Special Relativity and Quantum Mechanics began the era of “Modern Physics” just over 100 years ago, but are still frequently stumbling blocks for students who cling to the successes of Newtonian mechanics. Part of the confusion is perhaps that SR is often taught as a collection of counter-intuitive claims rather than as a single logical entity. The very good reason for that is that the universe is counter-intuitive if ones intuition has been trained on purely Newtonian grounds. In this series, I hope to cover the facts and logic behind special relativity and the stumbling blocks that some get stuck on.
Special Relativity, SR for short, was founded on two principles:
The laws of physics, formerly written in terms of a hypothetical absolute space, will turn out to be equally valid for all systems of spatial and temporal coordinates for which the law of inertia is valid -- or “all inertial frames are valid to do physics in”
Any ray of light in vacuum moves in such system of co-ordinates with a velocity c, whether the ray be emitted by a stationary or by a moving body -- or “the speed of light in vacuum is always c”
From those, we get a number of predictions:
The relation, in any inertial frame, between a free object’s energy, E; momentum, p; and velocity, v are as follows: E² = ( m c² )² + ( p⃗ c )² and E v⃗ = c² p⃗. Thus massless objects which carry energy and momentum always move at the speed of light in vacuum and massive objects always travel slower than c.
Elapsed time for things, Δτ, which is path-dependent, is distinct from the imaginary quantity of elapsed coordinate time, Δt. The time elapsed on a physical clock which follows a path is given by Δτ = ∫ √[ 1 − ( v⃗(t) / c )² ] dt. The relation between clock time and coordinate time holds true only if the clock is forever “at rest” with respect to the inertial coordinate system.
The quantity of spatial separation between two objects with the same state of motion turns out to be an imaginary consequence of choosing a coordinate system. Specifically, if two objects are separated by spatial distance L in a frame where they are considered “at rest” and a third object has non-zero motion parallel to that spatial separation, in another frame where the third object is at rest, the spatial separation between the first two objects is L’ = √[ 1 − v⃗²/c² ] L where v⃗ is the state of motion of the first two objects.
If u⃗ and v⃗ are parallel velocities of two objects in one inertial coordinate system, then in a frame where the first object is at rest, the speed of the second object is given by v′ = ( v − u ) / ( 1 − uv/c² ) . Expressed another way, if u is the velocity of B in a coordinate system where A is at rest and v is the velocity of C in a coordinate system where B is at rest, and u and v are parallel, then the velocity of C in the coordinate system of A is ( v + u ) / ( 1 + uv/c² ), the law of composition of parallel velocities.
While time and space seem to be imaginary constructs of our coordinate system, there is an underlying geometry of space time which preserves the space-time interval between space-time events. For any two inertial coordinate systems and any four events (A,B,C,D) the following relation holds: c² ( tB − tA ) ( tD − tC ) − ( x⃗B − x⃗A ) ⋅ ( x⃗D − x⃗C ) = c² ( t′B − t′A ) ( t′D − t′C ) − ( x⃗′B − x⃗′A ) ⋅ ( x⃗′D − x⃗′C ), while if A=C and B=D the same relation for just two events can be written as c² ( Δt )² − ( Δx⃗ )² = c² ( Δt′ )² − ( Δx⃗′ )².
These are not independent predictions, but all follow from the Lorentz-FitzGerald equations relating coordinates in one frame with coordinates in another. Einstein in 1905 showed that these equations in form of a Lorentz transform follow from his assumptions and in 1908 Minkowski showed that this was a geometry of space-time as fundamental as Euclid’s. Einstein and Poincaré would later build on this to construct General Relativity.
