# Writing Components of a Metric Tensor

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• kent davidge
In summary, a metric tensor is a mathematical object used to measure distances and angles in a space. It has components that make up its matrix representation, which can be calculated using the inner product of basis vectors. In physics, the metric tensor is important in understanding curved spaces and plays a crucial role in general relativity. Its components can change in different coordinate systems, but this does not affect its physical properties and can be accounted for using transformation rules.
kent davidge
I wonder if it is possible to write the components of the metric tensor (or any other tensor) as a summ of functions of the coordinates? Like this:

$g^{\mu\nu} = \sum_{\mu}^{D}\sum_{\nu}^{D} g_{_1}(x^{\mu}) g_{_2}(x^{\nu})$

where g1 and g2 are functions of one variable alone and D is the dimension of the Manifold. I hope you understand my poor English. Thanks in advance.

If no, then what would be a way of writing the components of a tensor? I don't like just gμν... It would be better if there were a deeper way of representing that.

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It is always possible, at least in GR, to find, for any given point P on the manifold, a coordinate system in which the metric is diagonal at P. That metric will trivially satisfy your requirement [EDIT: on reflection, I think that's wrong. A diagonal metric does not satisfy the condition], but only in that coordinate system, and only at that point.

Conversely, for any coordinate system in which your condition holds at a point P, there will be other coordinate systems and/or other points for which it does not hold - in fact probably most of them.

For your condition to be true for all points in all coordinate systems, it would have to be the case that the metric tensor is everywhere the tensor product of two vectors, that is
$$\mathbf g=\vec u\otimes\vec v=(u^\alpha\vec e_\alpha)\otimes(v^\beta\vec e_\beta)=u^\alpha v^\beta(\vec e_\alpha\otimes\vec e_\beta)=:u^\alpha v^\beta\mathbf e_{\alpha\beta}$$
The set of tensors that are tensor products of two vectors forms a generating set for the space of order 2 tensors. But they are only a small part of that space, which is formed from all linear combinations of tensor products. It might help to read this page explaining tensor products and how tensor spaces are constructed from them.

Metric tensors are a somewhat restricted subset of the set of all order 2 tensors, because they are required to be symmetric. But the class of symmetric tensors is much larger than the class of tensors that are tensor products of vectors.

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andrewkirk said:
It is always possible, at least in GR, to find, for any given point P on the manifold, a coordinate system in which the metric is diagonal at P. That metric will trivially satisfy your requirement
Why must the metric be diagonal at P to my condition be satisfied?

kent davidge said:
Why must the metric be diagonal at P to my condition be satisfied?
It doesn't have to. He's saying that if it is diagonal it will satisfy your condition, not that that's the only way of satisfying it.

kent davidge
Nugatory said:
It doesn't have to. He's saying that if it is diagonal it will satisfy your condition, not that that's the only way of satisfying it.
Thank you.

Nugatory said:
It doesn't have to. He's saying that if it is diagonal it will satisfy your condition, not that that's the only way of satisfying it.
Can you show me with equations why a diagonal metric will satisfy that condition?

kent davidge said:
Can you show me with equations why a diagonal metric will satisfy that condition?
Actually, on second thoughts, I don't think a diagonal metric does satisfy that condition. The rest of the post is correct though.

## 1. What is a metric tensor?

A metric tensor is a mathematical object that is used to quantify the distances and angles in a space. It is a symmetric, second-order tensor that assigns a length to each vector in the space. It is a fundamental tool in the study of geometry and is used in various areas of physics, including general relativity and differential geometry.

## 2. What are the components of a metric tensor?

The components of a metric tensor are the numbers that make up the matrix representation of the tensor. In a space with n dimensions, the metric tensor will have n² components. These components are typically denoted as gij, where i and j represent the indices of the matrix.

## 3. How are the components of a metric tensor calculated?

The components of a metric tensor can be calculated using the inner product of two vectors in the space. The formula for calculating the component gij is: gij = vi · vj, where vi and vj are the basis vectors of the space. This formula can also be written in terms of the metric tensor as gij = g(vi, vj).

## 4. What is the significance of the metric tensor in physics?

The metric tensor plays a crucial role in physics, particularly in the theory of general relativity. It is used to define the concept of distance in curved spaces, which is necessary for understanding the effects of gravity. The metric tensor is also used in the calculation of physical quantities, such as energy and momentum, in a curved spacetime.

## 5. Can the components of a metric tensor change in different coordinate systems?

Yes, the components of a metric tensor can change in different coordinate systems. This is known as a change of basis. However, the tensor itself remains the same object and its physical properties do not change. The change in components is accounted for by using transformation rules, which allow for the tensor to be expressed in different coordinate systems.

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