Generalized time dilation and length contraction in multiple framework

In this research paper, we present generalized equations for time dilation and length contraction by extending the conventional theory of special relativity. The analysis incorporates three inertial frames: (at rest), (moving with velocity relative to ), and (moving with velocity relative to ). The transformation equations (7.c) and (7.d), previously derived in Supplementary file of the paper “The Generalized theory of special relativity” (Zenodo, 2025), serve as the foundation of this theory. Using these transformations, we derive a novel form of time dilation and length contraction that depends on both and . As a result, if any frame moves with velocity , the time dilation and length contraction observed in that frame do not depend solely on , but also on the velocity of another moving third frame. This dual dependence goes beyond the limitations of classical special relativity and provides a more realistic and generalized framework. A remarkable feature of this study is that even when , i.e., the frame moves at the speed of light , the transformation equations remain consistent and meaningful. In this special case, our theory yields a non-singular, well-defined form that may play an important role in understanding transformations for light-like particles. Additionally, when , the generalized length contraction and time dilation equations naturally reduce to the standard Einstein length contraction and time dilation equation. This approach offers a broader and refined understanding of relativistic kinematics and space-time geometry in multi-frame systems, potentially providing new insights into the motion of high-speed particles, light-like models, and the fundamental structure of space-time.