# A Divergence of (covaraint) energymomentum tensor

1. Dec 25, 2017

### 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

2. Dec 25, 2017

### Orodruin

Staff Emeritus
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.

3. Dec 25, 2017

### Torg

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

Last edited by a moderator: Dec 25, 2017
4. Dec 25, 2017

### Stavros Kiri

Symbols need fixing.
I agree

5. Dec 25, 2017

### Torg

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

6. Dec 25, 2017

### Stavros Kiri

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

7. Dec 25, 2017

### Orodruin

Staff Emeritus
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.

8. Dec 25, 2017

### Orodruin

Staff Emeritus
$\nabla_b T^{ab}$ 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).

9. Dec 25, 2017

### 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

$\begin{array}{l} {T^{0b}}_{;b} = {T^{0b}}_{,b} + {\Gamma _{bc}}^0{T^{bc}} + {\Gamma _{bd}}^b{T^{0d}} = {T^{00}}_{,0} + {\Gamma _{11}}^0{T^{11}} + {\Gamma _{22}}^0{T^{22}} + {\Gamma _{33}}^0{T^{33}} + {\Gamma _{01}}^0{T^{00}} + {\Gamma _{02}}^0{T^{00}} + {\Gamma _{03}}^0{T^{00}} \\ {T_{0b;b}} = {T_{0b;b}} - {\Gamma _{0b}}^c{T_{bc}} - {\Gamma _{bb}}^d{T_{0d}} = {T_{00}}_{,0} - {\Gamma _{01}}^1{T_{11}} - {\Gamma _{02}}^2{T_{22}} - {\Gamma _{03}}^3{T_{33}} - {\Gamma _{11}}^0{T_{00}} - {\Gamma _{22}}^0{T_{00}} - {\Gamma _{33}}^0{T_{00}} \\ \end{array}$

${T^{ab}}_{;b} \ne {T_{ab;b}}$

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10. Dec 25, 2017

### Torg

they don't give the same answer

11. Dec 25, 2017

### Stavros Kiri

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

12. Dec 25, 2017

### Orodruin

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

13. Dec 25, 2017

### Torg

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

14. Dec 25, 2017

### 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.

15. Dec 25, 2017

### 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.

16. Dec 25, 2017

### Torg

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

17. Dec 25, 2017

### Stavros Kiri

It's not supposed to.
I agree with the others. It means nothing

18. Dec 25, 2017

### Orodruin

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

19. Dec 25, 2017

### 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.

20. Dec 25, 2017

### Orodruin

Staff Emeritus
Taking the divergence is a contraction.