Postulates of Special Relativity

The Two Postulates

Einstein founded Special Relativity (1905) on two fundamental axioms:

The Principle of Relativity

The laws of physics are the same in all inertial frames of reference (frames moving at constant velocity relative to each other). There is no “absolute” rest frame.

Constancy of the Speed of Light

The speed of light in a vacuum, , is invariant. It is the same for all observers, regardless of the motion of the source or the observer.

To transition between frames, we define dimensionless quantities.

Normalized Velocity & Lorentz Factor

We define the following dimensionless quantities:

• Normalized Velocity:
• The Lorentz Factor:

Since , , as , .

The Lorentz Transformations

The above postulates lead to the Lorentz transformations, which replace the classical Galilean transformations.

The Lorentz transformation relates coordinates in frame to coordinates in frame moving with velocity along the axis. The following is the inverse Lorentz transformations (switch and primes):

Link to original

For the derivation, refer to The Lorentz Transformations.

Consequences of Transformations

Relativity of Simultaneity (Clock Synchronization)

Events that are simultaneous in frame () are not simultaneous in frame if the events are separated spatially (): This leads clocks lag behind.

Time Dilation

A clock moving relative to an observer ticks slower than a clock at rest.

• Proper Time (): Time interval measured in the frame where the clock is at rest (same spatial location).
• Coordinate Time (): Time interval measured by the moving observer. Since , .

Length Contraction

An object moving relative to an observer appears shorter along the direction of motion.

• Proper Length (): Length measured in the frame where the object is at rest.
• Observed Length (): Length measured in the frame where the object is moving.

V. Relativistic Velocity Addition

If an object moves with velocity in frame , and moves at relative to , the velocity observed in is:

Note: If or , the result is always .

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VI. Relativistic Dynamics

1. Momentum

Newtonian momentum is replaced to preserve conservation laws.

( is the invariant “rest mass”).

2. Energy

• Total Energy:

• Rest Energy:

• Kinetic Energy:

3. The Energy-Momentum Relation

A vital invariant equation (independent of velocity ):

For massless particles (photons), , so .

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