Divergence of (covaraint) energymomentum tensor

[Torg]

whyT^[ab][;b]≠T_[ab][;b] for spatially flat FLWR cosmology ((ds)^2=(c^2)* (dt)^2-a(t)^2[(dx)^2+(dy)^2+(dz)^2])?
τ[ab][/;b] gives the right answer, but not τ[ab][/;b].

(T^(ab) or T_(ab)) contra-variant and co-variant energy momentum tensor of perfect fluid
(;) covariant derivative,
(c) spped of light in vacuum

 

[Orodruin]

Because covariant components generally are not the same as contravariant components. This should not come as a surprise.

τ[ab][/;b] gives the right answer, but not τ[ab][/;b].

Here you just wrote the same thing twice. I suggest using LaTeX to better transmit the meaning of your post.

 

[Torg]

$${T}_{ab;b}=0$$ \ne $${T}^{ab}_{;b}=0$$ for flat FLWR cosmology line element $${ds}^{2}=(c^{2}(dt)^{2}-a(t)^{2}[(dx)^{2}+(dy)^{2}+(dz)^{2}])$$

 

[Stavros Kiri]

Symbols need fixing.

Because covariant components generally are not the same as contravariant components. This should not come as a surprise.Here you just wrote the same thing twice. I suggest using LaTeX to better transmit the meaning of your post.

I agree

 

[Torg]

But the equations o motion should be the same whether i use covariant or contrvariant. They mathematical
construct!

 

[Stavros Kiri]

But the equations o motion should be the same whether i use covariant or contrvariant. They mathematical
construct!

The equations of motion yes, but not every tensor ... etc.

 

[Orodruin]

${T}_{ab;b}=0$ \ne ${T}^{ab}_{;b}=0$ for flat FLWR cosmology line element ${ds}^{2}=(c^{2}(dt)^{2}-a(t)^{2}[(dx)^{2}+(dy)^{2}+(dz)^{2}])$

See https://www.physicsforums.com/help/latexhelp/

Regardless, it does not change the answer. You have not explained why you would expect contravariant and covariant components to be the same.

 

[Orodruin]

But the equations o motion should be the same whether i use covariant or contrvariant. They mathematical
construct!

$\nabla_b T^{ab}$ is a nice expression that transforms as a contravariant vector. $\nabla_b T_{ab}$ is not and it makes no sense to write it down as it does not transform covariantly. You could write $\nabla^b T_{ab}=0$, which would be equivalent to $\nabla_b T^{ab} = 0$, but the left-hand sides would be different (although the entire system of equations would be equivalent).

 

[Torg]

\[\begin{array}{l}

{T^{ab}}_{;b} = {T^{ab}}_{,b} + {\Gamma _{bc}}^a{T^{bc}} + {\Gamma _{bd}}^b{T^{ad}} \\

{T_{ab;b}} = {T_{ab;b}} - {\Gamma _{ab}}^c{T_{bc}} - {\Gamma _{bb}}^d{T_{ad}} \\

\end{array}\]



The zero components give

\[\begin{array}{l}

{T^{0b}}_{;b} = {T^{0b}}_{,b} + {\Gamma _{bc}}^0{T^{bc}} + {\Gamma _{bd}}^b{T^{0d}} = {T^{00}}_{,0} + {\Gamma _{11}}^0{T^{11}} + {\Gamma _{22}}^0{T^{22}} + {\Gamma _{33}}^0{T^{33}} + {\Gamma _{01}}^0{T^{00}} + {\Gamma _{02}}^0{T^{00}} + {\Gamma _{03}}^0{T^{00}} \\

{T_{0b;b}} = {T_{0b;b}} - {\Gamma _{0b}}^c{T_{bc}} - {\Gamma _{bb}}^d{T_{0d}} = {T_{00}}_{,0} - {\Gamma _{01}}^1{T_{11}} - {\Gamma _{02}}^2{T_{22}} - {\Gamma _{03}}^3{T_{33}} - {\Gamma _{11}}^0{T_{00}} - {\Gamma _{22}}^0{T_{00}} - {\Gamma _{33}}^0{T_{00}} \\

\end{array}\]
\[{T^{ab}}_{;b} \ne {T_{ab;b}}\]

 

[Torg]

they don't give the same answer

 

[Stavros Kiri]

they don't give the same answer

They are not supposed to.
[Why would they? Explain]

 

[Orodruin]

they don't give the same answer

As you have already been told, one of the expressions is fine and the other is essentially nonsense.

 

[Torg]

Because both are zero when energymomentum is conserved in General Relativity and they should give the same answer for equation of motion.

 

[Torg]

The upper expression gives the true Frieman's equations, but the lower one doesn't. That what amazes me. I have been working on it for days couldn't get right.

 

[Ibix]

Not true. $T^{ab}{}_{;b}$ represents conservation of energy, yes. $T_{ab;b}$ doesn't mean anything. You can't contract a lower index with a lower index - it doesn't mean anything.

 

[Torg]

shall i just forget about it? I use the lower expression a lot.

 

[Stavros Kiri]

but the lower one doesn't.

It's not supposed to.

shall i just forget about it? I use the lower expression a lot.

I agree with the others. It means nothing

 

[Orodruin]

shall i just forget about it? I use the lower expression a lot.

You should forget about using that expression. It is just wrong.

 

[Torg]

Why do I need to contact it? when i use EFEs in their covariant form and when I differentiate both sides covariantly and put the divergence of the energymomentum tensor equals to zero the left hand side should give the same set of equation of motion.

 

[Orodruin]

Taking the divergence is a contraction.

 

[Ibix]

A lower and upper index can be contracted over because their product is a Lorentz scalar. A vector is a map from the space of co-vectors to the reals.

But a co-vector does not map a co-vector to a real. So contracting them isn't expected to produce a consistent result.

Ben Crowell gives an example of a frequency (a covariant quantity) and a time (a contravariant quantity). The product is the number of cycles in the time period, independent of the units used (a scalar). You can convert a frequency to a period by taking the reciprocal (lower an index). And you can multiply that period by the time. But the result is unit-dependent and physically meaningless.

 

[Torg]

I am really sad! to give up using EFEs in their covariant form when it comes to apply the conservation of energymomentum tensor.

 

[Ibix]

You can always raise an index on the covariant derivative operator, as Orodruin noted. $\nabla^bT_{ab}$ is fine and each element is a linear combination of the elements of $\nabla_bT^{ab}$.

 

[Torg]

I got it, but I will take it with a grain of salt.

 

[Torg]

I have to checked that if it doesn't affect the Bianchi second identity.

 

[Torg]

I understand it now. Thank you all very much for the help. You have been excellent in explaining it.

 

[Orodruin]

If you struggle with this kind of manipulations I would strongly suggest that you read up on differential geometry using a dedicated text that is more detailed than the summary you would typically find in a GR textbook.

 

[Stavros Kiri]

I understand it now. Thank you all very much for the help. You have been excellent in explaining it.

And again welcome to PF! I hope it works for you now. I'll go back to the chat later (it's still there).

Note: within 4hrs you can edit your last post (in general) instead of making many consecutive ones, if no one is in between.

 

[Dale]

shall i just forget about it? I use the lower expression a lot.

Younshould not use the lower expression ever. It is syntactically wrong. Never contract a lower index with a lower index.

 

[Torg]

Thank you all again.
I will ask for much today :-)
I would like to know name of a good reference in introductory Differential Geometry.
I struggled whole night with the second expression below, I couldn't figure out the positions of the indices

\[\begin{array}{l}
{T^{ab}}_{;b} = {T^{ab}}_{,b} + {\Gamma _{bc}}^a{T^{bc}} + {\Gamma _{bd}}^b{T^{ad}} \\
{T_{ab}}^{;b} = {T_{ab}}^{,b} - {\Gamma _{..}}^{.}{T_{..}} - {\Gamma _{..}}^{.}{T_{..}} \\
\end{array}\]
but happily I could write latex expression in PF :-)

I need a paper which I don't have and I couldn't afford and my institution too! one of these is about non-conserved of energy momentum tensor by Rastall.