The geodesics around a spherical mass (Schwarzschild solution) in G.R. can be described by(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\frac{1}{2}\left(\frac{dr}{d\lambda}\right)^2 + V(r) = \mathcal{E}[/tex]

where V(r) is the effective potential

[tex]\frac{1}{2}\epsilon - \epsilon\frac{GM}{r} + \frac{L^2}{2r^2} - \frac{GML^2}{r^3}[/tex]

and

[tex]\mathcal{E} = \frac{1}{2}E^2[/tex]

For photons [tex]\epsilon = 0[/tex] and for massive particles [tex]\epsilon = 1[/tex].

To illustrate I have tried to draw the potential for different values of L. Lets start with the null geodesics:

http://www.dianajuncher.dk/geodesic_light.png [Broken]

The red line indicates the Newtonian potential. This is easy enough: for larger L it just becomes steeper and lies closer to the y-axis meaning that photons with large L (e.i. photons that point more away from the source of gravity) don't get too close, while photons with smaller L (pointing more directly at the source) get closer before moving away again.

For the G.R. lines it gets a bit more tricky. Now the effective potential goes to minus infinity which means, that you can acutally 'hit' the center. I just don't understand the whole energy-barrier-thing.

L is again just the direction of the photon, right? The closer it's path points at the source, the smaller the L?

And then there is the energy. Apparently, if the energy is larger than the barrier, the light can reach the center. But what exactly is this energy? It can hardly be the frequency of the photon or something like that. I find it strange, that the barrier is high for high L - with both a high energy and high L I would think, that the photon just escaped. I thought high energy prevented things from being 'sucked in'?

And then there is the massive particles:

http://www.dianajuncher.dk/geodesic_particle.png [Broken]

Here the L depends on both the direction of the path and the velocity of the particle, right?

Again I don't get the whole energy-barrier-thing. My instinct tells me, that the higher energy you have, the better you can escape. But the higher your energy is, the better you can get over the barrier?

Bonus question: if you actually do manage to cross the barrier, you don't just go straight to the center, right? There must be a spiralling motion for both photons and massive particles.

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# Effective potential and geodesics in G.R.

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