How to represent the reduced mass in the curved space form, while studying a two-body system in GR?

The motion of a two-body system under their internal forces (assuming no external forces) can be studied using the Lagrangian analysis.

We assume that both the bodies experience equal and opposite forces and rotate around the common centre of mass. Then, the differential equations describing the orbits (see (1), (2) and (3) in image) have the reduced mass terms to represent combined motion of both the masses.

In these equations, if we assume that the central mass is very large compared to the planet mass, then the planet mass (m2) term vanishes and orbit equations become dependent only on the mass of the central body (m1). This situation is similar to the Schwarzschild solution results (eq. (4) in image) for a large central mass. The GR analysis suggests that the space-time around the large central mass gets curved and therefore the planets have to necessarily follow a curved path. The nature of the curved path is independent of the mass of the planet. Therefore, even a photon (which has zero mass) path has to bend as the space itself is curved.

But, we may also have a situation where the planet mass is not very small compared with the central star. Then according to the classical analysis, the rotation of both the bodies around the common centre will have to be considered. In a classical analysis, this two-body system can be represented mathematically as a single reduced mass rotating around the centre. Similarly, an analysis for the two-body system, based on GR should also consider the resultant curvature created by the both masses.

The question is: How to include the reduced mass term in the GR analysis? This is necessary to ensure that the results of the classical analysis and GR are consistent with each other. The short answer is that the full two-body problem is insanely hard in GR.

Certainly, we can take the Newtonian point-particle limit of GR. In this case, the problem reduces to Newtonian gravity with a series of relativistic corrections; and in the limit where the speeds of the particles are much less than $$c$$ and their separations are large compared to their Schwarzschild radii, all of these corrections vanish, leaving the expected Newtonian limit $$\ddot{\vec{r}}_i = - \sum_{j \neq i} \frac{m_j(\vec{r}_j - \vec{r}_i)}{|\vec{r}_j - \vec{r}_i|^3}$$ (in units where $$G = c = 1$$.) So in this regime, the notion of reduced mass will continue to be useful and valid; and in this limit, Newtonian gravity agrees with GR, as it must.

However, if you want to do a situation where two fundamentally non-Newtonian objects are orbiting around each other, that requires quite a lot of computer code and expertise. The main example that has consumed a lot of blood, sweat, and tears in the GR field over the past 20 years or so is the simulation of two black holes orbiting each other and spiraling inwards due to gravitational radiation. In a strongly curved spacetime like this one, familiar Newtonian notions like the center of mass of a system and the total mass of a system (let alone the reduced mass) are remarkably difficult, if not impossible, to define. But these are regimes in which we don't expect the results of Newtonian gravity to hold anyhow.

• Thanks. It will be interesting to know, how the GR results (using the Schwarzschild metric) can match the classical results (using the reduced mass) for a large planet like Jupiter? The mass of the planet term does not appear in the geodesic equations. I believe that the classical results will have a better match with the experimental observations, as it takes motion of both masses into consideration. Alternatively, we can replace mass of the star M by the total mass (M+m) in the geodesic equation (4) (see the image attached) so as to match with equation (1) of classical analysis.
– NSRG
May 18 '21 at 9:30

We may (?) conduct the GR analysis of two-body problem in two steps:

1. Find out geodesic equations (GR analysis) for the orbits of the second body (a big planet like the Jupiter), assuming that the first body (Sun) is fixed at the centre. Plot them on a graph. As per relativity, working in any frame should be allowed. In any case, an observer making measurements (on earth) is neither on the star nor at the centre of mass frame of that planet-star combination.

2. Geometrically transfer (or transform) the orbits to the centre of mass frame. We can draw another graph by measuring the distance between various points on the orbit and the centre of mass using the first graph. The precession of the orbits, if any, will also get transferred to the centre of mass frame graph.

This approach can be easier, if a direct mathematical transformation/analysis of a two-body system is difficult.

• The problem is "what is the center of mass of a Schwarzschild black hole and a massive body?" It doesn't have a good defininition in full GR. If you're interested in approximations to the two-body problem, with relativsistic corrections, I'd suggest studying the Post-Newtonian Framework, which starts at newtonian mechanics, and adds GR corrections as a series. Sep 11 '21 at 16:43