Module 1: Relativistic Mechanics - Complete Notes
Module 1: Relativistic Mechanics
Complete Notes with Formulas, Derivations & Physical Understanding
1. Frame of Reference
- Frame of reference: A coordinate system (x, y, z, t) with clocks and measuring rods to describe events.
- Inertial Frame: Frame where Newton’s 1st law holds without fictitious forces.
Examples: Earth (approximately), train moving with constant velocity. - Non-Inertial Frame: Accelerating or rotating frame → fictitious forces appear (centrifugal, Coriolis, etc.).
2. Galilean Transformations (Classical Physics)
S′ moves with constant velocity v along +x relative to S.
| Quantity | Galilean Transformation |
|---|---|
| x′ | x − v t |
| y′ | y |
| z′ | z |
| t′ | t ← Time is absolute |
Velocity addition: u′x = ux − v
Valid only when v ≪ c.
3. Michelson-Morley Experiment (1887)
- Purpose: Detect luminiferous aether.
- Method: Split light beam into two perpendicular arms → expected time difference due to “aether wind”.
- Result: Null result — no fringe shift observed in any orientation.
- Conclusion: No aether exists; speed of light is same in all directions → contradicts Galilean transformation.
4. Einstein’s Postulates of Special Relativity (1905)
- Principle of Relativity: Laws of physics are identical in all inertial frames.
- Constancy of Speed of Light Light: c = 3×10⁸ m/s is same for all inertial observers, independent of source/observer motion.
These destroy absolute time and absolute simultaneity.
5. Lorentz Transformations
S′ moves with velocity v along +x relative to S.
Forward (S → S′):
x′ = γ (x − v t)
y′ = y
z′ = z
t′ = γ (t − v x / c²)
where γ = 1⁄√(1 − v²/c²) = 1⁄√(1 − β²), β = v/c
x′ = γ (x − v t)
y′ = y
z′ = z
t′ = γ (t − v x / c²)
where γ = 1⁄√(1 − v²/c²) = 1⁄√(1 − β²), β = v/c
Inverse (S′ → S):
x = γ (x′ + v t′)
t = γ (t′ + v x′ / c²)
x = γ (x′ + v t′)
t = γ (t′ + v x′ / c²)
6. Length Contraction
- Proper length L₀: Length in rest frame of object.
- Observed length (parallel to motion):
L = L₀ √(1 − v²/c²) = L₀⁄γ - Only along direction of motion. Perpendicular lengths unchanged.
- Example: 100 m spaceship at 0.8c → appears 60 m long.
7. Time Dilation
- Proper time ∆t₀: Time between two events at same location (single clock).
- Dilated time:
∆t = γ ∆t₀ = ∆t₀⁄√(1 − v²/c²) - Moving clocks run slow.
- Twin paradox resolved by acceleration of travelling twin (breaks symmetry).
8. Relativistic Velocity Addition
Parallel: u = u′ + v⁄1 + u′v/c²
General:
ux = u′x + v⁄1 + v u′x/c²
uy = u′y⁄γ (1 + v u′x/c²)
uz = u′z⁄γ (1 + v u′x/c²)
General:
ux = u′x + v⁄1 + v u′x/c²
uy = u′y⁄γ (1 + v u′x/c²)
uz = u′z⁄γ (1 + v u′x/c²)
Guarantees nothing exceeds c. If u′ = c → u = c.
9. Variation of Mass with Velocity
Relativistic momentum: p = γ m₀ v
Relativistic mass (older concept): m = γ m₀ = m₀⁄√(1 − v²/c²)
Relativistic mass (older concept): m = γ m₀ = m₀⁄√(1 − v²/c²)
Modern approach: avoid “relativistic mass”, just write p = γ m₀ v, E = γ m₀ c².
10. Mass–Energy Equivalence
Total energy: E = γ m₀ c²
Rest energy: E₀ = m₀ c²
Kinetic energy: K = (γ − 1) m₀ c²
Rest energy: E₀ = m₀ c²
Kinetic energy: K = (γ − 1) m₀ c²
Mass is a form of stored energy.
11. Relativistic Energy–Momentum Relation
E² = p² c² + (m₀ c²)²
Valid for all particles (massive and massless).
12. Massless Particles (Photons, etc.)
E = p c
E = hν = h c⁄λ, p = h⁄λ
E = hν = h c⁄λ, p = h⁄λ
Always travel at exactly c.
Summary Table of Key Formulas
| Effect | Formula | Proper Quantity |
|---|---|---|
| Lorentz factor | γ = 1/√(1−v²/c²) | — |
| Time dilation | ∆t = γ ∆t₀ | ∆t₀ |
| Length contraction | L = L₀ / γ | L₀ |
| Velocity addition (parallel) | u = (u′ + v)/(1 + u′v/c²) | — |
| Relativistic momentum | p = γ m₀ v | m₀ |
| Total energy | E = γ m₀ c² | m₀ |
| Rest energy | E₀ = m₀ c² | m₀ |
| Kinetic energy | K = (γ − 1) m₀ c² | m₀ |
| Energy-momentum | E² = p²c² + (m₀c²)² | m₀ |
| Photon | E = pc = hc/λ | — |
Special Relativity fully replaces Newtonian mechanics at high speeds but reduces to it when v ≪ c.
Experimentally confirmed in GPS, particle accelerators, cosmic-ray muons, etc.