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Physics · Document 12

Special Relativity

Spacetime, Lorentz transformations, and the geometry of causality

1. Einstein's Postulates

Special relativity rests on two simple postulates that revolutionized our understanding of space and time:

1. Principle of Relativity

The laws of physics are the same in all inertial reference frames. No experiment can detect "absolute motion."

2. Constancy of Light Speed

The speed of light c is the same for all observers, regardless of the motion of the source.

These two postulates seem contradictory with Newtonian physics. The resolution requires abandoning absolute time and absolute simultaneity!

2. Time Dilation

Moving clocks run slow! A clock moving at velocity v relative to you ticks slower by a factor gamma:

gamma = 1 / sqrt(1 - v^2/c^2)
Lorentz Factor
🎮 Interactive: Time Dilation Calculator
Your clock (at rest)
1.00 second
Moving clock (v = 50% c)
0.866 seconds
(appears to run slower)
Lorentz Factor
gamma = 1.155
Time Dilation
Delta t = gamma * Delta tau
Proper Time
tau = t/gamma

The Twin Paradox

If you travel to a star 10 light-years away at 0.99c, only about 1.4 years pass for you, but about 10 years pass on Earth! This is not symmetric: the traveling twin accelerates and decelerates, breaking the symmetry.

3. Length Contraction

Objects contract along the direction of motion. A meter stick moving past you appears shorter:

L = L0 / gamma = L0 * sqrt(1 - v^2/c^2)
🎮 Interactive: Length Contraction
Object at rest (proper length = 100m)L0 = 100mSame object moving at v = 60% cL = 80.0mv
Length Contraction Formula
L = L0 / gamma = L0 * sqrt(1 - v^2/c^2)
Contraction Factor
L/L0 = 0.800 = 80.0%
Only the direction of motion contracts! Lengths perpendicular to motion are unchanged. This is why we get ellipses, not smaller spheres.

4. Spacetime and the Light Cone

Space and time are unified into spacetime. The light cone divides spacetime into causally connected and disconnected regions.

🎮 Interactive: Light Cone and Causality
xtFuturePastElsewhere
Spacetime Interval
s^2 = t^2 - x^2 = 0.00
Causal Relationship
Lightlike (null)
Timelike
s^2 > 0: Events can be causally connected
Lightlike
s^2 = 0: Connected by light ray
Spacelike
s^2 < 0: No causal connection possible

The Invariant Interval

While space and time separately depend on the observer, the spacetime interval is invariant:

ds^2 = c^2 dt^2 - dx^2 - dy^2 - dz^2
Minkowski Metric

This is the signature of Minkowski spacetime with metric signature (+,-,-,-).

5. Lorentz Transformations

The transformations that preserve the spacetime interval are the Lorentz transformations:

t' = gamma(t - vx/c^2), x' = gamma(x - vt)
Lorentz Boost in x-direction
TransformationMatrix FormPreserves
Rotation (3D)SO(3)Spatial distance
BoostHyperbolic rotationSpacetime interval
Full LorentzSO(3,1)Minkowski metric
Z²
SO(3,1)
The Lorentz group has 4 disconnected components, related by P (parity) and T (time reversal)

6. Energy-Momentum Relation

The famous equation E = mc^2 is actually a special case of a more general relation:

E^2 = (pc)^2 + (mc^2)^2
Energy-Momentum Relation
Rest mass (p = 0)
E = mc^2
Photon (m = 0)
E = pc
Energy and momentum form a 4-vector (E/c, p). The "length" of this 4-vector is the rest mass: m^2 c^2 = E^2/c^2 - p^2

7. Connection to Z^2 Framework

Special relativity is the foundation of relativistic quantum field theory:

Z²
ds^2 = g_mu,nu dx^mu dx^nu
The Minkowski metric is the flat limit of curved spacetime in general relativity
Z²
SO(3,1) -> Spin(3,1)
Spinors require the double cover of the Lorentz group
Z²
CPT Theorem
All Lorentz-invariant QFTs are invariant under combined C, P, T

Why Special Relativity Matters for Z^2

  • * Causality: The light cone structure determines what can influence what
  • * Lorentz invariance: All fundamental physics must respect this symmetry
  • * Spinors: Fermions transform under the double cover Spin(3,1)
  • * E = mc^2: Mass and energy are interchangeable in particle physics

Exercises

  1. Calculate gamma for v = 0.6c and verify gamma = 1.25.
  2. A muon (lifetime 2.2 microseconds) travels at 0.99c. How far does it travel in the lab frame?
  3. Show that if s^2 > 0, there exists a frame where the two events occur at the same place.
  4. Derive the velocity addition formula: u' = (u - v)/(1 - uv/c^2).
  5. Verify that the Lorentz transformation preserves ds^2.