Frame Drag and Orbits

QUESTION REGARDING FRAME DRAG AND ORBITAL DECAY. I don't do math beyond basic geometry and algebra so please bear with me. Two identical objects drift parallel to each other through space; a line drawn between them is perpendicular to their path of motion. The objects are captured by the gravity of a rotating planet and fall into very slowly decaying orbits which initially are mirror images of each other with one object orbiting spinward and the other antispinward (ignore the inevitable collision and assume that all numerical values are of measurable scale). From the perspective of a hypothetical observer at the planet's center of gravity would frame drag cause the orbits to decay at different rates? Would observers on each object measure a decay rate different from the other? In my head I can visualize spacetime being compressed for one object and stretched for the other yielding differing slopes of gravitational field strength versus altitude but my math breaks down and can't deal with dilation of measurements. Or does frame drag distort the geometry of the planet in spacetime to exactly compensate for differences in altitude measurement yielding identical decay rates? I'm trapped in a vision of all three observers calculating different decay rates right up to the moment the objects impact at exactly the same time. But I can't make sense of it. I would greatly appreciate your help.
- Patrick McMillan DDS (age 65)
Lewisville, NC,USA

A very interesting question!

Let's start with a little clearer Newtonian picture. Without some way of getting rid of energy, those objects are either in elliptical orbits or not, they won't get "captured". We don't want to assume there's dust or anything like that for ordinary friction to dissipate energy, because that would interact both with the spinning planet and the orbiters, making a huge difference between the two. So let's say they are both in elongated elliptical orbits. If that were the end of the story those orbits would never decay.

Now we'll use a little General Relativity to say that the gravitational radiation from those oscillating mass quadrupole moments would radiate some energy away, very gradually causing the orbiters to spiral in. Now we'll take into account frame-dragging from the spinning planet. It makes one of the orbiters see a very slightly higher effective field than the other when they're close to the planet, so it will swing around a little faster and radiate a little more. By now we're talking about very tiny effects but in principle one orbit will decay a little faster than the other.

Mike W.

(published on 09/20/2015)