Lorentz transformation

The Lorentz transformations are important equations in physics that describe how space and time coordinates change when switching between different frames of reference moving at constant speeds relative to each other. Named after the Dutch physicist Hendrik Lorentz, these transformations are essential for understanding Einstein's theory of special relativity. They show how measurements of distance and time by observers in different moving frames are related, especially when those speeds are close to the speed of light.

Unlike the older Galilean transformation used in Newtonian physics, which works well for everyday speeds, the Lorentz transformations take into account that the speed of light is the same for all observers, no matter how they move. This leads to some surprising results, such as time passing at different rates and distances appearing shorter for observers in motion relative to each other. These effects become noticeable only when speeds approach the speed of light.

The transformations apply only to inertial frames—reference frames moving at steady velocities without acceleration. They are linear equations that can be used to convert coordinates from one frame to another, preserving the spacetime interval between events. The Lorentz transformations form the foundation for much of modern physics, particularly in the study of high-speed particles and the behavior of light.

History

Main article: History of Lorentz transformations

Many scientists studied the ideas behind the Lorentz transformations starting in 1887. They tried to explain experiments and theories about light and motion. In 1905, Albert Einstein used these ideas to create his theory of special relativity, which 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 point in space and time. We can describe this point using numbers for time and position. 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 certain important relationships between time and space the same, just like rotations 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)}}}

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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 describe how space and time coordinates change when you switch between two frames that move at a steady speed relative to each other. These transformations keep certain important rules of physics the same, no matter how you move.

There are different kinds of Lorentz transformations. One type, called a Lorentz boost, happens when the frames move at a constant speed without turning. Another type is simply a rotation, where the frames 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 are a way to describe how space and time change when moving from one steady viewpoint to another. These transformations keep certain important rules the same, like the distance measured in space and time together, called the spacetime interval.

They can include simple shifts in viewpoint, as well as more complex changes that mix space and time. These transformations are very important in physics, especially in understanding how things move at very high speeds, 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 can describe how different physical quantities change between frames moving at constant speeds. These transformations apply to things that can be represented as vectors in four-dimensional spacetime, such as coordinates and certain fields.

The transformations also work for more complex objects called tensors, which are used to describe many physical properties. This means that the rules for changing between frames apply broadly, helping us understand how observations change when moving from one viewpoint to another in the universe.

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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