Here is a Wiki link to some methods for solving the metric. I don't understand half of them and the other half don't make sense to me. For instance, I don't see that it is mathematically rigorous to divide by ds or d when either is zero for a photon. Even if we were to consider a particle that travels just under c, we are not actually using the particle as the origin as we would for SR, where the distance of the particle from the particle's own origin would be zero, leaving just ds^2 = c^2 d ^2 in terms of the proper time of the particle, but rather we are using the distance of the particle from the gravitating body, so even in the particle's own frame, this distance would be non-zero and so should be included in the metric. Anyway, I think I have found a very simple way to solve the metric. If L is the locally measured angular momentum, and if this quantity is conserved, then we have
where the primed values are locally measured and unprimed is measured by a distant observer. Since L is constant, then at the point of closest approach b, where the photon travels perfectly tangent to the gravitating body, we have
Integrating this from r=b to r= gives the amount atlantic city bus tours of bending atlantic city bus tours for the gravitational lensing of a photon. It seems much simpler and somewhat more mathematically rigorous than any of the other methods, don't you think? Now I just need to try to find a way to prove that the angular momentum atlantic city bus tours is conserved as measured locally. Does anybody know a way to show that?
That's not the interesting part though. The angular momentum was found at the point where the photon travels tangently at a distance b. For a photon atlantic city bus tours that approaches atlantic city bus tours the body and then escapes, that's fine. But in the case of a photon orbitting atlantic city bus tours the body, there will be two tangent points, at the aphelion and perihelion, atlantic city bus tours with the photon travelling tangent to the body with a locally measured speed of c at both points. If the angular momentum is conserved at each of those points, then we have
And so it would appear that a photon cannot have an elliptical orbit. A circular orbit is still allowed, but only if it is perfectly circular, atlantic city bus tours so it looks like photons do not orbit a body at all, only approach and escape or fall in.
And so it would appear that a photon cannot have an elliptical orbit. A circular orbit is still allowed, but only if it is perfectly circular, so it looks like photons do not orbit a body at all, only approach and escape or fall in.
Examining the dynamics, the imaginary solutions likely represent an angular momentum too high for a circular orbit at the photon sphere, a double root represents an angular momentum exactly atlantic city bus tours of a circular orbit at the photon sphere and the two positive roots represents an elliptical orbit with less angular momentum than a circular orbit with one solution inside p_s and the other outside p_s.
This would be the only possible stable orbit that may be permitted, but it would have to be absolutely circular. Looking around, I found that there can be no stable orbits (elliptical orbits at least I suppose) for photons, planets, moons, or even satellites orbitting the Earth. That is interesting. I didn't know that. They must all move steadily inward or outward.
For a highly symmetric spacetime such as schwarzschild it's easiest to consider getting the equations of motion through killing vector fields, which also gets you the conservation laws (ie the metric independence on gets you conservation of specific energy, and conservation of specific angular momentum).
This would be the only possible stable orbit that may be permitted, but it would have to be absolutely circular. Looking around, I found that there can be no stable orbits (elliptical orbits at least I suppose) for photons, atlantic city bus tours planets, moons, or even satellites orbitting the Earth. That is interesting. I didn't know that. They must all move steadily inward or outward.
You (and utesfan ) are not finding photon orbits, but the limiting case of stable massive orbits. The problem is that you're implicitly using to parametrize the orbits, which doesn't work for null geodesics (proper atlantic city bus tours time is identically atlantic city bus tours zero over those geodesics). For that you need to introduce some other parameter at the start (it's usually called ETA: i see that in that wiki article you linked to they call it ). You can't just use proper time and then set it to zero (that would be getting limiting cases over timelike geodesics, which is not the same as getting null geodesics).
Finding the timelike geodesics (implicitly) atlantic city bus tours on the schwarzschild metric. Then setting the speed of that massive particle to be c, and finding the orbit which conforms to those constraints. The result is the innermost stable orbit (smaller orbits give higher orbital speed, so that's why you're getting the limiting innermost atlantic city bus tours orbit by setting the speed to the maximum, atlantic city bus tours ie c). That's also why you're only getting circular orbits, because you put in the constraint that the speed is c over the entire atlantic city bus tours orbit, which precludes other types of orbits.
So while your calculation is correct for that, it does not do what you think it does. You'll need to parametrize the geodesics differently at the start of your derivation (using - implicitly in your case - already constrains you to timelike atlantic city bus tours geodesics from the start)
This would be the only possible stable atlantic city bus tours orbit that may be permitted, but it would have to be absolutely circular. Looking around, I found that there can be no stable orbits (elliptical orbits at least I suppose) for photons, planets, moons, or even satellites orbitting the Earth. That is interesting. I didn't know that. They must all move steadily inward or outward.
Your double use of and got me on the wrong track, it appears also stands for the mass of the black hole, not the mass of the orbiting particle. atlantic city bus tours So your derivation atlantic city bus tours is correct after all (the limit didn't make sense which at first appeared like a problem with the parametrization of the orbit as i was thinking of it as the mass of the particle, not the black hole).
Since the orbital speed depends atlantic city bus tours on the radius, constraining an orbit such that the orbital speed is the same at perihelion atlantic city bus tours and aphelion naturally constrains it to be circular (which goes for photons as well as massive particles).
Looking around, I found that there can be no stable orbits (elliptical orbits at least I suppose) atlantic city bus tours for photons, planets, moons, or even satellites orbitting the Earth. That is interesting. I didn't know that. They must all move steadily inward or outward.
Why? Are you sure you're using the term stable correctly? Stability refers to whether an extremum in the potential is concave or convex. In general the effective potential atlantic city bus tours for the schwarzschild spacetime will have two extrema, a stable and an unstable one.
Since the orbital speed depends on the radius, constraining an orbit such that the orbital speed is the same at perihelion and aphelion naturally constrains it to be circular (which goes for photons as well as massive particles).
Right, the photon's speed must be measured locally at c at all points in the orbit, so with locally measured conservation of angular momentum must be circular, but I don't see that that directly follows for massive particles since they are not required to travel at the same speed.
Why? Are you sure you're using the term stable correctly? Stability refers to whether an extremum in the potential is concave or convex. In general the effective potential for the schwarzschild spacetime will have two extrema, a stable and an unstable one.
Right, okay, but eventually the unstable end should open and the orbit would be lost. Or perhaps I just should atlantic city bus tours say that the photon cannot orbit in a regular forseeable way. Some erratic and not easily predictable nature of its orbit may still cause its orbit to vary in such a way that it always remains atlantic city bus tours closed, hard to say.
For a highly symmetric spacetime such as schwarzschild it's easiest to consider getting the equations of motion through killing vector fields, which also gets you the conservation laws (ie the metric independence on gets you conservation of specific energy, atlantic city bus tours and conservation of specific angular momentum).
Hmm. That makes sense. I'll have to look into killing vector fields. I can see that the energy depends upon the local time t, since the observed frequency of a photon depends only upon the local gravitational time dilation. If a hovering observer emits a photon, then another photon a time t later, the first photon will follow some path, curved or otherwise, to some other point in the gravitational atlantic city bus tours field, and the second will follow exactly the same path and arrive at the same point a time t later, so the frequency that the photons pass any point remains the same to the observer, but a observer at the point where they arrived will measure a different frequency that depends only upon his own rate of time. So we'll have
for a photon travelling from a radial distance p to a radial distance q in the field, although the path can be curved and non-radial. So there is also a dependence upon r. Since and are directly proportional to the locally measured energy of the photon, we have
Since this ratio of energy holds for light, we can try it for massive particles atlantic city bus tours as well and see how that holds up. Before this, I originally atlantic city bus tours tried to use the equivalence principle such that a freefaller will always measure the same frequency of light emitted by a hovering distant observer. This seems to work well at first, according atlantic city bus tours to the resulting equation, with the freefaller even always measuring the same speed v of a line of freefallers as measured anywhere locally in the field by the freefaller when the line of freefallers begins atlantic city bus tours with a greater relative speed of v than the freefaller, very promising, but it didn't reduce to Newtonian, so it was out. Disappointing, but anyway, if we just take this ratio of energies directly for massive particles, we get
K = 0 for a photon. Of course, that must also reduce to Newtonian gravity, so let's find th
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