Carter constant

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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 ${\displaystyle p_{t}}$, axial angular momentum ${\displaystyle p_{\phi }}$, and particle rest mass ${\displaystyle {\sqrt {|p_{\mu }p^{\mu }|}}}$ provide the four conserved quantities necessary to uniquely determine all orbits in the Kerr–Newman spacetime (even those of charged particles).

Formulation

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:

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

where ${\displaystyle p_{\theta }}$ is the latitudinal component of the particle's angular momentum, ${\displaystyle E=p_{t}}$ is the conserved energy of the particle, ${\displaystyle L_{z}=p_{\phi }}$ is the particle's conserved axial angular momentum, ${\displaystyle m={\sqrt {|p_{\mu }p^{\mu }|}}}$ is the rest mass of the particle, and ${\displaystyle a}$ is the spin parameter of the black hole which satisfies ${\displaystyle 0\leq a\leq M}$.^[2] Note that here ${\displaystyle p_{\mu }}$ denotes the covariant components of the four-momentum in Boyer-Lindquist coordinates which may be calculated from the particle's position ${\displaystyle X^{\mu }=(t,r,\theta ,\phi )}$ parameterized by the particle's proper time ${\displaystyle \tau }$ using its four-velocity ${\displaystyle U^{\mu }=dX^{\mu }/d\tau }$ as ${\displaystyle p_{\mu }=g_{\mu \nu }p^{\nu }}$ where ${\displaystyle p^{\mu }=mU^{\mu }}$ is the four-momentum and ${\displaystyle g_{\mu \nu }}$ is the Kerr metric. Thus, the conserved energy constant and angular momentum constant are not to be confused with the energy ${\displaystyle U_{\rm {obs}}^{\mu }p_{\mu }}$ measured by an observer and the angular momentum ${\displaystyle \mathbf {L} ={\boldsymbol {x}}\wedge {\boldsymbol {p}}=rp_{\theta }{\boldsymbol {dr}}\wedge {\boldsymbol {d\theta }}+rp_{\phi }{\boldsymbol {dr}}\wedge {\boldsymbol {d\phi }}=mrg_{\theta \theta }{\dot {\theta }}{\boldsymbol {dr}}\wedge {\boldsymbol {d\theta }}+mr(g_{t\phi }{\dot {t}}+g_{\phi \phi }{\dot {\phi }}){\boldsymbol {dr}}\wedge {\boldsymbol {d\phi }}}$. The angular momentum component along ${\displaystyle z}$ is ${\displaystyle L_{xy}}$ which coincides with ${\displaystyle p_{\phi }}$.

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

${\displaystyle K=C+(L_{z}-aE)^{2}}$

in place of ${\displaystyle C}$. The quantity ${\displaystyle K}$ 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 ${\displaystyle a=0}$ limit, ${\displaystyle C=L^{2}-L_{z}^{2}}$ and ${\displaystyle K=L^{2}}$, where ${\displaystyle L}$ is the norm of the angular momentum vector, see Schwarzschild limit below. Note that while ${\displaystyle C=L_{x}^{2}+L_{y}^{2}\geq 0}$ and ${\displaystyle K=L^{2}\geq 0}$ in the Schwarzschild case, ${\displaystyle C}$ may be either positive or negative in the general case with ${\displaystyle a>0}$. For example, purely radially infalling or outgoing timelike geodesics have ${\displaystyle L_{z}=p_{\theta }=0}$ and a strictly negative ${\displaystyle C}$.

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 ${\displaystyle K}$ (different ${\displaystyle K}$ than used above). In component form:

${\displaystyle C=K^{\mu \nu }u_{\mu }u_{\nu }}$,

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

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

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

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

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

with

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

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

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

Asymptotics

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

${\displaystyle {\frac {C}{m^{2}}}=g_{\theta \theta }{\dot {\theta }}^{2}+\cos ^{2}\theta {\Bigg (}a^{2}(1-(g_{tt}{\dot {t}}+g_{t\phi }{\dot {\phi }})^{2})+\left({\frac {g_{\phi t}{\dot {t}}+g_{\phi \phi }{\dot {\phi }}}{\sin \theta }}\right)^{2}{\Bigg )}}$

Asymptotically for large ${\displaystyle r\gg M}$, this tends to

${\displaystyle {\frac {C}{m^{2}}}=(r^{2}+a^{2}\cos ^{2}\theta )^{2}{\dot {\theta }}^{2}+a^{2}\cos ^{2}\theta \left[1-\left[\left(1-{\frac {2M}{r}}\right){\dot {t}}+{\frac {2Ma\sin ^{2}\theta }{r}}{\dot {\phi }}\right]^{2}\right]+\left[(r^{2}+a^{2})\,{\dot {\phi }}-{\frac {2Ma}{r}}{\dot {t}}\right]^{2}\sin ^{2}\theta \cos ^{2}\theta }$

${\displaystyle C=m^{2}\gamma ^{2}r^{2}(v_{\theta }^{2}+v_{\phi }^{2}\cos ^{2}\theta )-4m^{2}Ma\gamma ^{2}\cos ^{2}\theta \sin \theta \,v_{\phi }+m^{2}\gamma ^{2}a^{2}\cos ^{2}\theta \,(v_{\theta }^{2}+v_{\phi }^{2}-v_{r}^{2})}$

${\displaystyle \quad =L^{2}-L_{z}^{2}-4\gamma {\frac {mM}{r}}a\cos ^{2}\theta \,L_{z}+m^{2}\gamma ^{2}a^{2}\cos ^{2}\theta \,(v_{\theta }^{2}+v_{\phi }^{2}-v_{r}^{2})}$

where ${\displaystyle U^{\mu }=dx^{\mu }/d\tau }$, ${\displaystyle v^{i}=dx^{i}/dt}$, ${\displaystyle v_{r}=dr/dt}$, ${\displaystyle v_{\theta }=r\,d\theta /dt}$, ${\displaystyle v_{\phi }=r\sin \theta \,d\phi /dt}$, ${\displaystyle \gamma =dt/d\tau =(1-v^{2})^{-1/2}}$, ${\displaystyle v=(v_{i}v^{i})^{1/2}}$, ${\displaystyle \mathbf {L} =\gamma m\mathbf {x} \times \mathbf {v} }$, ${\displaystyle {L}=\gamma mr{\sqrt {v_{\theta }^{2}+v_{\phi }^{2}}}}$, ${\displaystyle L_{z}=\gamma mr\sin \theta \,v_{\phi }}$ valid asymptotically for ${\displaystyle r\gg 2M}$. Given that ${\displaystyle C}$ and ${\displaystyle L_{z}}$ are conserved this shows that ${\displaystyle L^{2}}$ is conserved only up to ${\displaystyle a/r}$ 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 ${\displaystyle a}$.

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 ${\displaystyle E}$, ${\displaystyle L_{z}}$, and ${\displaystyle m}$ to determine the motion; however, the symmetry leading to Carter's constant still exists. Carter's constant for Schwarzschild space is:

${\displaystyle C=p_{\theta }^{2}+L_{z}^{2}\cot ^{2}\theta }$.

To see how this is related to the angular momentum two-form ${\displaystyle L_{ij}=x_{i}\wedge p_{j}}$ in spherical coordinates where ${\displaystyle {\boldsymbol {x}}=r{\boldsymbol {dr}}}$ and ${\displaystyle {\boldsymbol {p}}=p_{r}{\boldsymbol {dr}}+p_{\theta }{\boldsymbol {d\theta }}+p_{\phi }{\boldsymbol {d\phi }}}$, where ${\displaystyle (p_{r},p_{\theta },p_{\phi })=(g_{rr}p^{r},g_{\theta \theta }p^{\theta },g_{\phi \phi }p^{\phi })=m((1-2M/r)^{-1}{\dot {r}},r^{2}{\dot {\theta }},r^{2}\sin ^{2}\theta \,{\dot {\phi }})}$, and where ${\displaystyle {\dot {\phi }}=d\phi /d\tau }$ and similarly for ${\displaystyle {\dot {\theta }}}$, we have

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

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

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

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

${\displaystyle L^{2}=g^{\theta \theta }r^{2}p_{\theta }^{2}+g^{\phi \phi }r^{2}p_{\phi }^{2}=g_{\theta \theta }r^{2}(p^{\theta })^{2}+g_{\phi \phi }r^{2}(p^{\phi })^{2}=m^{2}r^{4}{\dot {\theta }}^{2}+m^{2}r^{4}\sin ^{2}\theta \,{\dot {\phi }}^{2}}$.

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

${\displaystyle C=m^{2}r^{4}{\dot {\theta }}^{2}+m^{2}r^{4}\sin ^{2}\theta \cos ^{2}\theta \,{\dot {\phi }}^{2}=m^{2}r^{4}{\dot {\theta }}^{2}+m^{2}r^{4}\sin ^{2}\theta \,{\dot {\phi }}^{2}-m^{2}r^{4}\sin ^{4}\theta \,{\dot {\phi }}^{2}=L^{2}-L_{z}^{2}}$.

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

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

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

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

See also

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

References