Special relativity in multidimensional geometric frameworks

This work generalizes the theory of Special Relativity (SR) to multidimensional geometric frameworks (NG) by extending the invariance of the Minkowski distance to an arbitrary N-dimensional spacetime. A unified methodology is developed to derive Lorentz transformations for even and odd dimensional geometries using the roots of unity (N−1≡k) in the transformation matrices, ensuring the invariance of the generalized interval (ds^′)^k=(ds)^k. The corresponding Lorentz factor is obtained as [Formula presented], which reduces to the conventional form when k=2. The analysis reveals distinct alignment symmetries: face-to-face configurations for even-dimensional and side-by-side configurations for odd-dimensional frameworks. Increasing dimensionality suppresses relativistic effects and introduces a complex spacetime structure characterized by real (visible) and imaginary (dark) time components, providing a geometric interpretation of the dark energy. The results demonstrate that the Lorentz group symmetry and spacetime invariance remain valid in higher-dimensional frameworks, offering new insights into the geometric foundations of relativity, quantum mechanics, and cosmology.