How can a particle coming from infinity get on bound orbit around BH?

[Eudaimon]

Studying the movement of a particle on bound orbits around a black hole I found a fact that seemed a little strange for me -- that on these orbits particle's total energy on infinity must be less than unit (E<1). As far as I understrand, it is not so horrible, since here we deal with an unobservable quantity, because a particle never reaches infinity (namely because its orbit is bound). But in this case I have a question: what if a particle moves from infinity (where naturally its total energy cannot be less than 1) towards a black hole. Does the fact that its E≥1 on infinity implies that it can never move on bound orbits? How then is it possible to get on a bound orbit at all?

 

[The_Duck]

Black holes are a distraction here. Consider a simpler case such as the Sun. An object coming in from infinity will never fall into orbit around the Sun: it will always go back out to infinity (unless it crashes into the Sun). If you want to get this object into orbit around the Sun, you need to apply some force to it when it is close to the Sun. For example, you could decelerate it with rocket boosters or hit it with another object to slow it down so that it becomes bound to the Sun.

 

[Eudaimon]

Black holes are a distraction here. Consider a simpler case such as the Sun. An object coming in from infinity will never fall into orbit around the Sun: it will always go back out to infinity (unless it crashes into the Sun). If you want to get this object into orbit around the Sun, you need to apply some force to it when it is close to the Sun. For example, you could decelerate it with rocket boosters or hit it with another object to slow it down so that it becomes bound to the Sun.

I see. Thank you -- I thought nearly the same: so we need some extra force to put it on a bound orbit.

 

[pervect]

A Newtonian style gravitational slingshot can also capture an object falling from infinity. Energy is conserved, but is transferred to one orbiting body from the other.

 

[PAllen]

A Newtonian style gravitational slingshot can also capture an object falling from infinity. Energy is conserved, but is transferred to one orbiting body from the other.

With only two objects, and no collision, is this really possible? In the COM frame, both objects have escape velocity at all times, so how is capture possible [ other than collision] ?

 

[pervect]

With only two objects, and no collision, is this really possible? In the COM frame, both objects have escape velocity at all times, so how is capture possible [ other than collision] ?

I didnt write that clearly enough - I was referring to a three body capture.

 

[hilbert2]

Relativistic orbital dynamics don't always work the same way that nonrelativistic dynamics do. In classical Newtonian mechanics, a system of two point masses interacting gravitationally will never collide, if the masses have any nonzero initial angular momentum relative to each other. When special relativity is taken into account, the point masses can collide even if one of them has a nonzero initial velocity component orthogonal to the separation vector of the masses. (can someone find a source that verifies my claim?)

EDIT: here's the source: http://arxiv.org/pdf/physics/0405090.pdf

For example, a relativistic particle in a 1/r potential can spiral into the force center (while
conserving mechanical energy and angular momentum). This behavior occurs because a
small increase in the velocity near the speed of light c can lead to a large increase in the
mass m/(1−v₂/c₂)^1/2 so that an increase in the kinetic energy will compensate the decrease
in the potential energy as the radius decreases, thus keeping the total energy constant.

 

[mfb]

EDIT: here's the source: http://arxiv.org/pdf/physics/0405090.pdf

They ignore effects of general relativity. I doubt that this gives relevant results - a relativistic escape velocity usually means that (general) relativistic effects are important.

The theory of gravitation finds its relativistic form in general relativity and electrostatics has a natural extension into electrodynamics. However, we will not discuss these extended theories.

Rotating black holes, on the other hand, have very non-trivial trajectories for infalling particles.

 

[hilbert2]

They ignore effects of general relativity. I doubt that this gives relevant results - a relativistic escape velocity usually means that (general) relativistic effects are important.

I think if one takes into account GR effects, the falling point mass loses energy to gravitational radiation and spirals even faster to the force center...

 

[PAllen]

I think if one takes into account GR effects, the falling point mass loses energy to gravitational radiation and spirals even faster to the force center...

I believe this is true (in GR, two bodies falling from infinity can lose enough energy due to gravitational radiation that they can capture, and eventually spiral together) . My comment earlier reflected, as it stated, pure Newtonian physics. I don't believe it is meaningful to discuss SR plus Newtonian gravity. It is not a consistent theory because of conflict between action at a distance and SR causal structure. The only consistent SR gravity theory I know of was Nordstrom's theory, which, however, is just wrong as a description of reality.

 

[Bill_K]

I believe this is true (in GR, two bodies falling from infinity can lose enough energy due to gravitational radiation that they can capture, and eventually spiral together)

This does not depend on energy loss due to gravitational radiation. Unbound spiral orbits exist for a test mass in the Schwarzschild field.

 

[pervect]

Studying the movement of a particle on bound orbits around a black hole I found a fact that seemed a little strange for me -- that on these orbits particle's total energy on infinity must be less than unit (E<1). As far as I understrand, it is not so horrible, since here we deal with an unobservable quantity, because a particle never reaches infinity (namely because its orbit is bound). But in this case I have a question: what if a particle moves from infinity (where naturally its total energy cannot be less than 1) towards a black hole. Does the fact that its E≥1 on infinity implies that it can never move on bound orbits? How then is it possible to get on a bound orbit at all?

This does not depend on energy loss due to gravitational radiation. Unbound spiral orbits exist for a test mass in the Schwarzschild field.

I'm not quite sure what's being asked. Certainly orbits exist for E>1 that impact - some of them spiral orbits, for instance a photon (or anything else for that matter) that gets closer than the photon sphere (R=3M) might spiral (or it might impact without having time to make a full spiral). But the OP seems to exclude them by his definition of bound orbits.

There are a variety of ways things can loose energy to move into what's being defined as a bound orbit (E<1) from an unbound one (E>=1), but they all involve energy loss by the current working definition of "bound orbit".

Three-body interactions are one way to loose energy, collisions (whether with other solid bodies or with gasses) are another, I would suspect there are more but those are at least the first two I think of.