Einstein's velocity addition formula revolutionized our understanding of motion at high speeds. It shows that velocities don't simply add up like we thought. Instead, they combine in a way that keeps the speed of light as the ultimate limit.

This formula is crucial for understanding how things move in the universe. It applies to everything from particles in accelerators to distant galaxies, showing us the true nature of space and time at extreme speeds.

Velocity Addition

Classical and Relativistic Velocity Addition

• Classical velocity addition follows Galilean relativity where velocities are added linearly ($v = v_1 + v_2$)
• Relativistic velocity addition is non-linear and follows Einstein's velocity addition formula
• Einstein velocity addition formula: $v = \frac{v_1 + v_2}{1 + \frac{v_1v_2}{c^2}}$, where $c$ is the speed of light
• Relativistic velocity addition approaches classical velocity addition at low velocities compared to the speed of light

Parallel and Perpendicular Velocities

• Parallel velocities are velocities in the same direction and are added using Einstein's velocity addition formula directly
• Perpendicular velocities are velocities at right angles to each other
• Perpendicular velocities are added using the Pythagorean theorem first, then applying Einstein's velocity addition formula
• Example: A spaceship moving at 0.8c fires a missile at 0.6c perpendicular to its motion. The resultant velocity is $v = \sqrt{(0.8c)^2 + (0.6c)^2} = c\sqrt{0.8^2 + 0.6^2} = 0.986c$

Speed Limit

The Speed of Light as a Universal Speed Limit

• The speed of light in vacuum ($c$) is the universal speed limit according to special relativity
• No object with mass can reach or exceed the speed of light
• As an object's velocity approaches the speed of light, its kinetic energy approaches infinity, requiring infinite energy to accelerate further

The Lorentz Factor

• The Lorentz factor ($\gamma$) is a key component in relativistic equations
• Lorentz factor: $\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$, where $v$ is the object's velocity
• As velocity approaches the speed of light, the Lorentz factor approaches infinity
• The Lorentz factor is used to calculate relativistic mass, time dilation, and length contraction

Frames of Reference

Inertial Frames of Reference

• An inertial frame of reference is a frame in which Newton's first law of motion holds true
• In an inertial frame, an object at rest remains at rest, and an object in motion remains in motion with constant velocity unless acted upon by an external force
• Special relativity deals with inertial frames of reference moving at constant velocities relative to each other

Non-Inertial Frames of Reference

• A non-inertial frame of reference is a frame that accelerates or rotates relative to an inertial frame
• In a non-inertial frame, fictitious forces (e.g., centrifugal force) appear to act on objects
• General relativity extends the principles of special relativity to non-inertial frames of reference
• Example: An elevator accelerating upward is a non-inertial frame of reference, and a person inside experiences an apparent increase in weight due to the fictitious force