Symmetry plays an important role in Einstein’s derivation of the Lorentz transformations. From the outset he assumes that the transformations must be linear “because of the properties of homogeneity which we attribute to space and time”, and his light postulate is based on the isotropy of space. Minkowski also pays explicit attention to the role of global symmetries in his formulation of relativity theory. His insight was that these symmetries require a geometrical theory of a four-dimensional spacetime manifold.
Minkowski identified the Lorentz group, or rather the Poincaré group including time and space translations, as the symmetry group of Minkowski spacetime isometries, that is, transformations that leave invariant the Minkowski spacetime interval. Although these ideas originated in the context of electromagnetism, Minkowski surmised that, in fact, Poincaré transformations are exact symmetries of all basic laws of physics.
Nowadays, Poincaré invariance is the fundamental hypothesis in relativistic physics. The 10-parameter group contains translations, rotations and boosts, which are ten transformations of Poincaré symmetry. Noether's Theorem then states that, corresponding to these ten symmetry transformations, there are ten conservation laws :
- Invariance under space and time translations is the mathematical expression of the uniformity of physical laws in space and time. The associated conservation laws are those of momentum and of energy.
- The conservation of angular momentum is associated with invariance under rotations.
- The three conservation laws associated with boosts conserve the three components of the velocity of the center of mass.
- Elementary particles are associated with irreducible representations of Poincaré group (Eugene Wigner, 1939).
- Poincaré invariance also underlies the formal consistency of general relativity, which is essentially the local version of Lorentz invariance.