Calculating Metric/Metric Tensor for Curved Spaces

[marcusesses]

I'm having some troubles with a very basic definition of the metric tensor.
The metric is defined as

$$ds^2 =[f(x + dx, y+dy) - f(x,y)]^2$$

However, I can't see how this is equal to

$$\frac{\partial r} {\partial x} \frac{\partial r} {\partial x}dx^2 + 2 \frac{\partial r} {\partial x} \frac{\partial r} {\partial y}dxdy + \frac{\partial r} {\partial y} \frac{\partial r} {\partial y} dy^2$$

I can see it in the linear case, like when
$$r = x+y$$
since
$$ds^2 = dx^2 + 2 dxdy + dy^2$$
for example. But what if there is a non-linear relation, like
$$f(x,y) = x^2 - y$$
it will produce terms like $$dx^4$$ ...(I think, anyway).

Basically, what I'm asking is how do you calculate the metric tensor components
$$g_{\alpha\beta}$$?
Are they just found by making assumptions in the curved space you are in?How do you calculate the metric tensor from the metric?

 

[pmb_phy]

I'm having some troubles with a very basic definition of the metric tensor.
The metric is defined as

$$ds^2 =[f(x + dx, y+dy) - f(x,y)]^2$$

There is no such thing as the metric tensor. There are many metric tensors. I know of only two myself that I find in common use in geometry and physics and that is the Euclidean metric and the metric of spacetime (does this metric tensor have a name??). The metric is different from the distance function. The distance function induces a topology whereas the metric defines the scalar product of two vectors. They need not be the same and in those two I mentioned above are not. Distance let's one define neighborhoods and neighborhoods allow one open and closed sets which allows one to define the topology of the space.

The quantity you have above doesn't look like anything I recognize. What you have ican be viewed as the equation of a surface, i.e. z = f(x,y). The ds is then the difference in height (delta z) of neighboring points. It does not represent the Euclidean distance between two points as I would recognize it. Can you tell us what this f(x,y) is an how its supposed to fit into the definition of a metric?

Basically, what I'm asking is how do you calculate the metric tensor components
$$g_{\alpha\beta}$$?
Are they just found by making assumptions in the curved space you are in?

The space need not be curved to define a metric tensor on it. You can't simply say "Here is Rⁿ. What is the metric for this space?". CAn't be done since there is a lack of information here. Someone has to give you the metric or tell you how the metric tensor maps basis vectors to scalars.


Best wishes

Pete

 

[tacman]

The metric tensor is a mathematical object that encodes the information about the geometry of a curved space. It is defined as the matrix of coefficients g_{\alpha\beta} in the expression for the squared infinitesimal distance, ds^2 = g_{\alpha\beta}dx^\alpha dx^\beta. In other words, the metric tensor is a way of representing the relationship between infinitesimal changes in coordinates (dx^\alpha) and the corresponding changes in distance (ds) in a curved space.

To calculate the metric tensor components, g_{\alpha\beta}, you need to first have a metric, g, which describes the geometry of your curved space. This metric can be defined in terms of a coordinate system and a set of basis vectors. For example, in spherical coordinates, the metric is given by:

ds^2 = dr^2 + r^2d\theta^2 + r^2\sin^2\theta d\phi^2

Here, r, \theta, and \phi are the coordinates, and the basis vectors are (1,0,0), (0,1,0), and (0,0,1). From this metric, you can calculate the metric tensor components by using the formula:

g_{\alpha\beta} = \frac{\partial x^i}{\partial x^\alpha}\frac{\partial x^j}{\partial x^\beta}g_{ij}

Where x^i and x^j are the basis vectors, and g_{ij} are the components of the metric. This formula can be generalized to any coordinate system and metric.

In the case of a non-linear relation, such as f(x,y) = x^2 - y, the metric will still have the same form as above, but the components g_{ij} will be different. These components can be calculated by taking derivatives of the metric with respect to the coordinates.

To summarize, the metric tensor is a way of representing the geometry of a curved space, and its components can be calculated from the metric of that space. The metric tensor is not assumed, but rather derived from the underlying geometry of the space.