Carter constant

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 theory.{{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}} The Carter constant can be written as follows:

:$C\; =\; p\_\{\backslash theta\}^\{2\}\; +\; \backslash cos^\{2\}\backslash theta\; \backslash Bigg(\; a^\{2\}(m^\{2\}\; -\; E^\{2\})\; +\; \backslash left(\backslash frac\{L\_z\}\{\backslash sin\backslash theta\}\; \backslash right)^\{2\}\; \backslash Bigg)$,

where $p\_\{\backslash theta\}$ is the latitudinal component of the particle's angular momentum, $E=p\_t$ is the conserved energy of the particle, $L\_z=p\_\{\backslash phi\}$ is the particle's conserved axial angular momentum, $m=\backslash sqrt$

p_{\mu}p^{\mu}

is the rest mass of the particle, and $a$ is the spin parameter of the black hole which satisfies $0\backslash leq\; a\backslash leq\; M$.{{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}} Note that here $p\_\{\backslash mu\}$ denotes the covariant components of the four-momentum in Boyer-Lindquist coordinates which may be calculated from the particle's position $X^\{\backslash mu\}=(t,r,\backslash theta,\backslash phi)$ parameterized by the particle's proper time $\backslash tau$ using its four-velocity $U^\{\backslash mu\}=dX^\{\backslash mu\}/d\backslash tau$ as $p\_\{\backslash mu\}=g\_\{\backslash mu\backslash nu\}\; p^\{\backslash nu\}$ where $p^\{\backslash mu\}=m\; U^\{\backslash mu\}$ is the four-momentum and $g\_\{\backslash mu\backslash nu\}$ is the Kerr metric. Thus, the conserved energy constant and angular momentum constant are not to be confused with the energy $U\_\{\backslash rm\; obs\}^\{\backslash mu\}p\_\{\backslash mu\}$ measured by an observer and the angular momentum

$\backslash mathbf\{L\}=\backslash boldsymbol\{x\}\backslash wedge\; \backslash boldsymbol\{p\}\; =\; r\; p\_\{\backslash theta\}\backslash boldsymbol\{dr\}\backslash wedge\backslash boldsymbol\{d\; \backslash theta\}\; +\; r\; p\_\{\backslash phi\}\backslash boldsymbol\{dr\}\backslash wedge\backslash boldsymbol\{d\backslash phi\}$

= 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}. The angular momentum component along $z$ is $L\_\{xy\}$ which coincides with $p\_\backslash phi$.

Because functions of conserved quantities are also conserved, any function of $C$ and the three other constants of the motion can be used as a fourth constant in place of $C$. This results in some confusion as to the form of Carter's constant. For example, it is sometimes more convenient to use:

:$K\; =\; C\; +\; (L\_z\; -\; a\; E)^\{2\}$

in place of $C$. The quantity $K$ 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 $a=0$ limit, $C=L^2-L\_z^2$ and $K=L^2$, where $L$ is the norm of the angular momentum vector, see Schwarzschild limit below. Note that while $C=L\_x^2\; +\; L\_y^2\backslash geq\; 0$ and $K=L^2\backslash geq\; 0$ in the Schwarzschild case, $C$ may be either positive or negative in the general case with $a>0$. For example, purely radially infalling or outgoing timelike geodesics have $L\_z=p\_\backslash theta=0$ and a strictly negative $C$.

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 $K$ (different $K$ than used above). In component form:

:$C\; =\; K^\{\backslash mu\backslash nu\}u\_\{\backslash mu\}u\_\{\backslash nu\}$,

where $u$ is the four-velocity of the particle in motion. The components of the Killing tensor in Boyer–Lindquist coordinates are:

:$K^\{\backslash mu\backslash nu\}=2\backslash Sigma\backslash \; l^\{(\backslash mu\}n^\{\backslash nu)\}\; +\; r^2\; g^\{\backslash mu\backslash nu\}$,

where $g^\{\backslash mu\backslash nu\}$ are the components of the metric tensor and $l^\backslash mu$ and $n^\backslash nu$ are the components of the principal null vectors:

:$l^\backslash mu\; =\; \backslash left(\backslash frac\{r^2\; +\; a^2\}\{\backslash Delta\},1,0,\backslash frac\{a\}\{\backslash Delta\}\backslash right)$

:$n^\backslash nu\; =\; \backslash left(\backslash frac\{r^2\; +\; a^2\}\{2\backslash Sigma\},-\backslash frac\{\backslash Delta\}\{2\backslash Sigma\},0,\backslash frac\{a\}\{2\backslash Sigma\}\backslash right)$

with

: $\backslash Sigma\; =\; r^2\; +\; a^2\; \backslash cos^2\; \backslash theta\; \backslash \; ,\; \backslash \; \backslash \; \backslash Delta\; =\; r^2\; -\; r\_\{s\}\; \backslash \; r\; +\; a^2$.

The parentheses in $l^\{(\backslash mu\}n^\{\backslash nu)\}$ are notation for symmetrization:

:$l^\{(\backslash mu\}n^\{\backslash nu)\}\; =\; \backslash frac\{1\}\{2\}(l^\{\backslash mu\}n^\{\backslash nu\}\; +\; l^\{\backslash nu\}n^\{\backslash mu\})$

Carter constant per unit mass squared may be expressed with the four-velocity as

:$\backslash frac\{C\}\{m^2\}\; =\; g\_\{\backslash theta\backslash theta\}\backslash dot\{\backslash theta\}^\{2\}\; +\; \backslash cos^\{2\}\backslash theta\; \backslash Bigg(\; a^\{2\}(1\; -\; (g\_\{tt\}\; \backslash dot\{t\}\; +\; g\_\{t\backslash phi\}\backslash dot\{\backslash phi\})^2)\; +\; \backslash left(\backslash frac\{g\_\{\backslash phi\; t\}\; \backslash dot\{t\}\; +\; g\_\{\backslash phi\backslash phi\}\backslash dot\{\backslash phi\}\}\{\backslash sin\backslash theta\}\; \backslash right)^\{2\}\; \backslash Bigg)$

Asymptotically for large $r\; \backslash gg\; M$, this tends to

:$\backslash frac\{C\}\{m^2\}\; =$

(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

:$C\; =\; m^2\backslash gamma^2\; r^2\; (v\_\backslash theta^2+v\_\backslash phi^2\backslash cos^2\backslash theta)\; -\; 4m^2Ma\; \backslash gamma^2\; \backslash cos^2\backslash theta\backslash sin\backslash theta\backslash ,v\_\backslash phi\; +\; m^2\; \backslash gamma^2\; a^2\; \backslash cos^2\backslash theta\backslash ,\; (v\_\backslash theta^2+v\_\backslash phi^2\; -\; v\_r^2)$

:$\backslash quad=L^2-L\_z^2\; -\; 4\backslash gamma\backslash frac\{mM\}\{r\}a\; \backslash cos^2\backslash theta\backslash ,L\_z\; +\; m^2\; \backslash gamma^2\; a^2\; \backslash cos^2\backslash theta\backslash ,\; (v\_\backslash theta^2+v\_\backslash phi^2\; -\; v\_r^2)$

where $U^\backslash mu=dx^\{\backslash mu\}/d\backslash tau$, $v^i=dx^i/dt$, $v\_\{r\}=dr/dt$, $v\_\{\backslash theta\}=r\backslash ,d\backslash theta/dt$, $v\_\{\backslash phi\}=r\backslash sin\backslash theta\backslash ,d\backslash phi/dt$, $\backslash gamma=dt/d\backslash tau=(1-v^2)^\{-1/2\}$, $v=(v\_i\; v^i)^\{1/2\}$, $\backslash mathbf\{L\}=\backslash gamma\; m\; \backslash mathbf\{x\}\; \backslash times\; \backslash mathbf\{v\}$, $\{L\}=\backslash gamma\; m\; r\; \backslash sqrt\{v\_\backslash theta^2+v\_\{\backslash phi\}^2\}$, $L\_z=\backslash gamma\; m\; r\backslash sin\; \backslash theta\; \backslash ,v\_\{\backslash phi\}$ valid asymptotically for $r\backslash gg\; 2M$. Given that $C$ and $L\_z$ are conserved this shows that $L^2$ is conserved only up to $a/r$ corrections. This is similar to the behavior of the angular momentum for a particle moving in the gravitational potential of an exteded body of size $a$.

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 $E$, $L\_z$, and $m$ to determine the motion; however, the symmetry leading to Carter's constant still exists. Carter's constant for Schwarzschild space is:

:$C\; =\; p\_\{\backslash theta\}^\{2\}\; +\; L\_z^2\; \backslash cot^2\backslash theta$.

To see how this is related to the angular momentum two-form $L\_\{ij\}=x\_i\; \backslash wedge\; p\_j$ in spherical coordinates where $\backslash boldsymbol\{x\}=r\backslash boldsymbol\{dr\}$ and $\backslash boldsymbol\{p\}=p\_r\backslash boldsymbol\{dr\}+p\_\backslash theta\backslash boldsymbol\{d\backslash theta\}+p\_\backslash phi\backslash boldsymbol\{d\backslash phi\}$, where $(p\_\{r\},\; p\_\{\backslash theta\},\; p\_\{\backslash phi\})\; =\; (g\_\{rr\}p^\{r\},\; g\_\{\backslash theta\backslash theta\}p^\{\backslash theta\},\; g\_\{\backslash phi\backslash phi\}p^\{\backslash phi\})\; =\; m(\; (1-2M/r)^\{-1\}\; \backslash dot\{r\},\; r^2\; \backslash dot\{\backslash theta\},\; r^2\; \backslash sin^2\backslash theta\backslash ,\; \backslash dot\{\backslash phi\})$, and where $\backslash dot\backslash phi\; =\; d\backslash phi/d\backslash tau$ and similarly for $\backslash dot\backslash theta$, we have

:$\backslash mathbf\{L\}=\backslash boldsymbol\{x\}\backslash wedge\; \backslash boldsymbol\{p\}\; =\; r\; p\_\{\backslash theta\}\backslash boldsymbol\{dr\}\backslash wedge\backslash boldsymbol\{d\; \backslash theta\}\; +\; r\; p\_\{\backslash phi\}\backslash boldsymbol\{dr\}\backslash wedge\backslash boldsymbol\{d\backslash phi\}$

= m r^3 \dot{\theta}\boldsymbol{dr}\wedge\boldsymbol{d \theta} + mr^3 \sin^2\theta\, \dot\phi\,\boldsymbol{dr}\wedge\boldsymbol{d\phi}.

Since $\backslash boldsymbol\{\backslash hat\{\backslash theta\}\}=r\backslash boldsymbol\{d\; \backslash theta\}$ and $\backslash boldsymbol\{\backslash hat\{\backslash phi\}\}=r\backslash sin\backslash theta\backslash ,\backslash boldsymbol\{d\; \backslash phi\}$ represent an orthonormal basis, the Hodge dual of $\backslash mathbf\{L\}$ is the one form

:$\backslash boldsymbol\{L^*\}\; =\; m\; r^2\; \backslash dot\{\backslash theta\}\; \backslash hat\{\backslash boldsymbol\{\backslash phi\}\}\; +\; m\; r^2\; \backslash sin\backslash theta\backslash ,\; \backslash dot\{\backslash phi\}\backslash ,\; \backslash hat\{\backslash boldsymbol\{\backslash theta\}\}$

consistent with $\backslash vec\{\backslash boldsymbol\{r\}\}\backslash times\; m\backslash vec\{\backslash boldsymbol\{v\}\}$ although here $\backslash dot\{\backslash theta\}$ and $\backslash dot\{\backslash phi\}$ are with respect to proper time. Its norm is

:$L^2\; =\; g^\{\backslash theta\backslash theta\}\; r^2\; p\_\{\backslash theta\}^2\; +\; g^\{\backslash phi\backslash phi\}\; r^2\; p\_\{\backslash phi\}^2$

= 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 .

Further since $p\_\{\backslash theta\}\; =\; g\_\{\backslash theta\backslash theta\}p^\{\backslash theta\}\; =\; m\; r^2\; \backslash dot\backslash theta$ and $L\_z\; =\; p\_\{\backslash phi\}\; =\; g\_\{\backslash phi\backslash phi\}p^\{\backslash phi\}\; =\; m\; r^2\; \backslash sin^2\backslash theta\backslash ,\; \backslash dot\backslash phi$, upon substitution we get

:$C\; =\; m^2r^4\; \backslash dot\backslash theta^2\; +\; m^2\; r^4\; \backslash sin^2\backslash theta\; \backslash cos^2\backslash theta\; \backslash ,\backslash dot\backslash phi^2\; =\; m^2r^4\; \backslash dot\backslash theta^2\; +\; m^2\; r^4\; \backslash sin^2\backslash theta\backslash ,\backslash dot\backslash phi^2\; -m^2\; r^4\; \backslash sin^4\backslash theta\backslash ,\backslash dot\backslash phi^2\; =\; L^2\; -\; L\_z^2$.

In the Schwarzschild case, all components of the angular momentum vector are conserved, so both

$L^2$ and $L\_z^2$ are conserved, hence $C$ is clearly conserved. For Kerr, $L\_z=p\_\{\backslash phi\}$ is conserved but $p\_\{\backslash theta\}$ and $L^2$ are not, nevertheless $C$ is conserved.

The other form of Carter's constant is the always non-negative conserved quantity

:$K\; =\; C\; +\; (L\_z\; -\; a\; E)^\{2\}\; =\; (L^2\; -\; L\_z^2)\; +\; (L\_z\; -\; a\; E)^\{2\}\; =\; L^2$

since here $a=0$. This is also clearly conserved. In the Schwarzschild case both $C\backslash geq\; 0$ and $K\backslash geq\; 0$, where $K=\; 0$ are radial orbits and $C=0$ with $K>\; 0$ corresponds to orbits confined to the equatorial plane of the coordinate system, i.e. $\backslash theta=\backslash pi/2$ for all times.

{{Portal|Mathematics|Physics}}

• Kerr metric
• Kerr–Newman metric
• Boyer–Lindquist coordinates
• Hamilton–Jacobi equation
• Euler's three-body problem

{{reflist|1}}

Category:Black holes

Category:Conservation laws