General Relativity gravitational redshift

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SUMMARY

The discussion focuses on the derivation of the gravitational redshift formula, specifically the relationship between the frequency of light emitted from a gravitational body and its frequency measured at a distance. The key equation presented is \(\frac{\Delta{f}}{f_{0}}=\frac{-gh}{c^2}\), leading to the expression \(\frac{f}{f_{0}}\cong{1-\frac{GM}{Rc^2}}\) after integration. The challenge lies in expressing the gravitational acceleration \(g\) in terms of the gravitational force equation \(F_{g}=G\frac{Mm}{r^2}\), particularly considering the mass of a photon is zero. The integration with respect to height \(h\) is also a critical step in the derivation.

PREREQUISITES
  • Understanding of gravitational physics, particularly gravitational fields.
  • Familiarity with the concept of gravitational potential energy.
  • Basic knowledge of calculus, specifically integration techniques.
  • Knowledge of the properties of photons and their behavior in gravitational fields.
NEXT STEPS
  • Study the derivation of gravitational potential energy in the context of General Relativity.
  • Learn about the implications of gravitational redshift in astrophysics.
  • Explore the mathematical techniques for integrating functions related to gravitational fields.
  • Investigate the role of massless particles in gravitational interactions.
USEFUL FOR

Students of physics, particularly those studying General Relativity, astrophysicists, and anyone interested in the effects of gravity on light propagation.

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Homework Statement


The gravitational redshift tends to decrease the frequency of light as it travels upwards a distance h,\frac{\Delta{f}}{f_{0}}=\frac{-gh}{c^2}
integrate both sides of this equation (from the surface of the gravitation body out to infinity) to derive the expression for the change in frequency near a high gravitational field:\frac{f}{f_{0}}\cong{1-\frac{GM}{Rc^2}}

Homework Equations


Given above. A photon is emitted at the surface of the gravitational body (M) with radius R. It's frequency is measured distance h above the gravitational body to be f, while its frequency at the gravitational body is f0. g is the gravitational attraction of the body on the photon.

The Attempt at a Solution


Well, I've gotten far enough to see that \frac{f}{f_{0}}-1=\frac{-gh}{c^2}, which makes sense because gh is the increase in gravitational potential energy.
However, I don't know how to express g. I would use F_{g}=G\frac{Mm}{r^2}, but because a photon's mass is zero, I don't know what to do.
I guess I also need to integrate with respect to h.
Help!?
 
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