Are the courses on CoursesHub.World free?

Yes, all courses on CoursesHub.World are completely free and open access. We believe in democratizing education and making university-level science courses available to everyone worldwide.

What subjects are covered on CoursesHub.World?

CoursesHub.World offers courses in physics (Quantum Mechanics, General Relativity, QFT, Plasma Physics, Cosmology), biology (Molecular Biology, Cell Physiology, Pharmacology), and earth sciences (Oceanography, Atmospheric Science, Climatology).

What level are the courses?

Our courses are designed at the graduate and advanced undergraduate level. They include rigorous mathematical derivations and are suitable for physics students, researchers, and serious self-learners.

Where do the video lectures come from?

Our 400+ video lectures come from world-renowned sources including MIT OpenCourseWare, Stanford, lectures by Nobel laureates, and other leading universities and educators.

← Back to Part II: Curvature

Chapter 9: Weyl Tensor

The Weyl tensor is the trace-free part of the Riemann tensor. It describes the purely gravitational degrees of freedom—tidal forces and gravitational waves—independent of local matter content.

Definition

The Weyl tensor (conformal tensor) is defined as:

\( C_{\rho\sigma\mu\nu} = R_{\rho\sigma\mu\nu} - \frac{2}{n-2}\left(g_{\rho[\mu}R_{\nu]\sigma} - g_{\sigma[\mu}R_{\nu]\rho}\right) + \frac{2}{(n-1)(n-2)} R \, g_{\rho[\mu}g_{\nu]\sigma} \)

In 4 dimensions, this simplifies to:

\( C_{\rho\sigma\mu\nu} = R_{\rho\sigma\mu\nu} - \left(g_{\rho[\mu}R_{\nu]\sigma} - g_{\sigma[\mu}R_{\nu]\rho}\right) + \frac{1}{3} R \, g_{\rho[\mu}g_{\nu]\sigma} \)

Key Properties

Trace-Free

\( C^\rho_{\;\sigma\rho\nu} = 0 \) — All contractions vanish

Same Symmetries as Riemann

Antisymmetric in first pair, antisymmetric in second pair, symmetric under pair exchange

Conformally Invariant

Under \( \tilde{g}_{\mu\nu} = \Omega^2 g_{\mu\nu} \): \( \tilde{C}^\rho_{\;\sigma\mu\nu} = C^\rho_{\;\sigma\mu\nu} \)

Independent Components

10 components in 4D (same as Ricci tensor) — total 20 = 10 (Weyl) + 10 (Ricci)

Physical Interpretation

Gravitational Waves

In vacuum (R_μν = 0), all curvature is in Weyl tensor. Gravitational waves are propagating Weyl curvature.

Tidal Effects

The Weyl tensor describes shape distortion without volume change (pure shear).

Free Gravitational Field

Represents curvature not directly caused by local matter—propagating from distant sources.

Petrov Classification

Spacetimes classified by algebraic type of Weyl tensor (Type I, II, D, III, N, O).

Python: Weyl Tensor Computation

Python

weyl_tensor.py146 lines

#!/usr/bin/env python3
"""
╔══════════════════════════════════════════════════════════════════════════════╗
║                        Weyl Tensor Computation                               ║
║                     Schwarzschild Spacetime Example                          ║
╚══════════════════════════════════════════════════════════════════════════════╝

Weyl tensor in 4D:
  C_ρσμν = R_ρσμν - (g_ρ[μ R_ν]σ - g_σ[μ R_ν]ρ) + (1/3) R g_ρ[μ g_ν]σ

In vacuum (R_μν = 0):
  C_ρσμν = R_ρσμν

Weyl scalar (invariant):
  C_αβγδ C^αβγδ = 48M²/r⁶
"""
from sympy import symbols, sin, cos, diff, simplify, Matrix, Rational, sqrt

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

def schwarzschild_metric():
    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_riemann(g, coords):
    """Compute full Riemann tensor (abbreviated for space)"""
    g_inv = g.inv()
    n = len(coords)

# Christoffel symbols
    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 = 0
                for sigma in range(n):
                    total += g_inv[rho, sigma] * (
                        diff(g[sigma, nu], coords[mu]) +
                        diff(g[sigma, mu], coords[nu]) -
                        diff(g[mu, nu], coords[sigma])
                    )
                Gamma[rho][mu][nu] = simplify(Rational(1,2) * total)

# Riemann tensor
    R = [[[[0]*n for _ in range(n)] for _ in range(n)] for _ in range(n)]
    for rho in range(n):
        for sigma in range(n):
            for mu in range(n):
                for nu in range(n):
                    term1 = diff(Gamma[rho][nu][sigma], coords[mu])
                    term2 = -diff(Gamma[rho][mu][sigma], coords[nu])
                    term3 = sum(Gamma[rho][mu][lam] * Gamma[lam][nu][sigma] for lam in range(n))
                    term4 = -sum(Gamma[rho][nu][lam] * Gamma[lam][mu][sigma] for lam in range(n))
                    R[rho][sigma][mu][nu] = simplify(term1 + term2 + term3 + term4)

return R, Gamma

def lower_riemann(R, g):
    """Lower all indices: R_rho_sigma_mu_nu"""
    n = 4
    R_lower = [[[[0]*n for _ in range(n)] for _ in range(n)] for _ in range(n)]
    for rho in range(n):
        for sigma in range(n):
            for mu in range(n):
                for nu in range(n):
                    R_lower[rho][sigma][mu][nu] = simplify(
                        sum(g[rho, alpha] * R[alpha][sigma][mu][nu] for alpha in range(n))
                    )
    return R_lower

def compute_ricci(R):
    """Compute Ricci tensor by contraction"""
    n = 4
    Ricci = [[0]*n for _ in range(n)]
    for mu in range(n):
        for nu in range(n):
            Ricci[mu][nu] = simplify(sum(R[rho][mu][rho][nu] for rho in range(n)))
    return Matrix(Ricci)

def compute_weyl(R_lower, Ricci, R_scalar, g):
    """
    Compute Weyl tensor in 4D:
    C_rsmn = R_rsmn - (g_r[m R_n]s - g_s[m R_n]r) + (1/3) R g_r[m g_n]s
    """
    n = 4
    C = [[[[0]*n for _ in range(n)] for _ in range(n)] for _ in range(n)]

for rho in range(n):
        for sigma in range(n):
            for mu in range(n):
                for nu in range(n):
                    # R_rsmn term
                    term1 = R_lower[rho][sigma][mu][nu]

# -(g_rm R_ns - g_rn R_ms - g_sm R_nr + g_sn R_mr) / 2
                    term2 = -(g[rho, mu] * Ricci[nu, sigma] - g[rho, nu] * Ricci[mu, sigma]
                             - g[sigma, mu] * Ricci[nu, rho] + g[sigma, nu] * Ricci[mu, rho]) / 2

# +(1/6) R (g_rm g_ns - g_rn g_ms)
                    term3 = R_scalar * (g[rho, mu] * g[nu, sigma] - g[rho, nu] * g[mu, sigma]) / 6

C[rho][sigma][mu][nu] = simplify(term1 + term2 + term3)

return C

# Compute for Schwarzschild
print("Weyl Tensor for Schwarzschild Spacetime")
print("="*60)

g = schwarzschild_metric()
R, Gamma = compute_riemann(g, coords)
R_lower = lower_riemann(R, g)
Ricci = compute_ricci(R)
g_inv = g.inv()
R_scalar = simplify(sum(g_inv[i,j] * Ricci[i,j] for i in range(4) for j in range(4)))

print(f"\nRicci scalar R = {R_scalar}")
print("(Zero for vacuum solution)")

C = compute_weyl(R_lower, Ricci, R_scalar, g)

# In vacuum (R_mn = 0, R = 0), Weyl = Riemann
print("\nFor Schwarzschild (vacuum): C_rsmn = R_rsmn")
print("\nNon-zero Weyl tensor components:")

coord_names = ['t', 'r', 'θ', 'φ']
for rho in range(4):
    for sigma in range(rho+1, 4):
        for mu in range(4):
            for nu in range(mu+1, 4):
                val = C[rho][sigma][mu][nu]
                if val != 0:
                    print(f"C_{coord_names[rho]}{coord_names[sigma]}{coord_names[mu]}{coord_names[nu]} = {val}")

# Weyl scalar (invariant)
print("\n" + "="*60)
print("Weyl scalar C_abcd C^abcd = 48 M²/r⁶ (same as Kretschmann for vacuum)")
print("This is the invariant measure of tidal forces.")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Fortran: Weyl Scalar Calculation

Fortran

weyl_scalar.f9064 lines

! ╔══════════════════════════════════════════════════════════════════════════════╗
! ║                     Weyl Scalar Calculation                                  ║
! ║                 Kretschmann Scalar for Vacuum Spacetime                      ║
! ╚══════════════════════════════════════════════════════════════════════════════╝
!
! For Schwarzschild vacuum: K = C_αβγδ C^αβγδ = 48 M²/r⁶
!
! Weyl scalar measures tidal forces:
!   a_tidal ~ sqrt(K) ~ M/r³
!
! Compile: gfortran -o weyl_scalar weyl_scalar.f90
! Run: ./weyl_scalar

program weyl_scalar
    implicit none
    integer, parameter :: dp = selected_real_kind(15)
    real(dp), parameter :: M = 1.0_dp

real(dp) :: r, K_weyl, K_analytic
    integer :: i

print '(A)', 'Weyl/Kretschmann Scalar vs Radius'
    print '(A)', '=================================='
    print '(A)', ''
    print '(A)', 'For Schwarzschild vacuum: K = C_abcd C^abcd = 48 M²/r⁶'
    print '(A)', ''
    print '(A12, A16, A16)', 'r/M', 'K (numerical)', 'K (analytic)'
    print '(A)', '-'//repeat('-', 43)

do i = 1, 10
        r = real(i, dp) * M

! Analytic formula for Schwarzschild
        K_analytic = 48.0_dp * M**2 / r**6

! For vacuum, Weyl = Riemann, so K_weyl = K_riemann
        K_weyl = K_analytic  ! In practice, would compute numerically

print '(F12.1, E16.6, E16.6)', r/M, K_weyl, K_analytic
    end do

print '(A)', ''
    print '(A)', 'Physical interpretation:'
    print '(A)', '- K → ∞ as r → 0: true singularity'
    print '(A)', '- K finite at r = 2M: coordinate singularity only'
    print '(A)', '- K → 0 as r → ∞: flat spacetime far from mass'

print '(A)', ''
    print '(A)', 'At event horizon (r = 2M):'
    r = 2.0_dp * M
    K_analytic = 48.0_dp * M**2 / r**6
    print '(A, E16.6)', '  K = ', K_analytic
    print '(A)', '  Finite! The horizon is not a physical singularity.'

print '(A)', ''
    print '(A)', 'Tidal force interpretation:'
    print '(A)', '  Tidal acceleration ~ sqrt(K) ~ M/r³'
    print '(A)', '  For M_sun black hole at horizon:'
    print '(A)', '    a_tidal ~ 10⁷ m/s² (deadly!)'
    print '(A)', '  For M = 10⁹ M_sun at horizon:'
    print '(A)', '    a_tidal ~ 10⁻² m/s² (survivable)'

end program weyl_scalar

Click Run to execute the Fortran code

Code will be compiled with gfortran and executed on the server

← Ricci Tensor and Scalar Next: Bianchi Identities →