Riemann Tensor Formula in Terms of Metric & Derivatives

In summary, the conversation revolves around obtaining the Riemann Tensor in a specific form and the possibility of disproving Birkhoff's Theorem. The person asking for help is hesitant to do the calculations themselves due to the risk of errors and is looking for a reliable source for the formula. The conversation ends with the expert dismissing the possibility of disproving Birkhoff's Theorem and closing the thread.
  • #1
Jogging-Joe
4
1
TL;DR Summary
Could someone please write out or post a link to the Riemann Tensor written out solely in terms of the metric and its first and second derivatives--i.e. with the Christoffel symbol gammas and their first derivatives not explicitly appearing in the formula.
Could someone please write out or post a link to the Riemann Tensor written out solely in terms of the metric and its first and second derivatives--i.e. with the Christoffel symbol gammas and their first derivatives not explicitly appearing in the formula.

Thanks.
 
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  • #2
This link has both the Riemann tensor and the Christoffel symbols. I am not inclined to retype everything just to substitute one into the other. Where are you headed with this? Even if you could find such an expression prechurned on the internet, it is one of those things I would not trust to be error free.
 
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  • #3
Thanks.

I know that a person can get the Riemann Tensor in the form I want by substituting the metric and its first derivatives into the Christofell gammas in the standard Riemann Tensor formula.. As you noted, it is easy to make errors and that is why I would not trust me doing it, and would want a second opinion calculation. Someone must have put it in that form, somewhere.
 
  • #4
Jogging-Joe said:
Thanks.

I know that a person can get the Riemann Tensor in the form I want by substituting the metric and its first derivatives into the Christofell gammas in the standard Riemann Tensor formula.. As you noted, it is easy to make errors and that is why I would not trust me doing it, and would want a second opinion calculation. Someone must have put it in that form, somewhere.
But why do you want it in that form?
 
  • #5
I can show that Birkhoff's Theorem is wrong, and I think I might be able to prove it in a second independent way, which would require me to use that particular form of the Riemann Tensor.
 
  • #6
Jogging-Joe said:
I can show that Birkhoff's Theorem is wrong

No, you can't. You may think you can, but you've made an error somewhere.

Jogging-Joe said:
I think I might be able to prove it in a second independent way, which would require me to use that particular form of the Riemann Tensor.

Sorry, we don't help people with personal theories on this site, particularly personal theories which we already know are wrong (see above).

Thread closed.
 
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1. What is the Riemann Tensor Formula in terms of metric and derivatives?

The Riemann Tensor Formula is a mathematical expression used in differential geometry to describe the curvature of a manifold. It is defined in terms of the metric tensor and its derivatives, and is used to calculate the curvature of a space.

2. How is the Riemann Tensor Formula related to Einstein's theory of general relativity?

The Riemann Tensor Formula plays a crucial role in Einstein's theory of general relativity, as it describes the curvature of spacetime. This curvature is what causes the gravitational force, and the Riemann Tensor Formula is used to calculate this curvature in terms of the metric tensor.

3. What are the components of the Riemann Tensor Formula?

The Riemann Tensor Formula has 4 indices, and each index can take on values from 0 to 3. This means that there are 256 components in total. However, due to the symmetries of the formula, there are only 20 unique components that need to be calculated.

4. How is the Riemann Tensor Formula used in physics?

The Riemann Tensor Formula is used in various areas of physics, including general relativity, quantum field theory, and cosmology. It is used to calculate the curvature of spacetime, which is crucial in understanding the behavior of gravity and the structure of the universe.

5. What are some applications of the Riemann Tensor Formula?

The Riemann Tensor Formula has many applications in mathematics and physics. It is used in the study of black holes, gravitational waves, and the evolution of the universe. It is also used in engineering, particularly in the design of structures and materials that can withstand high levels of stress and strain.

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