$\alpha \omega$ Gravity Frequently Asked Questions

Auxillary Derivations

The constant speed of light as a universal velocity transformation

It's important to note that in this model, all velocities transform in the same manner as to make them equivalent in each reference frame.

Consider a simplified one body system where we scale $T$ so that it's coordinate system matches $T_{B}$ at some instant in time. Let $B$ then go in motion some $\vec{u}$, which then gives $\vec{u}_{B} = \gamma ~ dt ~ \vec{u}_0$ in $T_{B}$.

Here, 'B' measures $dx = \left| \vec{u} \right|$ as he is measuring relative to $T_{B}$, where as 'A' measures the relativistic magnitude for 'B' in $T_A$.

Where $\vec{u}_{x,A}$ represents the $x$ component of the $\vec{u}$ vector in the coordinate system of 'A' in the lighthouse and the clock-tower thought experiment, and $\hat{u}$ represents the magnitude of velocity per unit 'time', giving the velocity in units of distance.

In the coordinate system of $B$, where for a simplified one-body system we find:

Then $\vec{u}_B$ in $T_{B}$ at $t_{0} + dt$, as measured in $T_{B}$, is equivalent $\vec{u}_A$, this same vector as measured in the coordinate system of 'A'. This reproduces the Newtonian magnitude of $\vec{u}$ in $T_{B}$, as measured using the unit vector magnitude of $T_{B}$, but the relativistic magnitude when considering the magnitude of $u_{B}$ in the coordinate system $T_{A}$.

This then gives that as $c_{B} = c_{A}$, where $c_{A}$ and $c_{B}$ represent the speed of light in the subscript's coordinate system, but we should then recognize that

Where $\left| \vec{z}_{\eta} \right|_\mu$ represents the magnitude of the $z$ vector, in the coordinate system of $\eta$ as measured proportional to the coordinate system of $\mu$. This demonstrates that an observer in relative motion will not only measure $c$ to be constant, he will measure all velocities to follow this pattern, dilating proportional to the underlying coordinate system.