What are the Christoffel symbols for the Schwarzschild metric?

In summary, the Schwarzschild metric has a vacuum solution, which tells you that the Ricci tensor and scalar should be zero. The method used to find the Christoffel symbols is by brute forcing the equation where they are equal to derivatives of g.
  • #1
Zeron_X25
7
0
I needed some help with the Christoffel symbols for the Schwarzschild metric. I used the metric in wikipedia with signature (+---). For some reason, I get different Christoffel symbols when I use Mathimatica so I'm not sure if it's my calculations that are wrong or not. This isn't homework or anything of the sorts. I'm just trying to find the Ricci tensor and scalar for this metric partly for fun, partly for getting used to the operations and partly to learn more about this metric. In the attached file you'll find the Christoffel symbols that were calculated by me.
Thanks in advance.
 

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  • #2
The Schwarzschild solution is a vacuum solution, what does that tell you about the Ricci tensor and scalar right off the bat?
 
  • #3
They should be zero. Sorry that part completely slipped my mind. But my overall goal remains the same nevertheless. I don't seem to be getting the correct Christoffels for some reason.
This is the third metric I'm solving and GR is very new to me so I apologize.
 
  • #4
Which method are you using to find the Christoffel symbols? Are you just brute forcing it from the equation where they are equal to derivatives of g?
 
  • #5
I don't know what you mean exactly but I am using the equation, the partials of g.
Is that a wrong approach to this?
 
  • #6
No, that's a perfectly valid approach, but sometimes other approaches are quicker. For example, you can use the Euler Lagrange equations on the integral [itex]\frac{1}{2}\int g_{\mu\nu}\dot{x}^\mu\dot{x}^\nu d\lambda[/itex] to get the geodesic equations. By comparing with the regular geodesic equations, one can often quickly read off all the Christoffel symbols.
 
  • #7
Yeah well the problem is that I have no knowledge of Lagrangian mechanics so my hands are quite tied.
 
  • #8
Interestingly enough, I can't seem to find a nice concise tabulation of the results online...and since I'm way too lazy to try to do this myself, I hope someone else will come along and help you. XD
 
  • #9
I have all the time in the world. I just want to make sure I get the Christoffel symbols right before I do the Riemann tensor calculation since getting non zero Ricci tensor would mean a lot of wasted time and I hate it when I get stuff wrong after working so hard for it :(
 
  • #10
There are maybe all wrong, for example [itex]\Gamma_{00}^1=-\frac{1}{2}g^{11}\partial_r g_{00}=\frac{c^2 r_s (r-r_s)}{2 r^3}[/itex]
So maybe something is wrong when you define the inverse, e.g. [itex]g^{11}=-(1-r_s/r)[/itex]
 
  • #11
[tex]
\begin{align}
{mcs}_{1,1,2}&=\frac{m\,\left( r-2\,m\right) }{{r}^{3}}\\
{mcs}_{1,2,1}&=\frac{m}{r\,\left( r-2\,m\right) }\\
{mcs}_{2,2,2}&=-\frac{m}{r\,\left( r-2\,m\right) }\\
{mcs}_{2,3,3}&=\frac{1}{r}\\
{mcs}_{2,4,4}&=\frac{1}{r}\\
{mcs}_{3,3,2}&=2\,m-r\\
{mcs}_{3,4,4}&=\frac{cos\left( \theta\right) }{sin\left( \theta\right) }\\
{mcs}_{4,4,2}&=-\left( r-2\,m\right) \,{sin\left( \theta\right) }^{2}\\
{mcs}_{4,4,3}&=-cos\left( \theta\right) \,sin\left( \theta\right)
\end{align}
[/tex]

where
[tex]
mcs_{abc}={\Gamma^c}_{ab}
[/tex]

and t=x1, r=x2 etc.

This is from Maxima, using the ctensor package.
 

1. What are Schwarzschild Christoffels?

Schwarzschild Christoffels are a set of mathematical symbols used to describe the curvature of space-time in the vicinity of a non-rotating, uncharged black hole. They are named after the German physicist Karl Schwarzschild and the mathematician Elwin Bruno Christoffel.

2. How are Schwarzschild Christoffels related to general relativity?

Schwarzschild Christoffels are a key component of the equations of general relativity, which is a theory of gravity that explains how massive objects affect the curvature of space-time. The Christoffel symbols are used to calculate the curvature of space-time in the presence of a massive object, such as a black hole.

3. Can Schwarzschild Christoffels be used to describe other objects besides black holes?

Yes, Schwarzschild Christoffels can be used to describe the curvature of space-time around any massive object, not just black holes. They are also used to study the behavior of light, particles, and other objects in the vicinity of massive objects.

4. What is the significance of the Schwarzschild radius in relation to Schwarzschild Christoffels?

The Schwarzschild radius is a critical value in the Schwarzschild metric, which is the mathematical model used to describe the curvature of space-time around a non-rotating, uncharged black hole. The Schwarzschild radius is equal to the radius of the event horizon of a black hole, and it plays a crucial role in the calculation of Schwarzschild Christoffels.

5. Are Schwarzschild Christoffels important in modern physics?

Yes, Schwarzschild Christoffels are an essential component of general relativity and have been used to make many predictions that have been confirmed by observations, such as the bending of light around massive objects. They are also used in other areas of physics, such as cosmology, to study the behavior of the universe at large scales.

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