Minkowski metric - to sperical coordinates transformation

[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

 

[stevendaryl]

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

The spherical form of the Minkowsky metric is just

[itex]ds^{2} = c^2 dt^2 - dr^2 - r^2 d\theta^2 - r^2 sin^2(\theta) d\phi^2[/itex]

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

The connection coefficients [itex]\Gamma_{uvw}[/itex] are computed in terms of the metric components via:

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

 

[soi]

OK, great thanks for your help.

To look if I understand it, i calculated it using formula
http://upload.wikimedia.org/wikipedia/en/math/f/f/d/ffdb897152259f912ad9c4d5ab3d474d.png

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?

 

[soi]

ok, nevermind, i found a paper that confirmed my result is okay:
http://web.ihep.su/library/pubs/prep1997/ps/97-36a.pdf

If anyone was looking for it