Origin of the eq. 5.141 in the Di francesco's book conformal field theory Recently I am reading Philippe Francesco's Book: Conformal Field Theory. Actually, I am having problems in understanding the origin of the initial part of the following equation:

$$\left\langle T^{00}\right\rangle =\left\langle T_{zz}\right\rangle +\left\langle T_{\overline{z}\overline{z}}\right\rangle .\tag{5.141} $$

I would also like to know which contravariant metric is necessary to be used in this special case.
I will be very grateful for your great help.
 A: The complex coordinates are defined by:
$$
\partial_z=\frac {1}{2}(\partial_0-i\partial_1)\,,\,\,\,\,\,\, \partial_{\bar {z}}=\frac {1}{2}(\partial_0+i\partial_1).
$$
$$
dz=dz^0+idz^1\,,\,\,\,\,\,\, d\bar{z}=dz^0-idz^1
$$
A good way to see how tensors transforms under change of coordinates is by the identification:
$$
T^{ab}=T^{\alpha\beta}\partial_{\alpha}\partial_{\beta}
$$
$$
T_{ab}=T_{\alpha\beta}dz^{\alpha}dz^{\beta}
$$
Then
$$
T_{00}dz^0dz^0+T_{01}dz^0dz^1+T^{10}dz^1dz^0+T^{11}dz^1dz^1=\\=T_{00}dz^0dz^0+...+T^{11}dz^1dz^1=T_{00}(dz^0dz^0-dz^1dz^1)+...=\\=\frac{1}{2}T_{00}(dzdz+d\bar {z}d\bar {z})+...\,\,\,\,\,
$$
In the second line we use the fact that the energy-momentum tensor is traceless. In the third line we use the fact that symmetrical combination of the product of $dzdz$ and $d\bar{z}d\bar{z}$ gives the difference between $dz^0dz^0$ and $dz^1dz^1$.
The tensor in $z$-coordinates, because the traceless condition, have $T_{z\bar{z}}=T_{\bar{z}z}=0$, and
$$
T_{ab}=T_{zz}dzdz+T_{\bar{z}\bar{z}}d\bar{z}d\bar{z}=\\=\frac{(T_{zz}+T_{\bar{z}\bar{z}})}{2}(dzdz+d\bar{z}d\bar{z})+\frac{(T_{zz}-T_{\bar{z}\bar{z}})}{2}(dzdz-d\bar{z}d\bar{z})
$$
Because $(dzdz-d\bar{z}d\bar{z})$ and $(dzdz+d\bar{z}d\bar{z})$ are linear independent derivatives, and $T_{00}=T^{00}$ then:
$$
T^{00}=T_{zz}+T_{\bar{z}\bar{z}}
$$ 
To calculate the metric tensor do the same thing. Express the tensor in terms of linear combination of partial derivatives or differentials and express the old derivatives  and differentials in terms of the new ones.
