Lagrange Points in General Relativity

In the Newtonian formulation of celestial mechanics it makes sense that a Lagrange point is a point where two gravitational forces of two bodies (and the centrifugal force of the rotating reference frame) cancel out each other. But how General relativity explain these points? If gravity is the curvature of spacetime? Does spacetime cancel out each other at certain points?

1 Answer

I will refer to the three bodies of the Lagrange point problem as 'the primary', 'the secondary', and 'the tertiary'.

The most interesting Lagrange points are L4 andL5, the discussion in this answer is for L4 and L5 only.

The mathematical analysis is simpler when the tertiary is so small that relative to the primary and secondary the mass of the tertiary is negligable.

In terms of newtonian mechanics:

The primary and the secondary have a common Center Of Mass (COM). The tertiary is orbiting that COM.

For both the primary and the secondary we can decompose the gravitational force into two perpendicular components: a component towards the COM, and a component away from the COM.

The respective components towards the COM add up; to find the total force towards the COM you add the components.

The respective components away from the COM act in oppposite direction. It's worth emphasizing that only the respective components acting in opposite direction can drop away against each other.

There is another aspect of the Lagrange points L4 and L5 that is worth emphasizing:
Motion of the tertiary in the vicinity of L4 (or L5) is not quite cyclic. It can be very close to a closed orbit, but it cannot quite be closed orbit.

This property manifests itself in a simulation where the motion is computed using numerical analysis. To verify I implemented that type of simulation. Today's PC's are capable of very high accuracy numerical performance, and there are efficient numerical analysis algorithms that are guaranteed to preserve overall kinetic energy.

When I start the simulation with the tertiary perfectly at the Lagrange point, with perfectly the velocity for orbiting there, then in a couple of orbits the tertiary drifts away from the Lagrange point. The tertiary never escapes, it does not have enough energy to escape, but the orbit has the following property: it is extraordinarily sensitive to minute differences in initial conditions.

Also, the mass ratio of the primary and the secondary is a factor. When the mass ratio of the primary to the secondary is less than about 25:1 then there is no long term persistent orbital motion for the tertiary. Inevitably there will come a point where the tertiary gets a significant gravity assist from the secondary, and then the tertiary is ejected from the system.

When trying simulation runs with different mass ratio: the closer the mass ratio came to that cut-off ratio of 25:1, the more sensitive the system. That is, the instability of the cases of mass ratio smaller than 25:1 is already quite noticable in the simulation runs with mass ratio a bit larger than that 25:1

So there is quite a contrast with the rock solid stable orbits of the two-body case.

All of those considerations transfer to the context of GR

The gravitational effects in the direction towards the COM add up.

I suppose: if you push the calculation hard enough eventually you can arrive at a slightly different value as compared to the newtonian calculation, since the equations of GR are non-linear. But at the level of gravitational field in our solar system the difference will be minute indeed.

The gravitational effects in the direction away the COM are in opposite direction.

General considirations:
When there are multiple sources of gravity at play then the result is a single effect. Locally you cannot distinguish whether you are subject to a single source of gravity, or multiple sources of gravity, locally that is not accessible to measurement.

In terms of newtonian mechanics the addition and subtraction of gravitational effect is linear. In terms of GR the combination of multiple sources is not quite linear, but for that to make a significant difference (as compared to newtonian mechanics) you need circumstances with far stronger gravitational source than what we have in our solar system.