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### [gr-qc/0512019] The Duality of Time Dilation and Velocity

Posted: March 23 2006
This paper argues that 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). I would be interested in reactions on the two points below.

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.

Two applications of this duality are presented. First, Einstein's velocity composition law $$u=\frac{v+w}{1+vw}$$ can be rewritten as a Doppler shift law $$\frac{1}{\sqrt{1-u^2}}=\frac{1+vw}{\sqrt{1-v^2}\sqrt{1-w^2}}$$. Under Schwarzschild coordinates, the Doppler shift formula is not $$\frac{\sqrt{1-2M/r_1}}{\sqrt{1-2M/r_2}}$$ but $$\frac{(1-2M/\sqrt{r_1 r_2})/\sqrt{1-2M/r_1}}{\sqrt{1-2M/r_2}}$$. The second order term $$vw$$ in the numerator provides a potential explanation of the Pioneer anomaly if it has been ignored.

Second, 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.