In general relativity, Schwarzschild geodesics describe the motion of test particles in the gravitational field of a central fixed mass that is, motion in the Schwarzschild metric. Schwarzschild geodesics have been pivotal in the validation of Einstein's theory of general relativity. For example, they provide accurate predictions of the anomalous precession of the planets in the Solar System and of the deflection of light by gravity.
Schwarzschild geodesics pertain only to the motion of particles of masses so small they contribute little to the gravitational field. However, they are highly accurate in many astrophysical scenarios provided that is many-fold smaller than the central mass , e.g., for planets orbiting their star. Schwarzschild geodesics are also a good approximation to the relative motion of two bodies of arbitrary mass, provided that the Schwarzschild mass is set equal to the sum of the two individual masses and . This is important in predicting the motion of binary stars in general relativity.
Historical context
The Schwarzschild metric is named in honour of its discoverer Karl Schwarzschild, who found the solution in 1915, only about a month after the publication of Einstein's theory of general relativity. It was the first exact solution of the Einstein field equations other than the trivial flat space solution.
In 1931, Yusuke Hagihara published a paper showing that the trajectory of a test particle in the Schwarzschild metric can be expressed in terms of elliptic functions.
Samuil Kaplan in 1949 has shown that there is a minimum radius for the circular orbit to be stable in Schwarzschild metric.
Schwarzschild metric
An exact solution to the Einstein field equations is the Schwarzschild metric, which corresponds to the external gravitational field of an uncharged, non-rotating, spherically symmetric body of mass . The Schwarzschild solution can be written asLandau and Lifshitz, pp. 299–301.
where , in the case of a test particle of small positive mass, is the proper time (time measured by a clock moving with the particle) in seconds, is the speed of light in meters per second, is, for , the time coordinate (time measured by a stationary clock at infinity) in seconds, is, for , the radial coordinate (circumference of a circle centered at the star divided by ) in meters, is the colatitude (angle from North) in radians, is the longitude in radians, and is the Schwarzschild radius of the massive body (in meters), which is related to its mass by
where is the gravitational constant. The classical Newtonian theory of gravity is recovered in the limit as the ratio 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 of the Earth is roughly 9 mm ( inch); 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 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 of test particles
We may simplify the problem by using symmetry to eliminate one variable from consideration. Since the Schwarzschild metric is symmetrical about , any geodesic that begins moving in that plane will remain in that plane indefinitely (the plane is totally geodesic). Therefore, we orient the coordinate system so that the orbit of the particle lies in that plane, and fix the coordinate to be so that the metric (of this plane) simplifies to
Two constants of motion (values that do not change over proper time ) can be identified (cf. the derivation given below). One is the total energy :
and the other is the specific angular momentum:
where is the total angular momentum of the two bodies, and is the reduced mass. When , the reduced mass is approximately equal to . Sometimes it is assumed that . In the case of the planet Mercury this simplification introduces an error more than twice as large as the relativistic effect. When discussing geodesics, can be considered fictitious, and what matters are the constants and . In order to cover all possible geodesics, we need to consider cases in which is infinite (giving trajectories of photons) or imaginary (for tachyonic geodesics). For the photonic case, we also need to specify a number corresponding to the ratio of the two constants, namely , which may be zero or a non-zero real number.
Substituting these constants into the definition of the Schwarzschild metric
yields an equation of motion for the radius as a function of the proper time :
The formal solution to this is
Note that the square root will be imaginary for tachyonic geodesics.
Using the relation higher up between and , we can also write
Since asymptotically the integrand is inversely proportional to , this shows that in the frame of reference if approaches it does so exponentially without ever reaching it. However, as a function of , does reach .
The above solutions are valid while the integrand is finite, but a total solution may involve two or an infinity of pieces, each described by the integral but with alternating signs for the square root.
When and , we can solve for and explicitly:
and for photonic geodesics () with zero angular momentum
(Although the proper time is trivial in the photonic case, one can define an affine parameter , and then the solution to the geodesic equation is .)
Another solvable case is that in which and and are constant. In the volume where this gives for the proper time
This is close to solutions with small and positive. Outside of the solution is tachyonic and the "proper time" is space-like:
This is close to other tachyonic solutions with small and negative. The constant tachyonic geodesic outside is not continued by a constant geodesic inside , but rather continues into a "parallel exterior region" (see Kruskal–Szekeres coordinates). Other tachyonic solutions can enter a black hole and re-exit into the parallel exterior region. The constant solution inside the event horizon () is continued by a constant solution in a white hole.
When the angular momentum is not zero we can replace the dependence on proper time by a dependence on the angle using the definition of
which yields the equation for the orbit
where, for brevity, two length-scales, and , have been defined by
Note that in the tachyonic case, will be imaginary and real or infinite.
The same equation can also be derived using a Lagrangian approachWhittaker 1937. or the Hamilton–Jacobi equationLandau and Lifshitz (1975), pp. 306–309. (see below). The solution of the orbit equation is
This can be expressed in terms of the Weierstrass elliptic function .
Local and delayed velocities
Unlike in classical mechanics, in Schwarzschild coordinates and are not the radial and transverse components of the local velocity (relative to a stationary observer), instead they give the components for the celerity which are related to by
for the radial and
for the transverse component of motion, with . The coordinate bookkeeper far away from the scene observes the shapiro-delayed velocity , which is given by the relation
and .
The time dilation factor between the bookkeeper and the moving test-particle can also be put into the form
where the numerator is the gravitational, and the denominator is the kinematic component of the time dilation. For a particle falling in from infinity the left factor equals the right factor, since the in-falling velocity matches the escape velocity in this case.
The two constants angular momentum and total energy of a test-particle with mass are in terms of
and
where
and
For massive testparticles is the Lorentz factor and is the proper time, while for massless particles like photons is set to and takes the role of an affine parameter. If the particle is massless is replaced with and with , where is the Planck constant and the locally observed frequency.
Exact solution using elliptic functions
The fundamental equation of the orbit is easier to solveThis substitution of for is also common in classical central-force problems, since it also renders those equations easier to solve. For further information, please see the article on the classical central-force problem. if it is expressed in terms of the inverse radius
The right-hand side of this equation is a cubic polynomial, which has three roots, denoted here as , , and
The sum of the three roots equals the coefficient of the term (see Vieta's formulas):
A cubic polynomial with real coefficients can either have three real roots, or one real root and two complex conjugate roots. If all three roots are real numbers, the roots are labeled so that . If instead there is only one real root, then that is denoted as ; the complex conjugate roots are labeled and . Using Descartes' rule of signs, there can be at most one negative root; is negative if and only if . As discussed below, the roots are useful in determining the types of possible orbits.
Given this labeling of the roots, the solution of the fundamental orbital equation is
where represents the function (one of the Jacobi elliptic functions) and is a constant of integration reflecting the initial position. The elliptic modulus of this elliptic function is given by the formula
Newtonian limit
To recover the Newtonian solution for the planetary orbits, one takes the limit as the Schwarzschild radius goes to zero. In this case, the third root becomes roughly , and much larger than or . Therefore, the modulus tends to zero; in that limit, becomes the trigonometric sine function
Consistent with Newton's solutions for planetary motions, this formula describes a focal conic of eccentricity
If is a positive real number, then the orbit is an ellipse where and represent the distances of furthest and closest approach, respectively. If is zero or a negative real number, the orbit is a parabola or a hyperbola, respectively. In these latter two cases, represents the distance of closest approach; since the orbit goes to infinity (), there is no distance of furthest approach.
Roots and overview of possible orbits
A root represents a point of the orbit where the derivative vanishes, i.e., where . At such a turning point, reaches a maximum, a minimum, or an inflection point, depending on the value of the second derivative, which is given by the formula
If all three roots are distinct real numbers, the second derivative is positive, negative, and positive at u1, u2, and u3, respectively. It follows that a graph of u versus φ may either oscillate between u1 and u2, or it may move away from u3 towards infinity (which corresponds to r going to zero). If u1 is negative, only part of an "oscillation" will actually occur. This corresponds to the particle coming from infinity, getting near the central mass, and then moving away again toward infinity, like the hyperbolic trajectory in the classical solution.
If the particle has just the right amount of energy for its angular momentum, u2 and u3 will merge. There are three solutions in this case. The orbit may spiral in to , approaching that radius as (asymptotically) a decreasing exponential in φ, , or . Or one can have a circular orbit at that radius. Or one can have an orbit that spirals down from that radius to the central point. The radius in question is called the inner radius and is between and 3 times rs. A circular orbit also results when is equal to , and this is called the outer radius. These different types of orbits are discussed below.