Visualization of metric tensor

Click For Summary

Discussion Overview

The discussion revolves around the visualization and interpretation of the metric tensor in the context of spacetime, particularly in relation to its components and their significance. Participants explore theoretical aspects, mathematical reasoning, and implications for specific scenarios such as the Schwarzschild black hole.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Barbour suggests that the metric tensor has ten components, with four reflecting the choice of coordinate system and six being significant for describing spacetime geometry.
  • Some participants argue that all ten components of the metric tensor are influenced by the coordinate system, and that only the derivatives of the metric tensor are truly relevant.
  • There is a question about whether the axes of the metric tensor must be rectangular, with one participant proposing that angles between axes can be measured and suggesting that derivatives might be ignored.
  • Another participant introduces the Schwarzschild black hole as a context for discussing the metric tensor, questioning whether the angles between axes can be assumed to be 90° and how this relates to the Minkowski metric.
  • Barbour's statement about the metric tensor determining the four-dimensional volume of spacetime is referenced, leading to speculation about whether this information is inherent in the Schwarzschild metric.
  • There is curiosity about Barbour's underlying ideas regarding the metric tensor and its components.

Areas of Agreement / Disagreement

Participants express differing views on the significance of the components of the metric tensor and the relevance of coordinate choices, indicating that multiple competing perspectives remain unresolved.

Contextual Notes

Participants highlight the dependence of the metric tensor's interpretation on coordinate choices and the potential implications for understanding spacetime geometry, but do not resolve these complexities.

exponent137
Messages
563
Reaction score
35
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?...
 
Physics news on Phys.org
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.
 
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?
 
Bill_K said:
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?
 

Similar threads

Undergrad How to calculate basis vectors from metric tensor (as a matrix)?
  • · Replies 6 ·
Replies
6
Views
1K
Graduate Weyl tensor and coordinate acceleration
  • · Replies 12 ·
Replies
12
Views
3K
Undergrad Coordinates for diagonal metric tensors
  • · Replies 12 ·
Replies
12
Views
3K
Undergrad Computing Riemann Tensor: 18 Predicted Non-Trivial Terms
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Can a Tensor be Represented by a 3-Dimensional Matrix?
  • · Replies 13 ·
Replies
13
Views
3K
Graduate Exploring the Stress-Energy Tensor of a Perfect Fluid
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Curvature and Schwarzschild metric
  • · Replies 11 ·
Replies
11
Views
3K
Graduate Comparing Gullstrand-Painleve & Lemaitre Coordinates
  • · Replies 8 ·
Replies
8
Views
3K
Graduate Is the Stress-Energy Tensor for a Scalar Field Phi Isotropic in FRW Metrics?
  • · Replies 7 ·
Replies
7
Views
2K
Graduate Coordinate transformation and metric tensor
  • · Replies 3 ·
Replies
3
Views
6K