Shell Coordinates
Imagine a series of spherical shells held stationary against the force of gravity by rockets outside the event horizon. The shell observer, standing on the spherical shell, can form his own shell frame. When constricted to small enough region, the space-time in the shell frame can be regarded as flat. Special relativity works in the shell frame. Consider two sequential clicks of a clock at rest on a spherical shell, dr, dθ, dφ of the Schwarzschild coordinates is 0. Therefore, dts, the time coordinate in the shell frame is the proper time. Substitute into the Schwarzschild metric,
Take the square root:
Similarly, by considering two simultaneous events located in the same radial direction on two spherical shells separated by distance dr, the equation for drs can be derived:
Locally, with the shell coordinates, the metric for the shell frame is thus expressed in hybrid form:
Read more about this topic: Gullstrand–Painlevé Coordinates
Famous quotes containing the word shell:
“I seem to have been only like a boy playing on the seashore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me.”
—Isaac Newton (1642–1727)