Two-body problem in general relativity

General relativity, special relativity and geometry

In the normal Euclidean geometry, triangles obey the Pythagorean theorem, which states that the square distance ds² between two points in space is the sum of the squares of its perpendicular components $${\displaystyle d{s}^2=d{x}^2+d{y}^2+d{z}^2}$$ where dx, dy and dz represent the infinitesimal differences between the x, y and z coordinates of two points in a Cartesian coordinate system. Now imagine a world in which this is not quite true; a world where the distance is instead given by $${\displaystyle d{s}^2=F(x,y,z)d{x}^2+G(x,y,z)d{y}^2+H(x,y,z)d{z}^2}$$ where F, G and H are arbitrary functions of position. It is not hard to imagine such a world; we live on one. The surface of the earth is curved, which is why it is impossible to make a perfectly accurate flat map of the earth. Non-Cartesian coordinate systems illustrate this well; for example, in the spherical coordinates (r, θ, φ), the Euclidean distance can be written $${\displaystyle d{s}^2=d{r}^2+r^{2}d{\theta }^2+r^2{\sin }^2\theta d{\phi }^2}$$

Another illustration would be a world in which the rulers used to measure length were untrustworthy, rulers that changed their length with their position and even their orientation. In the most general case, one must allow for cross-terms when calculating the distance ds $${\displaystyle d{s}^2=g_{xx}d{x}^2+g_{xy}dxdy+g_{xz}dxdz+\cdots +g_{zy}dzdy+g_{zz}d{z}^2}$$ where the nine functions g_xx, g_xy, ..., g_zz constitute the metric tensor, which defines the geometry of the space in Riemannian geometry. In the spherical-coordinates example above, there are no cross-terms; the only nonzero metric tensor components are g_rr = 1, g_θθ = r² and g_φφ = r² sin² θ.

In his special theory of relativity, Albert Einstein showed that the distance ds between two spatial points is not constant, but depends on the motion of the observer. However, there is a measure of separation between two points in space-time — called "proper time" and denoted with the symbol dτ — that is invariant; in other words, it does not depend on the motion of the observer. $${\displaystyle c^{2}d{\tau }^2=c^{2}d{t}^2-d{x}^2-d{y}^2-d{z}^2}$$ which may be written in spherical coordinates as $${\displaystyle c^{2}d{\tau }^2=c^{2}d{t}^2-d{r}^2-r^{2}d{\theta }^2-r^2{\sin }^2\theta d{\phi }^2}$$

This formula is the natural extension of the Pythagorean theorem and similarly holds only when there is no curvature in space-time. In general relativity, however, space and time may have curvature, so this distance formula must be modified to a more general form $${\displaystyle c^{2}d{\tau }^2=g_{\mu \nu }d{x}^{\mu }d{x}^{\nu }}$$ just as we generalized the formula to measure distance on the surface of the Earth. The exact form of the metric g_μν depends on the gravitating mass, momentum and energy, as described by the Einstein field equations. Einstein developed those field equations to match the then known laws of Nature; however, they predicted never-before-seen phenomena (such as the bending of light by gravity) that were confirmed later.

Geodesic equation

According to Einstein's theory of general relativity, particles of negligible mass travel along geodesics in the space-time. In uncurved space-time, far from a source of gravity, these geodesics correspond to straight lines; however, they may deviate from straight lines when the space-time is curved. The equation for the geodesic lines is $${\displaystyle \frac{d^2{x}^{\mu }}{d{q}^2}+{\mathrm{\Gamma }}_{\nu \lambda }^{\mu }\frac{d{x}^{\nu }}{dq}\frac{d{x}^{\lambda }}{dq}=0}$$ where Γ represents the Christoffel symbol and the variable q parametrizes the particle's path through space-time, its so-called world line. The Christoffel symbol depends only on the metric tensor g_μν, or rather on how it changes with position. The variable q is a constant multiple of the proper time τ for timelike orbits (which are traveled by massive particles), and is usually taken to be equal to it. For lightlike (or null) orbits (which are traveled by massless particles such as the photon), the proper time is zero and, strictly speaking, cannot be used as the variable q. Nevertheless, lightlike orbits can be derived as the ultrarelativistic limit of timelike orbits, that is, the limit as the particle mass m goes to zero while holding its total energy fixed.

Schwarzschild solution

An exact solution to the Einstein field equations is the Schwarzschild metric, which corresponds to the external gravitational field of a stationary, uncharged, non-rotating, spherically symmetric body of mass M. It is characterized by a length scale r_s, known as the Schwarzschild radius, which is defined by the formula $${\displaystyle r_{\text{s}}=\frac{2GM}{c^2}}$$ where G is the gravitational constant. The classical Newtonian theory of gravity is recovered in the limit as the ratio r_s/r goes to zero. In that limit, the metric returns to that defined by special relativity.

In practice, this ratio is almost always extremely small. For example, the Schwarzschild radius r_s of the Earth is roughly 9 mm; at the surface of the Earth, the corrections to Newtonian gravity are only one part in a billion. The Schwarzschild radius of the Sun is much larger, roughly 2953 meters, but at its surface, the ratio r_s/r is roughly 4 parts in a million. A white dwarf star is much denser, but even here the ratio at its surface is roughly 250 parts in a million. The ratio only becomes large close to ultra-dense objects such as neutron stars (where the ratio is roughly 50%) and black holes.

Orbits about the central mass

The orbits of a test particle of infinitesimal mass ${\displaystyle m}$ about the central mass ${\displaystyle M}$ is given by the equation of motion $${\displaystyle {\left(\frac{dr}{d\tau }\right)}^2={\left(\frac{E}{mc}\right)}^2-\left(1-\frac{r_{\text{s}}}{r}\right)\left(c^2+\frac{h^2}{r^2}\right).}$$ where ${\displaystyle h}$ is the specific relative angular momentum, $\mathbf{h}=\mathbf{r}\times \mathbf{v}=\frac{\mathbf{L}}{\mu }$ and ${\displaystyle \mu }$ is the reduced mass. This can be converted into an equation for the orbit $${\displaystyle {\left(\frac{dr}{d\phi }\right)}^2=\frac{r^4}{{\tilde{b}}^2}-\left(1-\frac{r_{\text{s}}}{r}\right)\left(\frac{r^4}{{\tilde{a}}^2}+r^2\right)}$$ where, for brevity, two length-scales, $\tilde{a}\equiv h/c$ and $\tilde{b}\equiv Lc/E,$ have been introduced. They are constants of the motion and depend on the initial conditions (position and velocity) of the test particle. Hence, the solution of the orbit equation is $${\displaystyle \phi =\int \frac{1}{r^2}{\left[\frac{1}{{\tilde{b}}^2}-\left(1-\frac{r_{\text{s}}}{r}\right)\left(\frac{1}{{\tilde{a}}^2}+\frac{1}{r^2}\right)\right]}^{-1/2}dr.}$$

Effective radial potential energy

The equation of motion for the particle derived above $${\displaystyle {\left(\frac{dr}{d\tau }\right)}^2={\left(\frac{E}{mc}\right)}^2-c^2+\frac{r_{\text{s}}{c}^2}{r}-{\left(\frac{h}{r}\right)}^2+\frac{r_{\text{s}}{h}^2}{r^3}}$$ can be rewritten using the definition of the Schwarzschild radius r_s as $${\displaystyle \frac{1}{2}m{\left(\frac{dr}{d\tau }\right)}^2=\frac{1}{2}\left[\frac{E^2}{m{c}^2}-m{c}^2\right]+\frac{GMm}{r}-\frac{L^2}{2\mu r^2}+\frac{G(M+m)L^2}{c^2\mu r^3}}$$ which is equivalent to a particle moving in a one-dimensional effective potential $${\displaystyle V(r)=-\frac{GMm}{r}+\frac{L^2}{2\mu r^2}-\frac{G(M+m)L^2}{c^2\mu r^3}.}$$ The first two terms are well-known classical energies, the first being the attractive Newtonian gravitational potential energy and the second corresponding to the repulsive "centrifugal" potential energy; however, the third term is an attractive energy unique to general relativity. As shown below and elsewhere, this inverse-cubic energy causes elliptical orbits to precess gradually by a small angle δω per revolution: $${\displaystyle \delta \omega \approx \frac{6\pi G\left(M+m\right)}{c^{2}a\left(1-e^2\right)},}$$ where a is the orbital semi-major axis and e is the eccentricity. Here, δω is not the change in the φ-coordinate from the spherical coordinates ( t, r, θ, φ ) but close to it: It's the change in the argument of periapsis of the classical closed orbit.

The third term is attractive and dominates at small r values, giving a critical inner radius r_inner at which a particle is drawn inexorably inwards to r = 0 ; this inner radius is a function of the particle's angular momentum per unit mass or, equivalently, the a length-scale defined above.

Circular orbits and their stability

The effective potential V can be re-written in terms of the length $A\equiv h/c$: $${\displaystyle V(r)=\frac{m{c}^2}{2}\left[-\frac{r_{\text{s}}}{r}+\frac{A^2}{r^2}-\frac{A^2{r}_{\text{s}}}{r^3}\right].}$$

Circular orbits are possible when the effective force is zero: $${\displaystyle \begin{array}{cc}F& =-\frac{dV}{dr}\\ & =-\frac{m{c}^2}{2r^4}\left[r_{\text{s}}{r}^2-2A^{2}r+3A^2{r}_{\text{s}}\right]=0;\end{array}}$$ i.e., when the two attractive forces – Newtonian gravity (first term) and the attraction unique to general relativity (third term) – are exactly balanced by the repulsive centrifugal force (second term). There are two radii at which this balancing can occur, denoted here as r_inner and r_outer: $${\displaystyle \begin{array}{cc}{r}_{\text{outer}}& =\frac{A^2}{r_{\text{s}}}\left(1+\sqrt{1-\frac{3{r_{\text{s}}}^2}{A^2}}\right),\\ r_{\text{inner}}& =\frac{A^2}{r_{\text{s}}}\left(1-\sqrt{1-\frac{3{r_{\text{s}}}^2}{A^2}}\right)=\frac{3A^2}{r_{\text{outer}}},\end{array}}$$ which are obtained using the quadratic formula. The inner radius r_inner is unstable, because the attractive third force strengthens much faster than the other two forces when r becomes small; if the particle slips slightly inwards from r_inner (where all three forces are in balance), the third force dominates the other two and draws the particle inexorably inwards to r = 0 . At the outer radius, however, the circular orbits are stable; the third term is less important and the system behaves more like the non-relativistic Kepler problem.

When A is much greater than r_s (the classical case), these formulae become approximately $${\displaystyle \begin{array}{cc}{r}_{\text{outer}}& \approx \frac{2A^2}{r_{\text{s}}},\\ r_{\text{inner}}& \approx \frac{3}{2}{r}_{\text{s}}.\end{array}}$$

Substituting the definitions of A and r_s into r_outer yields the classical formula for a particle of mass m orbiting a body of mass M.

The following equation $${\displaystyle r_{\text{outer}}^3=\frac{G\left(M+m\right)}{{\mathrm{\Omega }}_{\phi }^2}}$$ where ${\displaystyle {\mathrm{\Omega }}_{\phi }}$ is the orbital angular speed of the particle, is obtained in non-relativistic mechanics by setting the centrifugal force equal to the Newtonian gravitational force: $${\displaystyle \frac{GMm}{r^2}=\mu {\mathrm{\Omega }}_{\phi }^{2}r}$$ where ${\displaystyle \mu }$ is the reduced mass.

In our notation, the classical orbital angular speed equals $${\displaystyle \begin{array}{cc}{\mathrm{\Omega }}_{\phi }^2\approx \frac{GM}{r_{\text{outer}}^3}& =\frac{r_{\text{s}}{c}^2}{2r_{\text{outer}}^3}\\ & =\frac{r_{\text{s}}{c}^2}{2}\cdot \frac{r_{\text{s}}^3}{8A^6}=\frac{c^2{r_{\text{s}}}^4}{16A^6}\end{array}}$$

At the other extreme, when ${\displaystyle A^2}$ approaches 3 r_s² from above, the two radii converge to a single value $${\displaystyle r_{\text{outer}}\approx r_{\text{inner}}\approx 3r_{\text{s}}}$$ The quadratic solutions above ensure that r_outer is always greater than 3 r_s , whereas r_inner lies between ⁠ 3 / 2 ⁠ r_s and 3 r_s . Circular orbits smaller than ⁠ 3 / 2 ⁠ r_s are not possible. For massless particles, ${\displaystyle A}$ goes to infinity, implying that there is a circular orbit for photons at r_inner = ⁠ 3 / 2 ⁠ r_s . The sphere of this radius is sometimes known as the photon sphere.

Precession of elliptical orbits

The orbital precession rate may be derived using this radial effective potential V, which includes the non-Newtonian centrifugal force. A small radial deviation from a circular orbit of radius r_outer will oscillate in a stable manner with an angular frequency $${\displaystyle {\mathrm{\Omega }}_r^2=\frac{1}{m}{\frac{d^{2}V}{d{r}^2}|}_{r=r_{\text{outer}}}}$$ which equals $${\displaystyle {\mathrm{\Omega }}_r^2=\frac{c^2{r}_{\text{s}}}{2r_{\text{outer}}^4}(r_{\text{outer}}-r_{\text{inner}})={\mathrm{\Omega }}_{\phi }^2\sqrt{1-\frac{3r_{\text{s}}^2}{a^2}}}$$

Taking the square root of both sides and expanding using the binomial theorem yields the formula $${\displaystyle {\mathrm{\Omega }}_r={\mathrm{\Omega }}_{\phi }\left(1-\frac{3r_{\text{s}}^2}{4a^2}+\cdots \right)}$$ Multiplying by the period T of one revolution gives the precession of the orbit per revolution $${\displaystyle \begin{array}{cc}\delta \phi & =T({\mathrm{\Omega }}_{\phi }-{\mathrm{\Omega }}_r)\\ & \approx 2\pi \frac{3r_{\text{s}}^2}{4a^2}=\frac{3\pi m^2{c}^2}{2L^2}{r}_{\text{s}}^2\end{array}}$$ where we have used Ω_φ T ≈ 2 π (exact for circular orbits) and the definition of the length-scale set by a. Substituting the definition of the Schwarzschild radius r_s gives $${\displaystyle \delta \phi \approx \frac{3\pi m^2{c}^2}{2L^2}\frac{4G^2{M}^2}{c^4}=\frac{6\pi G^2{M}^2{m}^2}{c^2{L}^2}}$$

This may be simplified using the elliptical orbit's semi-major axis a, eccentricity e, and specific relative angular momentum h, which are related by the formula $${\displaystyle \frac{h^2}{G(M+m)}=a\left(1-e^2\right)}$$ to give the precession angle $${\displaystyle \delta \phi \approx \frac{6\pi G(M+m)}{c^{2}a\left(1-e^2\right)}.}$$

Since the closed classical orbit is generally an ellipse, the quantity a ( 1 − e²) is the semi-latus rectum ${\displaystyle \ell }$ of the ellipse.

Hence, the final formula of angular apsidal precession for a unit complete revolution is $${\displaystyle \delta \phi \approx \frac{6\pi G(M+m)}{\ell c^2}.}$$

Beyond the Schwarzschild solution

Post-Newtonian expansion

In the Schwarzschild solution, it is assumed that the larger mass M is stationary and it alone determines the gravitational field (i.e., the geometry of space-time) and, hence, the lesser mass m follows a geodesic path through that fixed space-time. This is a reasonable approximation for photons and the orbit of Mercury, which is roughly 6 million times lighter than the Sun. However, it is inadequate for binary stars, in which the masses may be of similar magnitude.

The metric for the case of two comparable masses cannot be solved in closed form and therefore one has to resort to approximation techniques such as the post-Newtonian approximation or numerical approximations. In passing, we mention one particular exception in lower dimensions (see R = T model for details). In (1+1) dimensions, i.e. a space made of one spatial dimension and one time dimension, the metric for two bodies of equal masses can be solved analytically in terms of the Lambert W function. However, the gravitational energy between the two bodies is exchanged via dilatons rather than gravitons which require three-space in which to propagate.

The post-Newtonian expansion is a calculational method that provides a series of ever more accurate solutions to a given problem. The method is iterative; an initial solution for particle motions is used to calculate the gravitational fields; from these derived fields, new particle motions can be calculated, from which even more accurate estimates of the fields can be computed, and so on. This approach is called "post-Newtonian" because the Newtonian solution for the particle orbits is often used as the initial solution.

The theory can be divided into two parts: first one finds the two-body effective potential that captures the GR corrections to the Newtonian potential. Secondly, one should solve the resulting equations of motion.

Modern computational approaches

Einstein's equations can also be solved on a computer using sophisticated numerical methods. Given sufficient computer power, such solutions can be more accurate than post-Newtonian solutions. However, such calculations are demanding because the equations must generally be solved in a four-dimensional space. Nevertheless, beginning in the late 1990s, it became possible to solve difficult problems such as the merger of two black holes, which is a very difficult version of the Kepler problem in general relativity.

Gravitational radiation

If there is no incoming gravitational radiation, according to general relativity, two bodies orbiting one another will emit gravitational radiation, causing the orbits to gradually lose energy.

The formulae describing the loss of energy and angular momentum due to gravitational radiation from the two bodies of the Kepler problem have been calculated. The rate of losing energy (averaged over a complete orbit) is given by $${\displaystyle -⟨\frac{dE}{dt}⟩=\frac{32G^4{m}_1^2{m}_2^2(m_1+m_2)}{5c^5{a}^5{\left(1-e^2\right)}^{7/2}}\left(1+\frac{73}{24}{e}^2+\frac{37}{96}{e}^4\right)}$$ where e is the orbital eccentricity and a is the semimajor axis of the elliptical orbit. The angular brackets on the left-hand side of the equation represent the averaging over a single orbit. Similarly, the average rate of losing angular momentum equals $${\displaystyle -⟨\frac{d{L}_z}{dt}⟩=\frac{32G^{7/2}{m}_1^2{m}_2^2\sqrt{m_1+m_2}}{5c^5{a}^{7/2}{\left(1-e^2\right)}^2}\left(1+\frac{7}{8}{e}^2\right)}$$

The rate of period decrease is given by $${\displaystyle -⟨\frac{d{P}_b}{dt}⟩=\frac{192\pi G^{5/3}{m}_1{m}_2}{5c^5{\left(1-e^2\right)}^{7/2}{\left(m_1+m_2\right)}^{1/3}}\left(1+\frac{73}{24}{e}^2+\frac{37}{96}{e}^4\right){\left(\frac{P_b}{2\pi }\right)}^{-5/3}}$$ where P_b is orbital period.

The losses in energy and angular momentum increase significantly as the eccentricity approaches one, i.e., as the ellipse of the orbit becomes ever more elongated. The radiation losses also increase significantly with a decreasing size a of the orbit.