Divergence of curvature scalars * metric

[JustinLevy]

Thanks atyy! That is great!

They show the method would work for an even larger class of theories. Unfortunately this does not include the scalar R_{ab}R^{ab}.

However, I have been able to get "Tensor Tools for Calculus" to simplify the relation obtained by my method. And it verified that the method works in that case as well!

As that paper you found mentions, there is good heuristic reason to expect this method to always work. It falls short of proving it in complete generality though.

I guess that answers the openning question of the thread, as well as (to current best ability) the question about the validity of the method I used to obtain further divergence relations. And at the very least, we now know the relation I obtained for the divergence of g^{ab} R_{cd}R^{cd} is correct.

Thanks everyone for your help!

 

[Altabeh]

Thanks atyy! That is great!

They show the method would work for an even larger class of theories. Unfortunately this does not include the scalar R_{ab}R^{ab}.

And it verified that the method works in that case as well!

I'm so curious to see how the method is verified. Put the results in this thread or otherwise I sure know that if I can call this a "method", it does not work at all! The reason I already gave umpteen times before is that the approach does not include any appropriate function or parameter like, $$f(R)$$ in the $$f(R)$$ gravity theories, which has this advantage that cancels out with the extra non-vanishing terms of the divergence of the field equations on a purely geometrical (or mathematical) basis. (See for example the explanation below the equation 8 in http://arxiv.org/abs/gr-qc/0505128). This does not occur by accident or by the use of a cheap guess as in your theory.

In order to make the theory obey the conservation law, the proposed Lagrangian must be in a suitable form, meaning that after taking the divergence of its field equation all the terms appearing in the equation have to vanish somehow. You don't put the non-vanishing terms equal to zero by hand, as for the same paper this is obvious from the equation 8 that authors do not seek out some relation between non-zero terms such as $$(\nabla^{\mu}F) R_{\mu\nu}$$ and $$(\square\nabla_{\nu}-\nabla_{\nu}\square)F$$, to get a conserved matter tensor but rather this happens all by a mathematical reasoning where your theory seems to have a hole in it: We cannot write a relation or relations between the terms to get this result. You have to show us after taking the divergence that all terms vanish or cancel out each other so you're done then with a claim regarding the success of the theory or method or whatever.

AB

 

[JustinLevy]

I'm so curious to see how the method is verified.

Then how about you reread the line you purposely deleted when quoting me in your post. "Tensor Tools for Calculus" is a freely available program. If you have access to Mathematica, give it a try yourself.

To make sure this isn't another reading comprehension issue, please note I said it verified the method in this case (ie. the divergence term I was working on). Nowhere did I claim I have been able to verify the method in generality.

Not trusting my math, and not trusting a computational package is one thing, but this comment is just offensive:

Put the results in this thread or otherwise I sure know that if I can call this a "method", it does not work at all!

You are sure the method doesn't work here (or at all)!?

You have repeatedly stated how easy it is to do the calculation I am struggling with (the divergence of $$g^{ab}R_{cd}R^{cd}$$). Yet instead of showing me how to do it, you deridingly give me "hints" which seem to lead no-where, or just insult me.

Look, I can't show the math here because I had to resort to using a computational package. I still don't know how to work it out by hand, except for using the method I suggested, which I can't prove in generality either. But if you are going to disagree with my result even after a computational package verified it... then please PLEASE, just show some math here. If it is so simple for you, and you are so sure the result is wrong, then please just work out the divergence of $$g^{ab}R_{cd}R^{cd}$$ as that will actually answer the openning question of this thread.

The reason I already gave umpteen times before is that the approach does not include any appropriate function or parameter like, $$f(R)$$ in the $$f(R)$$ gravity theories, which has this advantage that cancels out with the extra non-vanishing terms of the divergence of the field equations on a purely geometrical (or mathematical) basis. (See for example the explanation below the equation 8 in http://arxiv.org/abs/gr-qc/0505128). This does not occur by accident or by the use of a cheap guess as in your theory.

I stated that I worked out that my method works in general for f(R) theory, and you instead claimed that f(R) theory leads to violation of energy-momentum conservation unless careful choices of the function f(R) are made. I then wrote out the math explicitly showing that my method worked for f(R) theory in general. You called the math complete nonsense and dismissed my result without consideration. Now you've come all the way around and seem to agree with me on f(R) theory, but still after all this time you are not listening.

In trying to work out that divergence relation, I was using what amounts to f(R,P,Q) = P - R, in the paper mentioned previously http://arxiv.org/PS_cache/astro-ph/pdf/0410/0410031v2.pdf.

What you are claiming amounts to:
There are choices of f(R,P,Q) that lead to field equations which violate energy-momentum conservation, despite the action having no explicit time or spatial dependence.

I find that claim much MUCH more unlikely than the claim I made to arrive at the divergence relation:

Applying the method, we then get:
$$\nabla_b\left(g^{ab}R_{cd}R^{cd}\right) = 2\nabla_b\left(2R^{ca}R_c{}^{b} -2 \nabla^c\nabla^d R_c{}^{(a} \delta_d{}^{b)} + \nabla^c \nabla_c R^{ab} + g^{ab} \nabla^c \nabla^d R_{cd} \right)$$

If you believe the divergence relation I obtained is wrong, then finally stop leading us around and actually show the math which makes you so "sure" I am wrong. Showing the math will also answer the opening question of this thread, so will be directly useful as well.

 

[Altabeh]

then please PLEASE, just show some math here. If it is so simple for you, and you are so sure the result is wrong, then please just work out the divergence of $$g^{ab}R_{cd}R^{cd}$$ as that will actually answer the openning question of this thread.

Oh God, I never ever again get involved with the threads whose OP is JustinLevy!

Look, the divergence of $$g^{ab}R_{cd}R^{cd}$$ is going to nowhere because you're not doing any index-related action on the pair $$R_{cd}R^{cd}$$ and that $$g^{ab}$$ is just a spectator is the calculation of the divergence. Neither me nor any other physicist can do this by any means unless a contraction of one of the indices of the Ricci tensor with the only index of the divergence operator, $$\nabla_{a}$$, is produced which in this case is impossible! I want to make this point here that in calculating the divergence of the Ricci tensor, remember the Bianchi Identity, we are not getting Christoffel symbols involved directly as you can see http://en.wikipedia.org/wiki/Proofs_involving_Christoffel_symbols#Proof_1"! So if you were even an avreage physicist, you would be able to realize what choice of correction terms in Einstein-Hilbert action could be giving a divergence-free field equations and this is just a matter of realization to ignore $$g^{ab}R_{cd}R^{cd}$$ for not being influenced by the $$\nabla_{a}$$!

On the contrary, the situation is totally different for the Ricci scalar curvature: This has a relation with the Ricci tensor which together form the divergence-free Einstein tensor and this is taken advantage of in the paper http://arxiv.org/abs/gr-qc/0505128 to prove a same property for the proposed field equations. Besides, if they are claiming any conservation of energy-momentum in the paper http://arxiv.org/PS_cache/astro-ph/pdf/0410/0410031v2.pdf for the general field equations (15), it just falls upon the choice of $$f(R)$$ to determine whether the equation is conserved or not. As in this paper, I also believe and have been saying all this time in this thread that

"... it is that it is crucial to pick, not only the action, but also the fundamental fields which will be varied and assign to them a precise meaning."

I strongly suggest you to first learn the preliminaries of the f(R) theories (see the latest paper I cited above) and then try something new! All you're doing is wasting everyone's time and your own and if you don't take this seriously, I think I don't have anything more to say!

Period.

AB

 

[JustinLevy]

Neither me nor any other physicist can do this by any means

BY ANY MEANS!?
Again we seem to have some serious miscommunication issues. You seem not to even read what anyone writes. I've stated multiple times now that I have verified the divergence relation with a computational package.

You are still arguing against my result without showing any math. You again insult me without backing up your claims. This is unacceptable.

So if you were even an avreage physicist, you would be able to realize what choice of correction terms in Einstein-Hilbert action could be giving a divergence-free field equations and this is just a matter of realization to ignore $$g^{ab}R_{cd}R^{cd}$$ for not being influenced by the $$\nabla_{a}$$!

LISTEN! Will you please just actually listen.
I have verified that the field equations containing that term are indeed divergence-free. So don't be deriding me with phrases like "if you were even an average physicist", when I actually have proof behind my statements.

Please, you need to read more carefully. It seems the vast majority of our communication issues can be traced to you completely ignoring or misreading what has been written.

Besides, if they are claiming any conservation of energy-momentum in the paper http://arxiv.org/PS_cache/astro-ph/pdf/0410/0410031v2.pdf for the general field equations (15), it just falls upon the choice of $$f(R)$$ to determine whether the equation is conserved or not.

You are STILL claiming these actions, despite having no explicit time or spatial dependence, can lead to violation of local energy momentum conservation!?

This is just wrong.
We can debate about whether the assumption in my method is generally valid, but there can be NO debate about your claim. You are just WRONG. It is in direct violation of Noether's theorem.

As in this http://arxiv.org/PS_cache/arxiv/pdf/0810/0810.5594v1.pdf" , I also believe and have been saying all this time in this thread that

"... it is that it is crucial to pick, not only the action, but also the fundamental fields which will be varied and assign to them a precise meaning."

The thing being varied in ALL the actions we discussed in this thread, is the metric. These were ALL metric theories. And the metric has the same physical meaning in all these theories. Please read more carefully.If you are going to continue to claim that a computational package is giving incorrect results, or continue claiming these actions yield field equations which violate local energy-momentum conservation. Provide the math which backs up your statements, or stopping claiming them. For it appears clear to me at this point in time that you are just plain wrong on multiple accounts.

 

[atyy]

I think JustinLevy was asking about actions containing terms beyond f(R), such as the one in Eqn 1 of http://arxiv.org/abs/0911.3165 ? The divergence of g#R#R# was just incidental to that?