The special theory of relativity

Time Dilation: Moving clocks run slower than stationary clocks.

Mathematical Formula: $\Delta t' = \frac{\Delta t}{\sqrt{1-v^2/c^2}} = \gamma \Delta t$

Where:

• $\Delta t$ = proper time (time measured in frame where events occur at same location)
• $\Delta t'$ = improper time (time measured in frame where events occur at different locations)
• $\gamma = \frac{1}{\sqrt{1-v^2/c^2}}$ (Lorentz factor, always > 1)

Physical Interpretation:

• The proper time interval between two events is always the smallest possible measured time
• For a moving observer, time passes more slowly by a factor of γ
• This is a real physical effect, not an illusion or measurement error
• Time dilation is reciprocal: each observer sees the other's clock running slower
• This effect becomes significant only at speeds approaching c
• Verified experimentally with muon decay rates and atomic clocks on aircraft

Example: A spaceship traveling at 0.8c relative to Earth will experience only 6 years of elapsed time during what Earth observers measure as 10 years.

Length Contraction: Objects moving relative to an observer appear shortened along their direction of motion.

Formula: $L' = L\sqrt{1-v^2/c^2} = \frac{L}{\gamma}$

Where:

• $L$ = proper length (length measured in rest frame of object)
• $L'$ = contracted length (length measured in frame where object is moving)
• $\gamma = \frac{1}{\sqrt{1-v^2/c^2}}$ (Lorentz factor)

Crucial Constraint: Length contraction occurs ONLY in the direction parallel to motion. Dimensions perpendicular to motion remain unchanged.

Key aspects:

• Only affects the dimension along the direction of relative motion
• The proper (rest) length is always the maximum possible measured length
• Length contraction is reciprocal between reference frames
• This is a real physical effect, not an optical illusion
• No "stress" is involved—the object appears perfectly normal in its own rest frame

Example: A 10-meter rod moving at 0.87c (where γ = 2) would measure only 5 meters in length to a stationary observer.

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.

A light-beam clock consists of two mirrors separated by distance L with light bouncing between them:

• In a stationary reference frame S, light travels distance 2L at speed c, with time interval Δt = 2L/c
• In a moving reference frame S' (velocity v), light follows a longer diagonal path while mirrors move
• The light must travel further but still moves at speed c, requiring more time

Mathematical relationship: $\Delta t' = \frac{\Delta t}{\sqrt{1-v^2/c^2}} = \gamma \Delta t$

Where:

• Δt = proper time (measured in frame where events occur at same location)
• Δt' = improper time (measured in frame where events occur at different locations)
• γ = Lorentz factor = 1/√(1-v²/c²)

This demonstrates the central principle: moving clocks run slower than stationary clocks by a factor of γ.

• Events that are simultaneous in one reference frame are generally NOT simultaneous in another reference frame that moves relative to the first.

• For two events occurring at different spatial locations:

  • If simultaneous in reference frame S
  • Will NOT be simultaneous in reference frame S' moving at velocity v relative to S

• The time difference between events in the moving frame is: $\Delta t' = \frac{Lv}{c^2}\gamma$ Where:

  • L = spatial separation between events in the stationary frame
  • v = relative velocity between frames
  • γ = 1/√(1-v²/c²)

• Example: Light emitted from the center of a box reaches both ends simultaneously for a stationary observer, but for a moving observer, the light reaches the rear end before the front end.

Note: This directly contradicts our intuitive notion that simultaneity is absolute and demonstrates that time ordering depends on the observer's reference frame.

• Analysis from different reference frames reveals contradictory but equally valid door-opening sequences:

• Ground frame:

  • Box of length L with doors D₁ and D₂ at ends
  • Light source at center C emits signals simultaneously
  • Both doors open simultaneously (t = L/2c)

• Train T₁ frame (moving left at velocity v):

  • Box appears length-contracted
  • Door D₁ opens first
  • Door D₂ opens later by time interval Δt = Lv/c²γ
  • Observer concludes: "D₁ definitely opened first"

• Train T₂ frame (moving right at velocity v):

  • Box appears length-contracted
  • Door D₂ opens first
  • Door D₁ opens later by time interval Δt = Lv/c²γ
  • Observer concludes: "D₂ definitely opened first"

• This demonstrates that the temporal ordering of spatially separated events is not absolute but depends on the observer's state of motion.

Example: A referee standing midfield sees two football players touch boundary lines simultaneously, while a moving observer sees one player touch first—both observations are correct in their respective frames.

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

The ordering of events differs between reference frames due to: • Events that are simultaneous in one frame are not simultaneous in another • The relativity of simultaneity means that temporal ordering can change between frames • Spatially separated events with timelike separation maintain their order in all frames • Events with spacelike separation can have different ordering in different frames

This is critical to resolving the Twin Paradox because: • When Ram changes frames, he must account for different event orderings • In Earth's frame, the Planet and Earth clocks read zero simultaneously • In S₁-frame, "Planet's clock reading zero" occurred years before "Earth's clock reading zero" • When switching to S₂, the Earth's clock is leading rather than lagging • These ordering differences account for the final age discrepancy, resolving the paradox

The situation is asymmetrical because:

• One twin (Ram) changes reference frames while the other (Balram) remains in a single inertial frame • The twin who changes frames experiences proper acceleration, which is absolute and frame-independent • This acceleration breaks the symmetry of special relativity's inertial reference frames • The twin who stays in one frame follows a geodesic (straight worldline) in spacetime • The traveling twin follows a non-geodesic path with direction changes • Direction changes require accounting for different synchronization conventions between frames • The twin who changes frames must account for the discontinuity in how distant clocks are synchronized

This fundamental asymmetry is why the paradox is only apparent, not real.