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← Back to Part II: Curvature

Chapter 11: Symmetries and Killing Vectors

Killing vectors generate symmetries of spacetime—isometries that leave the metric unchanged. Each Killing vector corresponds to a conserved quantity along geodesics, providing powerful tools for solving the geodesic equation.

Killing's Equation

A vector field ξ^μ is a Killing vector if the Lie derivative of the metric vanishes:

\( \mathcal{L}_\xi g_{\mu\nu} = 0 \)

Equivalently: \( \nabla_\mu \xi_\nu + \nabla_\nu \xi_\mu = 0 \)

This means the metric is unchanged when we "flow" along the direction of ξ^μ. The spacetime looks the same from any point along this flow.

Common Killing Vectors

Time Translation (Schwarzschild)

\( \xi^\mu = (1, 0, 0, 0) = \partial_t \)

Metric doesn't depend on t → Energy conservation

Axial Rotation (Schwarzschild)

\( \psi^\mu = (0, 0, 0, 1) = \partial_\phi \)

Metric doesn't depend on φ → Angular momentum conservation

Minkowski Space (10 Killing vectors)

4 translations + 3 rotations + 3 boosts = 10 generators of Poincaré group

Conserved Quantities

For a geodesic with tangent u^μ, each Killing vector gives a conserved quantity:

\( Q = \xi_\mu u^\mu = \text{constant along geodesic} \)

Energy

\( E = -\xi_\mu u^\mu = -g_{tt} \frac{dt}{d\tau} \)

From time translation symmetry

Angular Momentum

\( L = \psi_\mu u^\mu = g_{\phi\phi} \frac{d\phi}{d\tau} \)

From axial rotation symmetry

Killing Tensors

Some spacetimes have higher-rank Killing tensors K_μν satisfying:

\( \nabla_{(\rho} K_{\mu\nu)} = 0 \)

Carter Constant (Kerr Spacetime)

The Kerr metric has a non-trivial Killing tensor giving the Carter constant Q:

\( Q = K_{\mu\nu} u^\mu u^\nu = \text{constant} \)

This "hidden symmetry" makes geodesics in Kerr spacetime integrable.

Python: Killing Vector Analysis

Python

killing_vectors.py155 lines

#!/usr/bin/env python3
"""
╔════════════════════════════════════════════════════════════════════════════╗
║                   KILLING VECTOR ANALYSIS FOR SCHWARZSCHILD                ║
║                                                                            ║
║  Killing Equation: ∇_μ ξ_ν + ∇_ν ξ_μ = 0                                 ║
║                                                                            ║
║  For Schwarzschild metric in coordinates (t, r, θ, φ):                   ║
║    ds² = -(1-2M/r)dt² + (1-2M/r)⁻¹dr² + r²dθ² + r²sin²θ dφ²             ║
║                                                                            ║
║  Time Translation Killing Vector:  ξ = ∂_t → Energy Conservation          ║
║  Axial Rotation Killing Vector:    ψ = ∂_φ → Angular Momentum             ║
║                                                                            ║
║  Conserved Quantities: Q = ξ_μ u^μ = constant along geodesics             ║
╚════════════════════════════════════════════════════════════════════════════╝
"""
from sympy import symbols, sin, cos, diff, simplify, Matrix, Rational

# Coordinates
t, r, theta, phi, M = symbols('t r theta phi M', real=True, positive=True)
coords = [t, r, theta, phi]

def schwarzschild_metric():
    """Schwarzschild metric: ds² = -(1-2M/r)dt² + dr²/(1-2M/r) + r²dΩ²"""
    g = [[0]*4 for _ in range(4)]
    f = 1 - 2*M/r
    g[0][0] = -f
    g[1][1] = 1/f
    g[2][2] = r**2
    g[3][3] = r**2 * sin(theta)**2
    return Matrix(g)

def compute_christoffel(g, coords):
    """Compute Christoffel symbols: Γ^ρ_μν = (1/2)g^ρσ(∂_μg_σν + ∂_νg_σμ - ∂_σg_μν)"""
    n = len(coords)
    g_inv = g.inv()
    Gamma = [[[0]*n for _ in range(n)] for _ in range(n)]
    for rho in range(n):
        for mu in range(n):
            for nu in range(n):
                total = sum(g_inv[rho, sigma] * (
                    diff(g[sigma, nu], coords[mu]) +
                    diff(g[sigma, mu], coords[nu]) -
                    diff(g[mu, nu], coords[sigma])
                ) for sigma in range(n))
                Gamma[rho][mu][nu] = simplify(Rational(1,2) * total)
    return Gamma

def verify_killing(xi, g, Gamma, coords):
    """
    Verify Killing equation: ∇_μ ξ_ν + ∇_ν ξ_μ = 0
    where covariant derivative: ∇_μ ξ_ν = ∂_μ ξ_ν - Γ^ρ_μν ξ_ρ
    """
    n = len(coords)

# Lower index: ξ_μ = g_μν ξ^ν
    xi_lower = [sum(g[mu, nu] * xi[nu] for nu in range(n)) for mu in range(n)]

# Covariant derivative
    nabla_xi = [[0]*n for _ in range(n)]
    for mu in range(n):
        for nu in range(n):
            partial = diff(xi_lower[nu], coords[mu])
            connection = -sum(Gamma[rho][mu][nu] * xi_lower[rho] for rho in range(n))
            nabla_xi[mu][nu] = simplify(partial + connection)

# Check Killing equation: symmetrized covariant derivative = 0
    killing_check = [[simplify(nabla_xi[mu][nu] + nabla_xi[nu][mu])
                      for nu in range(n)] for mu in range(n)]

return Matrix(killing_check)

# Setup
g = schwarzschild_metric()
Gamma = compute_christoffel(g, coords)

print("╔════════════════════════════════════════════════════════════════╗")
print("║      KILLING VECTORS OF SCHWARZSCHILD SPACETIME               ║")
print("╚════════════════════════════════════════════════════════════════╝")
print()

# Time translation Killing vector ξ = ∂_t
xi_t = [1, 0, 0, 0]
print("1. Time Translation Killing Vector")
print("   ξ = ∂_t = (1, 0, 0, 0)")
print()
killing_t = verify_killing(xi_t, g, Gamma, coords)
print("   Killing equation ∇_(μ ξ_ν) = 0?")
if all(killing_t[i,j] == 0 for i in range(4) for j in range(4)):
    print("   ✓ YES - This is a Killing vector!")
    print()
    print("   Physical Interpretation:")
    print("   • Metric is independent of t → Time translation symmetry")
    print("   • Conserved quantity: E = -ξ_μ u^μ = -(1-2M/r) dt/dτ")
    print("   • Energy is conserved along geodesics")
else:
    print("   ✗ NO")
    print(killing_t)

print()
print("─" * 66)
print()

# Axial rotation Killing vector ψ = ∂_φ
xi_phi = [0, 0, 0, 1]
print("2. Axial Rotation Killing Vector")
print("   ψ = ∂_φ = (0, 0, 0, 1)")
print()
killing_phi = verify_killing(xi_phi, g, Gamma, coords)
print("   Killing equation ∇_(μ ψ_ν) = 0?")
if all(killing_phi[i,j] == 0 for i in range(4) for j in range(4)):
    print("   ✓ YES - This is a Killing vector!")
    print()
    print("   Physical Interpretation:")
    print("   • Metric is independent of φ → Rotational symmetry")
    print("   • Conserved quantity: L = ψ_μ u^μ = r²sin²θ dφ/dτ")
    print("   • Angular momentum is conserved along geodesics")
else:
    print("   ✗ NO")

print()
print("─" * 66)
print()

# Non-Killing vector test (radial direction)
xi_bad = [0, 1, 0, 0]
print("3. Radial Direction (Counter-example)")
print("   ∂_r = (0, 1, 0, 0)")
print()
killing_r = verify_killing(xi_bad, g, Gamma, coords)
print("   Killing equation ∇_(μ ξ_ν) = 0?")
if all(killing_r[i,j] == 0 for i in range(4) for j in range(4)):
    print("   ✓ YES")
else:
    print("   ✗ NO - Not a Killing vector!")
    print()
    print("   • Metric depends on r → No radial translation symmetry")
    print("   • Schwarzschild has only 2 Killing vectors (not 4)")

print()
print("╔════════════════════════════════════════════════════════════════╗")
print("║           CONSERVED QUANTITIES ALONG GEODESICS                 ║")
print("╚════════════════════════════════════════════════════════════════╝")
print()
print("For a particle with 4-velocity u^μ = (ṫ, ṙ, θ̇, φ̇):")
print()
print("Energy (from ξ = ∂_t):")
print("  E = -ξ_μ u^μ = -g_tt ṫ = (1 - 2M/r) dt/dτ")
print()
print("Angular Momentum (from ψ = ∂_φ):")
print("  L = ψ_μ u^μ = g_φφ φ̇ = r²sin²θ dφ/dτ")
print()
print("These conserved quantities reduce the geodesic equation from 4 coupled")
print("ODEs to 2, making the problem tractable!")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Fortran: Conserved Quantities

Fortran

conserved_quantities.f90150 lines

! conserved_quantities.f90
! ════════════════════════════════════════════════════════════════════════════
!                    CONSERVED QUANTITIES ALONG GEODESICS
!
!  Killing vectors generate conserved quantities:
!    Q = xi_mu u^mu = constant along geodesic
!
!  For Schwarzschild spacetime:
!    Energy:           E = -xi_mu u^mu = -(1-2M/r) dt/dtau
!    Angular Momentum: L = psi_mu u^mu = r^2 sin^2(theta) dphi/dtau
!
!  Verify these are constant for a circular orbit at r = 6M
! ════════════════════════════════════════════════════════════════════════════

program conserved_quantities
    implicit none
    integer, parameter :: dp = selected_real_kind(15)
    real(dp), parameter :: M = 1.0_dp
    real(dp), parameter :: pi = 4.0_dp * atan(1.0_dp)

real(dp) :: r0, E, L
    real(dp) :: r, theta, phi
    real(dp) :: u(4), g(4,4)
    real(dp) :: E_check, L_check, norm
    integer :: i

print '(A)', '╔════════════════════════════════════════════════════════════════╗'
    print '(A)', '║     CONSERVED QUANTITIES IN SCHWARZSCHILD SPACETIME            ║'
    print '(A)', '╚════════════════════════════════════════════════════════════════╝'
    print '(A)', ''

! Initial conditions for circular orbit at r = 6M
    r0 = 6.0_dp * M
    theta = pi / 2.0_dp  ! Equatorial plane
    phi = 0.0_dp

print '(A,F8.4,A)', 'Circular orbit at \(r = ', r0/M, 'M\)'
    print '(A)', ''

! For circular orbit at radius r0, the conserved quantities are:
    ! E = (1 - 2M/r) / sqrt(1 - 3M/r)
    ! L = sqrt(M*r) / sqrt(1 - 3M/r)
    E = (1.0_dp - 2.0_dp*M/r0) / sqrt(1.0_dp - 3.0_dp*M/r0)
    L = sqrt(M*r0) / sqrt(1.0_dp - 3.0_dp*M/r0)

print '(A)', 'Expected Conserved Quantities:'
    print '(A,F12.8)', '  Energy \(E\) = ', E
    print '(A,F12.8)', '  Angular momentum \(L\) = ', L
    print '(A)', ''

! Get metric at this point
    r = r0
    call get_metric(r, theta, g)

! Four-velocity components for circular orbit
    ! From the geodesic equation with dr/dtau = dtheta/dtau = 0:
    !   u^t = dt/dtau = E / (1 - 2M/r)
    !   u^r = dr/dtau = 0
    !   u^theta = dtheta/dtau = 0
    !   u^phi = dphi/dtau = L / r^2
    u(1) = E / (1.0_dp - 2.0_dp*M/r)  ! dt/dtau
    u(2) = 0.0_dp                       ! dr/dtau
    u(3) = 0.0_dp                       ! dtheta/dtau
    u(4) = L / r**2                     ! dphi/dtau

print '(A)', '════════════════════════════════════════════════════════════════'
    print '(A)', '  FOUR-VELOCITY COMPONENTS'
    print '(A)', '════════════════════════════════════════════════════════════════'
    print '(A,F14.8)', '  \(u^0 = dt/d\tau\)   = ', u(1)
    print '(A,F14.8)', '  \(u^1 = dr/d\tau\)   = ', u(2)
    print '(A,F14.8)', '  \(u^2 = d\theta/d\tau\)   = ', u(3)
    print '(A,F14.8)', '  \(u^3 = d\phi/d\tau\)   = ', u(4)
    print '(A)', ''

! Verify normalization: g_munu u^mu u^nu = -1 (timelike geodesic)
    norm = g(1,1)*u(1)**2 + g(2,2)*u(2)**2 + g(3,3)*u(3)**2 + g(4,4)*u(4)**2
    print '(A)', 'Normalization Check:'
    print '(A,F14.8)', '  \(g_{\mu\nu} u^\mu u^\nu\) = ', norm
    print '(A)', '  (Should be \(-1\) for timelike geodesic)'
    print '(A)', ''

! ════════════════════════════════════════════════════════════════
    ! Verify Energy Conservation
    ! ════════════════════════════════════════════════════════════════
    ! Time translation Killing vector: xi = partial_t = (1, 0, 0, 0)
    ! Lowered: xi_mu = g_munu xi^nu = (g_tt, 0, 0, 0)
    ! Conserved energy: E = -xi_mu u^mu = -g_tt u^t
    E_check = -(g(1,1) * u(1))

print '(A)', '════════════════════════════════════════════════════════════════'
    print '(A)', '  ENERGY CONSERVATION (from time translation symmetry)'
    print '(A)', '════════════════════════════════════════════════════════════════'
    print '(A)', '  Killing vector: \(\xi = \partial_t = (1, 0, 0, 0)\)'
    print '(A)', '  Conserved quantity: \(E = -\xi_\mu u^\mu = -\left(1-\frac{2M}{r}\right)\frac{dt}{d\tau}\)'
    print '(A)', ''
    print '(A,F14.8)', '  Computed \(E = -g_{tt} u^t\) = ', E_check
    print '(A,F14.8)', '  Expected \(E\)             = ', E
    print '(A,ES12.4)', '  |Difference|           = ', abs(E_check - E)
    print '(A)', ''

! ════════════════════════════════════════════════════════════════
    ! Verify Angular Momentum Conservation
    ! ════════════════════════════════════════════════════════════════
    ! Axial rotation Killing vector: psi = partial_phi = (0, 0, 0, 1)
    ! Lowered: psi_mu = g_munu psi^nu = (0, 0, 0, g_phiphi)
    ! Conserved angular momentum: L = psi_mu u^mu = g_phiphi u^phi
    L_check = g(4,4) * u(4)

print '(A)', '════════════════════════════════════════════════════════════════'
    print '(A)', '  ANGULAR MOMENTUM CONSERVATION (from rotational symmetry)'
    print '(A)', '════════════════════════════════════════════════════════════════'
    print '(A)', '  Killing vector: \(\psi = \partial_\phi = (0, 0, 0, 1)\)'
    print '(A)', '  Conserved quantity: \(L = \psi_\mu u^\mu = r^2\sin^2\theta\frac{d\phi}{d\tau}\)'
    print '(A)', ''
    print '(A,F14.8)', '  Computed \(L = g_{\phi\phi} u^\phi\) = ', L_check
    print '(A,F14.8)', '  Expected \(L\)            = ', L
    print '(A,ES12.4)', '  |Difference|          = ', abs(L_check - L)
    print '(A)', ''

print '(A)', '════════════════════════════════════════════════════════════════'
    print '(A)', '  SUMMARY'
    print '(A)', '════════════════════════════════════════════════════════════════'
    print '(A)', ''
    print '(A)', 'Each Killing vector generates a conserved quantity:'
    print '(A)', ''
    print '(A)', '  • \(\partial_t\) (time translation) → Energy \(E\)'
    print '(A)', '  • \(\partial_\phi\) (axial rotation) → Angular momentum \(L\)'
    print '(A)', ''
    print '(A)', 'These reduce the geodesic equation from 4 coupled ODEs to 2,'
    print '(A)', 'making orbital dynamics in Schwarzschild spacetime tractable!'
    print '(A)', ''

contains

subroutine get_metric(r, theta, g)
        ! Schwarzschild metric in (t, r, theta, phi) coordinates
        real(dp), intent(in) :: r, theta
        real(dp), intent(out) :: g(4,4)
        real(dp) :: f

f = 1.0_dp - 2.0_dp*M/r
        g = 0.0_dp
        g(1,1) = -f                      ! g_tt
        g(2,2) = 1.0_dp/f                ! g_rr
        g(3,3) = r**2                    ! g_thetatheta
        g(4,4) = r**2 * sin(theta)**2    ! g_phiphi
    end subroutine get_metric

end program conserved_quantities

Click Run to execute the Fortran code

Code will be compiled with gfortran and executed on the server

← Bianchi Identities Next: Part III - Einstein Field Equations →