components of Christoffel symbol

latentcorpse

i dont see how im missing those factors. take for example

[itex]\Gamma^2{}_{12}=\frac{1}{2} g^{22} \left( \frac{\partial g_{22}}{\partial r} + \frac{\partial g_{12}}{\partial \theta} - \frac{\partial g_{12}}{\partial \theta} \right) = \frac{1}{2} r^2 \frac{\partial}{\partial r} \left(r^2 \right) = r^3[/itex]

i can't see where i'm missing this factor?

gabbagabbahey

You seem to have used [itex]g^{ab}=g_{ab}[/itex] instead of using the inverse matrix you calculated...

[tex]\Gamma^2{}_{12}=\frac{1}{2} g^{22} \left( \frac{\partial g_{22}}{\partial r} + \frac{\partial g_{12}}{\partial \theta} - \frac{\partial g_{12}}{\partial \theta} \right) = \frac{1}{2} \frac{1}{r^2} \frac{\partial}{\partial r} \left(r^2 \right) = \frac{1}{r}[/tex]

latentcorpse

lol. i'm an idiot sometimes...thanks.

the next bit asks me to write out the copmonents of the geodesic equaiton in this coordinate system and verify the solutions correspond ot straight lines in Cartesian coordinates.

so i guess the eqn tehy're referring to is 3.3.5.

[tex]\frac{d^2 x^{\mu}}{dt^2} + \sum_{\sigma, \nu} \Gamma^{\mu}{}_{\sigma \nu} \frac{d x^{\sigma}}{dt} \frac{dx^{\nu}}{dt}=0[/tex]

am i required here to solve 17 different differential equations?
even if the Christoffelsymbol is zero there will still be that first term equal to zero so none of the equaitons are going to be trivial. or do i just solve for one term of the Christoffel symbol and show the solutino is a straight line?

gabbagabbahey

Once you sum over [itex]\sigma[/itex] and [itex]\nu[/itex] you will only have 3 ODEs (one for each value of [itex]\mu[/itex])...

latentcorpse

ok so i get my three equations as:

[tex]\frac{d^2 r}{dt^2}-r \frac{d^2 \theta}{dt^2} - r \sin^2{\theta} \frac{d^2 \phi}{dt^2}=0[/tex]

[tex]\frac{d^2 \theta}{dt^2} + \frac{2}{r} \frac{dr}{dt} \frac{d \theta}{dt} - \sin{\theta} \cos{\theta} \frac{d^2 \phi}{dt^2}=0[/tex]

[tex]\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[/tex]

do they look correct? how on earth do i solve them?

gabbagabbahey

Assuming you meant [itex]\frac{d^2 \theta}{dt^2} +\frac{2}{r} \frac{dr}{dt} \frac{d \theta}{dt} - \sin{\theta} \cos{\theta} \frac{d^2 \phi}{dt^2}=0[/itex] for the 2nd ODE, then yes those look right....Now, do you really need to solve them, in order to verify that the solutions are straight lines in Cartesian coordinates?...What is the general (parametrized in terms of [itex]t[/itex]) form of a straight line in Cartesian coords?...What is it when you convert to Spherical coords?...Does that form satisfy the ODEs? Could there be any other solutions?

latentcorpse

in cartesian isnt it just going to be

[itex]\vec{r}(t)=\vec{a}+\vec{b}t[/itex] with [itex]\vec{a}, \vec{b} \in \mathbb{R}^3[/itex].

i'm not sure how to convert that to sphericals though? have i used the wrong form above?

gabbagabbahey

No, the general parametrized form of a line in 3D, is [itex]ax(t)+by(t)+cz(t)=0[/itex]

latentcorpse

[tex]\frac{x-x_0}{a}=\frac{y-y_0}{b}=\frac{z-z_0}{c}[/tex]?

gabbagabbahey

Did you see my previous post?

latentcorpse

ok. so if that's the cartesian form i need to convert it to spherical polars:

[tex]ar(t) \sin{\theta (t) } \cos{\phi (t) } + br(t) \sin{\theta (t)} \sin{\phi (t)} + c r(t) \cos{\theta (t)}=0[/tex]

but i don't see where to substitute that into the geodesic eqn. or am i wanting to show

[tex] \left( \frac{d^2 x^{\mu}}{dt^2} + \sum_{\sigma, \nu} \Gamma^{\mu}{}_{\sigma \nu} \frac{d x^{\sigma}}{dt} \frac{d x^{\nu}}{dt} \right) \left( ar(t) \sin{\theta (t)} \cos{\phi (t)} + br(t) \sin{\theta (t)} \sin{\phi (t)} + c r(t) \cos{\theta (t)} \right) = 0[/tex]?

gabbagabbahey

Okay, now take the derivative of the equation with respect to time twice; do some jiggling around and see what you can come up with...

latentcorpse

sorry. i don't understand what's going on here.

am i substituting that equation into the geodesic equations?

if so what is r theta and phi?

or am i just multiplying the geodesic eqn with the straight line eqn?

gabbagabbahey

You want to show that [itex]r(t)[/itex], [itex]\theta(t)[/itex] and [itex]\phi(t)[/itex] satisfy each of your 3 ODEs, so long as they also satisfy your straight line equation..

latentcorpse

ok. so i did the double derivative with time by hand (is there a way to write a simple maple code to do this for me? i tried but couldn't get it to work) and got:

[tex]a \ddot{r} \sin{\theta} \cos{\phi} + a \dot{r} \dot{\theta} \cos{\theta} \cos{\phi} - a \dot{r} \dot{\phi} \sin{\theta} \sin{\phi}[/tex]

[tex] + a \dot{r} \dot{\theta} \cos{\theta} \cos{\phi} + a r \ddot{\theta} \cos{\theta} \cos{\phi}[/tex]

[tex] - a r \dot{\theta}^2 \sin{\theta} \cos{\phi} - a r \dot{\theta} \dot{\phi} \cos{\theta} \sin{\phi} - a \dot{r} \dot{\phi} \sin{\theta} \sin{\phi} - a r \ddot{\phi} \sin{\theta} \sin{\phi} - a r \dot{\theta} \dot{\phi} \cos{\theta} \sin{\phi}[/tex]

[tex] - a r \dot{\phi}^2 \sin{\theta} \cos{\phi} + b \ddot{r} \sin{\theta} \sin{\phi} + b \dot{r} \dot{\theta} \cos{\theta} \sin{\phi} + b \dot{r} \dot{\phi} \sin{\theta} \cos{\phi} + b \dot{r} \dot{\theta} \cos{\theta} \sin{\phi} + b r \ddot{\theta} \cos{\theta} \sin{\phi}[/tex]

[tex] - b r \dot{\theta}^2 \sin{\theta} \sin{\phi} + b r \dot{\theta} \dot{\phi} \cos{\theta} \cos{\phi} + b \dot{r} \dot{\phi} \sin{\theta} \cos{\phi} + b r \ddot{\phi} \sin{\theta} \cos{\phi} + b r \dot{\theta} \dot{\phi} \cos{\theta} \cos{\phi}[/tex]

[tex] - b r \dot{\phi}^2 \sin{\theta} \ubs{\phi} + c \ddot{r} \cos{\theta{ - c \dot{r} \dot{\theta} \sin{\theta} - c \dot{r} \dot{\theta} \sin{\theta} - c r \ddot{\theta} \sin{\theta} - cr \dot{\theta}^2 \cos{\theta} = 0[/tex]

latentcorpse

hmm. i don't know why that all posted on one line? can you read it by just clicking on the code?

anyway, i can group some terms together and what not but i'm still not sure what im doing...surely i want to take the second derivative of r, theta and phi not of the straight line eqn?

latentcorpse

sorry, i sorted out the LaTeX in post 32. see my previous 2 posts.