The Duality of Time Dilation and Velocity

Details The Duality of Time Dilation and Velocity The Duality of Time Dilation and Velocity

2005-12-05

arxiv.org/abs/gr-qc/0512019v5

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

7 pages, Latex2e

Scientific paper

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.

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