imported>Ira Leviton: exteded→extended - toolforge:typos

2025-06-21T18:02:30Z

New page

{{short description|Constant of motion in the Kerr-Newman spacetime}}
{{General relativity}}
The '''Carter constant''' is a [[conserved quantity]] for motion around [[black hole]]s in the [[general relativity|general relativistic]] formulation of gravity. Its [[SI base units]] are kg2⋅m4⋅s−2. Carter's constant was derived for a spinning, charged black hole by [[Australians|Australian]] [[theoretical physicist]] [[Brandon Carter]] in 1968. Carter's constant along with the [[energy]] <math>p_{t}</math>, axial [[angular momentum]] <math>p_{\phi}</math>, and particle [[rest mass]] <math>\sqrt{|p_{\mu}p^{\mu}|}</math> 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|doi = 10.1103/PhysRev.174.1559 | bibcode=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_z}{\sin\theta} \right)^{2} \Bigg)</math>,

where <math>p_{\theta}</math> is the latitudinal component of the particle's angular momentum, <math>E=p_t</math> is the conserved energy of the particle, <math>L_z=p_{\phi}</math> is the particle's conserved axial angular momentum, <math>m=\sqrt{|p_{\mu}p^{\mu}|}</math> is the rest mass of the particle, and <math>a</math> is the spin parameter of the black hole which satisfies <math>0\leq a\leq M</math>.<ref name="MTW_1973">{{cite book | last1 = Misner | first1 = Charles W. | authorlink1 = Charles Misner | last2 = Thorne | 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> Note that here <math>p_{\mu}</math> denotes the [[Covariance and contravariance of vectors|covariant]] components of the [[four-momentum]] in [[Boyer-Lindquist coordinates]] which may be calculated from the particle's position <math>X^{\mu}=(t,r,\theta,\phi)</math> parameterized by the particle's [[proper time]] <math>\tau</math> using its [[four-velocity]] <math>U^{\mu}=dX^{\mu}/d\tau</math> as <math>p_{\mu}=g_{\mu\nu} p^{\nu}</math> where <math>p^{\mu}=m U^{\mu}</math> is the [[four-momentum]] and <math>g_{\mu\nu}</math> is the [[Kerr metric]]. Thus, the conserved energy constant and angular momentum constant are not to be confused with the energy <math>U_{\rm obs}^{\mu}p_{\mu}</math> measured by an observer and the angular momentum
<math>\mathbf{L}=\boldsymbol{x}\wedge \boldsymbol{p} = r p_{\theta}\boldsymbol{dr}\wedge\boldsymbol{d \theta} + r p_{\phi}\boldsymbol{dr}\wedge\boldsymbol{d\phi}
= m r g_{\theta\theta} \dot{\theta}\boldsymbol{dr}\wedge\boldsymbol{d \theta} + mr (g_{t\phi}\dot{t}+g_{\phi\phi}\dot\phi)\boldsymbol{dr}\wedge\boldsymbol{d\phi}</math>. The angular momentum component along <math>z</math> is <math>L_{xy}</math> which coincides with <math>p_\phi</math>.

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_z - 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 family of spacetimes may be referred to as "Carter's constant". In the <math>a=0</math> limit, <math>C=L^2-L_z^2</math> and <math>K=L^2</math>, where <math>L</math> is the norm of the angular momentum vector, see [[Carter Constant#Schwarzschild limit|Schwarzschild limit]] below. Note that while <math>C=L_x^2 + L_y^2\geq 0</math> and <math>K=L^2\geq 0</math> in the Schwarzschild case, <math>C</math> may be either positive or negative in the general case with <math>a>0</math>. For example, purely radially infalling or outgoing timelike geodesics have <math>L_z=p_\theta=0</math> and a strictly negative <math>C</math>.

== As generated by a Killing tensor ==

[[Noether's theorem]] states that each conserved quantity of a system generates a [[continuous symmetry]] of that system. Carter's constant is related to a higher order symmetry of the Kerr metric generated by a second order [[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>

with

: <math>\Sigma = r^2 + a^2 \cos^2 \theta \ , \ \ \Delta = r^2 - r_{s} \ r + a^2</math>.

The parentheses in <math>l^{(\mu}n^{\nu)}</math> are notation for symmetrization:
:<math>l^{(\mu}n^{\nu)} = \frac{1}{2}(l^{\mu}n^{\nu} + l^{\nu}n^{\mu})</math>

== Asymptotics ==

Carter constant per unit mass squared may be expressed with the four-velocity as
:<math>\frac{C}{m^2} = g_{\theta\theta}\dot{\theta}^{2} + \cos^{2}\theta \Bigg( a^{2}(1 - (g_{tt} \dot{t} + g_{t\phi}\dot{\phi})^2) + \left(\frac{g_{\phi t} \dot{t} + g_{\phi\phi}\dot{\phi}}{\sin\theta} \right)^{2} \Bigg)</math>
Asymptotically for large <math>r \gg M</math>, this tends to
:<math>\frac{C}{m^2} =
(r^2+a^2\cos^2\theta)^2\dot{\theta}^{2} +
a^{2}\cos^{2}\theta\left[1 - \left[\left(1-\frac{2M}{r}\right)\dot{t} + \frac{2Ma\sin^2\theta}{r} \dot{\phi}\right]^2\right]
+ \left[(r^2+a^2)\,\dot{\phi} - \frac{2Ma}{r} \dot{t}\right]^2\sin^2\theta\cos^{2}\theta
</math>
:<math>C = m^2\gamma^2 r^2 (v_\theta^2+v_\phi^2\cos^2\theta) - 4m^2Ma \gamma^2 \cos^2\theta\sin\theta\,v_\phi + m^2 \gamma^2 a^2 \cos^2\theta\, (v_\theta^2+v_\phi^2 - v_r^2) </math>
:<math>\quad=L^2-L_z^2 - 4\gamma\frac{mM}{r}a \cos^2\theta\,L_z + m^2 \gamma^2 a^2 \cos^2\theta\, (v_\theta^2+v_\phi^2 - v_r^2)
</math>
where <math>U^\mu=dx^{\mu}/d\tau</math>, <math>v^i=dx^i/dt</math>, <math>v_{r}=dr/dt</math>, <math>v_{\theta}=r\,d\theta/dt</math>, <math>v_{\phi}=r\sin\theta\,d\phi/dt</math>, <math>\gamma=dt/d\tau=(1-v^2)^{-1/2}</math>, <math>v=(v_i v^i)^{1/2}</math>, <math>\mathbf{L}=\gamma m \mathbf{x} \times \mathbf{v}</math>, <math>{L}=\gamma m r \sqrt{v_\theta^2+v_{\phi}^2}</math>, <math>L_z=\gamma m r\sin \theta \,v_{\phi}</math> valid asymptotically for <math>r\gg 2M</math>. Given that <math>C</math> and <math>L_z</math> are conserved this shows that <math>L^2</math> is conserved only up to <math>a/r</math> corrections. This is similar to the behavior of the angular momentum for a particle moving in the gravitational potential of an extended body of size <math>a</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_z</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} + L_z^2 \cot^2\theta </math>.

To see how this is related to the [[relativistic angular momentum|angular momentum]] [[differential form|two-form]] <math>L_{ij}=x_i \wedge p_j</math> in [[spherical coordinates]] where <math>\boldsymbol{x}=r\boldsymbol{dr}</math> and <math>\boldsymbol{p}=p_r\boldsymbol{dr}+p_\theta\boldsymbol{d\theta}+p_\phi\boldsymbol{d\phi}</math>, where <math>(p_{r}, p_{\theta}, p_{\phi}) = (g_{rr}p^{r}, g_{\theta\theta}p^{\theta}, g_{\phi\phi}p^{\phi}) = m( (1-2M/r)^{-1} \dot{r}, r^2 \dot{\theta}, r^2 \sin^2\theta\, \dot{\phi})</math>, and where <math>\dot\phi = d\phi/d\tau</math> and similarly for <math>\dot\theta</math>, we have

:<math>\mathbf{L}=\boldsymbol{x}\wedge \boldsymbol{p} = r p_{\theta}\boldsymbol{dr}\wedge\boldsymbol{d \theta} + r p_{\phi}\boldsymbol{dr}\wedge\boldsymbol{d\phi}
= m r^3 \dot{\theta}\boldsymbol{dr}\wedge\boldsymbol{d \theta} + mr^3 \sin^2\theta\, \dot\phi\,\boldsymbol{dr}\wedge\boldsymbol{d\phi}</math>.

Since <math>\boldsymbol{\hat{\theta}}=r\boldsymbol{d \theta}</math> and <math>\boldsymbol{\hat{\phi}}=r\sin\theta\,\boldsymbol{d \phi}</math> represent an orthonormal basis, the [[Hodge dual]] of <math>\mathbf{L}</math> is the [[one form]]

:<math>\boldsymbol{L^*} = m r^2 \dot{\theta} \hat{\boldsymbol{\phi}} + m r^2 \sin\theta\, \dot{\phi}\, \hat{\boldsymbol{\theta}} </math>
consistent with <math>\vec{\boldsymbol{r}}\times m\vec{\boldsymbol{v}}</math> although here <math>\dot{\theta}</math> and <math>\dot{\phi}</math> are with respect to proper time. Its norm is
:<math>L^2 = g^{\theta\theta} r^2 p_{\theta}^2 + g^{\phi\phi} r^2 p_{\phi}^2
= g_{\theta\theta} r^2 (p^\theta)^2 + g_{\phi\phi} r^2 (p^\phi)^2
= m^2 r^4 \dot{\theta}^2 + m^2 r^4 \sin^2\theta\, \dot\phi^2 </math>.

Further since <math>p_{\theta} = g_{\theta\theta}p^{\theta} = m r^2 \dot\theta</math> and <math>L_z = p_{\phi} = g_{\phi\phi}p^{\phi} = m r^2 \sin^2\theta\, \dot\phi</math>, upon substitution we get

:<math>C = m^2r^4 \dot\theta^2 + m^2 r^4 \sin^2\theta \cos^2\theta \,\dot\phi^2 = m^2r^4 \dot\theta^2 + m^2 r^4 \sin^2\theta\,\dot\phi^2 -m^2 r^4 \sin^4\theta\,\dot\phi^2 = L^2 - L_z^2</math>.

In the Schwarzschild case, all components of the angular momentum vector are conserved, so both
<math>L^2</math> and <math>L_z^2</math> are conserved, hence <math>C </math> is clearly conserved. For Kerr, <math>L_z=p_{\phi}</math> is conserved but <math>p_{\theta}</math> and <math>L^2</math> are not, nevertheless <math>C</math> is conserved.

The other form of Carter's constant is the always non-negative conserved quantity
:<math> K = C + (L_z - a E)^{2} = (L^2 - L_z^2) + (L_z - a E)^{2} = L^2 </math>
since here <math>a=0</math>. This is also clearly conserved. In the Schwarzschild case both <math>C\geq 0</math> and <math>K\geq 0</math>, where <math>K= 0</math> are radial orbits and <math>C=0</math> with <math>K> 0</math> corresponds to orbits confined to the equatorial plane of the coordinate system, i.e. <math>\theta=\pi/2</math> for all times.

==See also==
{{Portal|Mathematics|Physics}}
* [[Kerr metric]]
* [[Kerr–Newman metric]]
* [[Boyer–Lindquist coordinates]]
* [[Hamilton–Jacobi equation]]
* [[Euler's three-body problem]]

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

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

Carter constant - Revision history

imported>Ira Leviton: exteded→extended - toolforge:typos