Visualization of metric tensor

[exponent137]

Barbour writes:
the metric tensor g. Being symmetric (g_uv = g_vu) it has ten independent components,
corresponding to the four values the indices u and v can each take: 0 (for the
time direction) and 1; 2; 3 for the three spatial directions. Of the ten components,
four merely reflect how the coordinate system has been chosen; only six
count. One of them determines the four-dimensional volume, or scale, of the
piece of spacetime, the others the angles between curves that meet in it. These
are angles between directions in space and also between the time direction and
a spatial direction.

I please to write more visually and more precisely.
I suppose that the first four values are on diagonal?...

 

[Bill_K]

Of the ten components, four merely reflect how the coordinate system has been chosen; only six count.

Nope. At a single point, ALL TEN components of the metric tensor reflect how the coordinate system has been chosen. You can always choose the coordinates so that the metric tensor at the origin is equal to the Minkowski metric. The only things that count are the derivatives of the metric tensor.

 

[exponent137]

As I understand, it is not necessary that axes are rectangular. So he measure their angles? I think that it is possible here to ignore next derivatives?

 

[exponent137]

Nope. At a single point, ALL TEN components of the metric tensor reflect how the coordinate system has been chosen. You can always choose the coordinates so that the metric tensor at the origin is equal to the Minkowski metric. The only things that count are the derivatives of the metric tensor.

Let us imagine Spacetime around Schwarchild black hole, which is written in the last equation:
http://en.wikipedia.org/wiki/Metric_tensor
It is described with four numbers. Thus, I suppose all angles between axes are 90°? (In one point it can be also described with metric of Minkowski, and it really is, in infinity.)

Barbour:
One of them determines the four-dimensional volume, or scale, of the piece of spacetime,
Thus, I suppose that this information is hidden in Sch. metric?
Probably he has his own idea
four merely reflect how the coordinate system has been chosen.
What is his idea?

 

[sardar]

Thank you for your question. I am glad to provide a response to the content regarding the visualization of metric tensor as described by Barbour. The metric tensor g is a mathematical object used in the field of differential geometry to describe the geometric properties of a space. It is a symmetric tensor, meaning that g_uv = g_vu, which indicates that it has ten independent components.

These components correspond to the four possible values that the indices u and v can each take, namely 0 (for the time direction) and 1, 2, or 3 for the three spatial directions. However, it is important to note that only six of these components are actually independent, as the remaining four simply reflect how the coordinate system has been chosen.

Out of these six independent components, one is responsible for determining the four-dimensional volume or scale of a particular piece of spacetime. This is an important aspect of the metric tensor, as it allows us to understand the overall size and shape of a space.

The other five independent components determine the angles between curves that meet in a given point in spacetime. These angles can be between directions in space, as well as between the time direction and a spatial direction. This means that the metric tensor not only describes the geometric properties of a space, but also the relationships between different directions within that space.

To help visualize this concept, it may be helpful to imagine the metric tensor as a matrix, with the first four values on the diagonal and the remaining six values off the diagonal. Each value in the matrix represents a different aspect of the geometry of the space, and together they provide a comprehensive understanding of its properties.

In summary, the metric tensor is a powerful mathematical tool that allows us to visualize and understand the geometric properties of a space. Its ten components, though seemingly complex, can be broken down into six independent components that provide valuable information about the size, shape, and relationships within a given space.