How Do You Calculate g(W,W) Using the Given Metric?

[MathematicalPhysicist]

Suppose, I have the next metric:
$$g = du^1 \otimes du^1 - du^2 \otimes du^2$$

And I want to calculate $$g(W,W)$$, where for example $$W=\partial_1 + \partial_2$$

How would I calculate it?

Thanks.

 

[WannabeNewton]

The metric tensor is bilinear so $g_p(\partial _{1} + \partial _{2}, \partial _{1} + \partial _{2}) = g_p(\partial _{1} + \partial _{2},\partial _{1}) + g_p(\partial _{1} + \partial _{2},\partial _{2}) = \\g_p(\partial _{1},\partial _{1}) + g_p(\partial _{2},\partial _{1}) + g_p(\partial _{1},\partial _{2}) + g_p(\partial _{2},\partial _{2}) = g_{11}(p) + 2g_{12}(p) + g_{22}(p) = 1 +0 - 1 = 0$.

Assuming by $\partial _{i}|_{p}$ you are talking about the coordinate basis vectors, $g_p(\partial _{i},\partial _{j}) = g_{ij}$.

 

[MathematicalPhysicist]

ok, thanks.

 

[HallsofIvy]

In matrix terms we can representn $g = du^1 \otimes du^1+ 0 du^1\otimes du^2+ 0 du^2\otimes du^1 - du^2 \otimes du^2$ as
$$\begin{pmatrix}1 & 0 \\ 0 & -1\end{pmatrix}$$

And I want to calculate $$g(W,W)$$, where for example $$W=\partial_1 + \partial_2$$

How would I calculate it?

Thanks.

$$\begin{pmatrix}1 & 1\end{pmatrix}\begin{pmatrix}1 & 0 \\ 0 & -1\end{pmatrix}\begin{pmatrix}1 \\ 1 \end{pmatrix}= \begin{pmatrix}1 & -1\end{pmatrix}\begin{pmatrix}1 \\ 1\end{pmatrix}= 1+ (-1)= 0$$

 

[blue_raver22]

I would say that calculating the metric tensor is an important step in understanding the geometry and curvature of a given space. In this case, the given metric g represents a two-dimensional space with coordinates u^1 and u^2. The tensor g is a mathematical object that describes the distance between points in this space.

To calculate g(W,W), we first need to understand what W represents. In this case, W is a vector field with components \partial_1 and \partial_2. This means that at any given point in the space, the vector W will have a magnitude and direction determined by the values of \partial_1 and \partial_2 at that point.

To calculate g(W,W), we need to use the properties of the metric tensor. One important property is that the metric tensor is a bilinear form, meaning it takes two vectors as inputs and produces a scalar as output. In this case, we have two copies of W as our inputs, so we can use the bilinearity property to break up g(W,W) into two parts: g(W,W) = g(W,W^1) + g(W,W^2).

The next step is to apply the metric g to each component of W separately. Since g is a tensor, it is defined by its action on any two vectors. In this case, we can use the definition of g to calculate g(W,W^1) and g(W,W^2) as follows:

g(W,W^1) = g(\partial_1 + \partial_2, \partial_1) = g(\partial_1, \partial_1) + g(\partial_1, \partial_2)

g(W,W^2) = g(\partial_1 + \partial_2, \partial_2) = g(\partial_1, \partial_2) + g(\partial_2, \partial_2)

To calculate each of these terms, we can use the given metric g = du^1 \otimes du^1 - du^2 \otimes du^2 and the fact that \partial_1 and \partial_2 are the basis vectors for this space. This will give us the following results:

g(\partial_1, \partial_1) = g(\partial_2, \partial_2) = 1
g(\partial_1, \partial_2) = g(\partial