How to Calculate Christoffel Symbols in Spherical Coordinates?

[gabbagabbahey]

Shouldn't you have \phi=\tan^{-1}(y/x)?

 

[latentcorpse]

ok so

\ddot{\phi}=\frac{\ddot{y} x^{-1} -2 \dot{y} x^{-2} \dot{x} - y x^{-2} \dot{x} +2y x^{-3} \dot{x}^2}{1+ \left( \frac{y}{x} \right)^2} - \frac{2y}{x} \frac{ \left( \dot{y} x^{-1} - y x^{-2} \dot{x} \right)^2}{\left( 1+ \left( \frac{y}{x} \right)^2 \right)^2}

 

[gabbagabbahey]

That doesn't look quite right to me...I'd start by simplifying \dot{\phi} as much as possible, and then converting it to spherical coordinates...what do you get when you do that?

 

[latentcorpse]

\dot{\phi}=\frac{1}{1+ \left( \frac{y}{x} \right)^2 } \left( \dot{y} x^{-1} - y x^{-2} \dot{x}} \right)=\frac{1}{1+ \left( \frac{a_y+b_yt}{a_x+b_xt} \right)^2} \left( \frac{b_y}{a_x+b_xt}-\frac{b_x(a_y+b_yt)}{(a_x+b_xt)^2} \right)
that ok?

 

[gabbagabbahey]

\dot{\phi}=\frac{1}{1+ \left( \frac{y}{x} \right)^2 } \left( \dot{y} x^{-1} - y x^{-2} \dot{x}} \right)

I'd multiply both the numerator and denominator by x^2, sub in \dot{x}=b_x and \dot{y}=b_y and then write x and y in terms of r, \theta and \phi...

 

[latentcorpse]

\dot{\phi} = \frac{r \sin{\theta}}{1+r^2 \sin^2{\theta} \sin^2{\phi}} \left( b_y \cos{\phi} - b_x \sin{\phi} \right)

is that going to help? shouldn't i take the second derivative before i switch to spherical coordinates?

 

[gabbagabbahey]

\dot{\phi} = \frac{r \sin{\theta}}{1+r^2 \sin^2{\theta} \sin^2{\phi}} \left( b_y \cos{\phi} - b_x \sin{\phi} \right)

is that going to help? shouldn't i take the second derivative before i switch to spherical coordinates?

It would help, if you did it properly

Try again, you have an error...and don't be afraid to simplify your result as much as possible.

 

[latentcorpse]

\dot{\phi}=\frac{1}{1+ \left( \frac{y}{x} \right)^2} \left( \dot{y} x^{-1} - y x^{-2} \dot{x} \right) = \frac{1}{1+y^2} \left( \dot{y} x - y \dot{x} \right)

\ddot{\phi} = \frac{ \left( \ddot{y} x + \dot{y} \dot{x} - \dot{y} \dot{x} - y \ddot{x} \right) \left( 1+ y^2 \right) - 2 y \dot{y} \left( \dot{y} x - y \dot{x} \right) }{ \left(1+y^2 \right)^2 } note the first term in the numerator will dissappear because \ddot{y}=\ddot{x}=0 and \dot{y}\dot{x}-\dot{x}\dot{y}=0.

\ddot{\phi} = - \frac{2r \sin{\theta} \sin{\phi} b_y \left( b_y r \sin{\theta} \cos{\phi} - b_x r \sin{\theta} \sin{\phi} \right) }{ \left(1+r^2 \sin^2{\theta} \sin^2{\phi} \right)^2}
\ddot{\phi}=-\frac{2 b_y r^2 \sin^2{\theta} \sin{\phi} \left( b_y \cos{\phi} - b_x \sin{\phi} \right)}{ \left(1+r^2 \sin^2{\theta} \sin^2{\phi} \right)^2}

i can't see how to simplify that further...

 

[gabbagabbahey]

When you multiply 1+\left(\frac{y}{x}\right)^2 by x^2, should you get x^2+y^2?

 

[latentcorpse]

\dot{\phi}=\frac{\dot{y}x-y \dot{x}}{x^2+y^2}

\ddot{\phi}=\frac{ \left( \ddot{y} x + \dot{y} \dot{x} - \dot{y} \dot{x} - y \ddot{x} \right) \left( x^2+y^2 \right) - \left( 2 x \dot{x} + 2 y \dot{y} \right) \left( \dot{y} x - y \dot{x} \right)}{ \left( x^2+y^2 \right)^2}

\ddot{\phi} = \frac{-2 \left( b_x r \sin{\theta} \cos{\phi} + b_y r \sin{\theta} \sin{\phi} \right) \left( b_y r \sin{\theta} \cos{\phi} - b_x r \sin{\theta} \sin{\phi} \right) }{ \left( r^2 \sin^2{\theta} \cos^2{\phi} + r^2 \sin^2{\theta} \sin^2{\phi} \right)^2}

\ddot{\phi}=\frac{-2r^2 \sin^2{\theta} \left(b_x \cos{\phi} + b_y \sin{\phi} \right) \left( b_y \cos{\phi} - b_x \sin{\phi} \right)}{r^4 \sin^4{\theta}}

\ddot{\phi} = \frac{-2 \left( b_x b_y \cos^2{\phi} + b_y^2 \sin{\phi} \cos{\phi} - b_x^2 \sin{\phi} \cos{\phi} - b_x b_y sin^2{\phi} \right)}{r^2 \sin^2{\theta}}

hopefully that looks better? i was hoping the numerator would simplify nicer though.

 

[gabbagabbahey]

Your \LaTeX contains a few typos, but yes, that's correct.

Also, it is worth writing \dot{\phi} and \dot{r} in spherical coordinates;

\dot{r}=b_x\sin\theta\cos\phi+ b_y\sin\theta\sin\phi+b_z\cos\theta

\dot{\phi}=\frac{b_y\cos\phi-b_x\sin\phi}{r\sin\theta}

These will come in handy when calculating \ddot{\theta} (Since \dot{\theta} will be a function of r, \theta and \phi; knowing these derivatives allows you to find \ddot{\theta} quickly, using the chain rule)

 

[Phrak]

I recall that a straight line in cartesian coordinates is \ \frac{d^2 x^i}{d \lambda^2} = 0 \ ,
where the curve is parametric in lambda. The parameter can be time.

\frac{d^2 x^i}{d t^2} = 0

 

[gabbagabbahey]

I recall that a straight line in cartesian coordinates is \ \frac{d^2 x^i}{d \lambda^2} = 0 \ ,
where the curve is parametric in lambda. The parameter can be time.

\frac{d^2 x^i}{d t^2} = 0

This is true; which is why the solution is of the form x^i=a_i+b_it in Cartesians.

 

[latentcorpse]

ok. so finally \ddot{\theta}

\theta=\cos^{-1} \left( \frac{z}{r} \right)

\dot{\theta} = - \frac{1}{\sqrt{1 - \left( \frac{z}{r} \right)^2 }} \left( \dot{z} r^{-1} - z r^{-2} \dot{r} \right)

\dot{\theta}=-\frac{1}{\sqrt{r^4-z^2r^2}} \left(\dot{z}r-z \dot{r} \right)

is that ok for \dot{\theta}?

 

[gabbagabbahey]

Looks fine so far; now express it in terms of spherical coordinates and b_x, b_y and b_z...

 

[latentcorpse]

\dot{\theta}=-\frac{1}{r^2 \sqrt{1-\cos^2{\theta}}} \left( \dot{z}r - z \dot{r} \right) = -\frac{1}{r^2 \sin{\theta}} \left( b_z r - \left( a_z + b_z t \right) \left(b_x \sin{\theta} \cos{\phi} + b_y \sin{\theta} \sin{\phi} + b_z \cos{\theta} \right)

looking ok?

 

[gabbagabbahey]

I'd use z=r\cos\theta instead of z=a_z+b_zt...

 

[latentcorpse]

\dot{\theta}=-\frac{1}{r \sin{\theta}} \left(b_z - \cos{\theta} \right) \left( b_x \sin{\theta} \cos{\phi} + b_y \sin{\theta} \sin{\phi} + b_z \cos{\theta} \right)
that's ok?

should i take the derivative of the above or should i take the derivative of something shorter like:

\dot{\theta}=-\frac{1}{r \sin{\theta}} \left( \dot{z} r - z \dot{r} \right)?

this gives

\ddot{\theta}=\frac{ \left( \ddot{z} r + \dot{z} \dot{r} - \dot{z} \dot{r} - z \ddot{r} \right) r \sin{\theta} - \left( \dot{r} \sin{\theta} + r \dot{\theta} \cos{\theta} \right) \left( \dot{z} r - z \dot{r} \right) }{r^2 \sin^2{\theta}}

\ddot{\theta}=\frac{-zr \ddot{r} \sin{\theta}- \left( \dot{r} \sin{\theta} + r \dot{\theta} \cos{\theta} \right) \left( \dot{z} r - z \dot{r} \right)}{r^2 \sin^2{\theta}}

on the right lines?

 

[gabbagabbahey]

\dot{\theta}=-\frac{1}{r \sin{\theta}} \left(b_z - \cos{\theta} \right) \left( b_x \sin{\theta} \cos{\phi} + b_y \sin{\theta} \sin{\phi} + b_z \cos{\theta} \right)
that's ok?

Do b_z and \cos\theta have the same units? If not, the appearance of the factor b_z-\cos\theta should be a dead giveaway that you've made an error somewhere...(It's always a good idea to check the units of your expression at each step of a complicated calculation, it will help cut down on the number of error you make)

 

[latentcorpse]

you said i was fine at \dot{\theta}=-\frac{1}{\sqrt{r^4-z^2r^2}} \left(\dot{z}r-z \dot{r} \right)

so i need to change the coords here

\dot{\theta}=-\frac{1}{\sqrt{r^4-r^4 \cos^2{\theta}}} \left(b_z r - r \cos{\theta} \dot{r} \right) = -\frac{1}{r \sin{\theta}} \left( b_z - \cos{\theta} \dot{r} \right)
ok that should be better

now

\ddot{\theta}=- \left( \frac{ \left( - \ddot{r} \cos{\theta} + \dot{r} \dot{\theta} \sin{\theta} \right) r \sin{\theta} - \left( \dot{r} \sin{\theta} + r \dot{\theta} \cos{\theta} \right) \left(b_z - \dot{r} \cos{\theta} \right)}{r^2 \sin^2{\theta}} \right)

is that ok? it doesn't look like it's going to get much simpler : i can bring the minus at the front in and cancel the r \sin{\theta} in the first term i guess...

 

[gabbagabbahey]

you said i was fine at \dot{\theta}=-\frac{1}{\sqrt{r^4-z^2r^2}} \left(\dot{z}r-z \dot{r} \right)

so i need to change the coords here

\dot{\theta}=-\frac{1}{\sqrt{r^4-r^4 \cos^2{\theta}}} \left(b_z r - r \cos{\theta} \dot{r} \right) = -\frac{1}{r \sin{\theta}} \left( b_z - \cos{\theta} \dot{r} \right)
ok that should be better

Why not substitute in your expression for \dot{r} here?

 

[latentcorpse]

i get

\dot{\theta}=-\frac{1}{r \sin{\theta}} \left(b_z-\cos{\theta} \dot{r} \right)

\dot{\theta}=-\frac{1}{r \sin{\theta}} \left( b_z - b_x \sin{\theta} \cos{\theta} \cos{\phi} - b_y \sin{\theta} \cos{\theta} \sin{\phi} - b_z \cos^2{\theta} \right)

\dot{\theta}=-\frac{1}{r \sin{\theta}} \left( b_z \sin^2{\theta} - b_x \sin{\theta} \cos{\theta} \cos{\phi} - b_y \sin{\theta} \cos{\theta} \sin{\phi} \right)

\dot{\theta}=-\frac{1}{r} \left(b_z \sin{\theta} - b_x \cos{\theta} \cos{\phi} - b_y \cos{\theta} \sin{\phi} \right)
now do i take the time derivative of this?

that gives, by product rule:

\ddot{\theta}=r^{-2} \dot{r} \left( b_z \sin{\theta} - b_x \cos{\theta} \cos{\phi} - b_y \cos{\theta} \sin{\phi} \right) - \frac{1}{r} \left( b_z \dot{\theta} \cos{\theta} + b_x \dot{\theta} \sin{\theta} \cos{\phi} + b_x \dot{\phi} \cos{\theta} \sin{\phi} + b_y \dot{\theta} \sin{\theta} \sin{\phi} - b_y \dot{\phi} \cos{\theta} \cos{\phi} \right)

\ddot{\theta} = \frac{\dot{r}}{r^2} \left(b_z \sin{\theta} - b_x \cos{\theta} \cos{\phi} - b_y \cos{\theta} \sin{\phi} \right) - \frac{1}{r} \left( \dot{\theta} \dot{r} + \dot{\phi} b_x \cos{\theta} \sin{\phi} - \dot{\phi} b_y \cos{\theta} \cos{\phi} \right)

is this looking ok? i can't seem to simplify it any further though.

 

[gabbagabbahey]

Looks fine to me...now realize that the first term is just

\frac{\dot{r}}{r^2} \left(b_z \sin{\theta} - b_x \cos{\theta} \cos{\phi} - b_y \cos{\theta} \sin{\phi} \right)=-\frac{\dot{r}\dot{\theta}}{r}

And

-\frac{1}{r}\left(\dot{\phi} b_x \cos{\theta} \sin{\phi} - \dot{\phi} b_y \cos{\theta} \cos{\phi} \right)=\sin\theta\cos\theta\dot{\phi}^2

So,

\ddot{\theta}=-\frac{2}{r}\dot{r}\dot{\theta}+ \sin\theta\cos\theta\dot{\phi}^2

Compare that to your second Geodesic equation...

Edit: Is there a reason you've interpreted \frac{dx^i}{dt}\frac{dx^i}{dt} as \frac{d^2x^i}{dt^2} when calculating your Geodesic equations in post#22?

 

[latentcorpse]

should the geodesic eqns be

\frac{d^2 r}{dt^2}-r \frac{d \theta}{dt} \frac{d \theta}{dt} - r \sin^2{\theta} \frac{d \phi}{dt} \frac{d \phi}{dt}=0
\frac{d^2 \theta}{dt^2} + \frac{2}{r} \frac{dr}{dt} \frac{d \theta}{dt} - \sin{\theta} \cos{\theta} \frac{d \phi}{dt} \frac{d \phi}{dt}=0
\frac{d^2 \phi}{dt^2} + \frac{2}{r} \frac{dr}{dt} \frac{d \phi}{dt} + 2 \cot{\theta} \frac{d \theta}{dt} \frac{d \phi}{dt}=0?

 

[gabbagabbahey]

Yup.

 

[latentcorpse]

okay so I am testing the equations now:
looking at the first one i have

\ddot{r}-r \dot{\theta}^2 - r \sin^2{\theta} \dot{\phi}^2
=\frac{b^2}{r}-\frac{ \left( b_x x + b_y y + b_z z \right)^2}{r^3} + b_z \sin{\theta} - b_x \cos{\theta} \cos{\phi} - b_y \cos{\theta} \sin{\phi} + \frac{2}{r} \left( b_x b_y \cos^2{\phi} + b_y^2 \sin{\phi} \cos{\phi} - b_x^2 \sin{\phi} \cos{\phi} -b_x b_y \sin^2{\phi} \right)

i can't get any constructive cancellation after this line though...

 

[gabbagabbahey]

It might be easier if you just use the chain rule to find \ddot{r} from this equation \dot{r}=b_x\sin\theta\cos\phi+ b_y\sin\theta\sin\phi+b_z\cos\theta...that way everything is in Spherical coords.

 

[latentcorpse]

so \ddot{r}= \dot{\theta} \left( b_x \cos{\theta} \cos{\phi} + b_y \cos{\theta} \sin{\phi} - b_z \sin{\theta} \right) + \dot{\phi} \left( b_y \sin{\theta} \cos{\phi} - b_x \sin{\theta} \sin{\phi} \right)

\ddot{r}=r \dot{\theta}^2 + \dot{\phi} \left( b_y \sin{\theta} \cos{\phi} - b_x \sin{\theta} \sin{\phi} \right)

\ddot{r}=r \dot{\theta}^2 + r \sin^2{\theta} \dot{\phi}^2
so geodesic eqns 1 and 2 are obviously satisfied.

the third one is giving me a bit of grief though:

\ddot{\phi} + \frac{2}{r} \dot{r} \dot{\phi} + 2 \cot{\theta} \dot{\theta} \dot{\phi}
=\frac{-2 \left(b_xb_y \cos^2{\phi} + b_y^2 \sin{\phi} \cos{\phi} - b_x^2 \sin{\phi} \cos{\phi} - b_x b_y \sin^2{\phi} \right)}{r^2 \sin^2{\theta}}
+\frac{2 \left(b_x b_y \sin{\theta} \cos^2{\phi} - b_x^2 \sin{\theta} \cos{\theta} \cos{\phi} + b_y^2 \sin{\theta} \cos{\phi} \sin{\phi} - b_x b_y \sin{\theta} \sin^2{\phi} + b_y b_z \cos{\theta} \cos{\phi} - b_x b_z \sin{\phi} \cos{\theta} \right) }{r^2 \sin{\theta}}
-\frac{2 \cos{\theta}}{r^2 \sin^2{\theta}} \left(b_y b_z \sin{\theta} \cos{\phi} - b_x b_y \cos{\theta} \cos^2{\phi} - b_y^2 \cos{\theta} \cos{\phi} \sin{\phi} - b_x b_z \sin{\theta} \sin{\phi} - b_x^2 \cos{\theta} \sin{\phi} \cos{\phi} + b_x b_y \cos{\theta} \sin{\phi} \cos{\phi} \right)

which is proving hard to simplify. in particular the second term has a denominator that is different from that of the the first two terms - should i multiply through by \frac{\sin{\theta}}{\sin{\theta}}?

 

[gabbagabbahey]

It shouldn't be too hard to simplify; just collect terms with b_x^2 in them, and terms with b_xb_y etc..

 

[latentcorpse]

that goes to

\frac{2 b_x b_y}{r^2 \sin^2{\theta}} \left( - \cos^2{\phi} + \sin^2{\phi} + \sin^2{\theta} \cos^2{\phi} - \sin^2{\theta} \sin^2{\phi} + \cos^2{\theta} \cos^2{\phi} - \cos^2{\theta} \sin{\phi} \cos{\phi} \right)
+\frac{2b_x b_z}{r^2 \sin^2{\theta}} \left(- \sin{\theta} \cos{\theta} \sin{\phi} + \sin{\theta} \sin{\phi} \right) = \frac{2 b_y b_z}{r^2 \sin^2{\theta}} \left( \sin{\theta} \cos{\theta} \cos{\phi} - \sin{\theta} \cos{\phi} \right)
+\frac{2 b_y^2}{r^2 \sin^2{\theta}} \left( - \sin{\phi} \cos{\phi} + \sin^2{\theta} \sin{\phi} \cos{\phi} + \cos^2{\theta} \sin{\phi} \cos{\phi} \right) + \frac{2 b_x^2}{r^2 \sin^2{\theta}} \left( \sin{\phi} \cos{\phi} - \sin^2{\theta} \cos{\theta} \cos{\phi} + \cos^2{\theta} \sin{\phi} \cos{\phi} \right)

looks like I've made a mistake somewhere but i can't see where.

 

[gabbagabbahey]

The quickest way to do this is probably to start with this expression (from post #60)

\ddot{\phi}=\frac{-2r^2 \sin^2{\theta} \left(b_x \cos{\phi} + b_y \sin{\phi} \right) \left( b_y \cos{\phi} - b_x \sin{\phi} \right)}{r^4 \sin^4{\theta}}=\frac{-2\dot{\phi} \left(b_x \cos{\phi} + b_y \sin{\phi} \right) }{r\sin{\theta}}

 

[latentcorpse]

got it finally! thank you so much for your help!

what did you mean in post 23 when you asked if there could be any other solutions?
how would i go about answering that?

 

[gabbagabbahey]

Wald discusses a certain uniqueness theorem right after equation 3.3.5...does that help you here?

 

[latentcorpse]

well the idea behind it would suggest that, no, there can be no other solutions.

however, the theorem says that solutions are only unique if we define a point p in hte manifold M and a tangent vector, T^a \in V_p. In the question, neither of these were specified so perhaps that does leave scope for alternative solutions?

 

[gabbagabbahey]

Realize that any potential solution \textbf{r}(t) can be expanded in a Taylor series as

\textbf{r}(t)=\textbf{a}+\textbf{b}t+\textbf{c}t^2+\ldots

If \textbf{a} and \textbf{b} are specified, the uniqueness theorem tells you there is a unique solution...but you just showed that \textbf{r}(t)=\textbf{a}+\textbf{b}t satisfies the geodesic equation for all \textbf{a} and \textbf{b}...therfore ____?

 

[latentcorpse]

therefore straight lines are the unique solution as the values of a and b in the taylor expansion are precisely those of the coefficients a and b in the straight line eqn given in Cartesian coordinates, is that ok?

how did you manage to get round the fact that the point p and the tangent weren't specified though?

 

[gabbagabbahey]

Doesn't \textbf{a} specify a point on the geodesic (where t=0) ?...And the tangent vector is___?

 

[latentcorpse]

the vector \vec{b}. great. thanks.