- #1

- 716

- 9

## Main Question or Discussion Point

This is still rather new to me so please pardon my ignorance. My introduction to tensor differentiation involved only manifolds that were embedded in higher-dimensional Euclidean spaces. To describe them, I was instructed to find basis vectors using partial derivatives, as in

[tex]

e_\theta = <\frac {\partial x}{\partial \theta},\frac {\partial y}{\partial \theta},\frac {\partial z}{\partial \theta}>

[/tex]

then to use those and the dot product to find both the metric tensor and the Christoffel symbols.

[tex]g_\phi_\theta =e_\phi \cdot e_\theta[/tex]

[tex] g_\theta_\theta =e_\theta \cdot e_\theta[/tex]

.

.

[tex]

\Gamma ^\theta _\theta _\phi = e^\theta \cdot \frac {\partial e_\theta}{\partial \phi}

[/tex]

etc.

Since then I've noticed that GR doesn't seem to bother with higher-dimensional flat spaces, and instead begins with the metric tensor, using that alone to compute the Christoffel symbols. That's fine with me, but what about basis vectors? Can those somehow also be computed using the metric tensor? or should I regard them simply as

[tex]

e_0 = <1,0,0,0>[/tex]

[tex] e_1 =<0,1,0,0>[/tex]

[tex] e_2 =<0,0,1,0>[/tex]

[tex] e_3 = <0,0,0,1>[/tex]

and move on? Does it matter?

[tex]

e_\theta = <\frac {\partial x}{\partial \theta},\frac {\partial y}{\partial \theta},\frac {\partial z}{\partial \theta}>

[/tex]

then to use those and the dot product to find both the metric tensor and the Christoffel symbols.

[tex]g_\phi_\theta =e_\phi \cdot e_\theta[/tex]

[tex] g_\theta_\theta =e_\theta \cdot e_\theta[/tex]

.

.

[tex]

\Gamma ^\theta _\theta _\phi = e^\theta \cdot \frac {\partial e_\theta}{\partial \phi}

[/tex]

etc.

Since then I've noticed that GR doesn't seem to bother with higher-dimensional flat spaces, and instead begins with the metric tensor, using that alone to compute the Christoffel symbols. That's fine with me, but what about basis vectors? Can those somehow also be computed using the metric tensor? or should I regard them simply as

[tex]

e_0 = <1,0,0,0>[/tex]

[tex] e_1 =<0,1,0,0>[/tex]

[tex] e_2 =<0,0,1,0>[/tex]

[tex] e_3 = <0,0,0,1>[/tex]

and move on? Does it matter?