big-bang-claude-gr

big-bang-claude-gr.md

The key is applying Einstein's field equations to a homogeneous, isotropic universe.

Starting Point: Einstein Field Equations

The foundation is Einstein's field equation:

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

Where:

• $R_{\mu\nu}$ is the Ricci curvature tensor
• $R$ is the Ricci scalar
• $g_{\mu\nu}$ is the metric tensor
• $\Lambda$ is the cosmological constant
• $T_{\mu\nu}$ is the stress-energy tensor

The Cosmological Principle and FRW Metric

We assume the universe is homogeneous and isotropic on large scales. This leads to the Friedmann-Lemaître-Robertson-Walker (FRW) metric:

$$ds^2 = -c^2dt^2 + a(t)^2\left[\frac{dr^2}{1-kr^2} + r^2(d\theta^2 + \sin^2\theta , d\phi^2)\right]$$

Where:

• $a(t)$ is the scale factor
• $k = 0, +1, -1$ for flat, closed, open universes respectively

Calculating the Curvature Tensors

For the FRW metric, the non-zero Christoffel symbols include:

$$\Gamma^0_{ij} = \frac{\dot{a}a}{c^2}g_{ij}, \quad \Gamma^i_{0j} = \frac{\dot{a}}{a}\delta^i_j$$

The Ricci tensor components are:

$$R_{00} = -3\frac{\ddot{a}}{a}$$

$$R_{ij} = \left(\frac{\ddot{a}}{a} + 2\frac{\dot{a}^2}{a^2} + \frac{2kc^2}{a^2}\right)g_{ij}$$

The Ricci scalar becomes:

$$R = \frac{6}{c^2}\left(\frac{\ddot{a}}{a} + \frac{\dot{a}^2}{a^2} + \frac{kc^2}{a^2}\right)$$

The Stress-Energy Tensor

For a perfect fluid (reasonable for the early universe):

$$T_{\mu\nu} = (\rho c^2 + p)u_\mu u_\nu + pg_{\mu\nu}$$

In the comoving frame: $u^0 = 1/c$, $u^i = 0$

This gives: $$T_{00} = \rho c^2, \quad T_{ij} = pg_{ij}$$

Deriving the Friedmann Equations

Substituting into Einstein's equations:

The (0,0) component gives the first Friedmann equation:

$$\left(\frac{\dot{a}}{a}\right)^2 = \frac{8\pi G\rho}{3} - \frac{kc^2}{a^2} + \frac{\Lambda c^2}{3}$$

The (i,j) components give the acceleration equation:

$$\frac{\ddot{a}}{a} = -\frac{4\pi G}{3c^2}(\rho c^2 + 3p) + \frac{\Lambda c^2}{3}$$

The Continuity Equation

From covariant conservation of energy-momentum ($\nabla_\mu T^{\mu\nu} = 0$):

$$\dot{\rho} + 3\frac{\dot{a}}{a}\left(\rho + \frac{p}{c^2}\right) = 0$$

The Path to Singularity

Step 1: Define the Hubble Parameter

$$H(t) = \frac{\dot{a}}{a}$$

Step 2: For different matter types

• Radiation: $p = \frac{\rho c^2}{3}$ → $\rho \propto a^{-4}$
• Matter: $p = 0$ → $\rho \propto a^{-3}$
• Dark energy: $p = -\rho c^2$ → $\rho = \text{constant}$

Step 3: The Critical Density

$$\rho_{\text{crit}} = \frac{3H^2}{8\pi G}$$

Step 4: Going Backward in Time

From the first Friedmann equation, if we ignore $\Lambda$ and $k$ for simplicity:

$$H^2 = \frac{8\pi G\rho}{3}$$

For a radiation-dominated universe in the early stages: $$\rho \propto a^{-4}$$

So: $$H^2 \propto a^{-4}$$

This gives: $$\frac{\dot{a}}{a} \propto a^{-2}$$

Solving: $$a(t) \propto t^{1/2}$$

As $t \to 0$, we get $a(t) \to 0$ and $\rho \to \infty$

The Raychaudhuri Equation

A more rigorous approach uses the Raychaudhuri equation for geodesic congruences:

$$\frac{d\theta}{d\tau} = -\frac{1}{3}\theta^2 - \sigma_{ab}\sigma^{ab} + \omega_{ab}\omega^{ab} - R_{ab}u^au^b$$

For our case with $\theta = 3H$, $\sigma_{ab} = 0$, $\omega_{ab} = 0$:

$$\frac{d\theta}{d\tau} = -\frac{1}{3}\theta^2 - R_{ab}u^au^b$$

Since $R_{ab}u^au^b = R_{00} = 3\frac{\ddot{a}}{a}$:

$$\frac{d}{dt}(3H) = -3H^2 - 3\frac{\ddot{a}}{a}$$

Using the acceleration equation with dominant matter/radiation: $$\frac{\ddot{a}}{a} = -\frac{4\pi G}{3c^2}(\rho c^2 + 3p) < 0$$

This shows that $\theta$ (expansion rate) was larger in the past, leading inevitably to a singularity.

Hawking-Penrose Singularity Theorems

The mathematical conditions for singularities:

1. Energy condition: $R_{ab}u^au^b \geq 0$ (satisfied by normal matter)
2. Causality condition: No closed timelike curves
3. Initial condition: Existence of a trapped surface or similar

These conditions, when satisfied, mathematically prove that spacetime is geodesically incomplete - singularities are unavoidable.

Temperature Evolution

The mathematical relationship for photon temperature: $$T \propto a^{-1}$$

Combined with $a(t) \propto t^{1/2}$ in radiation era: $$T \propto t^{-1/2}$$

As $t \to 0$: $T \to \infty$

Key Mathematical Results

1. Scale factor: $a(t \to 0) \to 0$
2. Density: $\rho(t \to 0) \to \infty$
3. Temperature: $T(t \to 0) \to \infty$
4. Curvature: $R(t \to 0) \to \infty$

The mathematics inexorably leads to a state where all these quantities become infinite at $t = 0$ - the Big Bang singularity.

The equations don't tell us what "caused" this state, but they rigorously demonstrate that if we accept General Relativity and reasonable energy conditions, then tracing the expanding universe backward in time leads mathematically to a singular beginning where the known laws of physics break down.

This is why the Big Bang emerges from the mathematics rather than being assumed - it's an inevitable consequence of applying Einstein's field equations to an expanding, homogeneous universe.