FLRW Metric Tensor: Calculating g^{ca}g_{ab}

[trv]

Homework Statement

Not really homework, but thought this might be the best place to get a quick answer.

Question
Calculate g^{ca}g{ab} for the FLRW metric.

I would have thought this would be

$g^{ca}g{ab}=\delta^c_b=4$

I thought 4 because I assumed there should be "1" for each non-zero element in the metric tensor g_{ab}.

Apparently however it should be g^{ca}g_{ab}=1

Can anyone please explain why?

Homework Equations

The Attempt at a Solution

 

[Cyosis]

This is just the Kronecker delta which is 1 if b=c and zero if not, there is no summation going on. If it had been $\delta_b^b$ Einstein summation would have been implied, which would yield 4.

Mathematically:

$$A^\gamma=g^{\gamma \alpha}A_\alpha= g^{\gamma \alpha}g_{\alpha \nu}A^\nu=\delta_\nu^\gamma A^\nu=A^\gamma$$

If it were four, you would get $A^\gamma=4A^\gamma$ which would make little sense.

 

[trv]

Thanks again Cyosis that clears things up.

 

[trv]

Actually, another quick question.

So the tensor, g^a_b for the FLRW metric, simply has components 1 along the diagonal?

Or for that matter, to generalise it further. Given any metric tensor r_{ab} with all elements being non-zero, r^a_b has components 1 everywhere?

To add to that, given,

$L=-0.5(\frac{d\phi}{dt})^2+V(\phi)$

does g^a_bL = L?

 

[Cyosis]

So the tensor, g^a_b for the FLRW metric, simply has components 1 along the diagonal?

Correct.

Or for that matter, to generalise it further. Given any metric tensor r_{ab} with all elements being non-zero, r^a_b has components 1 everywhere?

No, proof:

$$r^\alpha_{\;\;\beta}=r^{\alpha \gamma}r_{\gamma \beta}=\delta^\alpha_\beta$$

In words, any mixed metric is equal to the Kronecker delta. You basically have a matrix with 1's on the diagonal and zeros elsewhere.

 

[trv]

Oh yeh, true the kronecker delta would imply zeros everywhere apart from the diagonal. I think where I'm going wrong is assuming we multiply each element by its inverse. Giving 1 at all places where the metric tensor had non-zero components. Obviously that's not how it works.

 

[cristo]

Oh yeh, true the kronecker delta would imply zeros everywhere apart from the diagonal. I think where I'm going wrong is assuming we multiply each element by its inverse. Giving 1 at all places where the metric tensor had non-zero components. Obviously that's not how it works.

Think of these tensors as matrices, where g^{ab} is the inverse of g_{ab}. Just as multiplying a matrix with its inverse yields the identity matrix (AA^{-1}=I), multiplying the metric with its inverse yields the kronecker delta.

 

[trv]

Thanks the two of you.