Physics behind the project

The three-body problem is famous for being a chaotic system without analytical solution in newtonian physics. Even more dificult is the three-body problem in General Relativity, because even the two-body system is intratable for arbitrary masses of the couple. A simple example of three-body systems is the description of the Earth-Sun-Jupiter orbits. When it is possible, physicists usually ignore the least massive body (Earth, in this example), since both Jupiter and the Sun are more massive. Another three-body system that is usually approximated to a two body system is the Alpha Centauri star system. The system is composed of Alpha Centauri A, B and C. However, Alpha Centauri C (also known as Proxima Centauri) mass is only 0.122 solar masses (denoted by \(M_\odot\) ), while the masses of Alpha Centauri A and B are 1.1 \(M_\odot\) and 0.907 \(M_\odot\), respectively. Furthermore, Proxima Centauri appears to be far away from the first two, not influencing their dynamics significantly. The system is then approximated to a binary sistem of Alpha Centauri A and B (See ref [ 1] for more details).

In this project, we do not wish to work with this approximation. That means we need to solve numerically the system of coupled equations in order to extract the orbits of the bodies. The equation of motion of each \(i\)-th body in the system is given by the Einstein-Infeld-Hoffmann equation [2] \[ \vec{a}_i = \sum_{i \neq j}{G m_j \vec{n}_{ij} \over r_{ij}^2} + {1 \over c^2} \sum_{i \neq j} {G m_j \vec{n}_{ij} \over r_{ij}^2} {\left[v_{i}^2 + 2v_{j}^2 - 4 \left( \vec{v}_i \cdot \vec{v}_j \right) - {3 \over 2} \left( \vec{n}_{ij} \cdot \vec{v}_j \right)^2 - 4 \sum_{k \neq i} {G m_k \over r_{ik}} - \right.}\] \[{\left. - \sum_{j \neq k} {G m_k \over r_{jk}} + {1 \over 2} \left( \left( \vec{r}_j - \vec{r}_i \right) \cdot \vec{a}_j \right) \right]} + {1 \over c^2} \sum_{i \neq j} {G m_j \over r_{ij}^2} \left[ \vec{n}_{ij} \cdot \left(4 \vec{v}_i - 3 \vec{v}_j \right) \right] \left( \vec{v}_i - \vec{v}_j \right) + \] \[+ {7 \over 2 c^2} \sum_{i \neq j} {G m_j \vec{a}_{j} \over r_{ij}^2} + \mathcal{O}\left( c^{-4} \right)\]

This is a pretty complicated equation, the acceleration of the \(i\)-th body depends on the acceleration of the others \(j\)-th bodies and it also depends on the velocities and the positions of all other \(j\)-th bodies, where \(i \neq j\). In our case, the total number of orbiting bodies is 3. The \(\vec{n}_{ij}\) vector is the unit vector pointing from the \(i\)-th body to the \(j\)-th body. We then have 3 equations like this one, one for each body.

In order to solve all three equations, we set initial conditions and the constant parameters (\(G, m_1, m_2, m_3\)). Since this is a relativistic problem, we may expect some gravitation waves being emitted from the orbital motion of the three bodies (just like the two black holes merger that provided the first gravitational wave detection [3]). The emission of gravitational waves close to the emitting source is a very complex problem. In our project, we are interested in how we see this gravitational waves from a frame of reference far way from the source. We may think of this consideration as equivallent to measuring gravitational waves here on Earth from the other side of the galaxy, as we observe on observatories such as LIGO and LISA (in the future). Therefore, we may consider only the first order of approximation from the multipole expansion using perturbation theory. The space-time dynamics in General Relativity is specified by the Einstein's equation: \[G_{\mu \nu} = {8 \pi G \over c^4} T_{\mu \nu} \] where the tensor \(G_{\mu \nu}\) is a function of the metric. When considering the emission of gravitational waves, we are talking about a little perturbation on the metric tensor \(g_{\mu \nu}\) that describes the correspondent space-time.Therefore, the metric can be decomposed as \[g_{\mu \nu} = \eta_{\mu \nu} + h_{\mu \nu} \], where \eta_{\mu \nu} describes a flat spacetime and each component of \(h_{\mu \nu} \) is small. Under that first order circunstances of a very far source, the Einstein equation can be rewritten as \[h_{\mu \nu} = -{4 G \over c^4 r} \int T_{\mu \nu}(ct - r, \vec{y}) d^3 y , \] where r is the distance to the source. Integrating the right hand side of this equation and rearanging, we obtain the components of the perturbation: \[ h_{00} = -{4G M \over c^2 r} \] \[ h_{0i} = h_{i0} \propto P_{i} \] \[ h_{ij} = -{2G \over c^6 r} {d^2 \over dt^2} I_{ij}(ct-r) \] Where \(M\) is the total mass of the system, \(P\) is the total momentum and \(I_{ij}\) is the quadrupole tensor. By a frame diffeomorphism transformation we may set \(P = 0\) and the only components of \(h_{\mu \nu}\) that are non-zero and dynamic are the usual \(h_{ij}\) [5]. The quadrupole momentum tensor for a trio of point particles is given by the following sum over their trajectories \[I^{i j}(ct - r) = \sum_{k = 0}^{3} m_{k} x^{i}_{k}(ct - r) x^{j}_{k}(ct - r),\] and the components of the gravitational waves are finally given by \[h^{i j}(t,r) = -{2G \over c^6 r} \sum_{k = 0}^{3} m_{k} {d^2 \over dt^2} (x^{i}_{k}(ct - r) x^{j}_{k}(ct - r)). \] It is interesting to notice that the perturbation tensor calculation depends on \(ct - r\), and this is because the field equations imply that gravitational waves travels at the speed of light, so a point in spacetime at the instant \(t\) and point \(r\) can only be influenced by gravitational waves emitted at the surface of its past light cone, as we should expect from causality.

References

1. Wiegert, P. A., & Holman, M. (1996). The stability of planets in the Alpha Centauri system. arXiv preprint astro-ph/9609106;
2. Einstein, A., Infeld, L., & Hoffmann, B. (1938). The gravitational equations and the problem of motion. Annals of mathematics, 65-100. doi:10.2307/1968714;
3. Abbott, B. P., Abbott, R., Abbott, T. D., Abernathy, M. R., Acernese, F., Ackley, K., ... & Cavalieri, R. (2016). Observation of gravitational waves from a binary black hole merger. Physical review letters, 116(6), 061102.doi:10.1103/PhysRevLett.116.061102.
4. Li, X., Jing, Y., & Liao, S. (2018). Over a thousand new periodic orbits of a planar three-body system with unequal masses. Publications of the Astronomical Society of Japan, 70(4), 64.doi:10.1093/pasj/psy057.
5. Dmitrašinović, V., Šuvakov, M., & Hudomal, A. (2014). Gravitational waves from periodic three-body systems. Physical review letters, 113(10), 101102.doi:10.1103/PhysRevLett.113.101102