How to calculate the contraction of metric tensor g^ab g_ab

[yicong2011]

I wish I could calculate the contraction:

g^abg_ab

I wish someone could show me how to get n!

Unfortunately, I find it difficult, for I am not familiar with Tensor Algebra ...
My wrong way to calculate it:

g^abg_ab= g^abg_ba (since g^ab is symmetric)

= δ^a_a

= 1Why is it wrong?

 

[Matterwave]

So, you just sum over a and b...so it's like g00g00+g01g01+g02g02+g03g03+g10g10+g11g11+...all 16 terms

 

[yicong2011]

But why the following is wrong? I cannot figure it out...

g^abg_ab= g^abg_ba (since g^ab is symmetric)

= δ^a_a

= 1

Anyone can help?

 

[JustinLevy]

The problem is your last step.

In four spacetime dimensions
\delta^a{}_a = 4
because
\delta^a{}_a = \delta^0{}_0 + \delta^1{}_1 + \delta^2{}_2 + \delta^3{}_3 = 1 + 1 +1 +1 = 4

 

[yicong2011]

The problem is your last step.

In four spacetime dimensions
\delta^a{}_a = 4
because
\delta^a{}_a = \delta^0{}_0 + \delta^1{}_1 + \delta^2{}_2 + \delta^3{}_3 = 1 + 1 +1 +1 = 4

Ahh...Ja... \delta^a{}_a is not the components... I need to expand it and sum over the components...