# Wave equation for Schwarzschild metric

• Arman777
In summary, the conversation was about finding the $$\nabla_{\mu}\nabla^{\mu} \Phi$$ for a specific metric, and the use of the formula $$\nabla_{\mu}\nabla^{\mu}\Phi = \frac{1}{\sqrt{-g}}\partial_{\mu}(\sqrt{-g}g^{\mu \nu}\partial_{\nu}\Phi)$$. The speaker provided a calculation for the expression and asked for suggestions on simplifying it. Another participant suggested a book as a resource, and the original speaker expressed concern about violating forum rules by giving direct answers in homework forums.
Arman777
Gold Member
Homework Statement
Wave equation for Schwarzschild metric
Relevant Equations
GR equations
I am trying to find the $$\nabla_{\mu}\nabla^{\mu} \Phi$$ for $$ds^2 = (1 - \frac{2M}{r})dt^2 + (1 - \frac{2M}{r})^{-1}dr^2 + r^2d\Omega^2$$

I have did some calculations by using

$$\nabla_{\mu}\nabla^{\mu}\Phi = \frac{1}{\sqrt{-g}}\partial_{\mu}(\sqrt{-g}g^{\mu \nu}\partial_{\nu}\Phi)$$

and I have found

$$\nabla_{\mu}\nabla^{\mu} \Phi = [g^{tt}\partial^2_t + 2(\frac{1}{r} - \frac{M}{r^2})\partial_r + g^{rr}\partial^2_r + \frac{cot(\theta)}{r^2}\partial_{\theta} + g^{\theta \theta}\partial^2_{\theta} + g^{\phi \phi}\partial^2_{\phi}]\Phi$$

but I am not sure that is this true or it can be further simplified ? Any ideas

Hello, @Arman777 . I am recently studying this problem. I would suggest Spacetime and geometry by Sean M. Carroll, especialy in pages 395 to 400.

Will I violate any rules in this forum if I just suggest a book?

PeroK and vanhees71

vanhees71 said:
Thanks! Sometimes I hesitate to answer in homework forum, because not only there are too many professors, but I am afraid I would give the answer directly which would violate the rules.

vanhees71 and PeroK
Don't worry about the "professors". They are also just human beings. Not giving directly the answer is also my problem with the homework forums, but it's of course much better to give only hints first and let the student find the solution him or herself.

JD_PM and Haorong Wu

## 1. What is the Schwarzschild metric?

The Schwarzschild metric is a mathematical model that describes the curvature of spacetime around a non-rotating, spherically symmetric mass. It is a solution to Einstein's field equations in general relativity and is used to describe the gravitational field around massive objects, such as stars and black holes.

## 2. What is the wave equation for the Schwarzschild metric?

The wave equation for the Schwarzschild metric is a differential equation that describes the propagation of waves in the curved spacetime of the Schwarzschild metric. It is derived from the Klein-Gordon equation, which is a relativistic wave equation, and takes into account the effects of gravity on the propagation of waves.

## 3. How is the wave equation for the Schwarzschild metric derived?

The wave equation for the Schwarzschild metric is derived by applying the Klein-Gordon equation to the curved spacetime described by the Schwarzschild metric. This involves using the metric tensor and the Christoffel symbols to account for the effects of gravity on the wave equation. The resulting equation is then simplified to obtain the final form of the wave equation.

## 4. What are the applications of the wave equation for the Schwarzschild metric?

The wave equation for the Schwarzschild metric has various applications in astrophysics and cosmology. It is used to study the behavior of waves, such as gravitational waves, in the curved spacetime around massive objects. It is also used in the study of black holes and their properties, as well as in the search for gravitational waves in experiments such as LIGO.

## 5. Are there any limitations to the wave equation for the Schwarzschild metric?

Like any mathematical model, the wave equation for the Schwarzschild metric has its limitations. It is based on the assumption of a non-rotating, spherically symmetric mass, which does not accurately describe all objects in the universe. It also does not take into account quantum effects, which are important at very small scales. Additionally, the wave equation may not be applicable in extreme cases, such as near the event horizon of a black hole.

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