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Displaying The Christoffel Symbols in Terms of a Function "F"

asked 2022-12-24 06:42:40 +0100

Jack Zuffante gravatar image

When dealing with a metric-tensor that has a function "f" of the coordinates in it's terms, is it possible to express the christoffel symbols in terms of, say, df/dx, df/dy, etc, rather than displaying the entire output?

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answered 2022-12-24 07:13:26 +0100

tolga gravatar image

Try this:

Man = Manifold(4, 'Man', r'\mathcal{M}')
M = var('M')

BL.<t,r,th,ph> = Man.chart(r't r:(0,+oo) th:(0,pi):\theta ph:(0,2*pi):\phi')
g = Man.lorentzian_metric('g')

g[0,0] = f(r)
g[1,1] = -1/f(r)
g[2,2] = -r^2
g[3,3] = -(r*sin(th))^2

# Christoffel symbols
nab = g.connection()
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One can also use Chart Functions

f = BL.function(function('f')(t,r,th,ph))

and replace f(r) by f(t,r,th,ph)

achrzesz gravatar imageachrzesz ( 2022-12-24 11:51:10 +0100 )edit

Do either of these methods allow an arbitrary (unspecified) function?

Jack Zuffante gravatar imageJack Zuffante ( 2022-12-25 05:32:27 +0100 )edit

Never mind, answered my own question. Thank you!

Jack Zuffante gravatar imageJack Zuffante ( 2022-12-25 05:35:09 +0100 )edit

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Asked: 2022-12-24 06:42:40 +0100

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Last updated: Dec 24 '22