How do Orbits behave as orbital velocities become significantly relatavistic

[Soul Surfer]

I am reasonably familiar with conventional elliptical orbits under gravity but how do they change as orbital velocities get very fast and start approaching the speed of light? let us initially neglect any energy losses due to gravitational radiation. I have been searching the web for some time for help on this subject and have found nothing of any use except the statement that relativistic effects cause the perihelion of mercury to move slightly

An object with a particular mass will accelerate and become "heavier" (or more difficult to accelerate) as it aproches its closest point to the object it is orbiting. does this effective mass increase change the greavitational forces on the object and what about the angular momentum?

As nobody mentions any significant changes to the standard elliptical shape of an object orbiting another of much heavier mass I presume that this does not change and that the main changes are in the way the interplay between potential and kinetic energy operates as orbital velocites get significant to the velocity of light.

 

[pervect]

I'd suggest looking at

http://www.fourmilab.ch/gravitation/orbits/

which has both a java applet, and presents the associated equations of motion for a body orbiting a Schwarzschild black hole.

Basically, there is an "effective potential" in this case, and (ignoring gravitational radiation by assuming the body follows a geodesic) conserved quantities for the orbiting body analogous to its energy and its angular momentum.

The correct expression for the effective potential is not going to be arrived at by mucking around by replacing Newtonian mass with relativistic mass however - "that trick never works". A considerably more sophisticated analysis needs to be done. To get the details of the reasoning used to derive the equations, you'll probably need to read a textbook or two, even the simplest of which is at a fairly high level. But you can see the results with resources like this Java applet.

Even a brief glimpse at the Java applet will show that the orbits are definitely not elliptical or even close.

 

[Soul Surfer]

Thankyou very much that applet is a great help to my understanding of how extreme relatavistic orbits work. I should have realized that it's the space curvature that is the controlling factor and not any sort of "effective mass".

 

[Thrice]

Somewhat along these lines, how would the orbit (or just the general field) change as the singularity accelerates to near the speed of light? Does a gravitational wave undergo redshift?

And maybe I'm missing something, but does that applet show the whole story? What frame are we looking at the picture from & does it matter?

Thanks in advance.

 

[pervect]

Somewhat along these lines, how would the orbit (or just the general field) change as the singularity accelerates to near the speed of light? Does a gravitational wave undergo redshift?

And maybe I'm missing something, but does that applet show the whole story? What frame are we looking at the picture from & does it matter?

Thanks in advance.

The applet shows the orbit in Schwarzschild coordinates. The Schwarzschild coordinates R and Phi are represented by the polar coordinates r and theta, respectively (hope that makes sense to you).

Specifying the orbit of the body when the central mass is moving requires replacing the Schwarzschild coordinate system with a different coordinate system. Unfortunately, exactly what coordinate system to use isn't well-defined. In principle, if one could define the coordinate system exactly, one can answer the question.

If the body is far away from the central mass, one can use a standard "Lorentz boost' to transform coordinates. There is no standard as to what coordiantes to use if the body is near the central mass, however.

Basically, the easy to analyze case is the case where the central body has all the mass, and the other body is a very light "test particle". In this case it is by far the most convenient to analyze the system from the frame of the large mass, and attempt to do any coordinate conversions to a "moving frame" after the analysis is done from the frame in which the large mass is stationary. Exactly what coordinate conversions to use while the test mass is close to the large body is not clear. One can use the "naive boost" to define _a_ coordinate system, but not much physical significance should be attached to these results (as such a coordinate system will be highly distorted).

If the test particle has appreciable mass, the assumption that the metric is given by the Schwarzschild metric of the large mass fails, and the whole problem becomes extremely complicated.

 

[Thrice]

(hope that makes sense to you)

hah :)

I keep forgetting how little I know. But at least I'm working on it.

 

[Jorrie]

Thankyou very much that applet is a great help to my understanding of how extreme relatavistic orbits work. I should have realized that it's the space curvature that is the controlling factor and not any sort of "effective mass".

I'm new to this forum, but read it for some time now - very interesting.

In reply to the above, yes space curvature plays a significant role. But did you know that unless a particle orbits very close to an event horizon, extreme velocities (over curved space) play the bigger role? I will post some calculations if you are interested.

 

[Soul Surfer]

Yes I would be very interested to see how the equations work under these conditions because I am very interested in an analysis of what happens to the material in a black hole (assumed to contain some angular momentum) as it contracts towards the "singularity"

It is always important to remember that the initial conditions at the very centre of the object as it contracts within its event horizon are zero gravity ie zero space curvature even though the material may be very hot and dense either as a neutron star or a white dwarf or even as a very big protostar in the early stages of the formation of the universe.

It is also important to remember that the initial conditions inside the event horizon are similar to those outside particularly for a large black hole and that conventional physics will operate for quite a long time before any quantum gravity needs to be invoked.

I feel that the theoreticians are missing a trick that could lead them to some significant physical insights by rushing to the final state of the equation (the ring singularity) without thinking about what happens on the way there once a black hole has formed

 

[pervect]

I feel that the theoreticians are missing a trick that could lead them to some significant physical insights by rushing to the final state of the equation (the ring singularity) without thinking about what happens on the way there once a black hole has formed

The ring singularity of a rotating black hole is probably not stable. Physicists have done simulations on charged scalar fields to gain some physical insight as to what they expect to happen with rotating black holes.

The results are very different from the classical "ring singularity" for a rotating black hole.

I can't find my usual reference for this, but a search on arxiv found the following which comes to a similar conclusion:

http://arxiv.org/abs/gr-qc/9608034

Gravitational perturbations which are present in any realistic stellar collapse to a black hole, die off in the exterior of the hole, but experience an infinite blueshift in the interior. This is believed to lead to a slowly contracting lightlike scalar curvature singularity, characterized by a divergence of the hole's (quasi-local) mass function along the inner horizon.
The region near the inner horizon is described to great accuracy by a plane wave spacetime. While Einstein's equations for this metric are still too complicated to be solved in closed form it is relatively simple to integrate them numerically.
We find for generic regular initial data the predicted mass inflation type null singularity, rather than a spacelike singularity. It thus seems that mass inflation indeed represents a generic self-consistent picture of the black hole interior.

This isn't the paper I was looking for, so I'm not sure exactly which case the authors were simulating. It fits in with the general idea that ring singularities are not expected to occur in physical collapse scenarios.