Lorentz transformation

In physics, the Lorentz transformations are a one-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity (the parameter) relative to the former. The transformations are named after the Dutch physicist Hendrik Lorentz. The respective inverse transformation is then parametrized by the negative of this velocity. c 2 ( t 2 − t 1 ) 2 − ( x 2 − x 1 ) 2 − ( y 2 − y 1 ) 2 − ( z 2 − z 1 ) 2 = 0 (lightlike separated events 1, 2) {displaystyle c^{2}(t_{2}-t_{1})^{2}-(x_{2}-x_{1})^{2}-(y_{2}-y_{1})^{2}-(z_{2}-z_{1})^{2}=0quad { ext{(lightlike separated events 1, 2)}}}     (D1) c 2 ( t 2 − t 1 ) 2 − ( x 2 − x 1 ) 2 − ( y 2 − y 1 ) 2 − ( z 2 − z 1 ) 2 = c 2 ( t 2 ′ − t 1 ′ ) 2 − ( x 2 ′ − x 1 ′ ) 2 − ( y 2 ′ − y 1 ′ ) 2 − ( z 2 ′ − z 1 ′ ) 2 (all events 1, 2) . {displaystyle {egin{aligned}&c^{2}(t_{2}-t_{1})^{2}-(x_{2}-x_{1})^{2}-(y_{2}-y_{1})^{2}-(z_{2}-z_{1})^{2}\={}&c^{2}(t_{2}'-t_{1}')^{2}-(x_{2}'-x_{1}')^{2}-(y_{2}'-y_{1}')^{2}-(z_{2}'-z_{1}')^{2}quad { ext{(all events 1, 2)}}.end{aligned}}}     (D2) c 2 t 2 − x 2 − y 2 − z 2 = c 2 t ′ 2 − x ′ 2 − y ′ 2 − z ′ 2 or c 2 t 1 t 2 − x 1 x 2 − y 1 y 2 − z 1 z 2 = c 2 t 1 ′ t 2 ′ − x 1 ′ x 2 ′ − y 1 ′ y 2 ′ − z 1 ′ z 2 ′ {displaystyle {egin{aligned}&c^{2}t^{2}-x^{2}-y^{2}-z^{2}=c^{2}t'^{2}-x'^{2}-y'^{2}-z'^{2}\{ ext{or}}quad &c^{2}t_{1}t_{2}-x_{1}x_{2}-y_{1}y_{2}-z_{1}z_{2}=c^{2}t'_{1}t'_{2}-x'_{1}x'_{2}-y'_{1}y'_{2}-z'_{1}z'_{2}end{aligned}}}     (D3) ( a , a ) = ( a ′ , a ′ ) or a ⋅ a = a ′ ⋅ a ′ , {displaystyle (a,a)=(a',a')quad { ext{or}}quad acdot a=a'cdot a',}     (D4) ( a , a ) = ( Λ a , Λ a ) = ( a ′ , a ′ ) , Λ ∈ O ( 1 , 3 ) , a , a ′ ∈ M , {displaystyle (a,a)=(Lambda a,Lambda a)=(a',a'),quad Lambda in mathrm {O} (1,3),quad a,a'in M,}     (D5) t ′ = γ ( t − v n ⋅ r c 2 ) , r ′ = r + ( γ − 1 ) ( r ⋅ n ) n − γ t v n . {displaystyle {egin{aligned}t'&=gamma left(t-{frac {vmathbf {n} cdot mathbf {r} }{c^{2}}} ight),,\mathbf {r} '&=mathbf {r} +(gamma -1)(mathbf {r} cdot mathbf {n} )mathbf {n} -gamma tvmathbf {n} ,.end{aligned}}} t = γ ( t ′ + r ′ ⋅ v n c 2 ) , r = r ′ + ( γ − 1 ) ( r ′ ⋅ n ) n + γ t ′ v n , {displaystyle {egin{aligned}t&=gamma left(t'+{frac {mathbf {r} 'cdot vmathbf {n} }{c^{2}}} ight),,\mathbf {r} &=mathbf {r} '+(gamma -1)(mathbf {r} 'cdot mathbf {n} )mathbf {n} +gamma t'vmathbf {n} ,,end{aligned}}} U ( Λ , a ) Ψ p 1 σ 1 n 1 ; p 2 σ 2 n 2 ; ⋯ = e − i a μ [ ( Λ p 1 ) μ + ( Λ p 2 ) μ + ⋯ ] ( Λ p 1 ) 0 ( Λ p 2 ) 0 ⋯ p 1 0 p 2 0 ⋯ ( ∑ σ 1 ′ σ 2 ′ ⋯ D σ 1 ′ σ 1 ( j 1 ) [ W ( Λ , p 1 ) ] D σ 2 ′ σ 2 ( j 2 ) [ W ( Λ , p 2 ) ] ⋯ ) Ψ Λ p 1 σ 1 ′ n 1 ; Λ p 2 σ 2 ′ n 2 ; ⋯ , {displaystyle {egin{aligned}&U(Lambda ,a)Psi _{p_{1}sigma _{1}n_{1};p_{2}sigma _{2}n_{2};cdots }\={}&e^{-ia_{mu }left}{sqrt {frac {(Lambda p_{1})^{0}(Lambda p_{2})^{0}cdots }{p_{1}^{0}p_{2}^{0}cdots }}}left(sum _{sigma _{1}'sigma _{2}'cdots }D_{sigma _{1}'sigma _{1}}^{(j_{1})}leftD_{sigma _{2}'sigma _{2}}^{(j_{2})}leftcdots ight)Psi _{Lambda p_{1}sigma _{1}'n_{1};Lambda p_{2}sigma _{2}'n_{2};cdots },end{aligned}}}     ( 1) In physics, the Lorentz transformations are a one-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity (the parameter) relative to the former. The transformations are named after the Dutch physicist Hendrik Lorentz. The respective inverse transformation is then parametrized by the negative of this velocity. The most common form of the transformation, parametrized by the real constant v , {displaystyle v,} representing a velocity confined to the x-direction, is expressed as where (t, x, y, z) and (t′, x′, y′, z′) are the coordinates of an event in two frames, where the primed frame is seen from the unprimed frame as moving with speed v along the x-axis, c is the speed of light, and γ = ( 1 − v 2 c 2 ) − 1 {displaystyle gamma = extstyle left({sqrt {1-{frac {v^{2}}{c^{2}}}}} ight)^{-1}} is the Lorentz factor. When speed v is significantly lower than c, the factor is negligible, but as v approaches c, there is a significant effect. The value of v cannot exceed c, in current understanding. Expressing the speed as β = v c , {displaystyle eta ={frac {v}{c}},} an equivalent form of the transformation is Frames of reference can be divided into two groups: inertial (relative motion with constant velocity) and non-inertial (accelerating, moving in curved paths, rotational motion with constant angular velocity, etc.). The term 'Lorentz transformations' only refers to transformations between inertial frames, usually in the context of special relativity. In each reference frame, an observer can use a local coordinate system (usually Cartesian coordinates in this context) to measure lengths, and a clock to measure time intervals. An event is something that happens at a point in space at an instant of time, or more formally a point in spacetime. The transformations connect the space and time coordinates of an event as measured by an observer in each frame. They supersede the Galilean transformation of Newtonian physics, which assumes an absolute space and time (see Galilean relativity). The Galilean transformation is a good approximation only at relative speeds well below the speed of light. Lorentz transformations have a number of unintuitive features that do not appear in Galilean transformations. For example, they reflect the fact that observers moving at different velocities may measure different distances, elapsed times, and even different orderings of events, but always such that the speed of light is the same in all inertial reference frames. The invariance of light speed is one of the postulates of special relativity. Historically, the transformations were the result of attempts by Lorentz and others to explain how the speed of light was observed to be independent of the reference frame, and to understand the symmetries of the laws of electromagnetism. The Lorentz transformation is in accordance with special relativity, but was derived before Einstein developed special relativity. The Lorentz transformation is a linear transformation. It may include a rotation of space; a rotation-free Lorentz transformation is called a Lorentz boost. In Minkowski space, the mathematical model of spacetime in special relativity, the Lorentz transformations preserve the spacetime interval between any two events. This property is the defining property of a Lorentz transformation. They describe only the transformations in which the spacetime event at the origin is left fixed. They can be considered as a hyperbolic rotation of Minkowski space. The more general set of transformations that also includes translations is known as the Poincaré group.