Peter Diener's WWW-exhibition

Description and visualization of particle trajectories near black holes

This WWW-exhibition will try to describe the strange kinds of motions particles can perform in the strongly relativistic regime near a black hole.

Table of contents

  1. Short introduction to general relativity
  2. Geodesics
  3. Black Holes
  4. The equations of motion
  5. The effective potential
  6. Visualization examples
  7. Examine the potential barrier
  8. Visualize your own geodesics

3. Black holes

In this section a few general properties of black holes will be describedgif. The coordinate system used are the Boyer-Lindquist coordinate system, but distance is measured in units of tex2html_wrap_inline162 , where tex2html_wrap_inline164 is the mass of the black hole and time is measured in units of tex2html_wrap_inline166 . The rotation of the black hole is characterized by the parameter tex2html_wrap_inline168 , where tex2html_wrap_inline170 is the angular momentum of the black hole. In these coordinates the metric for a rotating black hole is

  equation33

where

equation52

The Schwarzschild metric for non-rotating black holes can be obtained from equation (21) by setting tex2html_wrap_inline172 with the result

  equation63

In both cases the metric reduces to the galilean one, when tex2html_wrap_inline174 . In the metric in equation (21) it can be seen that tex2html_wrap_inline176 vanishes when tex2html_wrap_inline178 , i.e. when tex2html_wrap_inline180 , where

equation83

This is called the static limit. Outside this limit it is possible for physical observers to be at rest with respect to distant observers. Inside everything is dragged into rotation about the black hole. However as long as tex2html_wrap_inline182 it is possible to move outward towards larger radial coordinate. The horizon is located where tex2html_wrap_inline184 , i.e. at tex2html_wrap_inline186 , where

equation92

The region between tex2html_wrap_inline188 and tex2html_wrap_inline190 is called the ergosphere. In the non-rotating case there is no ergosphere since in that case tex2html_wrap_inline192 . For more information on the properties of black holes consult Novikov & Frolov (1989), Misner, Thorne & Wheeler (1973) and Thorne, Price & Macdonald (1986).

References

1
C.W. Misner, K.S. Thorne, and J. Wheeler. Gravitation. Freeman, San Francisco, 1973.

2
I.D. Novikov and V.P. Frolov. Physics of Black Holes. Kluwer Academic Publishers, Dordrecht, 1989.

3
K.T. Thorne, R.H. Price, and D.A. Macdonald. Black Holes: The Membrane Paradigm. Yale University, New Haven, 1986.

...described
Only non-charged black holes will be considered here