General Relativity

Introduction

General Relativity, formulated by Albert Einstein in 1915, is a geometric theory of gravitation that describes gravity not as a force, but as a consequence of the curvature of spacetime caused by the presence of mass and energy.

1. Tensor Analysis Foundations

What is a Tensor?

A tensor is a mathematical object that generalizes scalars, vectors, and matrices to arbitrary dimensions. Tensors are fundamental to General Relativity because they describe physical quantities in a way that is independent of the coordinate system chosen.

Rank/Order of Tensors:

  • • Rank 0 (Scalar): A single number, e.g., temperature $T$, mass $m$
  • • Rank 1 (Vector): Has one index, e.g., position $x^\mu$, velocity $v^\mu$
  • • Rank 2 (Matrix): Has two indices, e.g., metric $g_{\mu\nu}$, stress-energy $T_{\mu\nu}$
  • • Rank 3+: Higher-order tensors, e.g., Riemann curvature $R^\rho_{\sigma\mu\nu}$

Contravariant vs Covariant Indices

Tensors can have upper indices (contravariant) and lower indices (covariant). This distinction is crucial for understanding how tensors transform under coordinate changes.

Contravariant vector (upper index): $V^\mu$
Covariant vector (lower index): $V_\mu$
Mixed tensor: $T^\mu_\nu$ (one upper, one lower)

Transformation Laws: Under a coordinate transformation $x^\mu \to x'^{\mu}$:

$$V'^{\mu} = \frac{\partial x'^{\mu}}{\partial x^\nu} V^\nu \quad \text{(contravariant)}$$
$$V'_\mu = \frac{\partial x^\nu}{\partial x'^{\mu}} V_\nu \quad \text{(covariant)}$$

The Metric Tensor: Raising and Lowering Indices

The metric tensor $g_{\mu\nu}$ and its inverse $g^{\mu\nu}$ are used to convert between contravariant and covariant indices (raising and lowering):

$$V_\mu = g_{\mu\nu} V^\nu \quad \text{(lowering an index)}$$
$$V^\mu = g^{\mu\nu} V_\nu \quad \text{(raising an index)}$$

The metric and its inverse satisfy the identity:

$$g^{\mu\lambda} g_{\lambda\nu} = \delta^\mu_\nu$$

where $\delta^\mu_\nu$ is the Kronecker delta (identity matrix): $\delta^\mu_\nu = 1$ if $\mu = \nu$, and $0$ otherwise.

Einstein Summation Convention

When an index appears twice in a term (once upper, once lower), it implies summation over all values of that index:

$$A^\mu B_\mu \equiv \sum_{\mu=0}^{3} A^\mu B_\mu = A^0 B_0 + A^1 B_1 + A^2 B_2 + A^3 B_3$$

This convention eliminates the need to write summation symbols explicitly, making equations more compact. Examples in 4D spacetime ($\mu, \nu \in \{0,1,2,3\}$):

$$x^\mu x_\mu = -(ct)^2 + x^2 + y^2 + z^2 \quad \text{(spacetime interval)}$$
$$g_{\mu\nu} dx^\mu dx^\nu = \text{(line element)}$$
$$R = g^{\mu\nu} R_{\mu\nu} \quad \text{(Ricci scalar)}$$

Tensor Operations

1. Tensor Addition: Only tensors of the same rank and type can be added:

$$T^{\mu\nu} = A^{\mu\nu} + B^{\mu\nu}$$

2. Tensor Product (Outer Product): Combines tensors to create higher-rank tensors:

$$T^{\mu\nu\rho} = A^{\mu\nu} B^\rho \quad \text{(rank 2 + rank 1 = rank 3)}$$

3. Contraction: Reduces tensor rank by summing over paired indices:

$$T^\mu_\mu = T^0_0 + T^1_1 + T^2_2 + T^3_3 \quad \text{(trace)}$$
$$R = g^{\mu\nu} R_{\mu\nu} \quad \text{(contracting Ricci tensor gives scalar)}$$

4. Symmetrization and Antisymmetrization:

$$T_{(\mu\nu)} = \frac{1}{2}(T_{\mu\nu} + T_{\nu\mu}) \quad \text{(symmetric part)}$$
$$T_{[\mu\nu]} = \frac{1}{2}(T_{\mu\nu} - T_{\nu\mu}) \quad \text{(antisymmetric part)}$$

Why Tensors in General Relativity?

Coordinate Independence: Physical laws must be the same for all observers, regardless of their coordinate system. Tensor equations maintain their form under arbitrary coordinate transformations, embodying the principle of general covariance.

"If tensor equation $A^{\mu\nu} = B^{\mu\nu}$ holds in one coordinate system, it holds in all coordinate systems."

Key Tensors in General Relativity:

  • • Metric tensor $g_{\mu\nu}$: Defines distances and angles in spacetime
  • • Riemann tensor $R^\rho_{\sigma\mu\nu}$: Measures spacetime curvature
  • • Ricci tensor $R_{\mu\nu}$: Contraction of Riemann tensor
  • • Einstein tensor $G_{\mu\nu}$: Left side of field equations
  • • Stress-energy tensor $T_{\mu\nu}$: Describes matter and energy
  • • Four-velocity $u^\mu$: Velocity in spacetime

Index Gymnastics: A Practical Example

Let's compute $g^{\mu\rho} R_{\mu\nu}$ (raising the first index of Ricci tensor):

$$R^\rho_\nu = g^{\mu\rho} R_{\mu\nu}$$

Step-by-step:

  1. 1. $\mu$ appears twice (once upper in $g^{\mu\rho}$, once lower in $R_{\mu\nu}$) → sum over $\mu$
  2. 2. Free indices are $\rho$ (upper) and $\nu$ (lower)
  3. 3. Result: $R^\rho_\nu$ is a rank-2 mixed tensor

The Ricci scalar is obtained by contracting both indices:

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

2. Einstein-Hilbert Action Approach

The Variational Principle

The Einstein field equations can be derived from a variational principle using the Einstein-Hilbert action. This approach reveals the deep connection between geometry and physics, showing that General Relativity follows from a fundamental principle: spacetime geometry evolves to extremize the action.

The Einstein-Hilbert Action

The action for the gravitational field is:

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

where $R$ is the Ricci scalar (trace of the Ricci tensor), $g = \det(g_{\mu\nu})$ is the determinant of the metric tensor, and the integral is over 4-dimensional spacetime.

Including matter, the total action is:

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

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

Why This Form?

The Einstein-Hilbert action is constructed to satisfy several key requirements:

  • • General covariance: The action must be a scalar under coordinate transformations
  • • Simplicity: $R$ is the simplest scalar constructed from the metric and its first two derivatives
  • • Newtonian limit: Must reduce to Newton's gravity in the weak-field, slow-motion limit
  • • Volume element: $\sqrt{-g}\,d^4x$ is the invariant spacetime volume element

The Stress-Energy Tensor from Matter

The stress-energy tensor is defined by varying the matter action with respect to the metric:

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

Or equivalently:

$$T_{\mu\nu} = \frac{-2}{\sqrt{-g}}\frac{\partial(\sqrt{-g}\mathcal{L}_M)}{\partial g^{\mu\nu}} + g_{\mu\nu}\mathcal{L}_M$$

Variation of the Metric

To derive the field equations, we vary the action with respect to the metric: $g_{\mu\nu} \to g_{\mu\nu} + \delta g_{\mu\nu}$. The principle of least action states:

$$\delta S_{total} = 0$$

Step 1: Variation of the Volume Element

First, we need:

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

Step 2: Variation of the Ricci Scalar

Since $R = g^{\mu\nu}R_{\mu\nu}$, we have:

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

The second term can be shown to be a total derivative (Palatini identity):

$$g^{\mu\nu}\delta R_{\mu\nu} = \nabla_\lambda V^\lambda$$

which vanishes upon integration (assuming suitable boundary conditions). Thus:

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

Deriving Einstein's Field Equations

Varying the Einstein-Hilbert action:

$$\delta S_{EH} = \frac{c^4}{16\pi G}\int \left[\delta R\sqrt{-g} + R\delta\sqrt{-g}\right]d^4x$$

Substituting our variations:

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

Using $\delta g^{\mu\nu} = -g^{\mu\rho}g^{\nu\sigma}\delta g_{\rho\sigma}$, and combining terms:

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

For matter:

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

Setting $\delta S_{total} = 0$ for arbitrary $\delta g^{\mu\nu}$ gives:

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

These are Einstein's field equations!

Including the Cosmological Constant

Einstein originally added a cosmological constant term to the action:

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

This modifies the field equations to:

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

The cosmological constant $\Lambda$ can be interpreted as the energy density of the vacuum and is essential for explaining the observed accelerated expansion of the universe.

Alternative Form: The Einstein Tensor

Define the Einstein tensor:

$$G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R$$

The field equations become:

$$G_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4}T_{\mu\nu}$$

The Einstein tensor $G_{\mu\nu}$ automatically satisfies the contracted Bianchi identity:

$$\nabla_\mu G^{\mu\nu} = 0$$

This ensures local energy-momentum conservation: $\nabla_\mu T^{\mu\nu} = 0$.

Comparison with Other Theories

The Einstein-Hilbert action can be compared with other approaches:

  • Newton's Gravity: Action $S \sim \int |\nabla \phi|^2 d^3x$ (Poisson equation)
  • Electromagnetism: Action $S \sim \int F_{\mu\nu}F^{\mu\nu}\sqrt{-g}\,d^4x$ (Maxwell equations)
  • Modified Gravity: $f(R)$ theories use $S \sim \int f(R)\sqrt{-g}\,d^4x$

Physical Interpretation

The variational principle has deep physical meaning:

  • • Spacetime geometry (metric) is a dynamical field, not a fixed background
  • • The field equations emerge from extremizing the total curvature
  • • Matter tells spacetime how to curve (via stress-energy tensor)
  • • Curved spacetime tells matter how to move (via geodesic equation)

This elegant formulation shows that gravity is fundamentally different from other forces: it is the dynamics of spacetime geometry itself, derived from a simple and beautiful action principle.

3. Metric Tensor and Line Element

The metric tensor $g_{\mu\nu}$ defines the geometry of spacetime. The infinitesimal proper time interval is:

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

In Minkowski spacetime (flat space), the metric is:

$$\eta_{\mu\nu} = \text{diag}(-1, 1, 1, 1)$$
$$ds^2 = -c^2dt^2 + dx^2 + dy^2 + dz^2$$

4. Christoffel Symbols (Connection Coefficients)

Derivation from Metric Compatibility

The Christoffel symbols describe how the basis vectors change from point to point in curved spacetime. We derive them from the condition of metric compatibility:

$$\nabla_\lambda g_{\mu\nu} = 0$$

Expanding the covariant derivative:

$$\partial_\lambda g_{\mu\nu} - \Gamma^\sigma_{\lambda\mu} g_{\sigma\nu} - \Gamma^\sigma_{\lambda\nu} g_{\mu\sigma} = 0$$

Step 1: Write this equation for three cyclic permutations of indices:

$$\partial_\lambda g_{\mu\nu} = \Gamma^\sigma_{\lambda\mu} g_{\sigma\nu} + \Gamma^\sigma_{\lambda\nu} g_{\mu\sigma} \quad (1)$$$$\partial_\mu g_{\nu\lambda} = \Gamma^\sigma_{\mu\nu} g_{\sigma\lambda} + \Gamma^\sigma_{\mu\lambda} g_{\nu\sigma} \quad (2)$$$$\partial_\nu g_{\lambda\mu} = \Gamma^\sigma_{\nu\lambda} g_{\sigma\mu} + \Gamma^\sigma_{\nu\mu} g_{\lambda\sigma} \quad (3)$$

Step 2: Add equations (2) and (3), then subtract (1):

$$\partial_\mu g_{\nu\lambda} + \partial_\nu g_{\lambda\mu} - \partial_\lambda g_{\mu\nu} = 2\Gamma^\sigma_{\mu\nu} g_{\sigma\lambda}$$

Step 3: Multiply both sides by $g^{\lambda\rho}$ (inverse metric) to isolate $\Gamma$:

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

Final result: Using standard notation $\frac{\partial}{\partial x^\mu} = \partial_\mu$:

$$\Gamma^\lambda_{\mu\nu} = \frac{1}{2}g^{\lambda\sigma}\left(\frac{\partial g_{\sigma\nu}}{\partial x^\mu} + \frac{\partial g_{\sigma\mu}}{\partial x^\nu} - \frac{\partial g_{\mu\nu}}{\partial x^\sigma}\right)$$

Important properties:

  • • Symmetric in lower indices: $\Gamma^\lambda_{\mu\nu} = \Gamma^\lambda_{\nu\mu}$
  • • Not a tensor (transforms inhomogeneously)
  • • Vanishes in locally inertial coordinates
  • • In 4D, there are $4 \times 10 = 40$ independent components

5. Riemann Curvature Tensor

Derivation from Commutator of Covariant Derivatives

The Riemann tensor measures the curvature of spacetime. It arises when we compute the commutator of two covariant derivatives acting on a vector field $V^\rho$:

$$[\nabla_\mu, \nabla_\nu]V^\rho = R^\rho_{\sigma\mu\nu} V^\sigma$$

Step 1: Apply first covariant derivative:

$$\nabla_\nu V^\rho = \partial_\nu V^\rho + \Gamma^\rho_{\nu\sigma} V^\sigma$$

Step 2: Apply second covariant derivative $\nabla_\mu$ to this result:

$$\nabla_\mu\nabla_\nu V^\rho = \partial_\mu(\partial_\nu V^\rho) + \partial_\mu(\Gamma^\rho_{\nu\sigma} V^\sigma) + \Gamma^\rho_{\mu\lambda}(\partial_\nu V^\lambda + \Gamma^\lambda_{\nu\sigma} V^\sigma) - \Gamma^\lambda_{\mu\nu}(\partial_\lambda V^\rho + \Gamma^\rho_{\lambda\sigma} V^\sigma)$$

Step 3: Expand and collect terms:

$$\nabla_\mu\nabla_\nu V^\rho = \partial_\mu\partial_\nu V^\rho + (\partial_\mu\Gamma^\rho_{\nu\sigma})V^\sigma + \Gamma^\rho_{\nu\sigma}\partial_\mu V^\sigma + \Gamma^\rho_{\mu\lambda}\partial_\nu V^\lambda + \Gamma^\rho_{\mu\lambda}\Gamma^\lambda_{\nu\sigma} V^\sigma - \Gamma^\lambda_{\mu\nu}\partial_\lambda V^\rho - \Gamma^\lambda_{\mu\nu}\Gamma^\rho_{\lambda\sigma} V^\sigma$$

Step 4: Compute the commutator $[\nabla_\mu, \nabla_\nu]V^\rho = \nabla_\mu\nabla_\nu V^\rho - \nabla_\nu\nabla_\mu V^\rho$. The second derivative $\partial_\mu\partial_\nu V^\rho$ cancels, leaving:

$$[\nabla_\mu, \nabla_\nu]V^\rho = \left(\partial_\mu\Gamma^\rho_{\nu\sigma} - \partial_\nu\Gamma^\rho_{\mu\sigma} + \Gamma^\rho_{\mu\lambda}\Gamma^\lambda_{\nu\sigma} - \Gamma^\rho_{\nu\lambda}\Gamma^\lambda_{\mu\sigma}\right)V^\sigma$$

Final Formula: The Riemann curvature tensor is defined as:

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

Symmetries of the Riemann Tensor

Lowering the first index with the metric: $R_{\rho\sigma\mu\nu} = g_{\rho\lambda}R^\lambda_{\sigma\mu\nu}$

  • • Antisymmetric in first pair: $R_{\rho\sigma\mu\nu} = -R_{\sigma\rho\mu\nu}$
  • • Antisymmetric in second pair: $R_{\rho\sigma\mu\nu} = -R_{\rho\sigma\nu\mu}$
  • • Symmetric under pair exchange: $R_{\rho\sigma\mu\nu} = R_{\mu\nu\rho\sigma}$
  • • First Bianchi identity: $R_{\rho\sigma\mu\nu} + R_{\rho\mu\nu\sigma} + R_{\rho\nu\sigma\mu} = 0$ (cyclic sum vanishes)
  • • Second Bianchi identity: $\nabla_\lambda R_{\rho\sigma\mu\nu} + \nabla_\mu R_{\rho\sigma\nu\lambda} + \nabla_\nu R_{\rho\sigma\lambda\mu} = 0$

In 4D spacetime, these symmetries reduce the number of independent components from $4^4 = 256$ to just 20 independent components.

6. Ricci Tensor and Scalar

The Ricci tensor is a contraction of the Riemann tensor:

$$R_{\mu\nu} = R^\lambda_{\mu\lambda\nu} = g^{\rho\sigma}R_{\rho\mu\sigma\nu}$$

The Ricci scalar (scalar curvature) is the trace of the Ricci tensor:

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

7. Einstein Tensor

The Einstein tensor combines the Ricci tensor and scalar to satisfy the contracted Bianchi identities:

$$G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R$$

This tensor is automatically divergence-free: $\nabla^\mu G_{\mu\nu} = 0$, which ensures conservation of energy-momentum.

8. Einstein Field Equations

Derivation from the Einstein-Hilbert Action

The Einstein field equations can be derived from the variational principle. Start with the Einstein-Hilbert action:

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

Step 1: Vary the action with respect to the metric $g^{\mu\nu}$:

$$\delta S = \frac{c^4}{16\pi G}\int \left(\delta R \sqrt{-g} + R \delta\sqrt{-g}\right)d^4x + \int \frac{\delta(\mathcal{L}_{\text{matter}}\sqrt{-g})}{\delta g^{\mu\nu}}\delta g^{\mu\nu}\,d^4x$$

Step 2: Use the variation formulas:

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

Step 3: The term $g^{\mu\nu}\delta R_{\mu\nu}$ integrates to a surface term (Gauss theorem):

$$\int g^{\mu\nu}\delta R_{\mu\nu}\sqrt{-g}\,d^4x = \int \nabla_\lambda K^\lambda\sqrt{-g}\,d^4x = \oint K^\lambda dS_\lambda = 0$$

Step 4: Define the stress-energy tensor from matter variation:

$$T_{\mu\nu} = -\frac{2}{\sqrt{-g}}\frac{\delta(\mathcal{L}_{\text{matter}}\sqrt{-g})}{\delta g^{\mu\nu}} = 2\frac{\partial\mathcal{L}_{\text{matter}}}{\partial g^{\mu\nu}} - g_{\mu\nu}\mathcal{L}_{\text{matter}}$$

Step 5: Collect all terms and set $\delta S = 0$:

$$\frac{c^4}{16\pi G}\left(R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R\right) = \frac{1}{2}T_{\mu\nu}$$

The Einstein Field Equations

The fundamental equations of General Relativity relate spacetime curvature to matter-energy content:

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

Equivalently, using the Einstein tensor $G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R$:

$$G_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4}T_{\mu\nu}$$

where:

  • • $G_{\mu\nu}$ is the Einstein tensor (describes spacetime curvature)
  • • $\Lambda$ is the cosmological constant (vacuum energy density)
  • • $T_{\mu\nu}$ is the stress-energy tensor (matter-energy content)
  • • $G$ is Newton's gravitational constant
  • • $c$ is the speed of light

Physical Interpretation: "Spacetime tells matter how to move; matter tells spacetime how to curve" (John Archibald Wheeler)

9. Stress-Energy Tensor

For a perfect fluid:

$$T_{\mu\nu} = (\rho + p)u_\mu u_\nu + pg_{\mu\nu}$$

where $\rho$ is energy density, $p$ is pressure, and $u^\mu$ is the four-velocity.

For electromagnetic fields:

$$T_{\mu\nu} = \frac{1}{\mu_0}\left(F_{\mu\alpha}F_\nu^\alpha - \frac{1}{4}g_{\mu\nu}F_{\alpha\beta}F^{\alpha\beta}\right)$$

10. Geodesic Equation

Free particles follow geodesics, the straightest possible paths in curved spacetime:

$$\frac{d^2x^\mu}{d\tau^2} + \Gamma^\mu_{\alpha\beta}\frac{dx^\alpha}{d\tau}\frac{dx^\beta}{d\tau} = 0$$

This can also be written using the covariant derivative along the path:

$$\frac{Du^\mu}{D\tau} = u^\nu\nabla_\nu u^\mu = 0$$

11. Schwarzschild Solution

Derivation of the Schwarzschild Metric

We seek a spherically symmetric, static, vacuum solution to Einstein's equations ($R_{\mu\nu} = 0$).

Step 1: Assume the most general spherically symmetric, static metric ansatz:

$$ds^2 = -e^{2\alpha(r)}c^2dt^2 + e^{2\beta(r)}dr^2 + r^2(d\theta^2 + \sin^2\theta d\phi^2)$$

where $\alpha(r)$ and $\beta(r)$ are unknown functions, and $d\Omega^2 = d\theta^2 + \sin^2\theta d\phi^2$ is the metric on the unit 2-sphere.

Step 2: The non-zero Christoffel symbols include:

$$\Gamma^t_{tr} = \alpha'(r), \quad \Gamma^r_{tt} = \alpha'e^{2(\alpha-\beta)}c^2, \quad \Gamma^r_{rr} = \beta'$$$$\Gamma^r_{\theta\theta} = -re^{-2\beta}, \quad \Gamma^r_{\phi\phi} = -re^{-2\beta}\sin^2\theta$$$$\Gamma^\theta_{r\theta} = \frac{1}{r}, \quad \Gamma^\theta_{\phi\phi} = -\sin\theta\cos\theta, \quad \Gamma^\phi_{r\phi} = \frac{1}{r}, \quad \Gamma^\phi_{\theta\phi} = \cot\theta$$

Step 3: Compute the non-zero components of Ricci tensor $R_{\mu\nu} = 0$:

$$R_{tt} = e^{2(\alpha-\beta)}\left(\alpha'' + \alpha'^2 - \alpha'\beta' + \frac{2\alpha'}{r}\right) = 0$$$$R_{rr} = -\alpha'' - \alpha'^2 + \alpha'\beta' + \frac{2\beta'}{r} = 0$$$$R_{\theta\theta} = e^{-2\beta}\left(r\beta' - r\alpha' - 1\right) + 1 = 0$$

Step 4: From $R_{tt} + R_{rr} = 0$:

$$\frac{2}{r}(\alpha' + \beta') = 0 \quad \Rightarrow \quad \alpha + \beta = \text{const}$$

Choose the constant to be zero (by rescaling time): $\beta = -\alpha$.

Step 5: Substituting $\beta = -\alpha$ into $R_{\theta\theta} = 0$:

$$e^{-2\beta}(1 + 2r\alpha') = 1$$$$e^{2\alpha}(1 + 2r\alpha') = 1$$$$\frac{d}{dr}(re^{2\alpha}) = 1$$

Step 6: Integrating:

$$re^{2\alpha} = r + C$$$$e^{2\alpha} = 1 + \frac{C}{r}$$

Step 7: Matching to Newtonian limit as $r \to \infty$ requires $\Phi = -GM/r$, giving:

$$e^{2\alpha} = 1 - \frac{2GM}{c^2r} = 1 - \frac{r_s}{r}$$

The Schwarzschild Metric

The unique spherically symmetric, static, vacuum solution to Einstein's equations:

$$ds^2 = -\left(1-\frac{2GM}{c^2r}\right)c^2dt^2 + \left(1-\frac{2GM}{c^2r}\right)^{-1}dr^2 + r^2(d\theta^2 + \sin^2\theta d\phi^2)$$

The Schwarzschild radius (event horizon for black holes):

$$r_s = \frac{2GM}{c^2} \approx 2.95 \text{ km} \left(\frac{M}{M_\odot}\right)$$

Key Properties:

  • • Asymptotically flat: $ds^2 \to \eta_{\mu\nu}dx^\mu dx^\nu$ as $r \to \infty$
  • • Coordinate singularity at $r = r_s$ (removable by coordinate transformation)
  • • Physical singularity at $r = 0$ (curvature invariants diverge)
  • • Describes spacetime outside any spherically symmetric mass distribution
  • • Valid for stars, planets, and black holes

12. Weak Field Limit and Gravitational Waves

For weak fields, we write $g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}$ where $|h_{\mu\nu}| \ll 1$. To first order, the field equations become:

$$\Box \bar{h}_{\mu\nu} = -\frac{16\pi G}{c^4}T_{\mu\nu}$$

where $\bar{h}_{\mu\nu} = h_{\mu\nu} - \frac{1}{2}\eta_{\mu\nu}h$ is the trace-reversed metric perturbation.

In vacuum ($T_{\mu\nu} = 0$), this gives the wave equation:

$$\Box \bar{h}_{\mu\nu} = 0$$

describing gravitational waves propagating at the speed of light.

13. Newtonian Limit

In the weak field, slow motion limit, GR reduces to Newton's theory. For $h_{00} = -2\Phi/c^2$:

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

This is Poisson's equation for the Newtonian gravitational potential.