Christoffel symbols for gravitational waves

In summary, the Christoffel symbol \Gamma^{t}_{xx} for the given metric is \frac{h\omega}{2} \cos(\omega t) in natural units. The equation of motion can be simplified using the equation for Christoffel symbols in terms of permutations of first derivatives of the metric.
  • #1
e^ipi=-1
5
0

Homework Statement


Determine the Christoffel symbol [tex]\Gamma^{t}_{xx}[/tex] for the metric [tex]ds^2 = -c^2dt^2 + (1+h\sin(\omega t))dx^2 + (1-h\sin(\omega t))dy^2 + dz^2[/tex]

The answer should be: [tex]\frac{h\omega}{2} \cos(\omega t)[/tex]

Homework Equations


For the evaluation we have to use [tex]\frac{d^2x^\alpha}{d\tau^2}+\Gamma^\alpha_{\mu\nu}\frac{dx^\mu}{d\tau}\frac{dx^\nu}{d\tau}[/tex]


The Attempt at a Solution


I keep getting c's where they shouldn't be. I calculated the Euler Lagrange Equation for the time to be:
[tex]-2\frac{d^2(ct)}{d\tau^2}-c^{-1}\omega h\cos(\omega t)((\frac{dx}{d\tau})^2 - (\frac{dy}{d\tau})^2) = 0[/tex]
Which leaves us with the equation of motion
[tex]\frac{d^2 t}{d\tau^2}+\frac{1}{2c^2}\omega h\cos(\omega t)((\frac{dx}{d\tau})^2 - (\frac{dy}{d\tau})^2) =0[/tex]
So the answer is:
[tex]\Gamma^{t}_{xx}=\frac{h\omega}{2c^2} \cos(\omega t)[/tex]
Where have I gone wrong? Also, I don't understand whether you are supposed to take t or ct as the zero'th coordinate and whether it gives a different answer.
 
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  • #2
are you sure you're not working in natural units because I get the same answer as you. i don't think it cancels. you could redefine t' = ct but that still gives you factors of c
 
  • #3
Don't know if you already figured it out but the problem probably wanted it in natural units. Also, when you only have one specific Christoffel symbol to calculate it is much easier to just use the equation for the Christoffel symbols in terms of permutations of first derivatives of the metric; I am sure you know which one this is (too lazy to do all the latex business).
 

1. What are Christoffel symbols for gravitational waves?

Christoffel symbols are mathematical objects used to describe the curvature of space-time in the theory of general relativity. They are used to calculate the path of objects in the presence of a gravitational field, such as that created by a massive object like a planet or star. In the context of gravitational waves, Christoffel symbols help us understand how the fabric of space-time is distorted by these ripples in the gravitational field.

2. How are Christoffel symbols related to gravitational waves?

Christoffel symbols are related to gravitational waves because they are used to calculate the effects of these waves on the curvature of space-time. In particular, the Christoffel symbols help us understand how the wave's amplitude and frequency affect the curvature of space-time in different directions. This information is crucial for predicting the behavior of objects in the presence of gravitational waves.

3. Can Christoffel symbols be used to detect gravitational waves?

While Christoffel symbols themselves cannot be used to detect gravitational waves, they are an important tool in the study of these waves. By understanding how gravitational waves affect the curvature of space-time, scientists can develop methods for detecting and measuring these waves. In particular, the Christoffel symbols are used in the analysis of data from gravitational wave detectors, such as LIGO and VIRGO.

4. How do Christoffel symbols change in the presence of strong gravitational waves?

In the presence of strong gravitational waves, the Christoffel symbols can change significantly. This is because the waves themselves are causing distortions in the curvature of space-time, which is reflected in the values of the symbols. The changes in the Christoffel symbols can help us understand the strength and direction of the gravitational waves, as well as their impact on nearby objects.

5. Are there any limitations to using Christoffel symbols for gravitational waves?

While Christoffel symbols are a useful tool for understanding gravitational waves, they do have some limitations. For example, they are based on the assumptions of general relativity, which may not fully describe the behavior of gravity in extreme conditions. Additionally, calculating the Christoffel symbols can be a complex and time-consuming process, so they may not be suitable for certain types of analyses. However, overall, the use of Christoffel symbols is a valuable technique for studying and predicting the behavior of gravitational waves.

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