Orbits of Photons Around a Massball: Explained

[sqljunkey]

I am reading MWT gravitation and on page 676, they are talking about orbits of photon, and I don't understand it very well. Energy and angular momentum of the photon are important as a ratio when calculating the orbit. But not energy alone or angular momentum alone. Why is that, and the energy is in the denominator, b = L/E meaning that b will be a small number once E is big. Does that mean one photon will have a different orbit depending on it's energy?

 

[PeterDonis]

Energy and angular momentum of the photon are important as a ratio when calculating the orbit. But not energy alone or angular momentum alone.

Yes.

the energy is in the denominator, b = L/E meaning that b will be a small number once E is big

Assuming angular momentum is held constant, yes.

Does that mean one photon will have a different orbit depending on it's energy?

No, as is stated explicitly right after equation 25.61. As noted there, the trajectory of the photon depends on its direction. So if you have two photons with different impact parameters $b$, it's because their directions are different. For example, if you have two photons with the same $L$ but different energies $E$, the one with the larger $E$ will have a smaller impact parameter $b$ because it must be moving in a direction that is closer to radially inward, meaning that it will pass closer to the central mass. That closer to radial direction means the angular momentum per unit energy of the photon is smaller--so the photon's $E$ can be bigger but its $L$ is still the same.

 

[sqljunkey]

Well I did read past equation 25.61. But because it was a ratio now that says that you can have a photon with a lot of energy and a photon with lower energy have same trajectory along the geodesic given you adjust their L accordingly, they can have same b.

I will read the chapter on Redshift and see how that all plays out. But would Redshift make a difference in trajectory of the two different energy photons?

 

[PeterDonis]

that says that you can have a photon with a lot of energy and a photon with lower energy have same trajectory along the geodesic given you adjust their L accordingly, they can have same b

Yes.

would Redshift make a difference in trajectory of the two different energy photons?

What do you mean by "redshift"? How is it different from energy?

 

[sqljunkey]

Depending on the r distance from the mass the photon's E would change right? does the L change too, so it keeps the same b along the trajectory?

 

[PeterDonis]

Depending on the r distance from the mass the photon's E would change right?

No. $E$ is a constant of the motion. So is $L$.

 

[pervect]

I am reading MWT gravitation and on page 676, they are talking about orbits of photon, and I don't understand it very well. Energy and angular momentum of the photon are important as a ratio when calculating the orbit. But not energy alone or angular momentum alone. Why is that, and the energy is in the denominator, b = L/E meaning that b will be a small number once E is big. Does that mean one photon will have a different orbit depending on it's energy?

Consider two photons with the same energy, starting out at the same event, but angled differently, so that one approaches the black hole closer at the point of closest approach than the other. The one that has a point of approach closest to the black hole will have a lower angular momentum than the one that passes further away.

I don't recall if the impact parameter b is the distance of closest approach, or whether that's b^-1 offhand. I could dig it up from the text, but so can you :).

Now consider two photons with the same distance of closest approach, starting out from the same spot, and pointed in the same direction. Such photons will have the same impact parameter, but different energies.

In this second case, they are still both photons, and will follow the same trajectory, even though one has more energy than the other. They both travel at the speed of light, being photons, and are starting out at the same event and moving in the same direction.

 

[PeterDonis]

Now consider two photons with the same distance of closest approach, starting out from the same spot, and pointed in the same direction. Such photons will have the same impact parameter, but different energies.

Yes, and they will also have different angular momenta, so that the ratio $L / E$ is the same for both (since that ratio is the impact parameter).

 

[sqljunkey]

You meant E measured locally, or E measured as though you are traveling with the photon?

 

[PeterDonis]

You meant E measured locally, or E measured as though you are traveling with the photon?

Neither. $E$ is a global constant of the motion.

("measured as though you were traveling with the photon" makes no sense anyway.)

 

[sqljunkey]

ok heh

 

[pervect]

You meant E measured locally, or E measured as though you are traveling with the photon?

E is sometimes called the energy at infinity. It's the energy that the photon would have in the flat space-time far away from the mess. It, and the angular momentum (defined in a similar manner), are constants for geodesic motion, motion without external forces other than gravity.

 

[sqljunkey]

So if that is true, the equation b = L/E can only be calculated at r=infinity and no where else? if I want to know the orbit of the photon at any other r from the mass, I would have to find another equation.

 

[Ibix]

So if that is true, the equation b = L/E can only be calculated at r=infinity and no where else? if I want to know the orbit of the photon at any other r from the mass, I would have to find another equation.

L and E are constants of the motion, so can be calculated anywhere. $L=r^2\frac{d\phi}{d\lambda}$ and $E=(1-\frac{R_S}{r})\frac{dt}{d\lambda}$, so $\frac LE=\frac{r^3}{r-R_S}\frac{d\phi}{dt}$. However, the direct physical meaning of $E$ is the amount of energy per unit mass that it has at infinity. This is greater than $c^2$ if the particle is in an unbound orbit, less than that if is bound, and equal in the limiting case where it tends towards zero velocity as it approaches infinity.

 

[sqljunkey]

Okay thanks ibix. Is lambda in those equations wavelength, or some other parameter?

 

[Ibix]

$\lambda$ is an affine parameter, which uniquely identifies each event on the worldline of the light pulse. We normally use $\tau$, proper time, for this purpose with timelike worldlines, but that doesn't work for null worldlines.

Carroll uses $\lambda$ and that's the notation I'm familiar with. I don't have MTW to hand to see what symbol they use.

 

[sqljunkey]

ok thx

 

[PeterDonis]

Carroll uses $\lambda$ and that's the notation I'm familiar with. I don't have MTW to hand to see what symbol they use.

They use $\lambda$ as a general affine parameter, and also as the affine parameter for null worldlines, since proper time cannot be defined. For timelike worldlines, they generally use $\tau$. Sometimes they also use $s$ as a general affine parameter.