The physical significance of tensors

One of the central tenets of physics is that experiments should be repeatable. 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. For instance, one experimentalist might align his $x$-axis with the North Star, whilst the other might align the same axis to point towards Mecca. 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. This set of transformations forms a group known as the Poincaré group. 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.

Consider Newton's first law of motion. These take the form of three differential equations,

$$m \,\frac{d^2 x}{dt^2} = f_x,$$ (75)

$$m \,\frac{d^2 y}{dt^2} = f_y,$$ (76)

$$m \,\frac{d^2 z}{dt^2} = f_z,$$ (77)

in a general inertial frame. However, we can also write them as a single vector differential equation,

$$m\, \frac{d^2{\bfm r}}{dt^2} = {\bfm f}.$$ (78)

What is the advantage of the vector notation? Many people would say that it is just a convenient form of shorthand. However, there is another, far more important, advantage. Before we can accept Newton's first law of physics as a proper law of physics we need to convince ourselves that it is coordinate independent; i.e., that it also holds in coordinate frames which are related to the original frame via a general translation or rotation of the coordinate axes. It is indeed possible to prove this, but the demonstration is rather tedious because a general rotation is a rather complicated transformation. A vector is a geometric object (in fact, it is a rank one tensor in three dimensional Euclidean space) whose three components transform under a general translation and rotation of the coordinate axes in an analogous manner to the difference in coordinates between two fixed points in space. This ensures that any vector equation which is true in one coordinate frame is also true in any other coordinate frame which is related to the original frame via a general rotation or translation of the axes. Thus, the main advantage of Eq. (2.50) is that it makes the coordinate independent nature of Newton's first law of motion manifestly obvious. Of course, we cannot deny that Newton's first law also looks simpler when it is expressed in terms of vectors. This is one example of a rather general feature of physical laws. Namely, when the laws of physics are expressed in a manner which makes their invariance under various transformation groups manifest then they tend to take a particularly simple form. In general, the larger the group of transformations the simpler the form taken by the laws of physics. One of the major goals of modern physics is to find the largest possible group of transformations under which the laws of physics are invariant, and then prove that when expressed in a manner which makes this invariance manifest these laws reduce to a single unifying principle.

We already know how to write the laws of physics in terms of vectors and vector fields. This means that these laws are automatically invariant under translations and rotations. However, according to the relativity principle, there is a third class of transformations under which the laws of physics must also be invariant; namely, Lorentz transformations. There are two ways in which we could verify that the laws of physics are Lorentz invariant. The direct method is extremely tedious, since Lorentz transformations are rather complicated. An alternative method is to write the laws of physics in terms of geometric objects which transform as tensors under translations, rotations, and Lorentz transformations. This method has the advantage that it makes the Lorentz invariant nature of the laws of physics obvious. We also expect that when the laws of physics are written in manifestly Lorentz invariant form then they will look even simpler than they do when written just in terms of vectors. The laws of electromagnetism provide a particularly good illustration of this effect.