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## The physical significance of tensors

In this course, we shall only concern ourselves with coordinate transformations which transform an inertial frame into another inertial frame. This limits us to four classes of transformations: displacements of the coordinate axes, rotations of the coordinate axes, parity reversals (i.e., ), and Lorentz transformations.

One of the central tenets of physics is that experiments should be reproducible. In other words, if somebody performs a physical experiment today, and obtains a certain result, then somebody else performing the same experiment next week ought to obtain the same result, within the experimental errors. Presumably, in performing these hypothetical experiments, both experimentalists find it necessary to set up a coordinate frame. Usually, these two frames do not coincide. After all, the experiments are, in general, performed in different places and at different times. Also, the two experimentalists are likely to orientate their coordinate axes differently. Nevertheless, we still expect both experiments to yield the same result. What exactly do we mean by this statement? We do not mean that both experimentalists will obtain the same numbers when they measure something. For instance, the numbers used to denote the position of a point (i.e., the coordinates of the point) are, in general, different in different coordinate frames. What we do expect is that any physically significant interrelation between physical quantities (i.e., position, velocity, etc.) which appears to hold in the coordinate system of the first experimentalist will also appear to hold in the coordinate system of the second experimentalist. We usually refer to such interrelationships as laws of physics. So, what we are really saying is that the laws of physics do not depend on our choice of coordinate system. In particular, if a law of physics is true in one coordinate system then it is automatically true in every other coordinate system, subject to the proviso that both coordinate systems are inertial.

Recall that tensors are geometric objects which possess the property that if a certain interrelationship holds between various tensors in one particular coordinate system, then the same interrelationship holds in any other coordinate system which is related to the first system by a certain class of transformations. It follows that the laws of physics are expressible as interrelationships between tensors. In special relativity, the laws of physics are only required to exhibit tensor behaviour under transformations between different inertial frames: i.e., translations, rotations, and Lorentz transformations. Parity inversion is a special type of transformation, and will be dealt with later on. In general relativity, the laws of physics are required to exhibit tensor behaviour under all non-singular coordinate transformations.

Next: Space-time Up: Relativity and electromagnetism Previous: Tensors
Richard Fitzpatrick 2006-02-02