What is transformation formula?

Answer 1

Lorentz transformation

In physics, the Lorentz transformation describes how, according to the theory of special relativity, two observers' varying measurements of space and time can be converted into each other's frames of reference. It is named after the Dutch physicistHendrik Lorentz. It reflects the surprising fact that observers moving at different velocities may measure different distances, elapsed times, and even different orderings of events.

The Lorentz transformation was originally 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. Albert Einstein later re-derived the transformation from his postulates of special relativity. The Lorentz transformation supersedes the Galilean transformation of Newtonian physics, which assumes an absolute space and time (see Galilean relativity). According to special relativity, this is a good approximation only at relative speeds much smaller than the speed of light.

If space is homogeneous, then the Lorentz transformation must be a linear transformation. Also, since relativity postulates that the speed of light is the same for all observers, it must preserve the spacetime interval between any two events in Minkowski space. The Lorentz transformation describes only the transformations in which the spacetime event at the origin is left fixed, so they can be considered as a rotation of Minkowski space. The more general set of transformations that also includes translations is known as the Poincaré group.

Lorentz transformation for frames in standard configurationAssume there are two observers O and Q, each using their own Cartesian coordinate system to measure space and time intervals. O uses (t, x, y, z) and Q uses (t', x', y', z'). Assume further that the coordinate systems are oriented so that the x-axis and the x' -axis are collinear, the y-axis is parallel to the y' -axis, as are the z-axis and the z' -axis. The relative velocity between the two observers is v along the common x-axis. Also assume that the origins of both coordinate systems are the same. If all these hold, then the coordinate systems are said to be in standard configuration. A symmetric presentation between the forward Lorentz Transformation and the inverse Lorentz Transformation can be achieved if coordinate systems are in symmetric configuration. The symmetric form highlights that all physical laws should be of such a kind that they remain unchanged under a Lorentz transformation.

The Lorentz transformation for frames in standard configuration can be shown to be: \begin{cases}

t' &= \gamma \left( t - v x/c^{2} \right) \\ x' &= \gamma \left( x - v t \right)\\ y' &= y \\ z' &= z \end{cases} where \ \gamma = \frac{1}{ \sqrt{1 - { \frac{v^2}{c^2} is called the Lorentz factor.

Matrix formThis Lorentz transformation is called a "boost" in the x-direction and is often expressed in matrix form as

\begin{bmatrix} c t' \\ x' \\ y' \\ z' \end{bmatrix}

\begin{bmatrix} \gamma&-\beta \gamma&0&0\\ -\beta \gamma&\gamma&0&0\\ 0&0&1&0\\ 0&0&0&1\\ \end{bmatrix} \begin{bmatrix} c\,t \\ x \\ y \\ z \end{bmatrix}\ . More generally for a boost in any arbitrary direction (\beta_{x}, \beta_{y}, \beta_{z}),

\begin{bmatrix} c\,t' \\ x' \\ y' \\ z' \end{bmatrix}

\begin{bmatrix} \gamma&-\beta_x\,\gamma&-\beta_y\,\gamma&-\beta_z\,\gamma\\ -\beta_x\,\gamma&1+(\gamma-1)\dfrac{\beta_{x}^{2}}{\beta^{2}}&(\gamma-1)\dfrac{\beta_{x}\beta_{y}}{\beta^{2}}&(\gamma-1)\dfrac{\beta_{x}\beta_{z}}{\beta^{2}}\\ -\beta_y\,\gamma&(\gamma-1)\dfrac{\beta_{y}\beta_{x}}{\beta^{2}}&1+(\gamma-1)\dfrac{\beta_{y}^{2}}{\beta^{2}}&(\gamma-1)\dfrac{\beta_{y}\beta_{z}}{\beta^{2}}\\ -\beta_z\,\gamma&(\gamma-1)\dfrac{\beta_{z}\beta_{x}}{\beta^{2}}&(\gamma-1)\dfrac{\beta_{z}\beta_{y}}{\beta^{2}}&1+(\gamma-1)\dfrac{\beta_{z}^{2}}{\beta^{2}}\\ \end{bmatrix} \begin{bmatrix} c\,t \\ x \\ y \\ z \end{bmatrix}\ , where \beta = \frac{v}{c}=\frac{c} and \gamma = \frac{1}{\sqrt{1-\beta^2}}.

Note that this transformation is only the "boost," i.e., a transformation between two frames whose x, y , and z axis are parallel and whose spacetime origins coincide (see The "Standard configuration" Figure). The most general proper Lorentz transformation also contains a rotation of the three axes, because the composition of two boosts is not a pure boost but is a boost followed by a rotation. The rotation gives rise to Thomas precession. The boost is given by a symmetric matrix, but the general Lorentz transformation matrix need not be symmetric.

The composition of two Lorentz boosts B(u) and B(v) of velocities u and v is given by: B(\mathbf{u})B(\mathbf{v})=B(\mathbf{u}\oplus\mathbf{v})Gyr[\mathbf{u},\mathbf{v}]=Gyr[\mathbf{u},\mathbf{v}]B(\mathbf{v}\oplus\mathbf{u}),

where u\oplusv is the velocity-addition, and Gyr[u,v] is the rotation arising from the composition, gyr being the gyrovector space abstraction of the gyroscopic Thomas precession, and B(v) is the 4x4 matrix that uses the components of v, i.e. v1, v2, v3 in the entries of the matrix, or rather the components of v/c in the representation that is used above.

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