Carter constant

The Carter constant is a conserved quantity for motion around black holes in the 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 Australian theoretical physicist Brandon Carter in 1968. Carter's constant along with the energy $p_{t}$ , axial angular momentum $p_{\phi }$ , and particle rest mass ${\sqrt {|p_{\mu }p^{\mu }|}}$ provide the four conserved quantities necessary to uniquely determine all orbits in the Kerr–Newman spacetime (even those of charged particles).

Formulation[edit]

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.^[1] The Carter constant can be written as follows:

C=p_{\theta }^{2}+\cos ^{2}\theta {\Bigg (}a^{2}(m^{2}-E^{2})+\left({\frac {L_{z}}{\sin \theta }}\right)^{2}{\Bigg )}

,

where $p_{\theta }$ is the latitudinal component of the particle's angular momentum, $E=p_{t}$ is the conserved energy of the particle, $L_{z}=p_{\phi }$ is the particle's conserved axial angular momentum, $m={\sqrt {|p_{\mu }p^{\mu }|}}$ is the rest mass of the particle, and $a$ is the spin parameter of the black hole.^[2] Note that here $p_{\mu }$ denotes the covariant components of the four-momentum in Boyer-Lindquist coordinates which may be calculated from the particle's position $X^{\mu }=(t,r,\theta ,\phi )$ parameterized by the particle's proper time $\tau$ using its four-velocity $U^{\mu }=dX^{\mu }/d\tau$ as $p_{\mu }=g_{\mu \nu }p^{\nu }$ where $p^{\mu }=mU^{\mu }$ is the four-momentum and $g_{\mu \nu }$ is the Kerr metric. Thus, the conserved energy constant and angular momentum constant are not to be confused with the energy $U_{\rm {obs}}^{\mu }p_{\mu }$ measured by an observer and the angular momentum $\mathbf {L} ={\boldsymbol {x}}\wedge {\boldsymbol {p}}=rp_{\theta }{\boldsymbol {dr}}\wedge {\boldsymbol {d\theta }}+rp_{\phi }{\boldsymbol {dr}}\wedge {\boldsymbol {d\phi }}=mr^{3}{\dot {\theta }}{\boldsymbol {dr}}\wedge {\boldsymbol {d\theta }}+mr^{3}\sin ^{2}\theta \,{\dot {\phi }}\,{\boldsymbol {dr}}\wedge {\boldsymbol {d\phi }}$ . The angular momentum component along $z$ is $L_{xy}$ which coincides with $p_{\phi }$ .

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}-aE)^{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.

As generated by a Killing tensor[edit]

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^{\mu \nu }u_{\mu }u_{\nu }

,

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

K^{\mu \nu }=2\Sigma \ l^{(\mu }n^{\nu )}+r^{2}g^{\mu \nu }

,

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

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

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

with

\Sigma =r^{2}+a^{2}\cos ^{2}\theta \ ,\ \ \Delta =r^{2}-r_{s}\ r+a^{2}

.

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

l^{(\mu }n^{\nu )}={\frac {1}{2}}(l^{\mu }n^{\nu }+l^{\nu }n^{\mu })

Schwarzschild limit[edit]

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_{\theta }^{2}+L_{z}^{2}\cot ^{2}\theta

.

To see how this is related to the angular momentum two-form $L_{ij}=x_{i}\wedge p_{j}$ in spherical coordinates where ${\boldsymbol {x}}=r{\boldsymbol {dr}}$ and ${\boldsymbol {p}}=p_{r}{\boldsymbol {dr}}+p_{\theta }{\boldsymbol {d\theta }}+p_{\phi }{\boldsymbol {d\phi }}$ , where $p_{\theta }=g_{\theta \theta }p^{\theta }=r^{2}m{\dot {\theta }}$ and $p_{\phi }=g_{\phi \phi }p^{\phi }=r^{2}\sin ^{2}\theta \,m{\dot {\phi }}$ and where ${\dot {\phi }}=d\phi /d\tau$ and similarly for ${\dot {\theta }}$ , we have

\mathbf {L} ={\boldsymbol {x}}\wedge {\boldsymbol {p}}=rp_{\theta }{\boldsymbol {dr}}\wedge {\boldsymbol {d\theta }}+rp_{\phi }{\boldsymbol {dr}}\wedge {\boldsymbol {d\phi }}=mr^{3}{\dot {\theta }}{\boldsymbol {dr}}\wedge {\boldsymbol {d\theta }}+mr^{3}\sin ^{2}\theta \,{\dot {\phi }}\,{\boldsymbol {dr}}\wedge {\boldsymbol {d\phi }}

.

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

{\boldsymbol {L^{*}}}=mr^{2}{\dot {\theta }}{\boldsymbol {\hat {\theta }}}+mr^{2}\sin \theta \,{\dot {\phi }}\,{\boldsymbol {\hat {\phi }}}

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

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}

.

Further since $p_{\theta }=g_{\theta \theta }p^{\theta }=mr^{2}{\dot {\theta }}$ and $L_{z}=p_{\phi }=g_{\phi \phi }p^{\phi }=mr^{2}\sin ^{2}\theta \,{\dot {\phi }}$ , upon substitution we get

C=m^{2}r^{4}{\dot {\theta }}^{2}+m^{2}r^{4}\sin ^{2}\theta \cos ^{2}\theta \,{\dot {\phi }}^{2}=m^{2}r^{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}

.

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, $L_{z}=p_{\phi }$ is conserved but $p_{\theta }$ and $L^{2}$ are not, nevertheless $C$ is conserved.

The other form of Carter's constant is

K=C+(L_{z}-aE)^{2}=(L^{2}-L_{z}^{2})+(L_{z}-aE)^{2}=L^{2}

since here $a=0$ . This is also clearly conserved. In the Schwarzschild case both $C\geq 0$ and $K\geq 0$ , 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. $\theta =\pi /2$ for all times.