Initial conditions for orbits around a wormhole

[happyparticle]

Homework Statement

Find the initial conditions for $u^r$ and $u^\theta$ if a shuttle is launched with velocity $\beta$ at $x(t=0) = 20$.

Relevant Equations

$(ds)^2 = -(dt)^2 + (dr)^2 + (r^2 + b^2)( (d\theta)^2 + sin^2 \theta (d\phi)^2 )$

$\varepsilon = \frac{1}{2} (\frac{dr}{d\tau})^2 + V_{eff}(r)$

Hi,
I had an exam and I completely messed up a problem. Especially one part which was necessary for the rest of the problem.

Basically, I have a wormhole metric: $$(ds)^2 = -(dt)^2 + (dr)^2 + (r^2 + b^2)( (d\theta)^2 + sin^2 \theta (d\phi)^2 )$$

Where $b=1$ with an orbit only in the equatorial plane.

We also know from the question that the orbit must satisfy this relationship: $$\varepsilon = \frac{1}{2} (\frac{dr}{d\tau})^2 + V_{eff}(r)$$

Ultimately, I was tasked to find the initial conditions for $u^r$ and $u^\theta$ if a shuttle is launched with velocity $\beta$ at $x(t=0) = 20$.
(the center of the wormhole is located at $x=0,y=0$).

Firstly, I had to find a relation between $\beta$ and one of the invariant.

To answer this question I used $\frac{dt}{d\tau} = e$ and I found $\beta = \sqrt{(1- \frac{1}{e^2}})$, since $e = -g_{tt}\frac{dt}{d\tau}$ and $g_{tt} = -1$.
Where e is one of the invariant.

Secondly , I had to find the ratio $\frac{u^r}{u^{\phi}}$ which is fully defined by the position ($r_0,\phi_0$) in the equatorial plane.

Then , using the relationships above I could find the relation $u^r(t=0)$ in function of $\beta$ and $\phi_0$.

Finally, the equation for $\varepsilon$ allows us to find the second invariant.

I'm wondering if the relationship for $\beta$ and e is correct.
Also, I was completely unable to continue the problem thereafter. I had to plot a graph of the orbits, but after the first part of the problem I have practically did trial and error and obviously my graph made no sense.

I hope my question is clear.

Thanks you!

 

[anuttarasammyak]

Thogh I am a poor layman in GR, I am interested in understanding the question. Initial position and magnitude of initial velocity of the test body given, all we can arrange is the direction of the initial velocity. By that arrangement the test bddy could or couldn't pass the neck of worm hole to escape into the backside world. Do I understand the quesiton correctly ?　　If yes, I assume that the angular momentum matters.

 

[happyparticle]

You do understand the question correctly. I'm not sure if I need to use the angular momentum. Basically, I have to find a relationship with $\beta$, both trajectory invariants and a relationship with $\frac{u^r}{u^{\phi}}$

 

[anuttarasammyak]

I guess that at the neck the wormhole the angular velocity of the body increases by the law of angular momentum conservation, but its velocity cannot exceed light speed. I have found a support to my guess in https://en.wikipedia.org/wiki/Ellis_wormhole There parameter h which is angular momentum per unit mass wrt phi decides go or not go into the other side; small h lets it go but large h does not.

 

[happyparticle]

I'm wondering if the ratio of $\frac{u^r}{u^{\phi}}$ is "simply" $-r_0 cos \phi_0$?

Knowing that $u^r = \gamma \frac{dr}{dt}$, $u^{\phi} = \gamma \frac{d\phi}{dt}$, and $\frac{dr}{dt} = \frac{x v_x + y v_y}{r}$ , $\frac{d\phi}{dt} = \frac{xv_x - yv_y}{r^2}$.

Also, if the object is moving horizontally and start at $x=20$, then $v_x = - \beta$ and $v_y = 0$ and plugging $x = r cos \phi$ and $y = r sin \phi$, we get the ratio above for $u^r(0)/ u^{\phi}(0)$.