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# Computing Christoffel Symbols HELP!! (Relativity / Maths)

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1. Hey everyone! Need help in understanding how these Christoffel Symbels were computed? I know they used the formula for the "connection" but still can't seem to get my head round how it works? Or does anyone have an alternative method? THANKSS!!
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2. Could someone also explain how the got the metrics. Many thankss
3. (Original post by Nazzy_HCrest)
Could someone also explain how the got the metrics. Many thankss
Well, from the very little that I've just learnt through Google - one of the ways to get the 2-sphere is to imaging the 3-sphere then restrict it to constant radius.

Since, the 3-sphere has then restricting (the unit 2-sphere):

, as a matrix, you can represent this as:

which reduces to your matrix if we take (i.e: the unit 2-sphere)

The inverse is readily obtainable as .

Although there are more intuitive ways to get this metic, such as specifying the symemtries you require, three independent rotations and then finding the 2-sphere via that. Or you could look for 2-dim spaces with constant curvature or compute the metric for a general 2-dim geometry and then impose constant curvature that gives a set of DE's which you can then use to get this metric.
4. (Original post by Zacken)
Well, from the very little that I've just learnt through Google - one of the ways to get the 2-sphere is to imaging the 3-sphere then restrict it to constant radius.

Since, the 3-sphere has then restricting (the unit 2-sphere):

, as a matrix, you can represent this as:

which reduces to your matrix if we take (i.e: the unit 2-sphere)

The inverse is readily obtainable as .

Although there are more intuitive ways to get this metic, such as specifying the symemtries you require, three independent rotations and then finding the 2-sphere via that. Or you could look for 2-dim spaces with constant curvature or compute the metric for a general 2-dim geometry and then impose constant curvature that gives a set of DE's which you can then use to get this metric.
That's fantastic! Could you possibly explain how you represented it as the matrix? Thanks a million for taking the time to help!!
5. (Original post by Nazzy_HCrest)
That's fantastic! Could you possibly explain how you represented it as the matrix? Thanks a million for taking the time to help!!
I'm only an A-Level student, so I'm likely to be extremely useless, but I believe that the matrix comes by considering the basis vectors of your metric. We, have: , does this seem familiar to you?

In general: but for the 2-sphere, we have , so we get: and which allows us to write down our matrix right away.

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