Visualization of metric tensor

  • #1
509
6

Main Question or Discussion Point

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?....
 

Answers and Replies

  • #2
Bill_K
Science Advisor
Insights Author
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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.
 
  • #3
509
6
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?
 
  • #4
509
6
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?
 

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