# Minkowski metric - to sperical coordinates transformation

1. Jul 7, 2012

### soi

I need to transform cartesian coordinates to spherical ones for Minkowski metric.
Taking:
(x0, x1, x2, x3) = (t, r, α, β)

And than write down all Christoffel symbols for it.

I really have no clue, but from other examples i've seen i should use chain rule in first and symmetry of Christoffel symbol Tab=Tba

2. Jul 8, 2012

### stevendaryl

Staff Emeritus
The spherical form of the Minkowsky metric is just

$ds^{2} = c^2 dt^2 - dr^2 - r^2 d\theta^2 - r^2 sin^2(\theta) d\phi^2$

So the metric components are
$g_{tt} = c^2$
$g_{rr} = -1$
$g_{\theta\theta} = -r^2$
$g_{\phi\phi} = -r^2 sin^2(\theta)$

The connection coefficients $\Gamma_{uvw}$ are computed in terms of the metric components via:

$\Gamma_{uvw} = \frac{1}{2} (\partial_{v} g_{uw} + \partial_{w} g_{vu} - \partial_{u} g_{vw})$

3. Jul 11, 2012

### soi

OK, great thanks for your help.

To look if I understand it, i calculated it using formula

And i got what you got (not surprisingly) but with -1 everywhere:

gtt=-1
grr=1
gθθ=r^2
gββ=r^2(sinθ)^2

And Christoffel symbols (nonzoro, numering metric matrix from 0 to 3):
T221=1/r
T122=-r
T331=1/r
T332=1/2(rsinθ)^2
T133=-r (sinθ)^2
T233=(sin2θ)/2

Is it okay?

4. Jul 11, 2012