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← Back to Part III: Einstein Field Equations

Chapter 15: Einstein-Hilbert Action

The Einstein-Hilbert action provides the variational foundation of general relativity. Einstein's field equations emerge naturally as the Euler-Lagrange equations for spacetime geometry, unifying gravity with the action principle of classical mechanics.

The Gravitational Action

$ S_{EH} = \frac{c^4}{16\pi G} \int R \sqrt{-g} \, d^4x $

R = Ricci scalar, g = det(g_μν)

This is the simplest diffeomorphism-invariant action built from the metric and its derivatives. The factor √(-g) d⁴x is the invariant volume element.

With Cosmological Constant

$ S = \frac{c^4}{16\pi G} \int (R - 2\Lambda) \sqrt{-g} \, d^4x $

Deriving Field Equations

Varying the action with respect to the metric g_μν:

$ \delta S_{EH} = \frac{c^4}{16\pi G} \int \left( R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R \right) \delta g^{\mu\nu} \sqrt{-g} \, d^4x $

Key Variations

$ \delta\sqrt{-g} = -\frac{1}{2}\sqrt{-g} g_{\mu\nu} \delta g^{\mu\nu} $

$ \delta R = R_{\mu\nu} \delta g^{\mu\nu} + g^{\mu\nu} \delta R_{\mu\nu} $

Palatini Identity

The term g^μνδR_μν becomes a total derivative and vanishes with appropriate boundary conditions.

Total Action with Matter

$ S = S_{EH} + S_{matter} = \frac{c^4}{16\pi G} \int R \sqrt{-g} \, d^4x + \int \mathcal{L}_m \sqrt{-g} \, d^4x $

The stress-energy tensor is defined by variation with respect to the metric:

$ T_{\mu\nu} = -\frac{2}{\sqrt{-g}} \frac{\delta S_m}{\delta g^{\mu\nu}} $

Setting δS/δg^μν = 0 gives Einstein's equations!

Gibbons-Hawking-York Boundary Term

For a well-defined variational problem with boundaries:

$ S_{GHY} = \frac{c^4}{8\pi G} \int_\partial K \sqrt{|h|} \, d^3x $

K = trace of extrinsic curvature, h = induced metric

This boundary term ensures the variational problem is well-posed when δg_μν = 0 on the boundary (Dirichlet conditions).

Interactive Simulation: Action Principle

Run this Python code to explore the Einstein-Hilbert action, calculate Planck units, and visualize scalar field dynamics. The simulation shows how different equation of state values arise from the potential/kinetic energy ratio.

Einstein-Hilbert Action Analysis

Python

Explore the variational foundation of general relativity

einstein_hilbert_action.py134 lines

#!/usr/bin/env python3
"""
einstein_hilbert_action.py - Derive Einstein equations from action principle
Demonstrates the variational derivation with numerical examples
"""
import numpy as np
import matplotlib.pyplot as plt

# Physical constants
c = 3e8      # Speed of light (m/s)
G = 6.674e-11  # Gravitational constant (N m²/kg²)
hbar = 1.055e-34  # Reduced Planck constant (J s)

print("=" * 65)
print("Einstein-Hilbert Action and Field Equations")
print("=" * 65)

print("""
The Einstein-Hilbert action is:

S_EH = (c⁴/16πG) ∫ R √(-g) d⁴x

where:
    R = Ricci scalar (trace of Ricci tensor)
    g = determinant of metric tensor g_μν
    √(-g) d⁴x = invariant volume element

Varying with respect to g^μν and setting δS = 0 gives:

R_μν - (1/2)g_μν R = (8πG/c⁴) T_μν

This is the Einstein Field Equation!
""")

# Calculate fundamental constants
prefactor = c**4 / (16 * np.pi * G)
print("=" * 65)
print("Numerical Values")
print("-" * 65)
print(f"  c⁴/(16πG) = {prefactor:.3e} J·m")
print(f"  8πG/c⁴   = {8*np.pi*G/c**4:.3e} m/J")

# Planck units
l_planck = np.sqrt(hbar * G / c**3)
t_planck = np.sqrt(hbar * G / c**5)
m_planck = np.sqrt(hbar * c / G)
E_planck = m_planck * c**2

print(f"\nPlanck Units (natural units of quantum gravity):")
print(f"  Planck length:  ℓ_P = {l_planck:.3e} m")
print(f"  Planck time:    t_P = {t_planck:.3e} s")
print(f"  Planck mass:    m_P = {m_planck:.3e} kg")
print(f"  Planck energy:  E_P = {E_planck/1.6e-19/1e9:.3e} GeV")

# Scalar field example
print("\n" + "=" * 65)
print("Example: Scalar Field (φ) Stress-Energy Tensor")
print("-" * 65)
print("""
For a scalar field φ with Lagrangian:

L = -(1/2)∂_μφ ∂^μφ - V(φ)

The stress-energy tensor is:

T_μν = ∂_μφ ∂_νφ - g_μν L

For a homogeneous field φ(t):
    ρ = (1/2)φ̇² + V(φ)    (energy density)
    p = (1/2)φ̇² - V(φ)    (pressure)

Equation of state: w = p/ρ
""")

# Plot equation of state for scalar field
V_over_kinetic = np.linspace(0.01, 100, 1000)  # V/(φ̇²/2) ratio
w = (1 - 2*V_over_kinetic) / (1 + 2*V_over_kinetic)

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

# Left plot: Equation of state
ax1 = axes[0]
ax1.plot(V_over_kinetic, w, 'b-', linewidth=2)
ax1.axhline(y=-1, color='red', linestyle='--', alpha=0.7, label='w = -1 (cosmological constant)')
ax1.axhline(y=0, color='green', linestyle='--', alpha=0.7, label='w = 0 (dust/matter)')
ax1.axhline(y=1/3, color='orange', linestyle='--', alpha=0.7, label='w = 1/3 (radiation)')
ax1.axhline(y=1, color='purple', linestyle='--', alpha=0.7, label='w = 1 (stiff matter)')
ax1.set_xlabel('V/(φ̇²/2) (potential/kinetic ratio)', fontsize=12)
ax1.set_ylabel('Equation of state w = p/ρ', fontsize=12)
ax1.set_title('Scalar Field Equation of State', fontsize=14)
ax1.set_xscale('log')
ax1.set_xlim(0.01, 100)
ax1.set_ylim(-1.2, 1.2)
ax1.legend(loc='right', fontsize=9)
ax1.grid(True, alpha=0.3)
ax1.fill_between(V_over_kinetic, -1, w, where=(w < -1/3), alpha=0.2, color='red', label='Accelerating')

# Right plot: Curvature contributions
ax2 = axes[1]
# Schematic of action contributions
r = np.linspace(0.1, 10, 100)
R_contribution = 1/r**2  # Schematic Ricci scalar contribution
Lambda_contribution = np.ones_like(r) * 0.1  # Cosmological constant
matter_contribution = np.exp(-r)  # Localized matter

ax2.plot(r, R_contribution, 'b-', linewidth=2, label='Curvature R ~ 1/r²')
ax2.plot(r, Lambda_contribution, 'r--', linewidth=2, label='Cosmological Λ')
ax2.plot(r, matter_contribution, 'g-', linewidth=2, label='Matter T_μν')
ax2.set_xlabel('Radial distance r/r_s', fontsize=12)
ax2.set_ylabel('Contribution to action', fontsize=12)
ax2.set_title('Schematic Action Contributions', fontsize=14)
ax2.legend(fontsize=10)
ax2.grid(True, alpha=0.3)
ax2.set_ylim(0, 2)

plt.tight_layout()
plt.savefig('einstein_hilbert.png', dpi=150, bbox_inches='tight')

print("\n" + "=" * 65)
print("Key Results")
print("-" * 65)
print("""
1. For V >> φ̇²/2: w → -1 (inflation/dark energy regime)
2. For V << φ̇²/2: w → +1 (stiff matter, kinetic dominated)
3. For V = φ̇²/2:  w → 0  (dust-like behavior)

The Einstein-Hilbert action unifies:
  • Geometry (through Ricci scalar R)
  • Matter (through T_μν from variation of S_matter)
  • Cosmological constant (as a geometric term -2Λ)
""")

print("✓ Plot saved as 'einstein_hilbert.png'")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Fortran: Action Integration

This Fortran code demonstrates that the bulk Einstein-Hilbert action vanishes for vacuum solutions like Schwarzschild spacetime. The mass is encoded in boundary terms!

Schwarzschild Volume & Action

Python

Volume element and proper volume integration for vacuum spacetime

action_plot.py112 lines

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

fortran_source = r"""
program action_volume
    implicit none
    integer, parameter :: dp = selected_real_kind(15)
    real(dp), parameter :: pi = 4.0_dp * atan(1.0_dp)
    real(dp), parameter :: M = 1.0_dp
    real(dp) :: r, f, sqrt_g, vol_elem, cum_vol
    real(dp) :: dr
    integer :: i, n_pts

n_pts = 300
    dr = 48.0_dp / real(n_pts, dp)
    cum_vol = 0.0_dp

print '(A)', 'DATA_START'
    do i = 1, n_pts
        r = 2.01_dp * M + real(i-1, dp) * dr
        f = 1.0_dp - 2.0_dp*M/r
        ! sqrt(-g) = r^2 sin(theta) for spatial part at theta=pi/2
        sqrt_g = r**2
        ! Proper volume element: sqrt(g_rr) * r^2 * sin(theta) * dr * dtheta * dphi
        vol_elem = r**2 / sqrt(f)
        cum_vol = cum_vol + vol_elem * dr * 4.0_dp * pi
        write(*,'(ES20.10,A,ES20.10,A,ES20.10,A,ES20.10)') &
            r/M, ',', sqrt_g, ',', vol_elem, ',', cum_vol
    end do
    print '(A)', 'DATA_END'
end program action_volume
"""

workdir = tempfile.mkdtemp()
src = os.path.join(workdir, 'action.f90')
exe = os.path.join(workdir, 'action')

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

comp = subprocess.run(['gfortran', '-O2', '-o', exe, src],
                      capture_output=True, text=True, timeout=30)
if comp.returncode != 0:
    print("Compilation error:", comp.stderr)
else:
    result = subprocess.run([exe], capture_output=True, text=True, timeout=30, cwd=workdir)
    output = result.stdout

lines = []
    in_data = False
    for line in output.split('\n'):
        if 'DATA_START' in line:
            in_data = True
            continue
        if 'DATA_END' in line:
            break
        if in_data and line.strip():
            lines.append(line.strip())

r_arr, sqrt_g, vol_elem, cum_vol = [], [], [], []
    for line in lines:
        vals = [float(x) for x in line.split(',')]
        r_arr.append(vals[0])
        sqrt_g.append(vals[1])
        vol_elem.append(vals[2])
        cum_vol.append(vals[3])

r_arr = np.array(r_arr)
    vol_elem = np.array(vol_elem)
    cum_vol = np.array(cum_vol)
    flat_vol = r_arr**2  # Flat space comparison

fig, axes = plt.subplots(1, 2, figsize=(14, 6))
    fig.patch.set_facecolor('#0f172a')
    fig.suptitle('Einstein-Hilbert Action: Schwarzschild Volume Elements', color='white', fontsize=14, fontweight='bold')

ax = axes[0]
    ax.set_facecolor('#1e293b')
    ax.plot(r_arr, vol_elem, color='#f472b6', linewidth=2, label='Curved: $r^2/\\sqrt{f}$')
    ax.plot(r_arr, flat_vol, color='#22d3ee', linewidth=2, linestyle='--', label='Flat: $r^2$')
    ax.axvline(x=2.0, color='#ef4444', linestyle='--', alpha=0.7, label='Horizon')
    ax.set_xlabel('r / M', color='#cbd5e1')
    ax.set_ylabel('Volume element', color='#cbd5e1')
    ax.set_title('Proper Volume Element vs Coordinate Volume', color='white')
    ax.legend(facecolor='#1e293b', edgecolor='#334155', labelcolor='#cbd5e1')
    ax.tick_params(colors='#cbd5e1')
    for spine in ax.spines.values():
        spine.set_color('#334155')
    ax.grid(True, color='#334155', alpha=0.4)

ax2 = axes[1]
    ax2.set_facecolor('#1e293b')
    ax2.plot(r_arr, cum_vol, color='#4ade80', linewidth=2)
    ax2.set_xlabel('r / M', color='#cbd5e1')
    ax2.set_ylabel('Cumulative proper 3-volume ($M^3$)', color='#cbd5e1')
    ax2.set_title('Integrated Proper Volume (R=0 for vacuum!)', color='white')
    ax2.tick_params(colors='#cbd5e1')
    for spine in ax2.spines.values():
        spine.set_color('#334155')
    ax2.grid(True, color='#334155', alpha=0.4)
    ax2.annotate('$S_{EH} = \\frac{1}{16\\pi G}\\int R\\sqrt{-g}\\,d^4x = 0$\n(R=0 everywhere for vacuum)',
                 xy=(0.5, 0.85), xycoords='axes fraction', color='#fbbf24',
                 fontsize=11, ha='center', bbox=dict(boxstyle='round', facecolor='#1e293b', edgecolor='#fbbf24', alpha=0.8))

plt.tight_layout()
    plt.savefig('action_plot.png', dpi=150, bbox_inches='tight', facecolor=fig.get_facecolor())
    plt.close()
    print("Bulk action S_EH = 0 for vacuum. Mass encoded in GHY boundary term!")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

← Stress-Energy Tensor Next: Weak Field Limit →