en>99of9: explain "close packed"

2013-03-25T04:52:18Z

explain "close packed"

New page

The '''Carter constant''' is a [[conserved quantity]] for motion around [[black hole]]s in the [[general relativity|general relativistic]] formulation of gravity. Carter's constant was derived for a spinning, charged black hole by [[Australian]] [[theoretical physicist]] [[Brandon Carter]] in 1968. Carter's constant along with the energy, axial angular momentum, and particle rest mass provide the four conserved quantities necessary to uniquely determine all orbits in the [[Kerr-Newman metric|Kerr-Newman]] spacetime (even those of charged particles).

==Formulation==

Carter noticed that the Hamiltonian for motion in Kerr spacetime was separable in [[Boyer-Lindquist coordinates]], allowing the constants of such motion to be easily identified using [[Hamilton–Jacobi equation|Hamilton-Jacobi theory]].<ref name="carter_1968">{{cite journal | last = Carter | first = Brandon | authorlink = Brandon Carter | year = 1968 | title = Global structure of the Kerr family of gravitational fields | journal = Physical Review | volume = 174 | issue = 5 | pages = 1559–1571|bibcode = 1968PhRv..174.1559C |doi = 10.1103/PhysRev.174.1559 | url=http://adsabs.harvard.edu/abs/1968PhRv..174.1559C}}</ref> The Carter constant can be written as follows:

::<math>C = p_{\theta}^{2} + \cos^{2}\theta \Bigg( a^{2}(m^{2} - E^{2}) + \left(\frac{L}{\sin\theta} \right)^{2} \Bigg)</math>,

where <math>p_{\theta}</math> is the latitudinal component of the particle's momentum, <math>E</math> is the energy of the particle, <math>L</math> is the particle's axial angular momentum, <math>m</math> is the rest mass of the particle, and <math>a</math> is the spin parameter of the black hole.<ref name="MTW_1973">{{cite book | last1 = Misner | first1 = Charles W. | authorlink1 = Charles Misner | last2 = Thorone | first2 = Kip S. | authorlink2 = Kip Thorne | last3 = Wheeler | first3 = John Archibald | authorlink3 = John Archibald Wheeler | year = 1973 | title = Gravitation | publisher = W. H. Freeman and Co. | location = New York | isbn = 0-7167-0334-3 | page = 899}}</ref> Because functions of conserved quantities are also conserved, any function of <math>C</math> and the three other constants of the motion can be used as a fourth constant in place of <math>C</math>. This results in some confusion as to the form of Carter's constant. For example it is sometimes more convenient to use:

::<math>K = C + (L - a E)^{2}</math>

in place of <math>C</math>. The quantity <math>K</math> is useful because it is always non-negative. In general any fourth conserved quantity for motion in the [[Kerr metric|Kerr]] family of spacetimes may be referred to as "Carter's constant".

== As generated by a Killing tensor ==

[[Noether's theorem]] states that all conserved quantities are related to [[spacetime symmetries]]. Carter's constant is related to a higher order symmetry of the Kerr metric generated by a second order [[Killing_vector#Generalizations|Killing tensor field]] <math>K</math> (different <math>K</math> than used above). In component form:

::<math> C = K^{\mu\nu}u_{\mu}u_{\nu} </math>,

where <math>u</math> is the [[four-velocity]] of the particle in motion. The components of the Killing tensor in [[Boyer-Lindquist coordinates]] are:

::<math>K^{\mu\nu}=2\Sigma\ l^{(\mu}n^{\nu)} + r^2 g^{\mu\nu}</math>,

where <math>g^{\mu\nu}</math> are the components of the metric tensor and <math>l^\mu</math> and <math>n^\nu</math> are the components of the principal null vectors:

:<math>l^\mu = \left(\frac{r^2 + a^2}{\Delta},1,0,\frac{a}{\Delta}\right)</math>
:<math>n^\nu = \left(\frac{r^2 + a^2}{2\Sigma},-\frac{\Delta}{2\Sigma},0,\frac{a}{2\Sigma}\right)</math>.

== Schwarzschild limit ==

The spherical symmetry of the [[Schwarzschild metric]] for non-spinning black holes allows one to reduce the problem of finding the trajectories of particles to three dimensions. In this case one only needs <math>E</math>, <math>L</math>, and <math>m</math> to determine the motion; however, the symmetry leading to Carter's constant still exists. Carter's constant for Schwarzschild space is:

::<math>C = p_{\theta}^{2} + \left(\frac{L}{\sin\theta}\right)^{2}</math>.

By a rotation of coordinates we can put any orbit in the <math>\theta=\pi/2</math> plane so <math>p_{\theta}=0</math>. In this case <math>C = L^2</math>, the square of the orbital angular momentum.

==See also==

* [[Kerr metric]]
* [[Kerr-Newman metric]]
* [[Boyer-Lindquist coordinates]]
* [[Hamilton–Jacobi equation]]

==References==
{{reflist|1}}

[[Category:Black holes]]
[[Category:Conservation laws]]

Laves phase - Revision history

en>99of9: explain "close packed"