Carter constant
Conserved quantity for black hole motion in general relativity.
The Carter constant is a conserved quantity in general relativity, describing motion around black holes. It has SI base units of kg²·m⁴·s⁻². Australian theoretical physicist Brandon Carter derived it in 1968 for a spinning, charged black hole. Together with the energy, axial angular momentum, and rest mass of a particle, the Carter constant forms a set of four conserved quantities that uniquely determine all orbits in Kerr–Newman spacetime, including those of charged particles.
Carter discovered that the Hamiltonian for motion in Kerr spacetime separates in Boyer–Lindquist coordinates, making the constants of motion easy to identify using Hamilton–Jacobi theory. The constant is expressed as \( C = p_\theta^2 + \cos^2\theta \left( a^2(m^2 - E^2) + \left( \frac{L_z}{\sin\theta} \right)^2 \right) \), with the convention \( G = c = 1 \). Here, \( p_\theta \) is the latitudinal component of the particle's angular momentum, \( E = p_t \) is the conserved energy per unit mass, \( L_z = p_\phi \) is the conserved axial angular momentum per unit mass, \( m = \sqrt{|p_\mu p^\mu|} \) is the rest mass, and \( a \) is the black hole's spin parameter, satisfying \( 0 \leq a \leq M \). The covariant components \( p_\mu \) of the four-momentum are given in Boyer-Lindquist coordinates, derived from the particle's position \( X^\mu = (t, r, \theta, \phi) \) and proper time \( \tau \) via the four-velocity \( U^\mu = dX^\mu / d\tau \), with \( p^\mu = mU^\mu \) and \( p_\mu = g_{\mu\nu} p^\nu \), where \( g_{\mu\nu} \) is the Kerr metric. The conserved energy and angular momentum constants are distinct from the observed energy \( U^\mu_{\text{obs}} p_\mu \).
- Si base units
- kg2⋅m4⋅s−2
- Derived by
- Brandon Carter
- Year derived
- 1968
- Field
- General relativity
- Nationality
- Australian
- Known for
- Carter constant
Lore & Background
Brandon 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. The Carter constant can be written as C = p_θ^2 + cos^2θ (a^2(m^2 - E^2) + (L_z / sinθ)^2), with the convention G = c = 1. Here p_θ is the latitudinal component of the particle's angular momentum, E is the conserved energy per unit mass, L_z is the conserved axial angular momentum per unit mass, m is the rest mass, and a is the spin parameter of the black hole (0 ≤ a ≤ M).
Reader's Guide
The Carter constant is significant because it provides a fourth conserved quantity for motion in the Kerr–Newman spacetime, which is otherwise not fully integrable with only energy, angular momentum, and rest mass. This allows the complete determination of all orbits, including those of charged particles. Because functions of conserved quantities are also conserved, any function of C and the three other constants can be used as a fourth constant, leading to 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 constant's SI base units are kg2⋅m4⋅s−2, and it was derived by Australian theoretical physicist Brandon Carter in 1968 for a spinning, charged black hole.
Did You Know?
- The Carter constant has SI base units of kg2⋅m4⋅s−2.
- The constant is one of four conserved quantities needed to uniquely determine all orbits in Kerr–Newman spacetime.
- Any function of the Carter constant and the other three constants can be used as a fourth constant in its place.
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