From Wikipedia, the free encyclopedia Photon sphere
Photon sphere
A photon sphere is a spherical region of space where
gravity is strong enough that photons are forced to travel
Derivation for a Schwarzschild
in orbits. The minimum radius for a stable orbit is: black hole
Since a Schwarzschild black hole has spherical symme-
try, all possible axes for a circular photon orbit are equiv-
The radius for the photon sphere is alent, and all circular orbits have the same radius.
This derivation involves using the Schwarzschild
metric, given by:
or in other words, one and a half times the Schwarz-
schild Radius. For a photon travelling at a constant radius r (i.e. in
This equation entails that photon spheres can only the Φ-coordinate direction), ds, dr and dθ all must equal
exist in the space surrounding an extremely compact ob- zero (the consequence of ds = 0 is a "light-like interval").
ject, such as a black hole or a neutron star. Setting ds, dr and dθ to zero, we have:
As photons travel near the event horizon of a black
hole they can escape being pulled in by the gravity of
a black hole by traveling at a nearly vertical direction
known as an exit cone. A photon on the boundary of this Re-arranging gives:
cone will not completely escape the gravity of the black
hole. Instead it orbits the black hole. These orbits are not
stable.
The photon sphere is located farther from the center where Rs is the Schwarzschild radius.
of a black hole than the event horizon and ergosphere. The thing we shall know to proceed is the relation
Within a photon sphere it is possible to imagine a photon
that starts at the back of your head and orbits around a
black hole only then be seen by your eyes. For non-ro- . To find it we should use radial geodesic equation
tating black holes, the photon sphere is a sphere of ra-
dius 3/2 Rs, where Rs denotes the Schwarzschild radius
(the radius of the event horizon) - see below for a deriva-
Not null Γ-connection coefficients are
tion of this result. No unaccelerated orbit with a semi-
major axis less than this distance is possible, but within
the photon sphere, a constant acceleration will allow a
spacecraft or probe to hover above the event horizon.
A rotating black hole has two photon spheres. As a
black hole rotates, it drags space with it. The photon , where .
sphere that is closer to the black hole is moving in the We treat photon radial geodesic with constant r and
same direction as the rotation, whereas the photon θ, therefore
sphere further away is moving against it. The greater
the angular velocity of the rotation of a black hole the
.
greater distance between the two photon spheres. Be-
Putting it all into r-geodesic equation we obtain
cause the black hole has an axis of rotation this only
holds true if approaching the black hole in the direction
of the equator. If approaching at a different angle, such as
one from the poles of the black hole to the equator, there
is only one photon sphere. This is because approaching at Comparing it with obtained previously, we have:
this angle the possibility of traveling with or against the
rotation does not exist.
1
From Wikipedia, the free encyclopedia Photon sphere
exist in the equatorial plane, and there are two of them
where we have inserted radians (imagine (prograde and retrograde), with different radii. All other
that the central mass, about which the photon is orbit- constant-radius orbits have more complicated paths that
ting, is located at the centre of the coordinate axes. Then, oscillate in latitude about the equator.[1]
as the photon is travelling along the φ-coordinate line,
for the mass to be located directly in the centre of the References
• General Relativity: An Introduction for Physicists
photon’s orbit, we must have radians). [1] Teo, Edward (2003). "Spherical Photon Orbits
Hence, rearranging this final expression gives: Around a Kerr Black Hole". General Relativity and
Gravitation 35 (11): 1909–1926. doi:10.1023/
A:1026286607562. ISSN 0001-7701.
http://www.physics.nus.edu.sg/~phyteoe/kerr/
which is the result we set out to prove.
paper.pdf.
Spherical photon orbits around External links
a Kerr black hole • Step by Step into a Black Hole
In contrast to a Schwarzschild black hole, a Kerr (spin- • Virtual Trips to Black Holes and Neutron Stars
ning) black hole does not have spherical symmetry, but • Guide to Black Holes
only an axis of symmetry, which has profound conse- • Spherical Photon Orbits Around a Kerr Black Hole
quences for the photon orbits. A circular orbit can only
Retrieved from "http://en.wikipedia.org/w/index.php?title=Photon_sphere&oldid=464859654"
Categories:
• Black holes
• General relativity
This page was last modified on 8 December 2011 at 23:37. Text is available under the Creative Commons Attribution-
ShareAlike License; additional terms may apply. See Terms of use for details. Wikipedia® is a registered trademark of
the Wikimedia Foundation, Inc., a non-profit organization.Contact us
Privacy policy About Wikipedia Disclaimers
2