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← Back to Part I: Differential Geometry

Chapter 3: The Metric Tensor

The metric tensor is the fundamental object in general relativity. It encodes all information about spacetime geometry: distances, angles, volumes, and ultimately gravity itself.

Definition and Properties

The metric tensor g_μν is a symmetric (0,2) tensor that defines the infinitesimal spacetime interval:

$ ds^2 = g_{\mu\nu} dx^\mu dx^\nu $

The line element — fundamental invariant of spacetime

Symmetry

$ g_{\mu\nu} = g_{\nu\mu} $

10 independent components in 4D

Non-degeneracy

$ \det(g_{\mu\nu}) \neq 0 $

Invertible everywhere

Signature

(-,+,+,+) or (+,-,-,-)

One time, three space dimensions

Inverse Metric

$ g^{\mu\nu} g_{\nu\rho} = \delta^\mu_\rho $

Raises indices

Important Spacetime Metrics

Minkowski (Flat Spacetime)

$ ds^2 = -c^2dt^2 + dx^2 + dy^2 + dz^2 $

Special relativity, no gravity

Schwarzschild (Spherical Mass)

$ ds^2 = -\left(1 - \frac{2GM}{rc^2}\right)c^2dt^2 + \frac{dr^2}{1 - \frac{2GM}{rc^2}} + r^2 d\Omega^2 $

Black holes, stars, planets

FLRW (Expanding Universe)

$ ds^2 = -c^2dt^2 + a(t)^2\left[\frac{dr^2}{1-kr^2} + r^2 d\Omega^2\right] $

Cosmology, Big Bang

Physical Interpretation

Proper Time

For timelike intervals: $ d\tau^2 = -ds^2/c^2 $ — time measured by a moving clock

Proper Length

For spacelike intervals: $ dl^2 = ds^2 $ — distance measured by a ruler

Null Intervals

$ ds^2 = 0 $ — light rays, photon worldlines

Python: Schwarzschild Metric Analysis

Python

schwarzschild_metric.py118 lines

#!/usr/bin/env python3
"""
╔══════════════════════════════════════════════════════════════════╗
║          Schwarzschild Metric Analysis                           ║
║          Analyzes the metric outside a spherical mass            ║
╚══════════════════════════════════════════════════════════════════╝

The Schwarzschild metric in standard coordinates:
  $ds^2 = -\\left(1 - \\frac{r_s}{r}\\right)c^2dt^2 + \\frac{dr^2}{1 - \\frac{r_s}{r}} + r^2 d\\Omega^2$

where $r_s = 2GM/c^2$ is the Schwarzschild radius.

Physical effects:
  • Time dilation: $\\frac{d\\tau}{dt} = \\sqrt{1 - \\frac{r_s}{r}}$
  • Escape velocity: $v_{esc} = c\\sqrt{\\frac{r_s}{r}}$
  • Event horizon at $r = r_s$
"""
import numpy as np
import matplotlib.pyplot as plt
import base64
from io import BytesIO

# Constants (geometrized units: G = c = 1)
M = 1.0  # Mass in units where r_s = 2M

def schwarzschild_metric(r, theta):
    """Return the Schwarzschild metric components at (r, theta)"""
    rs = 2 * M  # Schwarzschild radius

if r <= rs:
        raise ValueError("Inside horizon!")

g = np.zeros((4, 4))
    g[0, 0] = -(1 - rs/r)     # g_tt
    g[1, 1] = 1/(1 - rs/r)    # g_rr
    g[2, 2] = r**2            # g_theta_theta
    g[3, 3] = r**2 * np.sin(theta)**2  # g_phi_phi

return g

def proper_time_ratio(r):
    """Time dilation factor: dtau/dt for stationary observer"""
    rs = 2 * M
    if r <= rs:
        return 0
    return np.sqrt(1 - rs/r)

def escape_velocity(r):
    """Escape velocity as fraction of c"""
    rs = 2 * M
    if r <= rs:
        return 1.0
    return np.sqrt(rs/r)

# Calculate metric at r = 6M (ISCO - Innermost Stable Circular Orbit)
r_isco = 6 * M
theta = np.pi / 2  # equatorial plane

g = schwarzschild_metric(r_isco, theta)
print("╔══════════════════════════════════════════════════════════════════╗")
print("║ Schwarzschild Metric at r = 6M (ISCO)                           ║")
print("╚══════════════════════════════════════════════════════════════════╝")
print()
print("Metric tensor g_μν:")
for i in range(4):
    print(f"  [{g[i,0]:8.4f}, {g[i,1]:8.4f}, {g[i,2]:8.4f}, {g[i,3]:8.4f}]")

# Determinant
det_g = np.linalg.det(g)
print(f"\nDeterminant: det(g) = {det_g:.4f}")
print(f"sqrt(-det(g)) = {np.sqrt(-det_g):.4f}")
print(f"Expected: r² sin(θ) = {r_isco**2 * np.sin(theta):.4f}")

# Physical quantities at ISCO
print(f"\nPhysical quantities at ISCO (r = 6M):")
print(f"  Time dilation: dτ/dt = {proper_time_ratio(r_isco):.6f}")
print(f"  Escape velocity: v_esc/c = {escape_velocity(r_isco):.6f}")

# Plot time dilation and escape velocity
r_vals = np.linspace(2.01 * M, 20 * M, 500)
tau_ratio = [proper_time_ratio(r) for r in r_vals]
v_esc = [escape_velocity(r) for r in r_vals]

fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(14, 5))

# Time dilation plot
ax1.plot(r_vals / M, tau_ratio, 'b-', linewidth=2.5, label='Time dilation')
ax1.axhline(y=1, color='gray', linestyle='--', alpha=0.5)
ax1.axvline(x=2, color='red', linestyle='--', linewidth=2, label='Horizon (r = 2M)')
ax1.axvline(x=6, color='orange', linestyle=':', linewidth=2, label='ISCO (r = 6M)')
ax1.set_xlabel('r / M', fontsize=13)
ax1.set_ylabel(r'$d\tau / dt$', fontsize=13)
ax1.set_title('Gravitational Time Dilation', fontsize=15, fontweight='bold')
ax1.legend(fontsize=10)
ax1.grid(True, alpha=0.3)
ax1.set_ylim([0, 1.1])

# Escape velocity plot
ax2.plot(r_vals / M, v_esc, 'r-', linewidth=2.5, label='Escape velocity')
ax2.axhline(y=1, color='gray', linestyle='--', alpha=0.5, label='Speed of light')
ax2.axvline(x=2, color='red', linestyle='--', linewidth=2, label='Horizon')
ax2.axvline(x=6, color='orange', linestyle=':', linewidth=2, label='ISCO')
ax2.set_xlabel('r / M', fontsize=13)
ax2.set_ylabel(r'$v_{esc} / c$', fontsize=13)
ax2.set_title('Escape Velocity', fontsize=15, fontweight='bold')
ax2.legend(fontsize=10)
ax2.grid(True, alpha=0.3)
ax2.set_ylim([0, 1.1])

plt.tight_layout()

# Save to base64
buffer = BytesIO()
plt.savefig(buffer, format='png', dpi=150, bbox_inches='tight')
buffer.seek(0)
image_base64 = base64.b64encode(buffer.read()).decode()
print(f'\n<img src="data:image/png;base64,{image_base64}" alt="Schwarzschild Analysis" />')
plt.close()

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Fortran: Metric Determinant and Inverse

Fortran

metric_inverse.f9083 lines

! metric_inverse.f90 - Compute metric inverse and determinant
! Compile: gfortran -o metric_inverse metric_inverse.f90
! Run: ./metric_inverse

program metric_inverse
    implicit none
    integer, parameter :: dp = selected_real_kind(15)
    real(dp) :: g(4,4), g_inv(4,4), identity(4,4)
    real(dp) :: r, theta, M, det_g
    integer :: i, j

! Print header with metric definition
    print '(A)', '╔══════════════════════════════════════════════════════════════════╗'
    print '(A)', '║          Metric Tensor Inverse Calculator                       ║'
    print '(A)', '╚══════════════════════════════════════════════════════════════════╝'
    print '(A)', ''
    print '(A)', 'Computing inverse of Schwarzschild metric:'
    print '(A)', '  $g_{\mu\nu} = \text{diag}\left(-f(r), f(r)^{-1}, r^2, r^2\sin^2\theta\right)$'
    print '(A)', ''
    print '(A)', 'where $f(r) = 1 - 2M/r$ and we set $c = G = 1$.'
    print '(A)', ''
    print '(A)', 'The inverse metric raises indices: $g^{\mu\nu}g_{\nu\rho} = \delta^\mu_\rho$'
    print '(A)', ''
    print '(A)', '──────────────────────────────────────────────────────────────────'
    print '(A)', ''

M = 1.0_dp
    r = 6.0_dp * M  ! ISCO (Innermost Stable Circular Orbit)
    theta = acos(0.0_dp)  ! pi/2 (equatorial plane)

! Schwarzschild metric (diagonal)
    g = 0.0_dp
    g(1,1) = -(1.0_dp - 2.0_dp*M/r)
    g(2,2) = 1.0_dp / (1.0_dp - 2.0_dp*M/r)
    g(3,3) = r**2
    g(4,4) = r**2 * sin(theta)**2

print '(A)', 'Schwarzschild Metric at r = 6M, θ = π/2'
    print '(A)', '========================================'
    print '(A)', ''
    print '(A)', 'Metric g_μν:'
    do i = 1, 4
        print '(4F12.6)', (g(i,j), j=1,4)
    end do

! Determinant of diagonal matrix
    det_g = g(1,1) * g(2,2) * g(3,3) * g(4,4)
    print '(A)', ''
    print '(A,F12.6)', 'Determinant det(g) = ', det_g
    print '(A,F12.6)', 'sqrt(-det(g)) = ', sqrt(-det_g)
    print '(A)', ''
    print '(A)', 'Note: For Schwarzschild metric, $\sqrt{-\det(g)} = r^2\sin\theta$'

! Inverse of diagonal matrix
    g_inv = 0.0_dp
    do i = 1, 4
        g_inv(i,i) = 1.0_dp / g(i,i)
    end do

print '(A)', ''
    print '(A)', 'Inverse metric g^μν:'
    do i = 1, 4
        print '(4F12.6)', (g_inv(i,j), j=1,4)
    end do

! Verify: g * g_inv = identity
    print '(A)', ''
    print '(A)', 'Verification: g_μρ · g^ρν = δ_μ^ν'
    identity = matmul(g, g_inv)
    do i = 1, 4
        print '(4F12.6)', (identity(i,j), j=1,4)
    end do

! Physical quantities
    print '(A)', ''
    print '(A)', '──────────────────────────────────────────────────────────────────'
    print '(A)', ''
    print '(A)', 'Physical quantities at r = 6M (ISCO):'
    print '(A,F12.6)', '  Time dilation factor (dτ/dt): ', sqrt(-g(1,1))
    print '(A,F12.6)', '  Escape velocity (v_esc/c):    ', sqrt(2.0_dp*M/r)
    print '(A,F12.6)', '  Redshift factor (1+z):        ', 1.0_dp/sqrt(-g(1,1))

end program metric_inverse

Click Run to execute the Fortran code

Code will be compiled with gfortran and executed on the server

← Tensor Analysis Next: Connection and Covariant Derivative →