Lorentz transformation

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Diagram 1. Appearance of space-time along a universe line of an accelerated observer.

The vertical direction indicates the time, the horizontal indicates the spatial distance, the pointed line is the observer's path in time space. The lower room represents the set of past events visible to the observer. Points can represent any kind of events in space time.

The slope of the universe line or vertical path gives the relative speed of the observer.

The Lorentz transformations, within the theory of special relativity, are a set of relationships that account for how the measurements of a physical quantity obtained by two different observers are related. These relationships established the mathematical basis for Einstein's theory of spatial relativity, since Lorentz transformations specify the type of geometry of spacetime required by Einstein's theory.

Mathematically, the set of all Lorentz transformations forms the Lorentz group.

History

Historically, Lorentz transformations were introduced by Hendrik Antoon Lorentz (1853 - 1928), who had introduced them phenomenally to resolve certain inconsistencies between electromagnetism and classical mechanics. Lorentz had discovered in the year 1900 that Maxwell's equations were invariant under this set of transformations, now called Lorentz transformations. Like other physicists, before the development of the theory of relativity, he assumed that the invariant speed for the transmission of electromagnetic waves referred to the transmission through a privileged reference system, a fact known by the name ether hypothesis. However, after Albert Einstein's interpretation of these relationships as genuine coordinate transformations in a four-dimensional space-time, the ether hypothesis was called into question.

Lorentz transformations were published in 1904 but their initial mathematical formalism was incorrect. The French mathematician Poincaré developed the set of equations in the consistent form in which they are known today. The work of Minkowski and Poincaré showed that the Lorentz relations could be interpreted as the transformation formulas for rotation in four-dimensional space-time, which had been introduced by Minkowski.

See also: History of Special Relativity

Form of Lorentz transformations

The Lorentz transformations relate the measurements of a physical magnitude made by two different inertial observers, being the relativistic equivalent of the Galileo transformation used in physics until then.

The Lorentz transformation allows us to preserve the value of the constant speed of light for all inertial observers.

From the coordinates

One of the consequences of the fact that —unlike what happens in classical mechanics— in relativistic mechanics there is no absolute time, is that both the time interval between two events, as well as the effective distances measured by different observers in different states of motion are different. That implies that the time and space coordinates measured by two inertial observers differ from each other. However, due to the objectivity of physical reality, the measurements of one and the other observers are relatable by fixed rules: the Lorentz transformations for the coordinates.

To examine the specific form taken by these coordinate transformations, two inertial reference systems or inertial observers are considered: $O ,$ and ${bar {O}}$ and each of them is supposed to represent the same event S or space-time point (representable for an instant of time and three spatial coordinates) by two different coordinate systems:

$S_O = (t,x,y,z) qquad S_{bar{O}} = (bar{t}, bar{x}, bar{y}, bar{z})$

Since the two sets of four coordinates represent the same point of space-time, these must be interrelated somehow. Lorentz's transformations say if the system ${bar {O}}$ It's moving even at speed. $V,$ along the X-axis of the system $O,$ and in the initial moment ( $t = bar{t} = 0$ ) the origin of coordinates of both systems coincide, then the coordinates attributed by the two observers are related by the following expressions:

${displaystyle {bar {x}}={frac {x-Vt}{sqrt {1-{frac {V^{2}}{c^{2}}}}}}qquad {bar {t}}={frac {t-{frac {Vx}{c^{2}}}}{sqrt {1-{frac {V^{2}}{c^{2}}}}}}qquad {bar {y}}=yqquad {bar {z}}=z,}$

Or equivalently by the inverse relations of the previous ones:

${displaystyle x={frac {{bar {x}}+V{bar {t}}}{sqrt {1-{frac {V^{2}}{c^{2}}}}}}qquad t={frac {{bar {t}}+{frac {V{bar {x}}}{c^{2}}}}{sqrt {1-{frac {V^{2}}{c^{2}}}}}}qquad y={bar {y}}qquad z={bar {z}}}$

Where $c,$ is the speed of light in the void. Previous relationships can also be written in a matrix form:

$begin{bmatrix} cbar{t} \ bar{x} \ bar{y} \ bar{z} end{bmatrix} = begin{bmatrix} gamma & -betagamma & 0 & 0 \ -betagamma & gamma & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1 end{bmatrix}begin{bmatrix} ct \ x \ y \ z end{bmatrix} qquad begin{bmatrix} c t \ x \ y \ z end{bmatrix} = begin{bmatrix} gamma & betagamma & 0 & 0 \ betagamma & gamma & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1 end{bmatrix} begin{bmatrix} cbar{t} \ bar{x} \ bar{y} \ bar{z} end{bmatrix}$

Where the Lorentz factor and the relative speed with respect to light have been introduced to abbreviate the expressions:

gamma = frac{1}{sqrt{1-frac{V^2}{c^2}}} qquad beta = frac{V}{c}

The above Lorentz transformation takes this form assuming that the origin of coordinates of both reference systems is the same for t = 0; If this restriction is removed, the concrete form of the equations becomes more complicated. If, in addition, the restriction that the relative velocity between the two systems is given along the X axis and that the axes of both coordinate systems are parallel is removed, the Lorentz transformation expressions become even more complicated, being called the expression general Poincaré transformation.

For the moment and the energy

The covariance requirement of the theory of relativity requires that any vector quantity of Newtonian mechanics be represented in relativistic mechanics by a quadrivector or quadritensor in relativity theory. Thus, the linear momentum needs to be expanded to a four-vector called four-vector energy-momentum or four-momentum, which is given by four components, one temporal component (energy) and three spatial components (linear moments in each coordinate direction):

mathbf{P} = (P^0, P^1, P^2, P^3) = left(frac{E}{c},p_x, p_y, p_zright)

When two inertial observers are examined, they are both measured different components of the moment according to their velocity relative to the observed particle (something that also happens in Newtonian mechanics). If denoted by two inertial observers to the quadruple $O ,$ and ${bar {O}}$ with cartesian coordinate systems of parallel and moving axis relative to the X axis, such as those considered in the previous section, the quadruples measured by both observers are related by a transformation of Lorentz given by:

bar{p}_x = frac{p_x - Efrac{V}{c^2}}{sqrt{1 - frac{V^2}{c^2}}} qquad bar{E} = frac{E - V p_x}{sqrt{1 - frac{V^2}{c^2}}} qquad bar{p}_y = p_y qquad bar{p}_z = p_z

And the inverse transformation is similarly given by:

p_x = frac{bar{p}_x + bar{E}frac{V}{c^2}}{sqrt{1 - frac{V^2}{c^2}}} qquad E = frac{bar{E} + V bar{p}_x}{sqrt{1 - frac{V^2}{c^2}}} qquad p_y = bar{p}_y qquad p_z = bar{p}_z

Or equivalently in matrix form the previous two sets of equations are represented as:

begin{bmatrix} bar{E}/c \ bar{p}_x \ bar{p}_y \ bar{p}_z end{bmatrix} = begin{bmatrix} gamma & -betagamma & 0 & 0 \ -betagamma & gamma & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1 end{bmatrix}begin{bmatrix} E/c \ p_x \ p_y \ p_z end{bmatrix} qquad begin{bmatrix} E/c \ p_x \ p_y \ p_z end{bmatrix} = begin{bmatrix} gamma & betagamma & 0 & 0 \ betagamma & gamma & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1 end{bmatrix} begin{bmatrix} bar{E}/c \ bar{p}_x \ bar{p}_y \ bar{p}_z end{bmatrix}

Where the Lorentz factor and the relative speed with respect to light have been introduced again to abbreviate the expressions.

For quadrivectors

Until now, only inertial systems in relative motion with respect to the X axis have been considered, but systems with parallel axes with respect to the Y and Z axes could also have been considered and, in that case, the coordinate transformation matrices would be given by matrices similar to those considered in the previous sections of the form:

Lambda_{(X)} = begin{bmatrix} gamma_x & -beta_x gamma_x & 0 & 0 \ -beta_x gamma_x & gamma_x & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1 end{bmatrix} qquad Lambda_{(Y)} = begin{bmatrix} gamma_y & 0 & -beta_y gamma_y & 0 \ 0 & 1 & 0 & 0 \ -beta_y gamma_y & 0 & gamma_y & 0 \ 0 & 0 & 0 & 1 end{bmatrix} qquad Lambda_{(Z)} = begin{bmatrix} gamma_z & 0 & 0 & -beta_z gamma_z \ 0 & 1 & 0 & 0 \ 0 & 0 & 1 & 0 \ -beta_z gamma_z & 0 & 0 & gamma_z end{bmatrix} qquad [*]

The above transformations are sometimes called boosts, space-time rotations, or sometimes Lorentz transformations themselves. The product of any number of transformations of the above type is also a Lorentz transformation. All these products form a subgroup of the proper Lorentz group. In general, the proper Lorentz group is formed by:

Space-temporal rotations or boostswhich can be written as the product of a finite number boosts type ${displaystyle [*]}$ .
Space rotations, consisting of a shaft turn. This type of transformation is also part of the Galileo group.

The proper Lorentz group thus defined is a connected Lie group. If improper transformations such as time inversions and spatial reflections are added to these proper transformations, the result is the complete Lorentz group, made up of four connected components, each of them homeomorphic to the proper Lorentz group. Once the Lorentz group is defined, we can write the most general linear transformations possible between measurements taken by inertial observers whose coordinate axes coincide at the initial instant:

begin{bmatrix}bar{V}^0 \ bar{V}^1 \ bar{V}^2 \ bar{V}^3 end{bmatrix} = begin{bmatrix} & & & \ & & R(theta_1,theta_2,theta_3) & \ & & & \ & & & end{bmatrix} begin{bmatrix} gamma & -gammabeta & 0 & 0 \ -gammabeta & gamma & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1 end{bmatrix} begin{bmatrix} & & & \ & & R(varphi_1,varphi_2,varphi_3) & \ & & & \ & & & end{bmatrix} begin{bmatrix} V^0 \ V^1 \ V^2 \ V^3 end{bmatrix}

Where, in addition to the boost given by the coordinate transformation according to the relative separation velocity, the two rotations in terms of Euler angles have been included:

The matrix ${displaystyle R(varphi _{1},varphi _{2},varphi _{3})}$ aligns the first coordinate system in such a way that the transformed X-axis becomes parallel to the separation speed of the two systems.
The matrix ${displaystyle R(theta _{1},theta _{2},theta _{3})}$ is the reverse rotation of which would align the X axis of the second observer with the speed of separation.

More compactly we can write the last transformation in tensor form using the Einstein summation convention as:

bar{V}^alpha = Lambda_beta^alpha V^beta qquad Lambda_beta^alpha:= {[R(theta)]}_rho^alpha [Lambda_{(X)}]_sigma^rho {[R(varphi)]}_beta^sigma

General Tensor Form

Suppose now that instead of measuring vector quantities two observers measure the components of some other tensile magnitude, suppose that the observers $O,$ and ${bar {O}}$ they measure in their coordinate systems the same tense magnitude but each from their own coordinate system reaching:

{displaystyle mathbf {T} _{O}=T_{beta _{1}...beta _{n}}^{alpha _{1},...alpha _{m}}quad {frac {partial }{partial x^{alpha _{1}}}}otimes...{frac {partial }{partial x^{alpha _{m}}}}otimes dx^{beta _{1}}otimes...otimes dx^{beta _{n}}}

{displaystyle mathbf {T} _{bar {O}}={bar {T}}_{beta _{1}...beta _{n}}^{alpha _{1},...alpha _{m}}quad {frac {partial }{partial {bar {x}}^{alpha _{1}}}}otimes...{frac {partial }{partial {bar {x}}^{alpha _{m}}}}otimes d{bar {x}}^{beta _{1}}otimes...otimes d{bar {x}}^{beta _{n}}}

The postulate that there is an objective reality independent of the observers and that their measurements can be compared through the appropriate covariance transformations leads to the fact that if these observers are inertial their measurements will be related by the following relationships:

{displaystyle {bar {T}}_{beta _{1}...beta _{n}}^{alpha _{1},...alpha _{m}}={[Lambda ^{T}]}_{beta _{1}}^{beta '_{1}}...{[Lambda ^{T}]}_{beta _{n}}^{beta '_{n}}quad [Lambda ]_{alpha '_{1}}^{alpha _{1}}...[Lambda ]_{alpha '_{m}}^{alpha _{m}}quad T_{beta '_{1}...beta '_{n}}^{alpha '_{1},...alpha '_{m}}}

Where the matrices Λ are defined, as in the previous section, by the product of two spatial rotations and a simple temporary rotation (boost).

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