[Reposted from my PF Blog.]
I haven't yet had the pleasure of participating in a PF thread on this topic :wink:, although I have made at least one post that refers to it in passing. But I know there have been some in the past, such as this, so I wanted to post a quick treatment of the topic since it didn't seem like that previous thread ended in a clear answer.
The goal is to derive a simple equation for the proper acceleration of an observer traveling on a circular path around a Schwarzschild black hole at some constant radius [itex]r > 2M[/itex]. (Note that this is not the same as being in a stable circular orbit about the hole; we allow for "orbits" that require nonzero rocket thrust to maintain, and we won't look at stability under small perturbations since we'll assume the orbit can be corrected, if needed, by rocket thrust.) This is, of course, a restricted answer since it doesn't cover trajectories that change radius; I plan to go into that in a follow-up post. This is just a simple, quick analysis to illustrate at least one way in which centrifugal force does "reverse" close to a black hole.
The simplest way to study the worldlines we're interested in is to define a 4-velocity field for "orbiting observers", in terms of the standard Schwarzschild coordinate chart for the exterior region outside the black hole's horizon (as should be evident, we are working in units where G = c = 1):
[tex]u^a = \frac{1}{\sqrt{1 - \frac{2M}{r}} \sqrt{1 - v^2}} \partial_t + \frac{v}{r \sqrt{1 - v^2}} \partial_{\phi}[/tex]
where [itex]M[/itex] is, of course, the mass of the hole, and [itex]v[/itex] is the tangential velocity of the observer (which we'll assume is constant for any particular orbit), as measured in the local inertial frame of a "static" observer (i.e., one who is "hovering" at constant r with zero angular momentum about the hole). Note that we are assuming an equatorial "orbit", so [itex]\theta = \pi / 2[/itex] and we can ignore factors of [itex]sin \theta[/itex]. We can sanity check this by plugging in [itex]v = 0[/itex] and seeing that the resulting 4-velocity is indeed that of the static observer.
Now we simply compute the path curvature of this 4-velocity field; the 4-acceleration vector is:
[tex]a^a = u^b \nabla_b u^a = u^b \partial_b u^a + u^b \Gamma^a{}_{bc} u^c[/tex]
and the magnitude of the 4-vector, which is what will actually be measured by an accelerometer carried by the "orbiting observer", is
[tex]a = \sqrt{g_{ab} a^a a^b}[/tex]
Expanding out the vector components, we find that only the r component is nonzero, so the magnitude of the 4-acceleration becomes:
[tex]a = \sqrt{g_{rr}} a^r = \sqrt{g_{rr}} \left( \Gamma^r{}_{tt} u^t u^t + \Gamma^r{}_{\phi \phi} u^{\phi} u^{\phi} \right)[/tex]
Substituting, we obtain
[tex]a = \sqrt{\frac{r}{r - 2M}} \left( \frac{M}{r^2 \left( 1 - v^2 \right)} + \frac{\left( 2M - r \right) v^2}{r^2 \left( 1 - v^2 \right)} \right)[/tex]
We can refactor this into a form that will make the physical interpretation easier:
[tex]a = \frac{M}{r^2 \sqrt{1 - \frac{2M}{r}}} \left[ \frac{1 - \left( \frac{r}{M} - 2 \right) v^2}{1 - v^2} \right][/tex]
Now we can see what's going on here. If we again take [itex]v = 0[/itex], we recover the standard formula for the proper acceleration of a "hovering" observer, as we should (another sanity check). For nonzero [itex]v[/itex], we can see that there are three regimes that have different physical behavior:
r > 3M
In this regime, for small [itex]v[/itex], [itex]a[/itex] is positive; that is, an observer "orbiting" at low speed will need outward rocket thrust to maintain a constant altitude. As [itex]v[/itex] increases, the thrust required decreases, until a point is reached where the thrust goes to zero. Obviously, this corresponds to a free-fall orbit, and the free-fall orbital velocity is given by:
[tex]v_{orbit} = \sqrt{\frac{1}{\frac{r}{M} - 2}}[/tex]
As one more sanity check, we note that for large r, we can neglect the 2 under the square root and we get [itex]v_{orbit} = \sqrt{M / r}[/itex], the standard Newtonian result (or [itex]\sqrt{GM / r}[/itex] in conventional units).
For [itex]v[/itex] larger than the orbital velocity, [itex]a[/itex] is negative; that is, we now require *inward* rocket thrust to maintain altitude. (I believe this is called a "forced orbit" in astronautics.) The inward rocket thrust required goes to infinity as [itex]v[/itex] goes to 1 (i.e., as tangential speed approaches the speed of light).
All this, of course, is the standard "centrifugal force" behavior that matches our intuitions. (We note in passing that for [itex]r < 6M[/itex], the free-fall orbit is no longer stable; but as I said at the start, we aren't considering stability since we are allowing for rocket thrust to adjust the orbit. If you like, you can consider the value of [itex]a[/itex] we are calculating as an average over many orbits, allowing for small time-varying adjustments.) But in the other two regimes, things get more interesting.
r = 3M
At this radius, we find that [itex]a[/itex] is *independent* of [itex]v[/itex]! That is, *any* object orbiting the hole at this radius, regardless of its tangential velocity, must exert the same outward rocket thrust to maintain altitude. This may seem a bit confusing since we know that at this radius, photons can orbit the hole on null geodesics (i.e., "free-fall" orbits). But remember that we are working with a 4-velocity field, i.e., with a unit timelike vector, so our results don't apply to photons. This is an example, in fact, of a case where the behavior of photons is discontinuous with the behavior of timelike objects.
(There is another sense in which the photon behavior at [itex]r = 3M[/itex] is not discontinuous with timelike behavior; as [itex]r[/itex] approaches [itex]3M[/itex], the free-fall orbital velocity approaches 1, as can be seen from the above formula. So the "photon orbit" at [itex]r = 3M[/itex] is in this sense continuous with the free-fall orbits of timelike objects at slightly larger radius. Of course, this again ignores questions of stability.)
We could interpret this result as telling us that centrifugal force "goes to zero" at this radius. And inside this radius, it gets more interesting still.
2M < r < 3M
In this regime, [itex]a[/itex] has a *minimum* for [itex]v = 0[/itex]; that is, at *any* nonzero [itex]v[/itex], the outward rocket thrust required to maintain altitude is *larger* than it is for the "hovering" observer with zero [itex]v[/itex]. And the thrust increases without bound as [itex]v[/itex] goes to 1. So if you're piloting your spaceship close to a black hole, the best way to conserve fuel is to hover, with zero tangential velocity. This is what is referred to in a number of papers as "centrifugal force reversal", and it clearly is a "real" phenomenon. Whether a similar phenomenon also appears in more general orbits is a topic for another post.
I haven't yet had the pleasure of participating in a PF thread on this topic :wink:, although I have made at least one post that refers to it in passing. But I know there have been some in the past, such as this, so I wanted to post a quick treatment of the topic since it didn't seem like that previous thread ended in a clear answer.
The goal is to derive a simple equation for the proper acceleration of an observer traveling on a circular path around a Schwarzschild black hole at some constant radius [itex]r > 2M[/itex]. (Note that this is not the same as being in a stable circular orbit about the hole; we allow for "orbits" that require nonzero rocket thrust to maintain, and we won't look at stability under small perturbations since we'll assume the orbit can be corrected, if needed, by rocket thrust.) This is, of course, a restricted answer since it doesn't cover trajectories that change radius; I plan to go into that in a follow-up post. This is just a simple, quick analysis to illustrate at least one way in which centrifugal force does "reverse" close to a black hole.
The simplest way to study the worldlines we're interested in is to define a 4-velocity field for "orbiting observers", in terms of the standard Schwarzschild coordinate chart for the exterior region outside the black hole's horizon (as should be evident, we are working in units where G = c = 1):
[tex]u^a = \frac{1}{\sqrt{1 - \frac{2M}{r}} \sqrt{1 - v^2}} \partial_t + \frac{v}{r \sqrt{1 - v^2}} \partial_{\phi}[/tex]
where [itex]M[/itex] is, of course, the mass of the hole, and [itex]v[/itex] is the tangential velocity of the observer (which we'll assume is constant for any particular orbit), as measured in the local inertial frame of a "static" observer (i.e., one who is "hovering" at constant r with zero angular momentum about the hole). Note that we are assuming an equatorial "orbit", so [itex]\theta = \pi / 2[/itex] and we can ignore factors of [itex]sin \theta[/itex]. We can sanity check this by plugging in [itex]v = 0[/itex] and seeing that the resulting 4-velocity is indeed that of the static observer.
Now we simply compute the path curvature of this 4-velocity field; the 4-acceleration vector is:
[tex]a^a = u^b \nabla_b u^a = u^b \partial_b u^a + u^b \Gamma^a{}_{bc} u^c[/tex]
and the magnitude of the 4-vector, which is what will actually be measured by an accelerometer carried by the "orbiting observer", is
[tex]a = \sqrt{g_{ab} a^a a^b}[/tex]
Expanding out the vector components, we find that only the r component is nonzero, so the magnitude of the 4-acceleration becomes:
[tex]a = \sqrt{g_{rr}} a^r = \sqrt{g_{rr}} \left( \Gamma^r{}_{tt} u^t u^t + \Gamma^r{}_{\phi \phi} u^{\phi} u^{\phi} \right)[/tex]
Substituting, we obtain
[tex]a = \sqrt{\frac{r}{r - 2M}} \left( \frac{M}{r^2 \left( 1 - v^2 \right)} + \frac{\left( 2M - r \right) v^2}{r^2 \left( 1 - v^2 \right)} \right)[/tex]
We can refactor this into a form that will make the physical interpretation easier:
[tex]a = \frac{M}{r^2 \sqrt{1 - \frac{2M}{r}}} \left[ \frac{1 - \left( \frac{r}{M} - 2 \right) v^2}{1 - v^2} \right][/tex]
Now we can see what's going on here. If we again take [itex]v = 0[/itex], we recover the standard formula for the proper acceleration of a "hovering" observer, as we should (another sanity check). For nonzero [itex]v[/itex], we can see that there are three regimes that have different physical behavior:
r > 3M
In this regime, for small [itex]v[/itex], [itex]a[/itex] is positive; that is, an observer "orbiting" at low speed will need outward rocket thrust to maintain a constant altitude. As [itex]v[/itex] increases, the thrust required decreases, until a point is reached where the thrust goes to zero. Obviously, this corresponds to a free-fall orbit, and the free-fall orbital velocity is given by:
[tex]v_{orbit} = \sqrt{\frac{1}{\frac{r}{M} - 2}}[/tex]
As one more sanity check, we note that for large r, we can neglect the 2 under the square root and we get [itex]v_{orbit} = \sqrt{M / r}[/itex], the standard Newtonian result (or [itex]\sqrt{GM / r}[/itex] in conventional units).
For [itex]v[/itex] larger than the orbital velocity, [itex]a[/itex] is negative; that is, we now require *inward* rocket thrust to maintain altitude. (I believe this is called a "forced orbit" in astronautics.) The inward rocket thrust required goes to infinity as [itex]v[/itex] goes to 1 (i.e., as tangential speed approaches the speed of light).
All this, of course, is the standard "centrifugal force" behavior that matches our intuitions. (We note in passing that for [itex]r < 6M[/itex], the free-fall orbit is no longer stable; but as I said at the start, we aren't considering stability since we are allowing for rocket thrust to adjust the orbit. If you like, you can consider the value of [itex]a[/itex] we are calculating as an average over many orbits, allowing for small time-varying adjustments.) But in the other two regimes, things get more interesting.
r = 3M
At this radius, we find that [itex]a[/itex] is *independent* of [itex]v[/itex]! That is, *any* object orbiting the hole at this radius, regardless of its tangential velocity, must exert the same outward rocket thrust to maintain altitude. This may seem a bit confusing since we know that at this radius, photons can orbit the hole on null geodesics (i.e., "free-fall" orbits). But remember that we are working with a 4-velocity field, i.e., with a unit timelike vector, so our results don't apply to photons. This is an example, in fact, of a case where the behavior of photons is discontinuous with the behavior of timelike objects.
(There is another sense in which the photon behavior at [itex]r = 3M[/itex] is not discontinuous with timelike behavior; as [itex]r[/itex] approaches [itex]3M[/itex], the free-fall orbital velocity approaches 1, as can be seen from the above formula. So the "photon orbit" at [itex]r = 3M[/itex] is in this sense continuous with the free-fall orbits of timelike objects at slightly larger radius. Of course, this again ignores questions of stability.)
We could interpret this result as telling us that centrifugal force "goes to zero" at this radius. And inside this radius, it gets more interesting still.
2M < r < 3M
In this regime, [itex]a[/itex] has a *minimum* for [itex]v = 0[/itex]; that is, at *any* nonzero [itex]v[/itex], the outward rocket thrust required to maintain altitude is *larger* than it is for the "hovering" observer with zero [itex]v[/itex]. And the thrust increases without bound as [itex]v[/itex] goes to 1. So if you're piloting your spaceship close to a black hole, the best way to conserve fuel is to hover, with zero tangential velocity. This is what is referred to in a number of papers as "centrifugal force reversal", and it clearly is a "real" phenomenon. Whether a similar phenomenon also appears in more general orbits is a topic for another post.
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