5.1 Invariance of physical laws

Suppose you calculate the hypotenuse of a right triangle given the base angles and adjacent sides. Whether you calculate the hypotenuse from one of the sides and the cosine of the base angle, or from the Pythagorean theorem, the results should agree. Predictions based on different principles of physics must also agree, whether we consider them principles of mechanics or principles of electromagnetism.

Albert Einstein pondered a disagreement between predictions based on electromagnetism and on assumptions made in classical mechanics. Specifically, suppose an observer measures the velocity of a light pulse in the observer’s own rest frame    ; that is, in the frame of reference in which the observer is at rest. According to the assumptions long considered obvious in classical mechanics, if an observer measures a velocity $\overrightarrow{v}$ in one frame of reference, and that frame of reference is moving with velocity $\overrightarrow{u}$ past a second reference frame, an observer in the second frame measures the original velocity as $\overrightarrow{v^{\prime }}=\overrightarrow{v}+\overrightarrow{u}.$ This sum of velocities is often referred to as Galilean relativity    . If this principle is correct, the pulse of light that the observer measures as traveling with speed c travels at speed c + u measured in the frame of the second observer. If we reasonably assume that the laws of electrodynamics are the same in both frames of reference, then the predicted speed of light (in vacuum) in both frames should be $c=1\text{/}\sqrt{{\epsilon }_0{\mu }_0}.$ Each observer should measure the same speed of the light pulse with respect to that observer’s own rest frame. To reconcile difficulties of this kind, Einstein constructed his special theory of relativity    , which introduced radical new ideas about time and space that have since been confirmed experimentally.

Inertial frames

All velocities are measured relative to some frame of reference. For example, a car’s motion is measured relative to its starting position on the road it travels on; a projectile’s motion is measured relative to the surface from which it is launched; and a planet’s orbital motion is measured relative to the star it orbits. The frames of reference in which mechanics takes the simplest form are those that are not accelerating. Newton’s first law, the law of inertia, holds exactly in such a frame.

For example, to a passenger inside a plane flying at constant speed and constant altitude, physics seems to work exactly the same as when the passenger is standing on the surface of Earth. When the plane is taking off, however, matters are somewhat more complicated. In this case, the passenger at rest inside the plane concludes that a net force F on an object is not equal to the product of mass and acceleration, ma . Instead, F is equal to ma plus a fictitious force. This situation is not as simple as in an inertial frame. The term “special” in “special relativity” refers to dealing only with inertial frames of reference. Einstein’s later theory of general relativity deals with all kinds of reference frames, including accelerating, and therefore non-inertial, reference frames.