Poincare group

The Poincaré Group is the group of transformations of four-dimensional spacetime whose action does not affect the "spacetime interval". The "spacetime interval" is the analogue of the familiar, three-dimensional notional of distance in the four-dimensional "spacetime" of special relativity. The spacetime interval between two points $(x_{1},y_{1},z_{1},ct_{1})$ and $(x_{2},y_{2},z_{2},ct_{2})$ is spacetime is given by

$s^{2}=(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}+(z_{1}-z_{2})^{2}-c^{2}(t_{1}-t_{2})^{2}$.

The first three terms in this expression are the familiar Euclidean distance on the space coordinates, while the difference in time is introduced with a minus sign. Thus, in contrast to the usual notion of distance, the "spacetime interval" between two points may be negative! Such points are said to be timelike separated.

The Poincaré Group is the full set of transformations which leave the spacetime interval invariant. Just as in 3-dimensional space distance is preserved by translations and rotations, so in Minkowski space the transformations of the Poincaré Group preserve the spacetime interval.

There are ten basic transformations in the Poincaré Group, and all other transformations may be written as combinations of these (for example, a translation followed by a rotation). In the language of group theory, these ten elements are said to be generators of the Poincaré Group, which thus has the structure of a ten-dimensional Lie group. Six of these are the familiar translations and rotations of space. In additional to translations of time, the Poincaré Group has three additional generators, corresponding to the Lorentz transformations.

Spatial translations

• $T_{x}:(x,y,z,ct)=(x+\epsilon ,y,z,ct)$
• $T_{y}:(x,y,z,ct)=(x,y+\epsilon ,z,ct)$
• $T_{z}:(x,y,z,ct)=(x,y,z+\epsilon ,ct)$

Time Translations

• $T_{t}:(x,y,z,ct)=(x,y,z,ct+\epsilon )$

Spatial Rotations

• $R_{x}:(x,y,z,ct)=(x,cos\theta y+sin\theta z,-sin\theta y+cos\theta y,ct)$
• $R_{y}:(x,y,z,ct)=(cos\theta x+sin\theta z,y,-sin\theta x+cos\theta z,ct)$
• $R_{z}:(x,y,z,ct)=(cos\theta x+sin\theta y,-sin\theta x+sin\theta y,z,ct)$

Lorentzian Transformations

• $L_{x}:(x,y,z,ct)=(x',y,z,ct')$
• $L_{y}:(x,y,z,ct)=(x,y',z,ct')$
• $L_{z}:(x,y,z,ct)=(x,y,z',ct')$

Mathematically, the Poincaré Group may be described as the semidirect product of the translations with the Lorentz transformations. It is the full isometry group of the pseudoriemannian manifold $\mathbb {R} ^{4}$ with the Lorentz metric.