# Gravitational Redshift

I want to calculate the gravitational redshift for schwarzchild spacetime. If we take the emitter(source of light) and observer stationary, how can be redshift found?

Frequency seen by observer : $$w=-u^ik_i$$ u-four velocity, k-four wave vector

## 1 Answer

If the emitter and the observer both are stationary, residing at different levels in the same spherically symmetric gravitational field you can calculate the redshift (or blueshift) from the expression:

$$\lambda_o=\lambda_e\frac{\sqrt{1-\frac{2GM}{r_oc^2}}}{\sqrt{1-\frac{2GM}{r_ec^2}}}$$

here $$\lambda_o$$ is perceived wavelength by the observer, $$\lambda_e$$ is the perceived wavelength at the emitter $$r_o$$ is distance from the center of the gravitational field do the observer and $$r_e$$ is distance from the center of the gravitational field to the emitter.

In general relativity the energy of an object at rest in a spherically symmetric gravitational field can be written as: $$E=mc^2\sqrt{1-\frac{2GM}{rc^2}}$$. You can look at the graviational redshift/blueshift as a consequence of this. I guess you can derive the blueshift/redshift formally somehow.

• thank you for your answer. I also want to see how can I get this expression (derivation). – Ali Oz May 28 '19 at 13:22
• You get it from the metric. Note that stationary means dr = 0 in both cases. The time lapse dt then yields the redshift respectively the blueshift. – timm May 29 '19 at 8:23