Energy-Momentum Relation: $E^2 = (pc)^2 + (mc^2)^2$

Use: Unified relationship between energy, momentum, and rest mass. For massless particles (e.g., photons), $m=0$, so $E=pc$.

Relativistic Force: $\vec{F} = \frac{d\vec{p}}{dt}$ (Newton's 2nd Law still holds with relativistic momentum).

Use: Defines force in special relativity. Note that $\vec{F} \ne m\vec{a}$ at high speeds, as mass effectively depends on velocity.

Rocket Equation: $\Delta v = u \ln(M_0/M_f)$ (non-relativistic), $\Delta v = c \tanh(u/c \ln(M_0/M_f))$ (relativistic, for $u$ relative exhaust speed)

Use: Calculates the change in velocity of a rocket based on its exhaust speed and mass ratio.
Drawback: Relativistic version is more complex but necessary for high-speed interstellar travel concepts.

13. 4-Vectors

Position 4-Vector: $X^\mu = (ct, x, y, z)$
- Use: Combines space and time into a single entity in spacetime.
Velocity 4-Vector: $U^\mu = (\gamma c, \gamma \vec{v})$
- Use: Provides a relativistic definition of velocity that transforms simply under Lorentz transformations.
Momentum 4-Vector: $P^\mu = (E/c, \vec{p})$
- Use: Unifies energy and momentum into a single 4-vector. Conservation of $P^\mu$ implies conservation of both energy and momentum.
Force 4-Vector: $F^\mu = (\gamma/c \frac{dE}{dt}, \gamma \vec{F})$
- Use: Relativistic generalization of force.
Inner Product (Minkowski Metric): $A \cdot B = A_0B_0 - A_1B_1 - A_2B_2 - A_3B_3$ (invariant).
- Use: A scalar product in Minkowski spacetime that is invariant under Lorentz transformations.
- Drawback: The metric signature (e.g., $+---$ or $-+++$) must be consistent.
Norm of Momentum 4-Vector: $P \cdot P = (E/c)^2 - |\vec{p}|^2 = m^2c^2$ (invariant, $m$ is rest mass).
- Use: Proves that rest mass is a relativistic invariant.

14. General Relativity

Equivalence Principle: Locally, gravity is indistinguishable from acceleration.
- Use: Fundamental postulate of GR. Implies that free-falling frames are locally inertial.
- Drawback: "Locally" is key; tidal forces show that gravity is not perfectly uniform over large regions.
Gravitational Time Dilation: Clocks run slower in stronger gravitational fields. $\Delta t = \Delta t_0 (1 + \frac{gh}{c^2})$ (approximate for weak fields).
- Use: Explains why clocks on GPS satellites run faster than clocks on Earth, requiring correction.
- Drawback: The approximate formula is only for weak, static gravitational fields. Full GR requires metric tensors.
Maximal Proper Time Principle: Particles under gravity follow paths that maximize proper time.
- Use: An action principle for general relativity, analogous to minimizing action in classical mechanics.
- Drawback: Requires understanding of proper time and variational calculus in curved spacetime.
Tidal Forces: Differential gravitational forces across an object. Example: Longitudinal tidal force $\approx \frac{2GMm x}{R^3}$.
- Use: Explains stretc