- Photon sphere
A

**photon sphere**is aspherical region of space wheregravity is strong enough thatphotons of light are forced to travel in orbits. The formula to find the radius for a circular photon orbit is: r=3GM/C^{2}. Because of this equation photon spheres can only exist in the space surrounding an extremely compact object, such as ablack hole .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 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 further from the center of a black hole than the event horizon and

ergosphere . Within a photon sphere it is possible to imagine aphoton that starts at the back of your head and orbits around a black hole only then be seen by your eyes.For non-rotating black holes, the photon sphere is a sphere ofradius 3/2 "R"_{s}, where "R"_{s}denotes theSchwarzschild radius (the radius of the event horizon) - see below for a derivation of this result. No unaccelerated orbit with asemi-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 sphere that is closer to the black hole is moving in the same direction as the rotation, whereas the photon sphere further away is moving against it. The greater theangular velocity of the rotation of a black hole the greater distance between the two photon spheres. Because 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 this angle the possibility of traveling with or against the rotation does not exist.**Derivation of the Photon Sphere Radius**This derivation involves using the

Schwarzschild metric , given by:$ds^\{2\}\; =\; (1\; -\; frac\{2GM\}\{rc^\{2)c^\{2\}dt^\{2\}\; -\; (1\; -\; frac\{2GM\}\{rc^\{2)^\{-1\}dr^\{2\}\; -\; r^\{2\}(\; extrm\{sin\}^\{2\}\; heta\; dphi^\{2\}\; +\; d\; heta^\{2\})$

For a photon travelling at a constant radius r (ie. in the Φ-coordinate direction), ds, dr and dθ all must equal zero (the consequence of ds = 0 is a "light-like interval").

Setting ds, dr and dθ to zero, we have:

$(1\; -\; frac\{2GM\}\{rc^\{2)c^\{2\}dt^\{2\}\; =\; r^\{2\}\; extrm\{sin\}^\{2\}\; heta\; dphi^\{2\}$

Re-arranging gives:

$frac\{dphi\}\{dt\}\; =\; frac\{c\}\{r\; extrm\{sin\}\; heta\}sqrt\{1\; -\; frac\{R\_s\}\{r$

where "R

_{s}" is the Schwarzschild radiusNow, the speed of light in the Φ-coordinate direction is given by $rfrac\{dphi\}\{dt\}$, so we can write the speed of light along the orbital path as:

$v\; =\; frac\{c\}\{\; extrm\{sin\}\; heta\}sqrt\{1\; -\; frac\{R\_s\}\{r$

But, as a direct result of Newton's Law of Universal Gravitation for an object travelling in a circular orbit at a radius r about a central body, M, the speed of light may also be given by:

$v\; =\; sqrt\{frac\{GM\}\{r$

We can write this in terms of the Schwarzschild radius:

$v\; =\; csqrt\{frac\{R\_s\}\{2r$

Putting our two expressions for the speed of light, we have:

$csqrt\{frac\{R\_s\}\{2r\; =\; csqrt\{1\; -\; frac\{R\_s\}\{r$

where we have inserted $heta\; =\; frac\{pi\}\{2\}$ radians (imagine that the central mass, about which the photon is orbitting, is located at the centre of the coordinate axes. Then, as the photon is travelling along the Φ-coordinate line, for the mass to be located directly in the centre of the photon's orbit, we must have $heta\; =\; frac\{pi\}\{2\}$ radians).

Hence, rearranging this final expression gives:

$r\; =\; frac\{3\}\{2\}R\_s$

which is the result we set out to prove.

**External links*** [

*http://www.spacetimetravel.org/expeditionsl/expeditionsl.html Step by Step into a Black Hole*]

* [*http://antwrp.gsfc.nasa.gov/htmltest/rjn_bht.html Virtual Trips to Black Holes and Neutron Stars*]

* [*http://www.gothosenterprises.com/black_holes/rotating_black_holes.html Guide to Black Holes*]**References**General Relativity: An Introduction for Physicists

*Wikimedia Foundation.
2010.*

### Look at other dictionaries:

**Photon belt**— Infobox Paranormalterms Image Caption = A Hubble Space Telescope image of the Pleiades. Usage = Terminology Name = Photon Belt Origin = Paul Otto Hesse (1961) Der Jüngste Tag Short = Additional Names = Photon Ring, Manasic Ring, golden nebula… … Wikipedia**Photon mapping**— In computer graphics, photon mapping is a two pass global illumination algorithm developed by Henrik Wann Jensen that solves the rendering equation. Rays from the light source and rays from the camera are traced independently until some… … Wikipedia**Strömgren sphere**— In theoretical astrophysics, a Strömgren sphere is the sphere of ionized hydrogen (H II) around a young star of the spectral classes O or B. Its counterpart in the real word are the H II regions, a type of an emission nebula, the most prominent… … Wikipedia**Riemann sphere**— The Riemann sphere can be visualized as the complex number plane wrapped around a sphere (by some form of stereographic projection – details are given below). In mathematics, the Riemann sphere (or extended complex plane), named after the 19th… … Wikipedia**Black hole**— For other uses, see Black hole (disambiguation). Simulated view of a black hole (center) in front of the Large Magellanic Cloud. Note the gravitat … Wikipedia**Properties and features of black holes**— According to the No Hair theorem a black hole has only three independent physical properties: mass, charge and angular momentum. [citation|last=Heusler |first=M. |year=1998 |title=Stationary Black Holes: Uniqueness and Beyond |journal=Living Rev … Wikipedia**Kerr metric**— In general relativity, the Kerr metric (or Kerr vacuum) describes the geometry of spacetime around a rotating massive body. According to this metric, such rotating bodies should exhibit frame dragging, an unusual prediction of general relativity; … Wikipedia**Black hole electron**— In physics, there is a speculative notion that if there were a black hole with the same mass and charge as an electron, it would share many of the properties of the electron including the magnetic moment and Compton wavelength. Problems As a… … Wikipedia**Kepler problem in general relativity**— The Kepler problem in general relativity involves solving for the motion of two spherical bodies interacting with one another by gravitation, as described by the theory of general relativity.Typically, and in this article, one body is assumed to… … Wikipedia**Fate Testarossa**— Infobox animanga character name = Fate Testarossa Fate T. Harlaown series = Magical Girl Lyrical Nanoha caption = first = Nanoha Episode 4 + Nanoha The MOVIE 1st last = Nanoha StrikerS Episode 26 + Sound Stage 04 creator = voiced by = Nana Mizuki … Wikipedia