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.py102 lines

#!/usr/bin/env python3
"""
Metric Tensor - Schwarzschild metric, inverse, geodesic distances, light cones.
"""
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
plt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor='#0a0a1a')
plt.close()
print("Plot saved successfully.")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Fortran: Metric Determinant and Inverse

Fortran + Python: Schwarzschild Metric Visualization

Python

Compiles Fortran code and plots metric components vs radial coordinate

metric_plot.py114 lines

import subprocess, sys
import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

fortran_source = r"""
program metric_inverse
    implicit none
    integer, parameter :: dp = selected_real_kind(15)
    real(dp) :: M, r, theta, f_r
    real(dp) :: g_tt, g_rr, g_thth, g_phph, det_g
    integer :: i, npts
    real(dp) :: dr, r_val

M = 1.0_dp
    theta = acos(0.0_dp)
    r = 6.0_dp * M

f_r = 1.0_dp - 2.0_dp*M/r
    write(*,'(A)') 'Schwarzschild Metric at r = 6M'
    write(*,'(A)') '================================'
    write(*,'(A,F10.6)') 'g_tt  = ', -(f_r)
    write(*,'(A,F10.6)') 'g_rr  = ', 1.0_dp/f_r
    write(*,'(A,F10.6)') 'g_thth= ', r**2
    write(*,'(A,F10.6)') 'g_phph= ', r**2 * sin(theta)**2
    write(*,'(A)') ''
    write(*,'(A,F10.6)') 'det(g) = ', -(f_r) * (1.0_dp/f_r) * r**2 * r**2*sin(theta)**2
    write(*,'(A,F10.6)') 'sqrt(-det(g)) = ', r**2 * sin(theta)
    write(*,'(A)') ''
    write(*,'(A,F10.6)') 'Time dilation: ', sqrt(f_r)
    write(*,'(A,F10.6)') 'Escape vel/c:  ', sqrt(2.0_dp*M/r)
    write(*,'(A)') ''

npts = 200
    write(*,'(A)') 'DATA_START'
    write(*,'(A)') 'r_over_rs,g_tt,g_rr,sqrt_neg_det'
    do i = 1, npts
        r_val = 2.05_dp * M + real(i-1, dp) * (20.0_dp * M - 2.05_dp * M) / real(npts-1, dp)
        f_r = 1.0_dp - 2.0_dp * M / r_val
        g_tt = -f_r
        g_rr = 1.0_dp / f_r
        det_g = g_tt * g_rr * r_val**4
        write(*,'(F12.6,A,F12.6,A,F12.6,A,F12.6)') &
            r_val/(2.0_dp*M), ',', g_tt, ',', g_rr, ',', r_val**2
    end do
    write(*,'(A)') 'DATA_END'
end program metric_inverse
"""

with open('metric_inverse.f90', 'w') as f:
    f.write(fortran_source)

comp = subprocess.run(['gfortran', '-o', './metric_inverse', 'metric_inverse.f90'],
                      capture_output=True, text=True)
if comp.returncode != 0:
    print("Compilation error:", comp.stderr)
    sys.exit(1)

result = subprocess.run(['./metric_inverse'], capture_output=True, text=True, timeout=30)

lines = result.stdout.strip().split('\n')
data_lines = []
in_data = False
for line in lines:
    if 'DATA_START' in line:
        in_data = True
        continue
    if 'DATA_END' in line:
        in_data = False
        continue
    if in_data:
        data_lines.append(line.strip())
    else:
        print(line)

data = np.array([[float(x) for x in line.split(',')] for line in data_lines[1:]])
r_rs = data[:, 0]
g_tt = data[:, 1]
g_rr = data[:, 2]
sqrt_det = data[:, 3]

fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(14, 5))
fig.patch.set_facecolor('#0f172a')
for ax in [ax1, ax2]:
    ax.set_facecolor('#1e293b')
    ax.tick_params(colors='#cbd5e1')
    for spine in ax.spines.values():
        spine.set_color('#334155')
    ax.grid(True, color='#334155', alpha=0.4)

ax1.plot(r_rs, g_tt, color='#22d3ee', linewidth=2.5, label=r'$g_{tt}=-(1-r_s/r)$')
ax1.plot(r_rs, g_rr, color='#f87171', linewidth=2.5, label=r'$g_{rr}=(1-r_s/r)^{-1}$')
ax1.axhline(y=0, color='#475569', linestyle='--', linewidth=1)
ax1.axhline(y=-1, color='#475569', linestyle=':', linewidth=0.5)
ax1.axhline(y=1, color='#475569', linestyle=':', linewidth=0.5)
ax1.axvline(x=1, color='#ef4444', linestyle='--', linewidth=1.5, alpha=0.7, label='Event horizon')
ax1.set_xlim(1, 10)
ax1.set_ylim(-2, 5)
ax1.set_title('Schwarzschild Metric Components', color='#e2e8f0', fontsize=14, fontweight='bold')
ax1.set_xlabel(r'$r / r_s$', color='#94a3b8')
ax1.set_ylabel('Metric component', color='#94a3b8')
ax1.legend(facecolor='#1e293b', edgecolor='#334155', labelcolor='#cbd5e1', fontsize=9)

ax2.plot(r_rs, sqrt_det, color='#a78bfa', linewidth=2.5, label=r'$\sqrt{-\det g}=r^2\sin\theta$')
ax2.set_title('Volume Element', color='#e2e8f0', fontsize=14, fontweight='bold')
ax2.set_xlabel(r'$r / r_s$', color='#94a3b8')
ax2.set_ylabel(r'$\sqrt{-\det g}$', color='#94a3b8')
ax2.legend(facecolor='#1e293b', edgecolor='#334155', labelcolor='#cbd5e1')

plt.tight_layout()
plt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor='#0f172a')
print("\nVisualization saved to output.png")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

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