Schwarzschild geodesics - AbsoluteAstronomy.com

General relativity

General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

, the geodesics of the Schwarzschild metric describe the motion of particles of infinitesimal mass in the gravitational field of a central fixed mass M. The Schwarzschild geodesics have been pivotal in the validation

Tests of general relativity

At its introduction in 1915, the general theory of relativity did not have a solid empirical foundation. It was known that it correctly accounted for the "anomalous" precession of the perihelion of Mercury and on philosophical grounds it was considered satisfying that it was able to unify Newton's...

of the Einstein's theory of general relativity

General relativity

General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

. For example, they provide quantitative predictions of the anomalous precession of the planets in the Solar System, and of the deflection of light by gravity.

The Schwarzschild geodesics pertain only to the motion of particles of infinitesimal mass m, i.e., particles that do not themselves contribute to the gravitational field. However, they are highly accurate provided that m is many-fold smaller than the central mass M, e.g., for planets orbiting their sun. The Schwarzschild geodesics are also a good approximation to the relative motion of two bodies of arbitrary mass, provided that the Schwarzschild mass M is set equal to the sum of the two individual masses m₁ and m₂. This is important in predicting the motion of binary star

Binary star

A binary star is a star system consisting of two stars orbiting around their common center of mass. The brighter star is called the primary and the other is its companion star, comes, or secondary...

s in general relativity.

Historical context

The Schwarzschild solution was found by Karl Schwarzschild

Karl Schwarzschild

Karl Schwarzschild was a German physicist. He is also the father of astrophysicist Martin Schwarzschild.He is best known for providing the first exact solution to the Einstein field equations of general relativity, for the limited case of a single spherical non-rotating mass, which he accomplished...

shortly after Einstein published his field equations.
The Schwarzschild metric is named in honour of its discoverer Karl Schwarzschild

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 solution of the Einstein field equations other than the trivial flat space solution.

Schwarzschild metric

An exact solution to the Einstein field equations

Einstein field equations

The Einstein field equations or Einstein's equations are a set of ten equations in Albert Einstein's general theory of relativity which describe the fundamental interaction of gravitation as a result of spacetime being curved by matter and energy...

is the Schwarzschild metric

Schwarzschild metric

In Einstein's theory of general relativity, the Schwarzschild solution describes the gravitational field outside a spherical, uncharged, non-rotating mass such as a star, planet, or black hole. It is also a good approximation to the gravitational field of a slowly rotating body like the Earth or...

, which corresponds to the external gravitational field of an uncharged, non-rotating, spherically symmetric body of mass M. The Schwarzschild solution can be written as

where

τ is the proper time (time measured by a clock moving with the particle) in seconds,

c is the speed of light

Speed of light

The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...

in meters per second,

t is the time coordinate (measured by a stationary clock at infinity) in seconds,

r is the radial coordinate (circumference of a circle centered on the star divided by 2π) in meters,

θ is the colatitude

Colatitude

In spherical coordinates, colatitude is the complementary angle of the latitude, i.e. the difference between 90° and the latitude.-Astronomical use:The colatitude is useful in astronomy because it refers to the zenith distance of the celestial poles...

(angle from North) in radians,

φ is the longitude

Longitude

Longitude is a geographic coordinate that specifies the east-west position of a point on the Earth's surface. It is an angular measurement, usually expressed in degrees, minutes and seconds, and denoted by the Greek letter lambda ....

in radians, and

r_s is the Schwarzschild radius

Schwarzschild radius

The Schwarzschild radius is the distance from the center of an object such that, if all the mass of the object were compressed within that sphere, the escape speed from the surface would equal the speed of light...

(in meters) of the massive body, which is related to its mass M by

where G is the gravitational constant

Gravitational constant

The gravitational constant, denoted G, is an empirical physical constant involved in the calculation of the gravitational attraction between objects with mass. It appears in Newton's law of universal gravitation and in Einstein's theory of general relativity. It is also known as the universal...

. 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

Special relativity

Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...

In practice, this ratio is almost always extremely small. For example, the Schwarzschild radius r_s of the Earth

Earth

Earth is the third planet from the Sun, and the densest and fifth-largest of the eight planets in the Solar System. It is also the largest of the Solar System's four terrestrial planets...

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 r_s/r is roughly 4 parts in a million. A white dwarf

White dwarf

A white dwarf, also called a degenerate dwarf, is a small star composed mostly of electron-degenerate matter. They are very dense; a white dwarf's mass is comparable to that of the Sun and its volume is comparable to that of the Earth. Its faint luminosity comes from the emission of stored...

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 star

Neutron star

A neutron star is a type of stellar remnant that can result from the gravitational collapse of a massive star during a Type II, Type Ib or Type Ic supernova event. Such stars are composed almost entirely of neutrons, which are subatomic particles without electrical charge and with a slightly larger...

s (where the ratio is roughly 50%) and black hole

Black hole

A black hole is a region of spacetime from which nothing, not even light, can escape. The theory of general relativity predicts that a sufficiently compact mass will deform spacetime to form a black hole. Around a black hole there is a mathematically defined surface called an event horizon that...

Orbits of test particles

Dividing both sides by dτ², the Schwarzschild metric can be rewritten as

The orbit of a particle in this metric is defined by the geodesic equation, which may be solved by any of several methods (as outlined below). This equation yields three constants of motion. First, the motion of the particle is always in a plane, which is equivalent to fixing θ = π/2. The second and third constants of motion, derived below, are taken as two length-scales, a and b, defined by the equations

where c represents the speed of light. Incorporating these constants of motion into the metric yields the fundamental equation for the particle's orbit

Exact solution using elliptic functions

The fundamental equation of the orbit is easier to solveThis substitution of u for r 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

Classical central-force problem

In classical mechanics, the central-force problem is to determine the motion of a particle under the influence of a single central force. A central force is a force that points from the particle directly towards a fixed point in space, the center, and whose magnitude only depends on the distance...

. if it is expressed in terms of the inverse radius u = 1/r

The right-hand side of this equation is a cubic polynomial

Cubic function

In mathematics, a cubic function is a function of the formf=ax^3+bx^2+cx+d,\,where a is nonzero; or in other words, a polynomial of degree three. The derivative of a cubic function is a quadratic function...

, which has three roots, denoted here as u₁, u₂, and u₃

The sum of the three roots equals the coefficient of the u² term

A cubic polynomial with real coefficients can either have three real roots, or one real root and two complex conjugate

Complex conjugate

In mathematics, complex conjugates are a pair of complex numbers, both having the same real part, but with imaginary parts of equal magnitude and opposite signs...

roots. If all three roots are real number

Real number

In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

s, the roots are labeled so that u₁ < u₂ < u₃. If instead there is only one real root, then that is denoted as u₃; the complex conjugate roots are labeled u₁ and u₂. Using Descartes' rule of signs

Descartes' rule of signs

In mathematics, Descartes' rule of signs, first described by René Descartes in his work La Géométrie, is a technique for determining the number of positive or negative real roots of a polynomial....

, there can be at most one negative root; u₁ is negative if and only if b < a. 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 for the orbit can be expressed in elliptic function

Elliptic function

In complex analysis, an elliptic function is a function defined on the complex plane that is periodic in two directions and at the same time is meromorphic...

where sn represents one of the Jacobi elliptic functions and δ is a constant of integration reflecting the initial position. The modulus k of this elliptic function is given by the formula

Classical limit

To recover the Newtonian solution for the planetary orbits, one takes the limit as the Schwarzschild radius r_s goes to zero. In this case, the third root u₃ becomes roughly 1/r_s, and much larger than u₁ or u₂. Therefore, the modulus k tends to zero; in that limit, sn becomes the trigonometric sine function

Consistent with Newton's solutions for planetary motions, this formula describes a focal conic of eccentricity e

If u₁ is a positive real number, then the orbit is an ellipse

Ellipse

In geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis...

where u₁ and u₂ represent the distances of furthest and closest approach, respectively. If u₁ is zero or a negative real number, the orbit is a parabola

Parabola

In mathematics, the parabola is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface...

or a hyperbola

Hyperbola

In mathematics a hyperbola is a curve, specifically a smooth curve that lies in a plane, which can be defined either by its geometric properties or by the kinds of equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, which are mirror...

, respectively. In these latter two cases, u₂ represents the distance of closest approach; since the orbit goes to infinity (u=0), 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 du/dφ = 0. At such a turning point, u reaches a maximum, a minimum, or an inflection point, depending on the value of the second derivative, which is given by the formula

Therefore, 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 the system may either oscillate between u₁ and u₂, or it may move away from u₃ towards infinity. Circular orbits result if u₂ is equal to either u₁ or u₃. These different types of orbits are discussed in the following subsections.

Precession of orbits

The function sn and its square sn² have periods of 4K and 2K, respectively, where K is defined by the equationIn the mathematical literature, K is known as the complete elliptic integral of the first kind; for further information, please see the article on elliptic integral

Elliptic integral

In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse. They were first studied by Giulio Fagnano and Leonhard Euler...

Therefore, the change in φ over one oscillation of u (or, equivalently, one oscillation of r) equals

In the classical limit, u₃ approaches 1/r_s and is much larger than u₁ or u₂. Hence, k² is approximately

For the same reasons, the denominator of Δφ is approximately

Since the modulus k is close to zero, the period K can be expanded in powers of k; to lowest order, this expansion yields

Substituting these approximations into the formula for Δφ yields a formula for angular advance per radial oscillation

For an elliptical orbit, u₁ and u₂ represent the inverses of the longest and shortest distances, respectively. These can be expressed in terms of the ellipse's semiaxis A and its eccentricity e,

giving

Substituting the definition of r_s gives the final equation

Bending of light by gravity

In the limit as the particle mass m goes to zero (or, equivalently, as the length-scale a goes to infinity), the equation for the orbit becomes

Expanding in powers of r_s/r, the leading order term in this formula gives the approximate angular deflection δφ for a massless particle coming in from infinity and going back out to infinity:

Here, b can be interpreted as the distance of closest approach. Although this formula is approximate, it is accurate for most measurements of gravitational lensing, due to the smallness of the ratio r_s/r. For light grazing the surface of the sun, the approximate angular deflection is roughly 1.75 arcseconds

Minute of arc

A minute of arc, arcminute, or minute of angle , is a unit of angular measurement equal to one sixtieth of one degree. In turn, a second of arc or arcsecond is one sixtieth of one minute of arc....

, roughly one millionth part of a circle.

Effective radial potential energy

The equation of motion for the particle derived above

can be rewritten using the definition of the Schwarzschild radius

Schwarzschild metric

r_s as

which is equivalent to a particle moving in a one-dimensional effective potential

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

Centrifugal force

Centrifugal force can generally be any force directed outward relative to some origin. More particularly, in classical mechanics, the centrifugal force is an outward force which arises when describing the motion of objects in a rotating reference frame...

; however, the third term is an attractive energy unique to general relativity

General relativity

General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

. As shown below and elsewhere, this inverse-cubic energy causes elliptical orbits to precess gradually by an angle δφ per revolution

where A is the semi-major axis and e is the eccentricity.

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 lengths a and b

Circular orbits are possible when the effective force is zero

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

which are obtained using the quadratic formula. The inner radiusr_inner is unstable, because the attractive third force strengthens much faster than the other two forces whenr 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

Kepler problem

In classical mechanics, the Kepler problem is a special case of the two-body problem, in which the two bodies interact by a central force F that varies in strength as the inverse square of the distance r between them. The force may be either attractive or repulsive...

.

When a is much greater than r_s (the classical case), these formulae become approximately

Substituting the definitions of a and r_s into r_outer yields the classical formula for a particle orbiting a mass M in a circle

where ω_φ is the orbital angular speed of the particle. This formula is obtained in non-relativistic mechanics by setting the centrifugal force

Centrifugal force

equal to the Newtonian gravitational force:

Where

is the reduced mass

Reduced mass

Reduced mass is the "effective" inertial mass appearing in the two-body problem of Newtonian mechanics. This is a quantity with the unit of mass, which allows the two-body problem to be solved as if it were a one-body problem. Note however that the mass determining the gravitational force is not...

.

In our notation, the classical orbital angular speed equals

At the other extreme, when a² approaches 3r_s² from above, the two radii converge to a single value

The quadratic solutions above ensure that r_outer is always greater than 3r_s, whereas r_inner lies between r_s and 3r_s. Circular orbits smaller than r_s are not possible. For massless particles, a goes to infinity, implying that there is a circular orbit for photons at r_inner = 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. A small radial deviation from a circular orbit of radius r_outer will oscillate stably with an angular frequency

which equals

Taking the square root of both sides and expanding using the binomial theorem

Binomial theorem

In elementary algebra, the binomial theorem describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the power n into a sum involving terms of the form axbyc, where the exponents b and c are nonnegative integers with , and the coefficient a of...

yields the formula

Multiplying by the period T of one revolution gives the precession of the orbit per revolution

where we have used ω_φT = 2п and the definition of the length-scale a. Substituting the definition of the Schwarzschild radius

Schwarzschild metric

r_s gives

This may be simplified using the elliptical orbit's semiaxis A and eccentricity e related by the formula

to give the precession angle

Geodesic equation

According to Einstein's theory of general relativity, particles of negligible mass travel along geodesic

Geodesic

In mathematics, a geodesic is a generalization of the notion of a "straight line" to "curved spaces". In the presence of a Riemannian metric, geodesics are defined to be the shortest path between points in the space...

s in the space-time. In uncurved (flat) 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

where Γ represents the Christoffel symbol and the variable q parametrizes the particle's path through space-time, its so-called world line

World line

In physics, the world line of an object is the unique path of that object as it travels through 4-dimensional spacetime. The concept of "world line" is distinguished from the concept of "orbit" or "trajectory" by the time dimension, and typically encompasses a large area of spacetime wherein...

. The Christoffel symbol depends only on the metric tensor

Metric tensor

In the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean...

g_μν, or rather on how it changes with position. The variable q is a constant multiple of the proper time

Proper time

In relativity, proper time is the elapsed time between two events as measured by a clock that passes through both events. The proper time depends not only on the events but also on the motion of the clock between the events. An accelerated clock will measure a smaller elapsed time between two...

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

Photon

In physics, a photon is an elementary particle, the quantum of the electromagnetic interaction and the basic unit of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...

), 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

Ultrarelativistic limit

In physics, a particle is called ultrarelativistic when its speed is very close to the speed of light c.Max Planck showed that the relativistic expression for the energy of a particle whose rest mass is m and momentum is p is given by E^2 = m^2 c^4 + p^2 c^2...

of timelike orbits, that is, the limit as the particle mass m goes to zero while holding its total energy

Energy

In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...

fixed.

Therefore, to solve for the motion of a particle, the most straightforward way is to solve the geodesic equation, an approach adopted by Einstein and others. The Schwarzschild metric may be written as

where the two functions w(r) = (1 - r_s/r) and its inverse v(r) = 1/w(r) are be defined for brevity. From this metric, the Christoffel symbols Γ^λ_μν may be calculated, and the results substituted into the geodesic equations

It may be verified that θ=π/2 is a valid solution by substitution into the first of these four equations. By symmetry, the orbit must be planar, and we are free to arrange the coordinate frame so that the equatorial plane is the plane of the orbit. This θ solution simplifies the second and fourth equations.

To solve the second and third equations, it suffices to divide them by dφ/dq and dt/dq, respectively.

which yields two constants of motion.

Lagrangian approach

Because test particles follow geodesics in a fixed metric, the orbits of those particles may be determined using the calculus of variations, also called the Lagrangian approach. Geodesics in space-time are defined as curves for which small local variations in their coordinates (while holding their endpoints events fixed) make no significant change in their overall length s. This may be expressed mathematically using the calculus of variations

Calculus of variations

Calculus of variations is a field of mathematics that deals with extremizing functionals, as opposed to ordinary calculus which deals with functions. A functional is usually a mapping from a set of functions to the real numbers. Functionals are often formed as definite integrals involving unknown...

where τ is the proper time

Proper time

, s=cτ is the arc-length in space-time and T is defined as

in analogy with kinetic energy

Kinetic energy

The kinetic energy of an object is the energy which it possesses due to its motion.It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes...

. If the derivative with respect to proper time is represented by a dot for brevity

T may be written as

Constant factors (such as c or the square root of two) don't affect the answer to the variational problem; therefore, taking the variation inside the integral yields Hamilton's principle

Hamilton's principle

In physics, Hamilton's principle is William Rowan Hamilton's formulation of the principle of stationary action...

The solution of the variational problem is given by Lagrange's equations

Euler-Lagrange equation

In calculus of variations, the Euler–Lagrange equation, Euler's equation, or Lagrange's equation, is a differential equation whose solutions are the functions for which a given functional is stationary...

When applied to t and φ, these equations reveal two constants of motion

Constant of motion

In mechanics, a constant of motion is a quantity that is conserved throughout the motion, imposing in effect a constraint on the motion. However, it is a mathematical constraint, the natural consequence of the equations of motion, rather than a physical constraint...

which may be expressed in terms of two constant length-scales, a and b

As shown above, substitution of these equations into the definition of the Schwarzschild metric

Schwarzschild metric

yields the equation for the orbit.

Hamiltonian approach

A Lagrangian solution can be recast into an equivalent Hamiltonian form. In this case, the Hamiltonian H is given by

Once again, the orbit may be restricted to θ = π/2 by symmetry. Since t, θ and φ do not appear in the Hamiltonian, their conjugate momenta are constant; they may be expressed in terms of the speed of light c and two constant length-scales a and b

The derivatives with respect to proper time are given by

Dividing the first equation by the second yields the orbital equation

The radial momentum p_r can be expressed in terms of r using the constancy of the Hamiltonian H = c²/2; this yields the fundamental orbital equation

Hamilton–Jacobi approach

The orbital equation can be derived from the Hamilton–Jacobi equation. The advantage of this approach is that it equates the motion of the particle with the propagation of a wave, and leads neatly into the derivation of the deflection of light by gravity in general relativity

General relativity

General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

, through Fermat's principle

Fermat's principle

In optics, Fermat's principle or the principle of least time is the principle that the path taken between two points by a ray of light is the path that can be traversed in the least time. This principle is sometimes taken as the definition of a ray of light...

. The basic idea is that, due to gravitational slowing of time, parts of a wave-front closer to a gravitating mass move more slowly than those further away, thus bending the direction of the wave-front's propagation.

Using general covariance, the Hamilton–Jacobi equation for a single particle of unit mass can be expressed in arbitrary coordinates as

This is equivalent to the Hamiltonian formulation above, with the partial derivatives of the action taking the place of the generalized momenta. Using the Schwarzschild metric

Schwarzschild metric

g^μν, this equation becomes

where we again orient the spherical coordinate system with the plane of the orbit. The time t and azimuthal angle φ are cyclic coordinates, so that the solution for Hamilton's principal function S can be written

where p_t and p_φ are the constant generalized momenta. The Hamilton–Jacobi equation gives an integral solution for the radial part S_r(r)

Taking the derivative of Hamilton's principal function S with respect to the conserved momentum p_φ yields

which equals

Taking an infinitesimal variation in φ and r yields the fundamental orbital equation

where the conserved length-scales a and b are defined by the conserved momenta by the equations

Hamilton's principle

The action

Action (physics)

In physics, action is an attribute of the dynamics of a physical system. It is a mathematical functional which takes the trajectory, also called path or history, of the system as its argument and has a real number as its result. Action has the dimension of energy × time, and its unit is...

integral for a particle affected only by gravity is

where τ is the proper time

Proper time

and q is any smooth parameterization of the particle's world line. If one applies the calculus of variations

Calculus of variations

to this, one again gets the equations for a geodesic. To simplify the calculations, one first takes the variation of the square of the integrand. For the metric and coordinates of this case and assuming that the particle is moving in the equatorial plane (θ= π/2), that square is

Taking variation of this gives

Motion in longitude

Vary with respect to longitude φ only to get

Divide by

to get the variation of the integrand itself

Thus

Integrating by parts gives

The variation of the longitude is assumed to be zero at the end points, so the first term disappears. The integral can be made nonzero by a perverse choice of δφ unless the other factor inside is zero everywhere. So the equation of motion is