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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 Action Integration

Fortran

Numerical integration of the gravitational action for vacuum spacetimes

action_integral.f90132 lines

! action_integral.f90 - Evaluate Einstein-Hilbert action for Schwarzschild
! Demonstrates that bulk action vanishes for vacuum solutions

program action_integral
    implicit none
    integer, parameter :: dp = selected_real_kind(15)
    real(dp), parameter :: pi = 4.0_dp * atan(1.0_dp)
    real(dp), parameter :: G = 1.0_dp  ! Geometrized units
    real(dp), parameter :: c = 1.0_dp

real(dp) :: M, r_inner, r_outer
    real(dp) :: r, theta, phi, dr, dtheta, dphi
    real(dp) :: g_det, R_scalar, integrand
    real(dp) :: action, volume
    integer :: nr, ntheta, nphi, ir, ith, iph

M = 1.0_dp

print '(A)', '╔══════════════════════════════════════════════════════════════╗'
    print '(A)', '║   Einstein-Hilbert Action for Schwarzschild Spacetime        ║'
    print '(A)', '╚══════════════════════════════════════════════════════════════╝'
    print '(A)', ''
    print '(A)', '━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━'
    print '(A)', '  VACUUM SOLUTION PROPERTIES'
    print '(A)', '━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━'
    print '(A)', ''
    print '(A)', 'For Schwarzschild spacetime (vacuum solution):'
    print '(A)', ''
    print '(A)', '  Ricci tensor:   \( R_{\mu\nu} = 0 \)  (everywhere outside horizon)'
    print '(A)', ''
    print '(A)', '  Ricci scalar:   \( R = g^{\mu\nu} R_{\mu\nu} = 0 \)'
    print '(A)', ''
    print '(A)', '  Therefore the bulk action:'
    print '(A)', ''
    print '(A)', '  \( S_{EH} = \frac{c^4}{16\pi G} \int R \sqrt{-g} \, d^4x = 0 \)'
    print '(A)', ''
    print '(A)', '  ⟹  The BULK gravitational action VANISHES for vacuum!'
    print '(A)', ''

! Numerical integration to verify volume element
    print '(A)', '━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━'
    print '(A)', '  SCHWARZSCHILD METRIC'
    print '(A)', '━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━'
    print '(A)', ''
    print '(A)', 'Line element:'
    print '(A)', ''
    print '(A)', '  \( ds^2 = -\left(1-\frac{2M}{r}\right)dt^2 + \left(1-\frac{2M}{r}\right)^{-1}dr^2 + r^2 d\Omega^2 \)'
    print '(A)', ''
    print '(A)', 'Volume element:'
    print '(A)', ''
    print '(A)', '  \( \sqrt{-g} = r^2 \sin\theta \)'
    print '(A)', ''

r_inner = 2.0_dp * M + 0.1_dp  ! Just outside horizon
    r_outer = 10.0_dp * M

nr = 100
    ntheta = 50
    nphi = 100

dr = (r_outer - r_inner) / nr
    dtheta = pi / ntheta
    dphi = 2.0_dp * pi / nphi

volume = 0.0_dp
    action = 0.0_dp

do ir = 1, nr
        r = r_inner + (ir - 0.5_dp) * dr
        do ith = 1, ntheta
            theta = (ith - 0.5_dp) * dtheta
            do iph = 1, nphi
                phi = (iph - 0.5_dp) * dphi

! √(-g) for Schwarzschild spatial part
                g_det = r**4 * sin(theta)**2

! R = 0 for vacuum (Einstein equations satisfied)
                R_scalar = 0.0_dp

! Volume integration
                volume = volume + sqrt(g_det) * dr * dtheta * dphi

! Action integration
                integrand = R_scalar * sqrt(g_det)
                action = action + integrand * dr * dtheta * dphi

end do
        end do
    end do

print '(A)', '━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━'
    print '(A)', '  NUMERICAL INTEGRATION RESULTS'
    print '(A)', '━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━'
    print '(A)', ''
    print '(A,F6.2,A,F6.2,A)', '  Integration region:  \( r \in [', r_inner/M, 'M, \, ', r_outer/M, 'M] \)'
    print '(A,E14.6,A)', '  Proper 3-volume:     \( V = \)', volume, ' \( M^3 \)'
    print '(A,E14.6)', '  Bulk action:         \( S_{EH} = \)', action
    print '(A)', ''

! Boundary contributions
    print '(A)', '━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━'
    print '(A)', '  GIBBONS-HAWKING-YORK BOUNDARY TERM'
    print '(A)', '━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━'
    print '(A)', ''
    print '(A)', 'The GHY boundary term is required for a well-posed variational problem:'
    print '(A)', ''
    print '(A)', '  \( S_{GHY} = \frac{c^4}{8\pi G} \oint_{\partial\mathcal{M}} K \sqrt{|h|} \, d^3x \)'
    print '(A)', ''
    print '(A)', 'where:'
    print '(A)', '  • \( K \) = trace of extrinsic curvature'
    print '(A)', '  • \( h \) = induced metric on boundary'
    print '(A)', ''
    print '(A)', 'This provides non-zero contribution at spatial infinity,'
    print '(A)', 'encoding the ADM mass of the spacetime:'
    print '(A)', ''
    print '(A,F8.4,A)', '  \( M_{ADM} = \)', M, ' (geometrized units, \( G = c = 1 \))'
    print '(A)', ''
    print '(A)', '┌────────────────────────────────────────────────────────────────┐'
    print '(A)', '│  KEY INSIGHT                                                   │'
    print '(A)', '├────────────────────────────────────────────────────────────────┤'
    print '(A)', '│  Mass is encoded in the ASYMPTOTIC geometry (boundary terms),  │'
    print '(A)', '│  not in a local source — vacuum has no matter!                 │'
    print '(A)', '│                                                                │'
    print '(A)', '│  The total action is:                                          │'
    print '(A)', '│                                                                │'
    print '(A)', '│  \( S_{total} = S_{EH} + S_{GHY} = 0 + S_{GHY} \)                     │'
    print '(A)', '└────────────────────────────────────────────────────────────────┘'
    print '(A)', ''

end program action_integral

Click Run to execute the Fortran code

Code will be compiled with gfortran and executed on the server

← Stress-Energy Tensor Next: Weak Field Limit →