Hypersphere metric

[Royston]

This is isn't homework, just something I decided to have a crack at, alongside my current course, and forgot about it. Just need confirmation on the last step.

 

I wanted to find the connection coefficients of a hypersphere, so the line element is...

 

[math]dl^2=R^2[d\psi^2+sin^2\psi(d\theta^2 + sin^2\theta d\phi^2)][/math] so...

 

[math]dl^2=R^2d\psi^2+R^2sin^2\psi d\theta^2+R^2sin^2\psi sin^2\theta d\phi^2[/math]

 

Where...

 

[math]x^1=\psi[/math]

[math]x^2=\theta[/math]

[math]x^3=\phi[/math]

 

So the metric is...

 

[math]g_{ij} = \left[ \begin{array}{ccc} R^2 & 0 & 0 \\ 0 & R^2sin^2 (x^1) & 0 \\ 0 & 0 & R^2sin^2(x^1)sin^2(x^2) \end{array} \right][/math]

 

Where the dual metric [math]g^{ij}[/math], is the inverse matrix of the above.

 

Now this was the bit I was unsure about...

 

[math]\frac{\partial g_{22}}{\partial x^1}=2R^2sin(x^1)cos(x^1)[/math]

 

[math]\frac{\partial g_{33}}{\partial x^1}=2R^2sin(x^1)cos(x^1)sin^2(x^2)[/math]

 

[math]\frac{\partial g_{33}}{\partial x^2}=2R^2sin^2(x^1)cos(x^2)sin(x^2)[/math]

 

[math]\frac{\partial g_{ij}}{\partial x^k}=0[/math] for all other values of i,j,k.

 

Now is it a general rule to partially differentiate each [math]x^k[/math] term, for increasing values of k (see the second to last equation for example). I'm guessing it is, because these will crop up in the connection coefficent equations, and determine which ones are independent, and non zero et.c...I just wanted to be doubly sure.

 

BTW, it's coming up to 2am where I am, so I might be being a bit dumb here.

[ajb]

Now is it a general rule to partially differentiate each [math]x^k[/math] term, for increasing values of k (see the second to last equation for example).

 

I think you are referring to the Einstein summation convention.

[Royston]

Well being a GR course, Einstein notation is used extensively, so I'm probably missing something obvious...or I worded my question poorly.

 

Are my steps correct so far ? If so, then I know I'm on the right track at least.

[ajb]

It looks ok to me so far.

[timo]

I think you worded your question very poorly. I might have understood your question now: The g22 term depends on only one coordinate, the g33 term depends on two coordinates. Your question is if the gXX term will always depend on at most as many variables as the gYY term for Y>X? The answer would be "no": Just switch the names of x1 and x2 which is nothing but a renaming but will also swap g22 and g33. You can also have non-zero diagonal terms in general.

[Royston]

I think you worded your question very poorly.

 

By the time I got to answer the question, I was practically asleep, so sorry about that.

 

Your question is if the gXX term will always depend on at most as many variables as the gYY term for Y>X? The answer would be "no": Just switch the names of x1 and x2 which is nothing but a renaming but will also swap g22 and g33. You can also have non-zero diagonal terms in general.

 

Thanks, that's precisely what I was getting at, I just havn't come across an example yet (in my course) where the names of x, were switched, but that makes things a lot clearer.

[ajb]

Now you have it answered I understand your question!

 

In general there will be off-diagonal terms and every term can depend on all the coordinates.

[Royston]

In general there will be off-diagonal terms and every term can depend on all the coordinates.

 

Thanks ajb, this was what (in a clumsy way) I was alluding to...however, I'm having a real hard time trying to visualize what's going on, I guess I'm not content with just method.

 

In any case, I'll post the results of the connection coefficients, and geodesics et.c tomorrow, it's good practice to tackle stuff outside the scope of what you're being taught...at least I think so.