The special theory of relativity

1. First Postulate (Principle of Relativity): The laws of nature have identical form in all inertial frames of reference.

2. Second Postulate (Invariance of Light Speed): The speed of light in vacuum has the same value c (≈ 3×10⁸ m/s) in all inertial frames, regardless of the motion of the source or observer.

These postulates were revolutionary because:

• They resolved contradictions between Maxwell's equations and classical mechanics
• They eliminated the need for a preferred reference frame (ether)
• They required abandoning the intuitive concepts of absolute time and space
• They mathematically explained the null result of the Michelson-Morley experiment

The second postulate particularly contradicted common sense as it meant that two observers moving relative to each other would measure the same speed for light.

Mass-Energy Equivalence: Basic form: $E = mc^2$

Complete relativistic energy equation: $E^2 = (m_0c^2)^2 + (pc)^2$

Where:

• $E$ = total energy
• $m_0$ = rest mass
• $p$ = relativistic momentum
• $c$ = speed of light

For an object with velocity v: $E = \frac{m_0c^2}{\sqrt{1-v^2/c^2}} = \gamma m_0c^2$

Profound implications:

1. Mass and energy are fundamentally interchangeable forms of the same entity
2. Even objects at rest contain enormous energy ($E_0 = m_0c^2$)
3. Energy contributes to inertia and gravitational effects
4. Mass can be converted to other forms of energy (nuclear fission/fusion)
5. Pure energy can create particles with mass (pair production)
6. Conservation laws must include both mass and energy together
7. A small mass defect accounts for the binding energy in atomic nuclei

This equivalence revolutionized physics, leading to nuclear energy, explaining stellar processes, and forming a foundation for particle physics and cosmology.

Twin Paradox: One twin travels into space at relativistic speeds while the other remains on Earth. When the traveling twin returns, they find they've aged less than their Earth-bound sibling.

Apparent contradiction:

• From Earth twin's perspective: Traveler's clock runs slow due to time dilation, so traveler ages less
• From traveling twin's perspective: Earth appears to move away at high speed, so Earth twin's clock should run slow, suggesting Earth twin should age less
• This symmetry seems to create a paradox with contradictory predictions

Resolution:

1. The situation is NOT symmetric because:

   • Only the traveling twin experiences acceleration (during departure, turnaround, and return)
   • The traveling twin changes reference frames, while the Earth twin remains in a single inertial frame
   • The traveler's perspective requires at least three different inertial frames for analysis

2. During turnaround:

   • The traveler must accelerate to reverse direction
   • This acceleration breaks the symmetry between the twins
   • The traveling twin observes the Earth twin's clock suddenly "jump ahead"

3. Mathematical resolution:

   • The proper time (τ) along a worldline is an invariant quantity: $\tau = \int\sqrt{1-v^2/c^2}dt$
   • This integral is smaller along the traveling twin's path
   • The twin who follows a geodesic in spacetime ages the most

This paradox has been confirmed experimentally using precise atomic clocks on airplanes.

Relativistic Velocity Addition Formula:

For velocities along the same direction: $u' = \frac{u + v}{1 + \frac{uv}{c^2}}$

Where:

• $u$ = velocity of object in frame S
• $v$ = velocity of frame S' relative to frame S
• $u'$ = velocity of object in frame S'

Differences from classical addition:

• Classical: $u' = u + v$ (simple addition)
• Relativistic: Always produces $u' < c$ when both $u < c$ and $v < c$
• Denominator term $1 + \frac{uv}{c^2}$ ensures the result never exceeds c

Surprising consequences:

1. Speed limit preservation: No matter how many velocities you add, you cannot exceed c
2. Non-intuitive results: For large velocities, the sum is significantly less than expected:
   • Example: 0.6c + 0.6c = 0.88c (not 1.2c)
   • Example: 0.9c + 0.9c = 0.994c (not 1.8c)
3. Light speed invariance: If u = c, then u' = c regardless of

The invariance of the speed of light states that light travels at exactly the same speed c (≈ 3×10^8 m/s) in all inertial reference frames, regardless of the relative motion between observers or light sources.

This is counterintuitive because:

• It contradicts classical velocity addition (if you move at 0.5c toward a light beam, the light still approaches you at exactly c, not 1.5c)
• Two observers moving relative to each other will measure the same light pulse moving at exactly the same speed c
• Time and space must adjust (via time dilation and length contraction) to maintain this invariance

Example: If observer A is stationary and observer B moves at 0.8c relative to A, both will measure a light pulse passing them at exactly 3×10^8 m/s, despite their relative motion.

Mathematical analysis of events in different reference frames:

1. For two events at the same position in reference frame S:

   • Both events occur at position x₀
   • They occur at different times t₁ and t₂
   • The time interval in this frame is ∆t = t₂ - t₁ (proper time)

2. In a reference frame S' moving at velocity v relative to S:

   • The events occur at different positions x₁' and x₂'
   • They occur at different times t₁' and t₂'
   • The spatial separation is ∆x' = v∆t'
   • The time interval is ∆t' = ∆t/√(1-v²/c²) = γ∆t

3. This demonstrates time dilation:

   • The proper time interval ∆t is always the minimum possible time interval
   • Any reference frame where events occur at different locations measures a longer time interval
   • The dilated time ∆t' = γ∆t is related to the proper time by the Lorentz factor γ

This phenomenon explains why moving light-beam clocks run slower - the "ticks" (light reflections) occur at different positions in the observer's frame, requiring light to travel greater distances at the same speed c.

• The moving clock C measures proper time (t') between events E₁ and E₂
• The frame where events are separated by distance L measures improper time (t)
• The proper time is shorter by the Lorentz factor: t' = t/γ
• Specifically: t' = t·√(1-v²/c²)
• Since v = L/t, the clock moves at exactly the right speed to be present at both events
• This demonstrates time dilation - moving clocks run slower relative to stationary ones
• Example application: A muon traveling at 0.99c from the upper atmosphere to Earth's surface experiences much less proper time than the improper time measured on Earth

In relativity, clock synchronization is frame-dependent: • Events that are simultaneous in one frame are not simultaneous in another frame • In a frame where objects are moving, clocks at different positions appear desynchronized • For objects moving at velocity $v$, distant clocks appear offset by $\gamma vx/c^2$ where $x$ is their separation • Front clocks (in direction of motion) appear to lag behind rear clocks when viewed from a stationary frame • When changing reference frames, this synchronization difference must be accounted for • This effect explains why the "leading" clock in one frame becomes the "lagging" clock in another frame moving in the opposite direction

For objects moving at velocity $v = 0.8c$ relative to each other:

Time dilation factor: $\gamma = \frac{1}{\sqrt{1-v^2/c^2}} = \frac{1}{\sqrt{1-(0.8c)^2/c^2}} = \frac{1}{\sqrt{1-0.64}} = \frac{1}{\sqrt{0.36}} = \frac{1}{0.6} = 1.67$

Significance: • Moving clocks run slower by this factor (appear to tick at 0.6× normal rate) • 1 second in the moving frame corresponds to 1.67 seconds in the stationary frame • 12 years on a stationary clock corresponds to 12 × 0.6 = 7.2 years on the moving clock • This factor is reciprocal - each observer sees the other's clock running slow by the same factor • This creates the apparent paradox until reference frame changes are properly accounted for

When transitioning between frames moving in opposite directions:

1. For objects separated by distance $L$ moving at speed $v$ relative to an observer:

   • Synchronization offset = $\gamma vL/c^2$
   • For $v = 0.8c$ and $L = 4.8$ light-years: offset = $1.67 × 0.8c × 4.8$ ly ÷ $c^2$ = $6.4$ years

2. When switching frames:

   • The synchronization convention reverses
   • This causes an apparent "jump" of twice the synchronization offset
   • In the example, when Ram switches from S₁ to S₂, Earth's clock appears to jump from 3.6 years to 16.4 years
   • This 12.8-year jump is twice the synchronization offset (2 × 6.4 = 12.8 years)

3. The apparent jump is not an actual time discontinuity but a result of changing synchronization conventions between frames.