Reference: Moore, Thomas A., *A General Relativity Workbook*, University Science Books (2013) – Chapter 10; Boxes 10.1 – 10.2; Problem 10.2.

One form of the geodesic equation is

A geodesic is the path followed by a free particle in a given metric, so we can apply this equation to the Schwarzschild metric to discover what orbits a particle can have around a mass. The metric is

From the component, we get from 1:

Since the metric does not depend on the second term is zero. Also, since the metric is diagonal, the first term becomes

where is a constant.

The meaning of can be inferred from the following argument. At , the relation becomes

We’ve seen that the Schwarzschild coordinate is the same as the object’s proper time for an object at rest, when measured by an observer at infinity. Thus for an object at rest, . However, in general , the time component of the four-velocity. In this case, is the time as measured by an observer at rest at infinity, and is the proper time as measured by the object, which may be moving. The four-momentum‘s time component is the energy, and the four-velocity is the four-momentum per unit mass, so is the energy per unit mass of the object which remains constant as the object moves in from infinity.

Now look at the component of 1. Again, since the metric does not depend on and it is diagonal, we get

where is another constant. If we look at motion in the equatorial plane, and which is the angular speed of rotation, so we get

which is equivalent to the classical definition if is the angular momentum.

In fact, any initial velocity lies in *some* equatorial plane (merely redefine the location of the axis so that the velocity lies in the plane with ) so for any specific motion, we can assume without, as they say, any loss of generality. Also, because the metric is spherically symmetric, any motion that starts in an equatorial plane must stay in the plane, since there is no asymmetry that would push the object to one side or the other of that plane. Looking at the component of 1, we can see that an equatorial path is a geodesic:

If , then all three terms in the second line are zero, so the geodesic equation is satisfied. Note that this last conclusion doesn’t *prove* that all geodesics lie in an equatorial plane; it merely verifies that a planar path *is* a solution. We rely on the symmetry argument to state that all geodesics must lie in an equatorial plane. Of course, in a different metric where the mass is not spherically symmetric, non-planar paths are possible.

Finally, we consider the component of 1. Since all four components of the metric depend on , this takes a bit more work. We can start with the universal relation for the four-velocity , which in this metric is

From above, we know that, since

Therefore

Since the only dependent variable in this equation is ( and are constants), this is a first-order ODE for in terms of the proper time .

As a simple example of the use of this equation, suppose we start a particle at rest at infinity and let it fall in radially towards the mass. Since the motion is entirely radial, the angular momentum is . Also, since the particle starts off at rest (see above), so the equation becomes

where is a constant of integration. If the particle is falling inwards, then we would expect to increase as decreases, so we need to take the minus sign on the left. The proper time interval as measured by the falling object between two radii (say, and ) is

Pingback: Particles falling towards a mass | Physics tutorials

Pingback: Particle falling towards a mass: two types of velocity | Physics tutorials

Pingback: Vertical particle motion | Physics tutorials

Pingback: Particle orbiting a neutron star | Physics tutorials

Pingback: Circular orbits: relation between radius and angular momentum | Physics tutorials

Pingback: Orbit of a comet around a black hole | Physics tutorials

Pingback: Twin paradox with a black hole | Physics tutorials

Pingback: Perihelion shift in planetary orbits | Physics tutorials

Pingback: Perihelion shift – contribution of the time coordinate | Physics tutorials

Pingback: Perihelion shift: numerical solution | Physics tutorials

Pingback: Photon equations of motion | Physics tutorials

Pingback: Local flat coordinate systems; four-momentum of photons | Physics tutorials

Pingback: Apparent size of a black hole to a moving observer | Physics tutorials

Pingback: Circular orbit: appearance to a falling observer | Physics tutorials

Pingback: Local flat frame for a circular orbit | Physics tutorials

Pingback: Falling object observed near the event horizon | Physics pages

Pingback: Escape velocity near an event horizon | Physics pages

Pingback: Painlevé-Gullstrand (global rain) coordinates | Physics pages

Pingback: Black hole heat engine | Physics pages

Pingback: Black hole radiation: energy of emitted particles | Physics pages

Pingback: Circular orbits: Kepler’s law | Physics pages

Pingback: Schwarzschild metric with negative mass | Physics pages

Pingback: Schwarzschild metric with non-zero cosmological constant | Physics pages