# Frequency of Photon in Schwarzschild Metric?

• unscientific
In summary, the conversation discusses the Schwarzschild metric and a particle orbiting in circular motion at a given radius. The given geodesic equations and Christoffel symbols are used to solve for the frequency of a photon at infinity and its range of frequencies. The conversation then asks for help in finding the momentum and four-velocity of the photon.
unscientific

## Homework Statement

The schwarzschild metric is given by ##ds^2 = -Ac^2 dt^2 + \frac{1}{A} dr^2 + r^2\left( d\theta^2 + sin^2\theta d\phi^2 \right)##. A particle is orbiting in circular motion at radius ##r##.

(a) Find the frequency of photon at infinity ##\omega_{\infty}## in terms of when it is at ##r##, ##\omega_r##.
(b) Find the range of frequencies.

## The Attempt at a Solution

I have found the geodesic equations:
$$\frac{d}{ds}\left[A \dot t \right] = 0$$
$$0 = -Ac^2 \dot {t}^2 + \frac{1}{A} \dot {r}^2 + r^2 \dot {\phi}^2$$
$$\frac{d}{ds} \left[ r^2 \dot \phi \right] = 0$$

I have also found the christoffel symbols: ##\Gamma^r_{rr}=-\frac{A'}{2A}## and ##\Gamma^r_{tt}=-\frac{1}{2}A A'c^2## and ##\Gamma^r_{\theta \theta} = -Ar## and ##\Gamma^r_{\phi\phi}=-Ar## and ##\Gamma^t_{rt}=\Gamma^r_{tr}=\frac{A'}{2A}##.

Part(a)
The differential equation is simply the geodesic equation for a photon:
$$\dot p^\mu + \frac{1}{\hbar} \Gamma^\mu_{\alpha \beta} p^\alpha p^\beta = 0$$
For the temporal component we have exactly what the question wants:
$$\dot p^0 = - \frac{A'}{A}p^0 \dot r$$

The energy of the photon is given by ##E = p_\mu U^\mu =p_0 U^0+p_1 U^1+p_2 U^2+p_3 U^3 = \hbar \omega##.

How do I find ##p^\mu## and ##U^\mu##?

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## 1. What is the Schwarzschild Metric?

The Schwarzschild Metric is a mathematical solution to Einstein's field equations in general relativity, which describes the curvature of space and time around a spherical mass. It is commonly used to study the effects of gravity on light, as well as the behavior of objects near massive bodies like stars and black holes.

## 2. How does the Schwarzschild Metric affect the frequency of photons?

The Schwarzschild Metric predicts that the frequency of photons (light particles) will change as they travel through space near a massive object. This is due to the curvature of space and time caused by the massive object, which can cause the photon's wavelength to stretch or compress, resulting in a change in frequency.

## 3. Why is it important to study the frequency of photons in the context of the Schwarzschild Metric?

Studying the frequency of photons in the context of the Schwarzschild Metric can provide insights into the behavior of light near massive objects, which is crucial for understanding the effects of gravity. This can also help us better understand the properties of black holes, which have an extremely strong gravitational pull that can significantly affect the frequency of surrounding photons.

## 4. How is the frequency of a photon affected by the distance from the massive object?

The frequency of a photon is inversely proportional to the distance from the massive object. As the distance increases, the curvature of space and time decreases, resulting in a smaller change in the photon's frequency. In other words, the closer the photon is to the massive object, the more significant the change in frequency will be.

## 5. Can the frequency of photons be measured in the presence of a black hole using the Schwarzschild Metric?

Yes, the frequency of photons can be measured in the presence of a black hole using the Schwarzschild Metric. This has been demonstrated through observations of light from stars orbiting around supermassive black holes at the center of galaxies. The changes in the frequency of these photons can provide valuable information about the properties and behavior of black holes.

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