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
w’=\frac{w-v}{1-wv/c^2}.
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: w’ \approx w-v.
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 \frac{v+u}{1+\frac{vu}{c^2}}. 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 \mathbf{u}. Let K’ be an inertial system such that the relative velocity of K to K’ is \mathbf{v}, where \mathbf{u} and \mathbf{v} are now vectors in R3. An observer at rest in K’ will then measure the velocity of the object as [16]
where \mathbf{u}_{\parallel} and \mathbf{u}_{\perp} are the components of \mathbf{u} parallel and perpendicular, respectively, to \mathbf{v}, and \alpha_{\mathbf{v}}=\frac{1}{\gamma(\mathbf{v})}=\sqrt{1-\frac{|\mathbf{v}|^2}{c^2}}.
Einstein velocity addition is commutative only when \mathbf{v} and \mathbf{u} are parallel. In fact,
Special Relativity
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
w’=\frac{w-v}{1-wv/c^2}.
This equation can be derived from the space and time transformations above.
w’=\frac{dx’}{dt’}=\frac{\gamma(dx-v dt)}{\gamma(dt-v dx/c^2)}=\frac{(dx/dt)-v}{1-(v/c^2)(dx/dt)}
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: w’ \approx w-v.
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 \frac{v+u}{1+\frac{vu}{c^2}}. 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 \mathbf{u}. Let K’ be an inertial system such that the relative velocity of K to K’ is \mathbf{v}, where \mathbf{u} and \mathbf{v} are now vectors in R3. An observer at rest in K’ will then measure the velocity of the object as [16]
\mathbf{v} \oplus_E \mathbf{u}=\frac{\mathbf{v}+\mathbf{u}_{\parallel} + \alpha_{\mathbf{v}}\mathbf{u}_{\perp}}{1+\frac{\mathbf{v}\cdot\mathbf{u}}{c^2}},
where \mathbf{u}_{\parallel} and \mathbf{u}_{\perp} are the components of \mathbf{u} parallel and perpendicular, respectively, to \mathbf{v}, and \alpha_{\mathbf{v}}=\frac{1}{\gamma(\mathbf{v})}=\sqrt{1-\frac{|\mathbf{v}|^2}{c^2}}.
Einstein velocity addition is commutative only when \mathbf{v} and \mathbf{u} are parallel. In fact,
\mathbf{v} \oplus \mathbf{u}=gyr[\mathbf{v},\mathbf{u}](\mathbf{u} \oplus \mathbf{v}),
,where gyr is the mathematical abstraction of Thomas precession into an operator called Thomas gyration and given by
gyr[\mathbf{u},\mathbf{v}]\mathbf{w}=\ominus(\mathbf{u} \oplus \mathbf{v}) \oplus (\mathbf{u} \oplus (\mathbf{v} \oplus \mathbf{w}))
for all w.