Invariant Contraction Leads to Anisotropy-Covariant Expressions

The paper develops the mathematical expressions of momentum and energy as functions of speed within the framework of relativity with synchronization-induced anisotropy, using the invariant contractions of the velocity and momentum four-vectors. This form of anisotropy arises from the freedom in choosing the synchronization convention for distant clocks. In that case, the one-way speeds of light in opposite directions need not be equal, although the measured round-trip speed remains invariant. Einstein’s special relativity appears as the symmetric limit of a broader relativistic framework based on asynchronous synchronization. In Einstein’s isotropic relativity, the dependence of momentum and energy on speed is governed by the Lorentz factor, which approaches unity in the low-speed limit, u << c. However, it is shown that these expressions are invariant with respect to the synchronization anisotropy parameter, namely the governing equations remain structurally invariant, regardless of the value of the anisotropy parameter. We then extend the analysis to a relativistic framework formulated with complex four-vectors and repeat the derivation for complex anisotropy. Remarkably, the same momentum and energy relations as functions of speed are recovered, exactly as in Einstein’s isotropic theory. Hence, the resulting expressions remain independent of the synchronization convention, even when the synchronization anisotropy parameter is allowed to take complex values.