Friday, 6 May 2016

general relativity - Gravitational Redshift around a Schwarzschild Black Hole


Let's say that I'm hovering in a rocket at constant spatial coordinates outside a Schwarzschild black hole.


I drop a bulb into the black hole, and it emits some light at a distance of re from the center, with a wavelength of λe in the rest frame of the bulb.


What would the wavelength of the light be when it reaches me, at robs in terms of the radius at which it is emitted, re?


This is a subquestion from Sean Carroll's Spacetime and Geomtery. Earlier in the chapter, Carroll asserts that any stationary observer (Ui=0) measures the frequency of a photon following a null geodesic xμ(λ) to be



ω=gμνUμdxνdλ


I don't understand where this expression comes from. How does one even conceptualize things like wavelength and frequency of light in terms of general relativistic quantities like U,gμν,ds2, etc?




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