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Einstein's field equations are the cornerstone of General Relativity, describing how matter and energy curve spacetime. In this post, we'll derive these equations from first principles using the principle of least action.

The Foundations

1. Spacetime Metric

In General Relativity, spacetime is described by a 4-dimensional Lorentzian manifold with metric tensor $g_{\mu\nu}$ . The line element is:

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

where we use Einstein summation convention (repeated indices are summed over).

2. The Equivalence Principle

The equivalence principle states that locally, the effects of gravity are indistinguishable from acceleration. This leads us to seek a geometric description of gravity.

Building the Mathematical Framework

Christoffel Symbols

The connection coefficients (Christoffel symbols) describe how vectors change under parallel transport:

\Gamma^\lambda_{\mu\nu} = \frac{1}{2}g^{\lambda\rho}\left(\partial_\mu g_{\nu\rho} + \partial_\nu g_{\mu\rho} - \partial_\rho g_{\mu\nu}\right)

These symbols are not tensors but transform in a specific way under coordinate transformations.

Riemann Curvature Tensor

The curvature of spacetime is encoded in the Riemann tensor:

R^\rho_{\ \sigma\mu\nu} = \partial_\mu \Gamma^\rho_{\nu\sigma} - \partial_\nu \Gamma^\rho_{\mu\sigma} + \Gamma^\rho_{\mu\lambda}\Gamma^\lambda_{\nu\sigma} - \Gamma^\rho_{\nu\lambda}\Gamma^\lambda_{\mu\sigma}

This tensor satisfies several important symmetries:

$R_{\rho\sigma\mu\nu} = -R_{\sigma\rho\mu\nu}$
$R_{\rho\sigma\mu\nu} = -R_{\rho\sigma\nu\mu}$
$R_{\rho\sigma\mu\nu} = R_{\mu\nu\rho\sigma}$

Ricci Tensor and Scalar

By contracting the Riemann tensor, we obtain the Ricci tensor:

R_{\mu\nu} = R^\lambda_{\ \mu\lambda\nu}

Further contraction gives the Ricci scalar:

R = g^{\mu\nu}R_{\mu\nu}

The Action Principle

Einstein-Hilbert Action

The dynamics of spacetime are determined by the Einstein-Hilbert action:

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

where:

$c$ is the speed of light
$G$ is Newton's gravitational constant
$g = \det(g_{\mu\nu})$ is the determinant of the metric
$R$ is the Ricci scalar

Matter Action

Matter fields contribute their own action:

S_M = \int d^4x \sqrt{-g} \, \mathcal{L}_M

where $\mathcal{L}_M$ is the matter Lagrangian density.

Total Action

The total action is:

S = S_{EH} + S_M

Deriving the Field Equations

Variation of the Action

To find the equations of motion, we vary the total action with respect to the metric:

\delta S = \delta S_{EH} + \delta S_M = 0

Variation of the Einstein-Hilbert Action

The variation of the Einstein-Hilbert action involves several steps:

Variation of the determinant:
$\delta \sqrt{-g} = -\frac{1}{2}\sqrt{-g} \, g_{\mu\nu} \delta g^{\mu\nu}$
Variation of the Ricci scalar:
$\delta R = R_{\mu\nu} \delta g^{\mu\nu} + g^{\mu\nu} \delta R_{\mu\nu}$
The second term is a total derivative:
$g^{\mu\nu} \delta R_{\mu\nu} = \nabla_\lambda \left(g^{\mu\nu} \delta \Gamma^\lambda_{\mu\nu} - g^{\lambda\nu} \delta \Gamma^\rho_{\rho\nu}\right)$

This total derivative vanishes when integrated (assuming appropriate boundary conditions).

The Result

After careful calculation, the variation of the Einstein-Hilbert action yields:

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

Energy-Momentum Tensor

The variation of the matter action defines the energy-momentum tensor:

\delta S_M = -\frac{1}{2} \int d^4x \sqrt{-g} \, T_{\mu\nu} \delta g^{\mu\nu}

where:

T_{\mu\nu} = -\frac{2}{\sqrt{-g}} \frac{\delta(\sqrt{-g}\mathcal{L}_M)}{\delta g^{\mu\nu}}

Einstein's Field Equations

Setting $\delta S = 0$ and requiring this to hold for arbitrary variations $\delta g^{\mu\nu}$ , we obtain:

R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R = \frac{8\pi G}{c^4} T_{\mu\nu}

These are Einstein's field equations in their standard form.

Alternative Forms

Trace-reversed form: Taking the trace: $R - 2R = \frac{8\pi G}{c^4} T$ where $T = g^{\mu\nu}T_{\mu\nu}$

This gives: $R = -\frac{8\pi G}{c^4} T$

Substituting back:
$R_{\mu\nu} = \frac{8\pi G}{c^4} \left(T_{\mu\nu} - \frac{1}{2}g_{\mu\nu}T\right)$
With cosmological constant: Including a cosmological constant $\Lambda$ :
$R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}$

Key Properties

Conservation Laws

The Bianchi identity $\nabla^\mu R_{\mu\nu} = \frac{1}{2}\nabla_\nu R$ ensures:

\nabla^\mu T_{\mu\nu} = 0

This represents the conservation of energy and momentum.

Newtonian Limit

In the weak field limit with slow motion:

$g_{00} \approx -(1 + 2\Phi/c^2)$ where $\Phi$ is the Newtonian potential
$g_{ij} \approx \delta_{ij}$ (spatial components)
$T_{00} \approx \rho c^2$ (mass density)

The field equations reduce to Poisson's equation:

\nabla^2 \Phi = 4\pi G \rho

Geometric Interpretation

Einstein's equations can be written compactly as:

G_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}

where $G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R$ is the Einstein tensor.

This equation states:

Geometry (left side) = Energy-Momentum (right side)

The curvature of spacetime at a point is directly proportional to the energy and momentum content at that point.

Conclusion

Einstein's field equations emerge naturally from:

The principle of general covariance
The requirement that the field equations be second-order
The principle of least action applied to the Einstein-Hilbert action

These equations have been tested to extraordinary precision and form the foundation for our understanding of:

Black holes
Gravitational waves
Cosmology
GPS satellites (requiring relativistic corrections)

The derivation shows how profound physical insights can emerge from elegant mathematical principles.

DANIEL MARIN

Deriving Einstein's Field Equations from First Principles