Schwarzschild Geodesics - Orbits of Test Particles

Orbits of Test Particles

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

c^{2} = \left( 1 - \frac{r_{s}}{r} \right) c^{2} \left( \frac{dt}{d\tau} \right)^{2} - \frac{1}{1 - \frac{r_{s}}{r}} \left( \frac{dr}{d\tau} \right)^{2} - r^{2} \left( \frac{d\theta}{d\tau} \right)^{2} - r^{2} \sin^{2} \theta \, \left( \frac{d\varphi}{d\tau} \right)^{2},

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

\left( 1 - \frac{r_{s}}{r} \right) \left( \frac{dt}{d\tau} \right) = \frac{a}{b}
r^{2} \left( \frac{d\varphi}{d\tau} \right) = a c

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

\left( \frac{dr}{d\varphi} \right)^{2} = \frac{r^{4}}{b^{2}} - \left( 1 - \frac{r_{s}}{r} \right) \left( \frac{r^{4}}{a^{2}} + r^{2} \right)

Read more about this topic:  Schwarzschild Geodesics

Famous quotes containing the words orbits of, orbits, test and/or particles:

    “To me, however, the question of the times resolved itself into a practical question of the conduct of life. How shall I live? We are incompetent to solve the times. Our geometry cannot span the huge orbits of the prevailing ideas, behold their return, and reconcile their opposition. We can only obey our own polarity.”
    —Ralph Waldo Emerson (1803–1882)

    “To me, however, the question of the times resolved itself into a practical question of the conduct of life. How shall I live? We are incompetent to solve the times. Our geometry cannot span the huge orbits of the prevailing ideas, behold their return, and reconcile their opposition. We can only obey our own polarity.”
    —Ralph Waldo Emerson (1803–1882)

    “The face we see was never young,
    Nor could it ever have been old.

    For he, to whom we had applied
    Our shopman’s test of age and worth,
    Was elemental when he died,
    As he was ancient at his birth:”
    —Edwin Arlington Robinson (1869–1935)

    “The atoms of Democritus
    And Newton’s particles of light
    Are sands upon the Red Sea shore,
    Where Israel’s tents do shine so bright.”
    —William Blake (1757–1827)