The Duality of Time Dilation and Velocity

TL;DR

The paper explores the duality between time dilation and velocity in relativity, showing how they are observationally indistinguishable and imply phenomena like local inflation during gravitational collapse.

Contribution

It demonstrates the duality between time dilation and velocity in both special and general relativity, extending the concept to Fermi coordinates and implications for gravitational collapse.

Findings

Time dilation and velocity are observationally indistinguishable in relativity.

Duality implies local inflation during gravitational collapse.

Time dilation can be interpreted as receding velocity in certain metrics.

Abstract

Time dilation $\frac{1}{\sqrt{1-v^2}}$ and relative velocity $v$ are observationally indistinguishable in the special theory of relativity, a duality that carries over into the general theory under Fermi coordinates along a curve (in coordinate-independent language, in the tangent Minkowski space along the curve). For example, on a clock stationary at radius $r$, a distant observer sees time dilation of $\frac{1}{\sqrt{1-v^2}}=\frac{1}{\sqrt{1-2M/r}}$ under the Schwarzschild metric and sees the clock receding with a relative velocity of $v=\sqrt{2M/r}$ under the Painlev{\'e}-Gullstrand free fall metric. Duality implies that during gravitational collapse, the intensifying time dilation observed at the star's center from a fixed radius $r>0$ is indistinguishable (along a curve) from an increasing relative velocity at which the center recedes as seen from any direction, implying a local inflation.