Geodesics in Schwarzschild Space

The right-hand side of the graphic in this Demonstration shows a geodesic outside the Schwarzschild radius , while the left-hand side shows the effective radial potential. The latter depends on the value of the angular momentum , which you can vary with the first slider. The effective radial potential also depends on : a particle orbit has , a null geodesic (such as a light-ray) has , and a time-like geodesic has . You can specify the starting point of the integration for both energy and radius by clicking on an allowed value, on or above the curve on the left.
The two remaining controls, "maximum " and "zoom", specify the length of proper integration time and the zoom scale for the graphic on the right.



  • [Snapshot]
  • [Snapshot]
  • [Snapshot]
  • [Snapshot]


The right-hand pane in this Demonstration shows a geodesic outside the Schwarzschild radius , assuming a spherical mass distribution or point mass of total mass inside this radius. This would include the case of a black hole. On the left, we show the effective radial potential. As discussed in some detail in standard textbooks (e.g., [1]), this depends on the value of the angular momentum . It also depends on whether we are looking at a particle orbit (), a null geodesic (say, a light-ray, ) or a time-like geodesic ().
The snapshots show a couple of interesting cases. The first one shows a "plunge orbit", in which a particle falls into the singularity at the origin; the second shows a precessing bound-state orbit, notably different from the classical Keplerian case. The third shows that near the maximum of the potential, light evolves into an almost bound state orbit (there is an unstable bound state at the top). The final picture shows a time-like geodesic, which closely resembles the behavior of a light ray.
This work was influenced by David Saroff's Demonstration on black-hole orbits [2].
[1] J. B. Hartle, Gravity: An Introduction to Einstein's General Relativity, San Francisco, CA: Addison-Wesley, 2003.
[2] D. Saroff. "Orbits around Schwarzschild Black Holes" from the Wolfram Demonstrations Project—A Wolfram Web Resource. www.demonstrations.wolfram.com/OrbitsAroundSchwarzschildBlackHoles.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2014 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+