Composition of velocities
Main article: Velocity-addition formula If a ship is moving relative to the shore at velocity v, and a fly is moving with velocity u as measured on the ship, calculating the velocity of the fly as measured on the shore is what is meant by the addition of the velocities v and u. When both the fly and the ship are moving slowly compared to light, the addition law is a vector sum:If the observer in S sees an object moving along the x axis at velocity w, then the observer in the S' system, a frame of reference moving at velocity v in the x direction with respect to S, will see the object moving with velocity w' where
This equation can be derived from the space and time transformations above.
Notice that if the object were moving at the speed of light in the S system (i.e. w = c), then it would also be moving at the speed of light in the S' system. Also, if both w and v are small with respect to the speed of light, we will recover the intuitive Galilean transformation of velocities: .
The usual example given is that of a train (call it system K) travelling due east with a velocity v with respect to the tracks (system K'). A child inside the train throws a baseball due east with a velocity u with respect to the train. In classical physics, an observer at rest on the tracks will measure the velocity of the baseball as v + u.
In special relativity, this is no longer true. Instead, an observer on the tracks will measure the velocity of the baseball as . If u and v are small compared to c, then the above expression approaches the classical sum v + u.
In the more general case, the baseball is not necessarily travelling in the same direction as the train. To obtain the general formula for Einstein velocity addition, suppose an observer at rest in system K measures the velocity of an object as . Let K' be an inertial system such that the relative velocity of K to K' is , where and are now vectors in R3. An observer at rest in K' will then measure the velocity of the object as [16]
where and are the components of parallel and perpendicular, respectively, to , and .
Note that the Einstein velocity addition is commutative only when and are parallel. In fact,
where gyr is A.A. Ungar's gyration operator In mathematics and physics, gyrovectors are a tool for studying hyperbolic geometry in analogy to the way vector spaces are used in Euclidean geometry. This vector-based approach has been developed by Abraham Albert Ungar from the late 1980s onwards. These gyrovectors can be used to unify the study of Euclidean and hyperbolic geometry. The Bloch.[26]
<<Table of Contents Special relativity (also known as the special theory of relativity or STR) is the physical theory of measurement in inertial frames of reference proposed in 1905 by Albert Einstein (after the considerable and independent contributions of Hendrik Lorentz, Henri Poincaré and others) in the paper "On the Electrodynamics of Moving Bodies" | Next>> | Show All>>
