Schwarzschild Geodesics - Exact Solution Using Elliptic Functions

Exact Solution Using Elliptic Functions

The fundamental equation of the orbit is easier to solve if it is expressed in terms of the inverse radius u = 1/r

$\left( \frac{du}{d\varphi} \right)^{2} = \frac{1}{b^{2}} - \left( 1 - u r_{s} \right) \left( \frac{1}{a^{2}} + u^{2} \right)$

The right-hand side of this equation is a cubic polynomial, which has three roots, denoted here as u₁, u₂, and u₃

$\left( \frac{du}{d\varphi} \right)^{2} = r_{s} \left( u - u_{1} \right) \left( u - u_{2} \right) \left( u - u_{3} \right)$

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

$u_{1} + u_{2} + u_{3} = \frac{1}{r_{s}}$

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 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, 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 is

$u = u_{1} + \left( u_{2} - u_{1} \right) \, \mathrm{sn}^{2}\left( \frac{1}{2} \varphi \sqrt{r_{s} \left( u_{3} - u_{1} \right)} + \delta \right)$

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

$k = \sqrt{\frac{u_{2} - u_{1}}{u_{3} - u_{1}}}$

Read more about this topic: Schwarzschild Geodesics

Famous quotes containing the words exact, solution and/or functions:

“Danger lies in the writer becoming the victim of his own exaggeration, losing the exact notion of sincerity, and in the end coming to despise truth itself as something too cold, too blunt for his purpose—as, in fact, not good enough for his insistent emotion. From laughter and tears the descent is easy to snivelling and giggles.”
—Joseph Conrad (1857–1924)

“What is history? Its beginning is that of the centuries of systematic work devoted to the solution of the enigma of death, so that death itself may eventually be overcome. That is why people write symphonies, and why they discover mathematical infinity and electromagnetic waves.”
—Boris Pasternak (1890–1960)

“The English masses are lovable: they are kind, decent, tolerant, practical and not stupid. The tragedy is that there are too many of them, and that they are aimless, having outgrown the servile functions for which they were encouraged to multiply. One day these huge crowds will have to seize power because there will be nothing else for them to do, and yet they neither demand power nor are ready to make use of it; they will learn only to be bored in a new way.”
—Cyril Connolly (1903–1974)