Hi,
I have a disk in the (x,y) plane rotating with the angular velocity
omega. Now I'd like to play with differential geomtry on the disk, so
I start with the Euclidean metrics in my laboratory system and
transform it to the disk, I get this:
In [1]: var("t omega")
Out[1]: (t, ω)
In [2]: g_dd = Matrix([(-1+omega*(x**2+y**2),2*omega*y,-2*omega*x,0),
(2*omega*y, 1, 0, 0), (-2*omega*x, 0, 1, 0), (0, 0, 0, 1)])
In [4]: g_dd
Out[4]:
⎡ ⎛ 2 2⎞ ⎤
⎢-1 + ω*⎝x + y ⎠ 2*ω*y -2*ω*x 0⎥
⎢ ⎥
⎢ 2*ω*y 1 0 0⎥
⎢ ⎥
⎢ -2*ω*x 0 1 0⎥
⎢ ⎥
⎣ 0 0 0 1⎦
The g_dd stands for g_\mu_\nu (i.e. both indices down). Now I want to
calculate the Christoffel symbols from this metrics. For this, I need
to calculate the g^\mu^\nu, which is just the matrix
inversion:
In [3]: g_uu = g_dd.inv()
However, try to display this:
In [6]: g_uu
[many pages of output]
So let's just try to look at [0,0]:
In [9]: g_uu[0, 0]
[still too long]
In [9]: print g_uu[0, 0]
-1/(1 - 4*omega**2*x**2/(-1 + omega*(x**2 + y**2)) -
16*omega**4*x**2*y**2*(-1 + omega*(x**2 + y**2))**(-2)/(1 -
4*omega**2*y**2/(-1 + omega*(x**2 + y**2))))*(2*omega*x/(-1 +
omega*(x**2 + y**2)) + 8*x*omega**3*y**2*(-1 + omega*(x**2 +
y**2))**(-2)/(1 - 4*omega**2*y**2/(-1 + omega*(x**2 +
y**2))))*(-2*omega*x/(-1 + omega*(x**2 + y**2)) -
8*x*omega**3*y**2*(-1 + omega*(x**2 + y**2))**(-2)/(1 -
4*omega**2*y**2/(-1 + omega*(x**2 + y**2)))) + 1/(-1 + omega*(x**2 +
y**2)) + 4*omega**2*y**2*(-1 + omega*(x**2 + y**2))**(-2)/(1 -
4*omega**2*y**2/(-1 + omega*(x**2 + y**2)))
Any try to simplify this has failed (I mean simply that it didn't
simplify much). Is there any way to get this simplified, or is it just
so complex by nature? Maybe that's the reason to use polar
coordinates.
Ondrej
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