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Einstein–Hilbert Action and Palatini Variation

The variational foundation of general relativity: from the Ricci scalar action to the vacuum field equations via the Palatini identity

1. The Einstein–Hilbert Action

The starting point of the variational formulation of general relativity is the Einstein–Hilbert action, which is the simplest diffeomorphism-invariant functional of the metric that is second order in derivatives:

$$S_{EH}[g] = \frac{1}{16\pi G}\int_{\mathcal{M}}\sqrt{-g}\,R\,d^4x$$

Here $R = g^{\mu\nu}R_{\mu\nu}$ is the Ricci scalar, $g = \det(g_{\mu\nu})$ is the determinant of the metric, and the integral is taken over a four-dimensional manifold $\mathcal{M}$. The factor $1/(16\pi G)$ is fixed by matching to the Newtonian limit.

The Ricci scalar encodes the trace of the spacetime curvature. Since $R_{\mu\nu}$ contains second derivatives of $g_{\mu\nu}$, the action is second-order in the metric. This is the unique scalar density (up to boundary terms and the cosmological constant) that yields second-order field equations.

Including a cosmological constant and matter, the total action becomes:

$$S = \frac{1}{16\pi G}\int_{\mathcal{M}}\sqrt{-g}\,(R - 2\Lambda)\,d^4x + S_{\text{matter}}[g, \psi]$$

2. Variation of the Metric Determinant

To derive the field equations, we vary the action with respect to $g^{\mu\nu}$. The integrand$\sqrt{-g}\,R$ has three pieces that respond to $\delta g^{\mu\nu}$: the Ricci scalar, the inverse metric inside $R = g^{\mu\nu}R_{\mu\nu}$, and the determinant.

Using Jacobi's formula for the determinant, the variation of $\sqrt{-g}$ is:

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

This follows from $\delta(\ln\det M) = \text{tr}(M^{-1}\delta M)$, applied to $M = g_{\mu\nu}$ with the identity $g_{\mu\nu}\delta g^{\mu\nu} = -g^{\mu\nu}\delta g_{\mu\nu}$.

Combining with the explicit $g^{\mu\nu}$ in $R = g^{\mu\nu}R_{\mu\nu}$, the full variation is:

$$\delta(\sqrt{-g}\,R) = \sqrt{-g}\left(R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R\right)\delta g^{\mu\nu} + \sqrt{-g}\,g^{\mu\nu}\delta R_{\mu\nu}$$

The first term immediately gives the Einstein tensor $G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R$. The second term requires the Palatini identity to evaluate.

3. The Palatini Identity

The Palatini identity expresses the variation of the Riemann tensor in terms of covariant derivatives of the connection variation. Start from the definition of the Riemann tensor:

$$R^\mu_{\ \nu\rho\sigma} = \partial_\rho\Gamma^\mu_{\nu\sigma} - \partial_\sigma\Gamma^\mu_{\nu\rho} + \Gamma^\mu_{\alpha\rho}\Gamma^\alpha_{\nu\sigma} - \Gamma^\mu_{\alpha\sigma}\Gamma^\alpha_{\nu\rho}$$

The key observation is that $\delta\Gamma^\mu_{\nu\sigma}$ is a tensor, even though$\Gamma^\mu_{\nu\sigma}$ itself is not. This is because the difference of two connections transforms as a $(1,2)$-tensor. Taking the variation:

$$\delta R^\mu_{\ \nu\rho\sigma} = \nabla_\rho(\delta\Gamma^\mu_{\nu\sigma}) - \nabla_\sigma(\delta\Gamma^\mu_{\nu\rho})$$

This is the Palatini identity. The non-tensorial parts of the partial derivatives cancel against the connection terms, leaving purely covariant derivatives. To prove it explicitly, write$\nabla_\rho(\delta\Gamma^\mu_{\nu\sigma}) = \partial_\rho(\delta\Gamma^\mu_{\nu\sigma}) + \Gamma^\mu_{\alpha\rho}\delta\Gamma^\alpha_{\nu\sigma} - \Gamma^\alpha_{\nu\rho}\delta\Gamma^\mu_{\alpha\sigma} - \Gamma^\alpha_{\sigma\rho}\delta\Gamma^\mu_{\nu\alpha}$and verify term-by-term cancellation of the non-covariant pieces.

Contracting on $\mu$ and $\rho$ gives the variation of the Ricci tensor:

$$\delta R_{\nu\sigma} = \nabla_\mu(\delta\Gamma^\mu_{\nu\sigma}) - \nabla_\sigma(\delta\Gamma^\mu_{\nu\mu})$$

4. The Boundary Vector Field

Contracting the Ricci variation with $g^{\mu\nu}$ and using the metric compatibility$\nabla_\alpha g^{\mu\nu} = 0$:

$$g^{\mu\nu}\delta R_{\mu\nu} = \nabla_\alpha\left(g^{\mu\nu}\delta\Gamma^\alpha_{\mu\nu} - g^{\mu\alpha}\delta\Gamma^\beta_{\mu\beta}\right)$$

Define the boundary vector field:

$$V^\alpha = g^{\mu\nu}\delta\Gamma^\alpha_{\mu\nu} - g^{\mu\alpha}\delta\Gamma^\beta_{\mu\beta}$$

Then $g^{\mu\nu}\delta R_{\mu\nu} = \nabla_\alpha V^\alpha$ is a total divergence. To compute$V^\alpha$ explicitly in terms of $\delta g^{\mu\nu}$, use the formula for the connection variation:

$$\delta\Gamma^\alpha_{\mu\nu} = \frac{1}{2}g^{\alpha\beta}\left(\nabla_\mu\delta g_{\beta\nu} + \nabla_\nu\delta g_{\beta\mu} - \nabla_\beta\delta g_{\mu\nu}\right)$$

Substituting and simplifying yields the explicit expression:

$$V^\alpha = g^{\mu\nu}\nabla^\alpha\delta g_{\mu\nu} - \nabla_\beta\delta g^{\alpha\beta}$$

This vector field depends on the normal derivative of $\delta g_{\mu\nu}$ at the boundary, which is precisely why the Dirichlet problem for the Einstein–Hilbert action is not well-posed without an additional boundary term.

5. The Boundary Integral

By the divergence theorem, the total divergence term integrates to a boundary integral:

$$\int_{\mathcal{M}}\sqrt{-g}\,\nabla_\alpha V^\alpha\,d^4x = \oint_{\partial\mathcal{M}}\sqrt{|h|}\,n_\alpha V^\alpha\,d^3y$$

where $n_\alpha$ is the outward unit normal to the boundary $\partial\mathcal{M}$and $h$ is the determinant of the induced metric $h_{ij}$ on the boundary. Evaluating $n_\alpha V^\alpha$:

$$n_\alpha V^\alpha = n^\alpha g^{\mu\nu}\nabla_\alpha\delta g_{\mu\nu} - n^\alpha\nabla_\beta\delta g^{\alpha\beta}$$

This term involves the normal derivative $n^\alpha\nabla_\alpha\delta g_{\mu\nu}$ of the metric variation, which does not vanish even when we fix $\delta g_{\mu\nu}|_{\partial\mathcal{M}} = 0$. This means that setting Dirichlet boundary conditions on the metric alone does not yield a well-posed variational principle for $S_{EH}$.

Collecting all terms, the variation of the Einstein–Hilbert action is:

$$\delta S_{EH} = \frac{1}{16\pi G}\int_{\mathcal{M}}\sqrt{-g}\,G_{\mu\nu}\,\delta g^{\mu\nu}\,d^4x + \frac{1}{16\pi G}\oint_{\partial\mathcal{M}}\sqrt{|h|}\,n_\alpha V^\alpha\,d^3y$$

The vacuum Einstein equations $G_{\mu\nu} = 0$ follow from $\delta S_{EH} = 0$only if the boundary term is separately cancelled. This motivates the Gibbons–Hawking–York boundary term discussed in the next section.

6. Connection to Ricci Flow

The Einstein–Hilbert action has a deep structural parallel with Perelman's $\mathcal{F}$-functional for Ricci flow. The Ricci flow action is:

$$S_{RF}[g, f] = \int_0^T \mathcal{F}(g, f)\,d\tau, \quad \mathcal{F}(g, f) = \int_M (R + |\nabla f|^2)e^{-f}\,d\mu_g$$

The Euler–Lagrange equations of $\mathcal{F}$ with respect to $g_{ij}$and $f$ produce exactly the gradient Ricci flow system:

$$\partial_\tau g_{ij} = -2(R_{ij} + \nabla_i\nabla_j f), \qquad \partial_\tau f = -R - \Delta f$$

The analogy is precise: just as $S_{EH}$ yields the Einstein equations through its Euler–Lagrange equations, $\mathcal{F}$ yields Ricci flow through its gradient flow structure. The Ricci scalar$R$ appears in both actions, and both require careful treatment of boundary terms. In the Ricci flow case, the role of the GHY boundary term is played by the dilaton gradient $|\nabla f|^2$, which ensures that $\mathcal{F}$ is monotone along the flow.

Moreover, Perelman's variation of $\mathcal{F}$ mirrors the Palatini calculation:

$$\delta\mathcal{F} = -\int_M \left(R_{ij} + \nabla_i\nabla_j f\right)\delta g^{ij}\,e^{-f}\,d\mu_g + \text{boundary terms}$$

The tensor $R_{ij} + \nabla_i\nabla_j f$ is the Ricci flow analogue of the Einstein tensor $G_{\mu\nu}$, and its vanishing characterizes gradient Ricci solitons — the fixed points of the flow that play the role of vacuum solutions in this setting.

7. First-Order (Palatini) Formalism

In the first-order or Palatini formalism, the metric $g_{\mu\nu}$ and the connection $\Gamma^\alpha_{\mu\nu}$ are treated as independent variables. The action is the same Einstein–Hilbert functional, but now $R_{\mu\nu}$ depends only on the connection:

$$S_{\text{Pal}}[g, \Gamma] = \frac{1}{16\pi G}\int_{\mathcal{M}}\sqrt{-g}\,g^{\mu\nu}R_{\mu\nu}(\Gamma)\,d^4x$$

Varying with respect to $g^{\mu\nu}$ at fixed $\Gamma$ gives $R_{(\mu\nu)} - \frac{1}{2}g_{\mu\nu}R = 0$, where the parentheses denote symmetrization (since $\Gamma$ need not be symmetric a priori). Varying with respect to $\Gamma^\alpha_{\mu\nu}$ at fixed $g$:

$$\nabla_\alpha(\sqrt{-g}\,g^{\mu\nu}) - \frac{1}{2}\delta^\mu_\alpha\nabla_\beta(\sqrt{-g}\,g^{\beta\nu}) - \frac{1}{2}\delta^\nu_\alpha\nabla_\beta(\sqrt{-g}\,g^{\beta\mu}) = 0$$

This equation forces $\Gamma$ to be the Levi-Civita connection of $g_{\mu\nu}$, recovering metric compatibility $\nabla_\alpha g_{\mu\nu} = 0$ as an equation of motion rather than an assumption. The two formalisms are equivalent for pure gravity but differ when torsion or non-minimal couplings are present.

The Palatini formalism has a Ricci flow analogue: treating the connection on the frame bundle as independent of the metric leads to the DeTurck trick, where the modified Ricci flow$\partial_\tau g_{ij} = -2R_{ij} + \mathcal{L}_V g_{ij}$ with $V^i = g^{jk}(\Gamma^i_{jk} - \hat{\Gamma}^i_{jk})$is strictly parabolic. Here $\hat{\Gamma}$ is a reference connection, playing the role of the independent connection in the Palatini formalism.

8. Summary of the Variational Dictionary

The variational structure of the Einstein–Hilbert action establishes a precise dictionary between general relativity and Ricci flow that extends through the entire Lagrangian framework:

Action: $S_{EH} = \frac{1}{16\pi G}\int\sqrt{-g}\,R\,d^4x$ corresponds to $\mathcal{F}(g,f) = \int(R + |\nabla f|^2)e^{-f}d\mu$
Field equation: $G_{\mu\nu} = 0$ corresponds to $R_{ij} + \nabla_i\nabla_j f = 0$ (gradient soliton)
Boundary term: GHY $\int\sqrt{|h|}K\,d^3y$ corresponds to $\int|\nabla f|^2 e^{-f}d\mu$ (dilaton boundary)
Palatini identity: $\delta R^\mu_{\ \nu\rho\sigma} = \nabla_\rho\delta\Gamma - \nabla_\sigma\delta\Gamma$ corresponds to DeTurck linearization
Boundary vector: $V^\alpha = g^{\mu\nu}\delta\Gamma^\alpha_{\mu\nu} - g^{\mu\alpha}\delta\Gamma^\beta_{\mu\beta}$ corresponds to DeTurck vector $V^i$

This dictionary will be deepened in subsequent sections as we introduce the GHY term, the ADM decomposition, and Perelman's $\mathcal{F}$-functional as a gravitational action principle in its own right.

Simulation: Palatini Variation and Einstein Equations

Python

script.py101 lines

import numpy as np
import matplotlib.pyplot as plt

# Palatini Variation and Einstein Equations
# Verify G_uv = 0 for Schwarzschild by computing curvature numerically

M = 1.0  # Black hole mass (geometric units G=c=1)
rs = 2 * M  # Schwarzschild radius

# Radial grid outside the horizon
r = np.linspace(rs * 1.05, 10 * rs, 800)
dr = r[1] - r[0]

# Schwarzschild metric components
f_r = 1.0 - rs / r  # g_tt = -f(r), g_rr = 1/f(r)

# Numerical derivatives of f(r) using central differences
df = np.gradient(f_r, dr)
d2f = np.gradient(df, dr)

# Ricci scalar for Schwarzschild (should be 0 in vacuum)
# R = f'' + 2f'/r  (in the diagonal static metric)
# More precisely for Schwarzschild: R = 0 analytically
R_numerical = d2f + 2.0 * df / r

# Einstein tensor components (vacuum => should vanish)
# G_tt = (1/r^2)(1 - d/dr(r*f)) for Schwarzschild
d_rf = np.gradient(r * f_r, dr)
G_tt = (1.0 / r**2) * (1.0 - d_rf)

# G_rr = -(1/r^2)(1 - r*f'/f - f) / f^2
# For Schwarzschild: G_rr = (1/(r^2 * f^2)) * (f + r*f' - 1)
G_rr = (1.0 / (r**2 * f_r**2)) * (f_r + r * df - 1.0)

# G_theta_theta involves second derivatives
G_thth = (r / 2.0) * d2f + df

fig, axes = plt.subplots(2, 2, figsize=(12, 9))
fig.patch.set_facecolor('#0a0a1a')
fig.suptitle('Palatini Variation: Schwarzschild Vacuum Verification',
             color='#e2e8f0', fontsize=16, fontweight='bold', y=0.97)

for ax in axes.flat:
    ax.set_facecolor('#0a0a1a')
    ax.tick_params(colors='#94a3b8')
    for spine in ax.spines.values():
        spine.set_color('#334155')
    ax.grid(True, alpha=0.15, color='#334155')

# Panel 1: Ricci scalar
axes[0, 0].plot(r / rs, R_numerical, color='#38bdf8', linewidth=2, label='R(r) numerical')
axes[0, 0].axhline(y=0, color='#f59e0b', linestyle='--', alpha=0.5, label='R = 0 (exact)')
axes[0, 0].set_xlabel('r / r_s', color='#94a3b8')
axes[0, 0].set_ylabel('Ricci scalar R', color='#94a3b8')
axes[0, 0].set_title('Ricci Scalar R(r)', color='#e2e8f0', fontsize=13)
axes[0, 0].legend(facecolor='#0a0a1a', edgecolor='#334155', labelcolor='#94a3b8')
axes[0, 0].set_ylim(-0.05, 0.05)

# Panel 2: G_tt
axes[0, 1].plot(r / rs, G_tt, color='#a78bfa', linewidth=2, label='G_tt numerical')
axes[0, 1].axhline(y=0, color='#f59e0b', linestyle='--', alpha=0.5, label='G_tt = 0 (exact)')
axes[0, 1].set_xlabel('r / r_s', color='#94a3b8')
axes[0, 1].set_ylabel('G_tt', color='#94a3b8')
axes[0, 1].set_title('Einstein Tensor G_tt(r)', color='#e2e8f0', fontsize=13)
axes[0, 1].legend(facecolor='#0a0a1a', edgecolor='#334155', labelcolor='#94a3b8')
axes[0, 1].set_ylim(-0.05, 0.05)

# Panel 3: G_rr
axes[1, 0].plot(r / rs, G_rr, color='#fb923c', linewidth=2, label='G_rr numerical')
axes[1, 0].axhline(y=0, color='#f59e0b', linestyle='--', alpha=0.5, label='G_rr = 0 (exact)')
axes[1, 0].set_xlabel('r / r_s', color='#94a3b8')
axes[1, 0].set_ylabel('G_rr', color='#94a3b8')
axes[1, 0].set_title('Einstein Tensor G_rr(r)', color='#e2e8f0', fontsize=13)
axes[1, 0].legend(facecolor='#0a0a1a', edgecolor='#334155', labelcolor='#94a3b8')
axes[1, 0].set_ylim(-0.05, 0.05)

# Panel 4: All components together (log scale of |value|)
axes[1, 1].semilogy(r / rs, np.abs(R_numerical) + 1e-16, color='#38bdf8', linewidth=2, label='|R|')
axes[1, 1].semilogy(r / rs, np.abs(G_tt) + 1e-16, color='#a78bfa', linewidth=2, label='|G_tt|')
axes[1, 1].semilogy(r / rs, np.abs(G_rr) + 1e-16, color='#fb923c', linewidth=2, label='|G_rr|')
axes[1, 1].set_xlabel('r / r_s', color='#94a3b8')
axes[1, 1].set_ylabel('|Component|', color='#94a3b8')
axes[1, 1].set_title('Numerical Residuals (log scale)', color='#e2e8f0', fontsize=13)
axes[1, 1].legend(facecolor='#0a0a1a', edgecolor='#334155', labelcolor='#94a3b8')
axes[1, 1].set_ylim(1e-8, 1)

plt.tight_layout(rect=[0, 0, 1, 0.94])
plt.savefig('output.png', dpi=130, bbox_inches='tight', facecolor='#0a0a1a')

print("Palatini Variation: Schwarzschild Vacuum Verification")
print("=" * 55)
print(f"Schwarzschild radius: r_s = {rs:.1f} M")
print(f"Radial range: [{r[0]/rs:.2f}, {r[-1]/rs:.2f}] r_s")
print(f"Max |R(r)|:   {np.max(np.abs(R_numerical)):.2e}  (exact: 0)")
print(f"Max |G_tt|:   {np.max(np.abs(G_tt)):.2e}  (exact: 0)")
print(f"Max |G_rr|:   {np.max(np.abs(G_rr)):.2e}  (exact: 0)")
print()
print("All Einstein tensor components vanish to numerical precision,")
print("confirming G_uv = 0 for the Schwarzschild vacuum solution.")
print("The Palatini identity yields the correct vacuum field equations.")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

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