Coordinate Transformations | Research & Encyclopedia Articles

Coordinate Transformations

Physics is an experimental science, an therefore a science of measurement. Of all possible measurements, the most fundamental is position. To quantitatively locate an object in space (or space-time) requires a coordinate system (CS). The choosing of a coordinate system is nothing more than the assignment of a label to each point in space. (In general, such a label is an n-tuplet of numbers, where n is the dimension of the space. Thus it requires three numbers to label a point in space, and four numbers to label an event in space-time.) In order to compare measurements between coordinate systems a set of equations relating the two different labels assigned to the same physical point is required. These relations are known as "coordinate transformations."

The simplest CS is the well-known Cartesian system, with three orthogonal (mutually-perpendicular) axes, usually labeled x, y, and z. It is not always the best choice, however. It is usually most convenient (and calculatingly most simple) to choose a CS that possess the same symmetries as the situation being modeled. Hence, for problems possessing spherical symmetry, spherical coordinates, are chosen. A point is still labeled by three numbers, but they are now the distance from the origin r and two angular coordinates. The coordinate transformations relate the two systems.

Two coordinate systems can differ only in their choice of origin; one is then translated with respect to the other. If the origins coincide, the axis of one system can be rotated with respect to the corresponding axis of the other. Again, a simple set of equations relates the labels of a point in one CS to the labels of the same point in the other.

It is of very deep significance that the laws of physics are invariant (have the same mathematical form) under certain coordinate transformations. No matter what CS is chosen, physical law must be invariant under coordinate translations; that symmetry leads to momentum conservation. Invariance under time translations leads to energy conservation, while the invariance of the laws of mechanics under rotations is at the heart of angular momentum conservation.

Coordinate transformations between different CSs may involve one observer, or two. One observer may analyze a problem using spherical coordinates, because the mathematics will then be simpler, but then express the result in terms of Cartesian coordinates. Coordinate transformations between the two systems allow him to do this. In addition, two observers, each looking at the same situation, may each employ a different CS. This may occur for any of a number of reasons. Perhaps they are not at the same place (the origins of their CSs differ), or one observer must look in a different direction" than the other (his CS is rotated with respect to his colleague's). There is no physical significance attached to the use of these different systems. Each physicist will (if he analyzes the situation correctly) predict that the same thing will happen at the same place--although the coordinates used to label that place will differ. The coordinate transformations will show that the two sets of labels refer to the same point in space.

Another possibility is that the two physicists are in motion with respect to each other. Suppose one physicist moves with a constant velocity with respect to another. The space and time coordinates of the two physicist's CSs are related by a set of transformations named after the Dutch physicist Hendrik Lorentz. In 1905 German-American physicist Albert Einstein showed that the laws of physics must be invariant under the Lorentz transformations, which led directly to the Special theory of relativity. The requirement that physical law be invariant under general (arbitrary) coordinate transformations led Einstein to the General theory of relativity. In relativity theory the terms "Special" and "General" refer to the type of coordinate transformations under which the theories are invariant.