User:Mpatel/sandbox/Principle of 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, for example, 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. It is one of the defining features of general relativity. The principle of general covariance states that the laws of physics should take the same form in all coordinate systems. Other physical theories such as electrodynamics and special relativity also have a generally covariant formulation, although their classical formulations involve a privileged time variable. In fact, most physical laws can be written in a generally covariant way.

The principle of general covariance is a very useful tool in general relativity.

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


 * $$\ U(x, t) = f(kx + ct) + g(kx - ct)$$

for some functions f, g and some scalars k and c. Note how this explicitly uses coordinates t (time) and x (space).

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


 * $$\begin{matrix} X = (x_1, x_2) && K_1 = (k, c) && K_2 = (k, -c) \end{matrix}$$

Then the wave equation can be written as:


 * $$\ U(X) = f(K_1 \cdot X) + g(K_2 \cdot X)$$

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

Status
While the principle of general covariance is mathematically appealing, it's importance regarding the role it plays in general relativity is still disputed. Some authors argue that the principle is physically vacuous, whilst others maintain that it is a crucial feature of GR.