[gr-qc/0512019] The Duality of Time Dilation and Velocity
Posted: March 23 2006
This paper argues that time dilation [tex]\frac{1}{\sqrt{1-v^2}}[/tex] and relative velocity [tex]v[/tex] 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 [tex]r[/tex], a distant observer sees time dilation of [tex]\frac{1}{\sqrt{1-v^2}}=\frac{1}{\sqrt{1-2M/r}}[/tex] under the Schwarzschild metric and sees the clock receding with a relative velocity of [tex]v=\sqrt{2M/r}[/tex] under the Painlev\'e-Gullstrand free fall metric.
Two applications of this duality are presented. First, Einstein's velocity composition law [tex]u=\frac{v+w}{1+vw}[/tex] can be rewritten as a Doppler shift law [tex]\frac{1}{\sqrt{1-u^2}}=\frac{1+vw}{\sqrt{1-v^2}\sqrt{1-w^2}}[/tex]. Under Schwarzschild coordinates, the Doppler shift formula is not [tex]\frac{\sqrt{1-2M/r_1}}{\sqrt{1-2M/r_2}}[/tex] but [tex]\frac{(1-2M/\sqrt{r_1 r_2})/\sqrt{1-2M/r_1}}{\sqrt{1-2M/r_2}}[/tex]. The second order term [tex]vw[/tex] 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 [tex]r>0[/tex] is indistinguishable (along a curve) from an increasing relative velocity at which the center recedes as seen from any direction, implying a local inflation.
For example, on a clock stationary at radius [tex]r[/tex], a distant observer sees time dilation of [tex]\frac{1}{\sqrt{1-v^2}}=\frac{1}{\sqrt{1-2M/r}}[/tex] under the Schwarzschild metric and sees the clock receding with a relative velocity of [tex]v=\sqrt{2M/r}[/tex] under the Painlev\'e-Gullstrand free fall metric.
Two applications of this duality are presented. First, Einstein's velocity composition law [tex]u=\frac{v+w}{1+vw}[/tex] can be rewritten as a Doppler shift law [tex]\frac{1}{\sqrt{1-u^2}}=\frac{1+vw}{\sqrt{1-v^2}\sqrt{1-w^2}}[/tex]. Under Schwarzschild coordinates, the Doppler shift formula is not [tex]\frac{\sqrt{1-2M/r_1}}{\sqrt{1-2M/r_2}}[/tex] but [tex]\frac{(1-2M/\sqrt{r_1 r_2})/\sqrt{1-2M/r_1}}{\sqrt{1-2M/r_2}}[/tex]. The second order term [tex]vw[/tex] 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 [tex]r>0[/tex] is indistinguishable (along a curve) from an increasing relative velocity at which the center recedes as seen from any direction, implying a local inflation.