Lorentz transformation

The Lorentz transformations are important equations in physics. They show how space and time change when you look at them from different moving viewpoints. These equations are named after the Dutch physicist Hendrik Lorentz. They help us understand Einstein's ideas about how fast things can go, especially close to the speed of light.

The Lorentz transformations are different from older ideas used for everyday speeds. They tell us that the speed of light looks the same to everyone, no matter how they move. This can make time seem to pass differently and distances seem shorter when things move very fast.

These equations only work for inertial frames—viewpoints that move at steady speeds without speeding up or slowing down. They help change information from one moving viewpoint to another. The Lorentz transformations are important for studying fast-moving particles and light.

History

Main article: History of Lorentz transformations

Many scientists worked on ideas about light and motion starting in 1887. In 1905, Albert Einstein used these ideas to create his theory of special relativity. This changed how we understand space and time.

Derivation of the group of Lorentz transformations

Main articles: Derivations of the Lorentz transformations and Lorentz group

An event is something that happens at a specific place and time. We can describe this place and time using numbers. According to Einstein's ideas, the speed of light is the same for everyone, no matter how they are moving.

The Lorentz transformations tell us how to change these numbers from one viewpoint to another when the viewpoints are moving at steady speeds. These transformations keep important relationships between time and space the same, just like turns keep shapes the same in regular geometry. They include both simple turns in space and changes when moving quickly, called boosts. When we also include simple shifts in position and time, we get an even larger group called the Poincaré group.

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}=0\quad {\text{(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 {\begin{aligned}c^{2}(t_{2}-t_{1})^{2}-(x_{2}-x_{1})^{2}-(y_{2}-y_{1})^{2}-(z_{2}-z_{1})^{2}&=&\\[6pt]c^{2}(t_{2}'-t_{1}')^{2}-(x_{2}'-x_{1}')^{2}-(y_{2}'-y_{1}')^{2}-(z_{2}'-z_{1}')^{2}&&\quad {\text{(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 {\begin{aligned}c^{2}t^{2}-x^{2}-y^{2}-z^{2}&=c^{2}t'^{2}-x'^{2}-y'^{2}-z'^{2}&\quad {\text{or}}\\[6pt]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 {\text{or}}\quad a\cdot 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

Generalities

The Lorentz transformations tell us how space and time change when you look at them from two places that move at a steady speed compared to each other. These changes keep important physics rules the same, no matter how you move.

There are different kinds of Lorentz transformations. One type, called a Lorentz boost, happens when the places move at a constant speed without turning. Another type is a rotation, where the places are tilted but not moving. Mixing a rotation and a boost creates a special change called a homogeneous transformation. There are also transformations called reflections, which flip the signs of space or time coordinates.

Physical formulation of Lorentz boosts

Further information: Derivations of the Lorentz transformations

Imagine two observers. One stands still in a place we call frame F, and the other moves at a steady speed in frame F′. Both use special sets of coordinates — like maps — to note where and when things happen.

If the moving observer notes an event happening at coordinates t′, x′, y′, z′, the stationary observer will note the same event at different coordinates t, x, y, z. These coordinates change in a special way depending on how fast one frame moves compared to the other. This change is called a Lorentz boost, named after physicist Hendrik Lorentz.

One key idea is that the speed of light stays the same no matter how fast you move. This rule shapes how space and time mix together when you change from one moving view to another.

Mathematical formulation

Main article: Lorentz group

Further information: Matrix (mathematics), matrix product, linear algebra, and rotation formalisms in three dimensions

The Lorentz transformations tell us how space and time look different when you change your viewpoint. These changes keep some important rules the same, like the combined distance in space and time, called the spacetime interval.

They can include simple changes in where you stand, or more complex changes that mix space and time. These ideas are very useful in physics, especially when studying things that move very fast, close to the speed of light.

Intersection, ∩	Antichronous (or non-orthochronous) LTs L ↓ = { Λ : Γ ≤ − 1 } {\displaystyle {\mathcal {L}}^{\downarrow }=\{\Lambda :\Gamma \leq -1\}}	Orthochronous LTs L ↑ = { Λ : Γ ≥ 1 } {\displaystyle {\mathcal {L}}^{\uparrow }=\{\Lambda :\Gamma \geq 1\}}
Proper LTs L + = { Λ : det ( Λ ) = + 1 } {\displaystyle {\mathcal {L}}_{+}=\{\Lambda :\det(\Lambda )=+1\}}	Proper antichronous LTs L + ↓ = L + ∩ L ↓ {\displaystyle {\mathcal {L}}_{+}^{\downarrow }={\mathcal {L}}_{+}\cap {\mathcal {L}}^{\downarrow }}	Proper orthochronous LTs L + ↑ = L + ∩ L ↑ {\displaystyle {\mathcal {L}}_{+}^{\uparrow }={\mathcal {L}}_{+}\cap {\mathcal {L}}^{\uparrow }}
Improper LTs L − = { Λ : det ( Λ ) = − 1 } {\displaystyle {\mathcal {L}}_{-}=\{\Lambda :\det(\Lambda )=-1\}}	Improper antichronous LTs L − ↓ = L − ∩ L ↓ {\displaystyle {\mathcal {L}}_{-}^{\downarrow }={\mathcal {L}}_{-}\cap {\mathcal {L}}^{\downarrow }}	Improper orthochronous LTs L − ↑ = L − ∩ L ↑ {\displaystyle {\mathcal {L}}_{-}^{\uparrow }={\mathcal {L}}_{-}\cap {\mathcal {L}}^{\uparrow }}

Tensor formulation

Main article: Representation theory of the Lorentz group

The Lorentz transformations explain how different measurements change between places moving at steady speeds. These changes work for things we can picture as points in a four-part space, like locations and some forces.

The rules also work for more detailed descriptions called tensors, which help us understand many parts of the world. This shows that the same ideas help us see how what we observe shifts when we change where we are looking in space.

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 {\begin{aligned}&U(\Lambda ,a)\Psi _{p_{1}\sigma _{1}n_{1};p_{2}\sigma _{2}n_{2};\cdots }\\={}&e^{-ia_{\mu }\left[(\Lambda p_{1})^{\mu }+(\Lambda p_{2})^{\mu }+\cdots \right]}{\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})}\left[W(\Lambda ,p_{1})\right]D_{\sigma _{2}'\sigma _{2}}^{(j_{2})}\left[W(\Lambda ,p_{2})\right]\cdots \right)\Psi _{\Lambda p_{1}\sigma _{1}'n_{1};\Lambda p_{2}\sigma _{2}'n_{2};\cdots },\end{aligned}}}

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