Physics Relativity

# Physics Relativity

## 1. Special Relativity (SR)
Postulates: (1) Laws of physics are invariant in all inertial frames. (2) $c$ is constant. 
- **LT**: $x' = \gamma(x - vt)$, $t' = \gamma(t - vx/c^2)$.
- **Time Dilation**: $\Delta t = \gamma \Delta t_0$.
- **Length Contraction**: $L = L_0 / \gamma$.
- **Mass-Energy**: $E^2 = (pc)^2 + (mc^2)^2$. $E=mc^2$ at rest.

## 2. General Relativity (GR)
Moving beyond inertial frames to accelerated systems.
- **EP**: Gravity is indistinguishable from acceleration.
- **Spacetime Curvature**: Einstein Field Equations (EFE): $G_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}$.
- **Geodesics**: Objects follow the shortest path in curved 4D ST.
- **Predictions**: Gravitational lensing, time dilation in gravity (Pound-Rebka), black hole event horizons, frame dragging (Lense-Thirring effect).

## 3. Mathematical Foundations
- **Metric Tensor ($g_{\mu\nu}$)**: Defines distance in non-Euclidean geometry.
- **Christoffel Symbols**: Connection coefficients for covariant derivatives.
- **Curvature Tensor**: Riemann-Christoffel tensor $R^\rho_{\sigma\mu\nu}$ measures deviation from flatness.

## 4. Key Phenomena
- **Twin Paradox**: Resolution via acceleration/frame change.
- **Precession of Mercury**: First success of GR over Newtonian mechanics.
- **Gravitational Waves**: Ripples in ST metric caused by accelerating massive objects (detected by LIGO).