Confused about the metric tensor

In summary, a metric tensor is a mathematical object used in geometry and general relativity to describe the distance between points in a space. It is a specific type of tensor that is used to define the metric or distance function and is important in general relativity as it relates matter and energy to the curvature of space-time. The metric tensor is used in various calculations involving distance, angles, and volumes, and is also used in the formulation of Einstein's field equations. It cannot be directly visualized, but can be represented graphically through diagrams and matrices to aid in understanding.
  • #1
GarageDweller
104
0
Now let's say I have the metric for some curved two surface

ds^2=G(u,v)du^2+P(u,v)dv^2 ( the G and P functions being the 00 and 11 components, assuming the metric is diagonal)

Now my question is, since the metric defines the scalar product of two vectors, let's say
(1,0) and (0,1), for simplicity, which values do I take for u and v in G(u,v) and P(u,v)?
 
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  • #2
u and v would be the coordinates of wherever you are evaluating the scalar product.

In other words, the coordinates of where the vectors are located.
 

Related to Confused about the metric tensor

What is a metric tensor?

A metric tensor is a mathematical object used in the study of geometry and general relativity. It is a matrix that describes the distance between points in a given space.

How is a metric tensor different from a regular tensor?

A metric tensor is a specific type of tensor that is used to define the metric or distance function in a space. It is also known as a "metric tensor field" because its values can vary at different points in space.

Why is the metric tensor important in general relativity?

In general relativity, the metric tensor is used to describe the curvature of space-time. It is a fundamental concept in the theory, as it relates the presence of matter and energy to the curvature of space-time.

How is the metric tensor used in calculations?

The metric tensor is used in calculations involving distance, angles, and volumes in a given space. It is also used in the formulation of Einstein's field equations, which describe the relationship between matter and the curvature of space-time.

Can the metric tensor be visualized?

The metric tensor is a mathematical concept and cannot be directly visualized. However, it can be represented graphically through diagrams and matrices to aid in understanding its properties and calculations.

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