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

Authors:  Hunter Monroe
Abstract:  Because the equivalence principle holds as a theorem (of Iliev gr-qc/9709053) in the general theory of relativity along any non-self-intersecting curve, gravitational time dilation of 1/(1-v^2)^(1/2) is indistinguishable (along a curve) from time dilation caused by a relative velocity of v; time dilation and relative velocity are two sides of the same coin even in the general theory. Therefore, Einstein's velocity composition law u=(v+w)/(1+vw) can be rewritten to accumulate time dilation in the general theory according to 1/(1-u^2)^(1/2)=(1+vw)/[(1-v^2)(1-w^2)]^(1/2), not by direct multiplication of time dilation factors 1/(1-u^2)^(1/2)=1/[(1-v^2)(1-w^2)]^(1/2). The unbounded intensification of time dilation observed as an event horizon emerges at the center of a collapsing star is therefore indistinguishable (along a curve) from a relative velocity of the center approaching the speed of light, implying a bubble-like local inflation of the star's interior as the center recedes toward infinity even as the star continues to collapse as seen from the outside (as conjectured by Shatskiy astro-ph/0407222). This potentially provides a generic mechanism through which trapped surfaces and therefore singularities do not occur after gravitational collapse and before the big bang.
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Hunter Monroe
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Joined: November 17 2005
Affiliation: IMF

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

Post by Hunter Monroe » 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.

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