Effective Potential of a Black Hole Metric

mass

M

1

angular momentum

L

0

charge

Q

0

cosmological constant

Λ

0

null or time paths

ϵ

1

plot range

10

This Demonstration shows the effective potential of a black hole of given mass and charge as a function of the radius. The variable

ϵ

gives null paths for the value 0 and time-like paths for the value 1. If the angular momentum is zero, then by setting

ϵ=2

, the effective potential is the negative of the time component of the diagonal metric tensor.

Details

The formula explored is

1

2

ϵ-

2

L

2

r

1-

2M

r

-

Λ

2

r

6

+

2

Q

2

r

.

The convention of the cosmological constant is chosen to be two times larger than conventional in most literature on the topic. The effective potential is used in the geodesic equation, which is derived from minimizing the action of the path length given some metric.

External Links

Schwarzschild Black Hole (ScienceWorld)

Orbits around Scwarzschild Black Holes

Schwarzschild Space-Time Embedding Diagram

Permanent Citation

Timofei Petoukhov

"Effective Potential of a Black Hole Metric"
http://demonstrations.wolfram.com/EffectivePotentialOfABlackHoleMetric/
Wolfram Demonstrations Project
Published: November 1, 2013