General covariance

In theoretical physics, general covariance (also known as diffeomorphism invariance) is the invariance of physical laws (for example, the equations of general relativity) under arbitrary coordinate transformations. This means that such laws take the same mathematical form regardless of whether they are expressed in an accelerating or non-accelerating reference frame. Alternatively, one could say that a generally covariant theory is one which treats time-like and space-like coordinates in the same way.

The term "general covariance" was first introduced by Albert Einstein to describe the property he sought to obtain in the theory of general relativity, and indeed, it is the defining feature of general relativity. Other physical theories such as electrodynamics and special relativity also have a generally covariant formulation, although their classical formulations involve a priveleged time variable. In fact, most physical laws can be written in a generally covariant way.

An example: the wave equation

The wave equation (which describes the behavior of a vibrating string) is classically written as:

<math>\ U(x, t) = f(kx + ct) + g(kx – ct)<math>

for some functions f, g and some scalars k and c. Note how t (time) is set apart from x.

To put this in a generally covariant form, one could define the following vectors:

<math>\begin{matrix} X = (x_1, x_2) && K_1 = (k, c) && K_2 = (k, -c) \end{matrix}<math>

Then the wave equation can be written as:

<math>\ U(X) = f(K_1 \cdot X) + g(K_2 \cdot X)<math>

Now, <math>x_2<math> plays the role of the "time" variable, but it is not treated specially in the form of the equation.

One interpretation of general covariance is that space and time are not separate, as was thought prior to the 1905 introduction of special relativity; rather, they are linked in an aggregate concept called space-time. Another interpretation is simply that general covariance makes it easy to apply, for instance, Lorentz transformations to see how physical quantities change in different reference frames.

See also

• Lorentz covariance
• Covariant
• Covariance (This article is about statistics.)
• List of mathematical topics in relativity