Isometries in physics

Isometries are a special type of symmetry in which distances are preserved. They are a subclass of spacetime symmetries, and are represented by Killing vector fields.

In vector calculus, scalar are invariant under isometries, vectors and pseudovectors rotate with rotations, and vectors invert under inversion, but pseudovectors do not. This concerns scalar multiplication, the dot product, Euclidean norm, gradient, and divergence, and, producing pseudovectors and pseudoscalars, cross product, curl, and scalar triple product.

Typically physical laws are expressed in these operations and Euclidean distance, so symmetry with respect to Euclidean distance isometries plays a prominent role compared with generalisations, i.e. symmetry with respect to non-isometric transformations.

When discussing symmetry of a vector field one should distinguish between the case that the function does not change under some isometry, or that it changes accordingly.

Consider the gravitational field due to a mass distribution.

• A rotation of the mass distribution gives at a correspondingly rotated position a corresponding rotation of the gravity vector
• A reflection in a plane of the mass distribution gives at a correspondingly reflected position a correspondingly reflected gravity vector
• An improper rotation of the mass distribution gives at a correspondingly changed position the same improper rotation of the gravity vector
• A translation of the mass distribution at a correspondingly changed position does not change the gravity vector (a translation would mean addition of a constant vector; it would anyway not be clear what would be "corresponding", since the physical dimension is different).
• A glide reflection or a combination of a translation and a rotation of the mass distribution change the gravity vector at a correspondingly changed position according to the reflection and rotation, respectively, without translation.

Suppose the mass distribution has some geometrical symmetry; this can be expressed by the symmetry group. This gives a corresponding symmetry group for the gravitational field in a sense corresponding to the observations above. Suppose some isometry leaves the mass distribution unchanged, and compare gravity at some point and gravity at the point to which the isometry maps the first point. They are related by the linear part of the isometry: the possible rotation and reflection, but not the translation.

For example, the gravitational field due to a cube-shaped uniform mass has octahedral symmetry.

This is similar in many other cases where a physical quantity is associated with a situation with geometrical symmetry.

However, in the case of a pseudoscalar or pseudovector, it does not apply for indirect isometries.

For example, the magnetic field due to a current-carrying wire has cylindrical symmetry in the sense that a rotation of position gives a corresponding rotation of the field. A reflection in a plane through the wire, of position, neither leaves the magnetic field unchanged nor changes the field by the same reflection: it changes the field by a reflection in the perpendicular plane through the wire.

In general, for various symmetries:

• translation of position is not associated with any change of scalars, pseudoscalars, other vectors or pseudovectors
• rotation of position is associated with corresponding rotation of other vectors and pseudovectors, and no change of scalars and pseudoscalars
• inversion of position is associated with inversion of other vectors and of pseudoscalars, but no change of scalars and pseudovectors

Reflections and improper rotations are combinations of rotation and inversion and we can conclude:

• an isometry of position is associated with an isometry of other vectors and pseudovectors, which is the linear part of the isometry of position, combined with an inversion of pseudovectors in the case that the isometry changes orientation.

Consider physical situations with symmetry:

• In the case of n-fold rotational symmetry about an axis (n≥2), all vectors and pseudovectors at the axis are directed along it (or zero); a fortiori this applies for cylindrical symmetry.
• In the case of mirror image symmetry in a plane with respect to scalars and vectors, the vectors at the plane are directed in the plane (or zero), and the associated pseudovectors are at the plane directed perpendicular to it (or zero).
• In the case of inversion symmetry with respect to a point, with respect to scalars and vectors, all vectors at the point are zero. The pseudovectors can be arbitrary at the point.
• In the case of rotational symmetry about two intersecting axes, all vectors and pseudovectors at the point of intersection are zero; a fortiori this applies for spherical symmetry, at the center.
• In the case of mirror image symmetry in a plane with respect to scalars and vectors, and an axis of rotation in the plane, the vectors at the axis are directed along it (or zero), and the associated pseudovectors are zero.
• In the case of mirror image symmetry in a plane with respect to scalars and vectors, and an axis of rotation perpendicular to the plane, the vectors at the point of intersection are zero, and the associated pseudovectors are along the axis.

Examples:

• gravity due to a mass with a vertical symmetry plane containing the point of observation is vertical
• if a body rotating about an axis has a symmetry plane perpendicular to the axis, then the angular momentum with respect to the intersection of the plane and the axis is along the axis; if the body has cylindrical symmetry about the axis, the angular momentum with respect to any point on the axis is along the axis.

See also

• Field theory
• Isometry group
• Killing vector field
• Spacetime symmetries
• Symmetry group
• Symmetry in physics