Space-time symmetries and invariances are properties of space-time that do not change under certain transformations, such as rotations, translations, boosts, or scalings. They are important in physics because they imply the conservation of certain quantities, such as energy, momentum, angular momentum, or charge. They also simplify the solutions of physical equations, such as Einstein’s field equations of general relativity.

There are different types of space-time symmetries and invariances, depending on the nature and degree of the transformations involved. Some examples are:

• Killing symmetry: a symmetry that preserves the metric tensor, which measures the distance and angle between points in space-time. A Killing vector field is a vector field that generates such a symmetry. For example, a stationary space-time has a Killing vector field that corresponds to time translation.
• Homothetic symmetry: a symmetry that preserves the metric tensor up to a constant factor, which means that the space-time is self-similar. A homothetic vector field is a vector field that generates such a symmetry. For example, a flat space-time has a homothetic vector field that corresponds to scaling.
• Affine symmetry: a symmetry that preserves the geodesics of the space-time, which are the paths of free-falling particles or light rays. An affine vector field is a vector field that generates such a symmetry. For example, a conformally flat space-time has an affine vector field that corresponds to conformal transformation.
• Conformal symmetry: a symmetry that preserves the angles between vectors in space-time, but not their lengths. A conformal vector field is a vector field that generates such a symmetry. For example, a Minkowski space-time has a conformal vector field that corresponds to Lorentz transformation.
• Curvature symmetry: a symmetry that preserves the curvature tensor of the space-time, which measures the deviation from flatness and the tidal forces. A curvature vector field is a vector field that generates such a symmetry. For example, a de Sitter space-time has a curvature vector field that corresponds to de Sitter transformation.
• Matter symmetry: a symmetry that preserves the matter content of the space-time, such as the energy-momentum tensor or the electromagnetic field tensor. A matter vector field is a vector field that generates such a symmetry. For example, an electrovacuum space-time has a matter vector field that corresponds to gauge transformation.

There may be four-dimensional analogues to some of these space-time symmetries and invariances, depending on how one defines and generalizes them. For example, one can consider higher-dimensional space-times with extra dimensions or compactified dimensions, and look for symmetries and invariances in those settings. One can also consider internal symmetries and invariances that act on other physical fields or parameters besides space-time coordinates, such as Yang-Mills fields or scalar fields. One can also consider discrete symmetries and invariances that involve finite transformations rather than continuous ones, such as parity or time reversal.