distances. These equations, which have influenced the foundations of classical mechanics, introduce the concept of negative inertia, where mass tends to alter its dynamical state spontaneously. While Newton’s theory of gravitation effectively explains short-range cosmic interactions, its limitations become evident when applied to large-scale phenomena such as galactic rotation and cosmic expansion. Similarly, Einstein’s general relativity, originally formulated for local gravitational effects, struggles to describe the universe as a whole. These challenges lead to fundamental issues, including the instability of a static universe and the presence of negative energy in an expanding one. Although the ΛCDM model successfully explains cosmic expansion, the true nature of dark energy remains unresolved. By deriving higher-order equations of motion within a general relativistic framework, cosmic expansion can be explained in a way that accounts for dark energy as a manifestation of negative or spontaneous inertia rather than an unknown energy form. This approach provides a theoretical basis for the universe’s continuous growth, avoiding predictions of gravitational collapse and the resulting space-time incompleteness.