A GPS satellite travels at 14,000 kilometers per hour, and needs corrections for relativity to stay accurate.
Physics at the usual speeds found on Earth mostly follow our intuitive understanding of time and space. However, in the 20th century, very accurate measurements of the speed of light found that these could not be true. This puzzle was solved by a new theory of time and space by Albert Einstein, called the Special Theory of Relativity. This chapter covers the paradox of the speed of light, time dilation, length contraction, simultaneity, and the General Theory of Relativity.
1. Postulates of Special Relativity
Postulate 1: The laws of physics have the same form in all inertial frames of reference.
Postulate 2: The speed of light in vacuum is constant in all inertial frames of reference.
2. Time dilation:
Clocks moving relative to an observer who assumes himself “at rest,” run more slowly compared to the clocks in the “at rest” frame. Such a slowing of time is called time dilation. The transformation equation between the two frames is given by
The time is the proper time (also called the lab or at rest frame).
3. Length contraction
Objects in motion with respect to a proper frame (or “rest frame”) of reference are contracted in the direction of their motion. The length of the (moving) object as measured from the proper frame is given by
where is the length measured in the proper frame.
4. Newton’s Equations with the proper transformations
5. The total energy is defined as ,
where is the kinetic energy of an object and is its rest mass.
6. The correct relativistic addition of velocities equation which Einstein (and Lorentz) derived is
where is the velocity of an inertial frame measured in the proper frame and is the velocity of an object (or light wave) measured with respect . The relative velocity is with respect to frame .