Attempted proof of the Contracted Bianchi Identity

In summary, the conversation discusses the attempted proof of the identity ##R^{mn}_{;n} - \frac {1} {2} g^{mn} R_{;n} = 0## using the second Bianchi Identity and the inverse metric tensor. The steps involved include manipulating indices, contracting, summing, and using the symmetric property of the metric tensor and its inverse. The result obtained is ##2 R^{mn}_{;n} = g^{mn} R_{;n} ##, which is the desired identity.
  • #1
Vanilla Gorilla
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TL;DR Summary
I constructed an attempted proof of the Contracted Bianchi Identity, and cannot identify any errors myself. However, obviously, this does not mean that the proof is watertight. For any errors present, I am hoping to get details and explanations on where I went wrong, how to avoid the mistake in the future, etc.
Criticism welcome!
My Attempted Proof
##R^{mn}_{;n} - \frac {1} {2} g^{mn} R_{;n} = 0##
##R^{mn}_{;n} = \frac {1} {2} g^{mn} R_{;n}##
So, we want ##2 R^{mn}_{;n} = g^{mn} R_{;n} ##

Start w/ 2nd Bianchi Identity ##R_{abmn;l} + R_{ablm;n} + R_{abnl;m} = 0##
Sum w/ inverse metric tensor twice ##g^{bn} g^{am} (R_{abmn;l} + R_{ablm;n} + R_{abnl;m}) = g^{bn} g^{am} (0)##
Distributing ##g^{bn} (g^{am} R_{abmn;l} + g^{am} R_{ablm;n} + g^{am} R_{abnl;m}) = 0##
Index manipulating ##g^{bn} (R^{m}_{~~~bmn;l} + R^{m}_{~~~blm;n} + R^{m}_{~~~bnl;m}) = 0##
Contracting ##g^{bn} (R_{bn;l} - R_{bl;n} + R^{m}_{~~~bnl;m}) = 0##
Distributing and manipulating indices ## R^n_{n;l} - R^n_{l;n} + g^{bn} R^{m}_{~~~bnl;m} = 0##
Summing ##R_{;l} - R^n_{l;n} + g^{bn} R^{m}_{~~~bnl;m} = 0##
Sum W/ ##g^{lp}##: ##g^{lp} (R_{;l} - R^n_{l;n} + g^{bn} R^{m}_{~~~bnl;m} = 0##
Distributing ##g^{lp} R_{;l} - g^{lp} R^n_{l;n} + g^{lp} g^{bn} R^{m}_{~~~bnl;m} = 0##
Summing ##g^{lp} R_{;l} - R^{np}_{;n} + g^{lp} g^{bn} R^{m}_{~~~bnl;m} = 0##
Ricci Tensor (downstairs indices is symmetric, so I THINK it is valid to flip the 2 top indices here ##g^{lp} R_{;l} - R^{pn}_{;n} + g^{lp} g^{bn} R^{m}_{~~~bnl;m} = 0##
##g^{lp} R_{;l} + g^{lp} g^{bn} R^{m}_{~~~bnl;m} = R^{pn}_{;n}## becomes ##R^{pn}_{;n} = g^{lp} R_{;l} + g^{lp} g^{bn} R^{m}_{~~~bnl;m}##
##R^{pn}_{;n} = g^{lp} R_{;l} - g^{lp} g^{bn} R^{m}_{~~~bln;m}##
##R^{pn}_{;n} = g^{lp} R_{;l} - R^{mnp}_{~~~~~~~~n;m}##
##R^{pn}_{;n} + R^{mnp}_{~~~~~~~~n;m} = g^{lp} R_{;l} ##
The metric tensor and its inverse are symmetric ##R^{pn}_{;n} + R^{mnp}_{~~~~~~~~n;m} = g^{pl} R_{;l} ##
Summing over ##n##, ##R^{pn}_{;n} + R^{mp}_{;m} = g^{pl} R_{;l} ##
I've seen others just replace indices like I am about to, so I am about 60% sure the following operation is valid. However, I am not sure. Any insight and corrections would be much appreciated here especially! ##R^{mn}_{;n} + R^{mn}_{;m} = g^{mn} R_{;n} ##
Combining like terms ##2 R^{mn}_{;n} = g^{mn} R_{;n} ##, which is just the result we're looking for, assuming I did it right.

Any help is much appreciated!
P.S., I'm not always great at articulating my thoughts, so my apologies if this question isn't clear. Also, I know this isn't high school material, but I am currently in high school, which is why I made my level "Basic/high school level."
Lastly, I know that this isn't the typical way one would go about proving this identity, but I was wondering if this was still a valid avenue to do so (I did it from scratch).
 
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  • #2
Moderator's note: Thread moved to the Differential Geometry forum.
 

1. What is the Contracted Bianchi Identity?

The Contracted Bianchi Identity is a mathematical equation that relates the covariant derivative of the metric tensor to the Riemann curvature tensor. It is a fundamental equation in Einstein's theory of general relativity.

2. Why is it important to attempt to prove the Contracted Bianchi Identity?

Proving the Contracted Bianchi Identity is important because it provides a rigorous mathematical foundation for Einstein's theory of general relativity. It also allows for further developments and applications of the theory in physics and cosmology.

3. What are some challenges in attempting to prove the Contracted Bianchi Identity?

One challenge is the complexity of the mathematical equations involved. Another challenge is the need for a deep understanding of differential geometry and tensor calculus. Additionally, the proof may require advanced techniques and mathematical tools.

4. Has the Contracted Bianchi Identity been proven before?

Yes, the Contracted Bianchi Identity has been proven by several mathematicians and physicists. However, new and alternative proofs are still being sought after to deepen our understanding of the equation and its implications.

5. What are the potential implications of a successful proof of the Contracted Bianchi Identity?

A successful proof would not only solidify the mathematical foundation of general relativity, but it could also lead to new insights and developments in the field of physics. It may also have implications for other areas of mathematics, such as differential geometry and topology.

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