Misplaced Pages

Revision as of 07:41, 8 June 2005

Pedantic Quibble

The author of this article says that

${\displaystyle U(x,t)=f(kx+ct)+g(kx-ct)}$

is the wave equation. But of course this is the D'Alembert general solution to the one-dimensional wave equation, not the equation itself, is often written in coordinate-free notation as

${\displaystyle \Box U=0}$

(where ${\displaystyle \Box }$ is the Laplace/Beltrami operator) or in conventional PDE notation (using a Cartesian coordinate chart) as

${\displaystyle U_{tt}=U_{xx}}$

I propose to modify this page to correct this, if no-one objects.

C.H.

Clean-up tag

Would someone mind explaining how this article needs to be cleaned up? Other than an extraneous paragraph (which I have removed), I see nothing wrong with the structure of this article. I wonder if a wave equation is the right example myself, but I will leave it up to Hillman's judgement as to what to do with that.

It is my opinion that this article is of the right size and structure, being a coherent explanation of general covariance and a simple example. I think that the best way of cleaning it up is to drop the needs-cleaning-up tag.

--EMS | Talk 19:09, 2 Jun 2005 (UTC)

Clarification of Critique

Hi, EMS, I am still very new to Misplaced Pages, so please bear with me.

I did not add the "clean-up" tag, and I guess I didn't read the article very carefully first time around, because now I see some more objections, in addition to the one mentioned above. So I'd have to agree with the other critic that the article should be rewritten essentially from scratch. Some points to bear in mind:

0. The sentence

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

${\displaystyle U(x,t)=f(kx+ct)+g(kx-ct)}$

for some functions f, g and some scalars k and c."

is seriously misleading. In fact, ${\displaystyle U(x,t)=f(kx+ct)+g(kx-ct)}$ is not the wave equation! Rather, this is D'Alembert's general solution to the one-dimnensional wave equation.

1. The article should be rewritten to discuss the notion of covariance of a differential equation under a transformation group. Thus, in modern physics/math, we can have Lorentz covariance, diffeomorphism covariance, ${\displaystyle SU(2)}$ covariance (with the group action being understood), etc. For example, the original formulation of Maxwell's equations turns out to be Lorentz covariant; this is obvious when one writes the equations in modern form as

${\displaystyle F_{,a}^{ab}=4\pi J^{a}}$

${\displaystyle F_{ab,c}+F_{bc,a}+F_{ca,b}}$

However, these equations are not diffeomorphism covariant, because if you apply a more general difffeomorphism than a Lorentz transformation, they assume a new form. But if we change the partial derivatives to covariant derivatives, we do get a set of diffeomorphism invariant equations,

${\displaystyle F_{;a}^{ab}=4\pi J^{a}}$

${\displaystyle F_{ab;c}+F_{bc;a}+F_{ca;b}}$

It turns out that this formulation can be used to define EM on curved spacetimes.

2. The term "general covariance" is in fact archaic, so the article should really be called "diffeomorphism covariance".

3. The EFE is a tensor equation, hence automatically diffeomorphism covariant, but while this is a very important property, it is not "the defining characteristic" of GTR. Indeed, competitors such as scalar-tensor theories are also diffeomorphism covariant.

4. You said "classical formulations involve a privileged time variable". I think you might mean that Maxwell was not aware of the Lorentz covariance of his field equations, and did not know either Einstein's kinematic or Minkowski's geometric interpretation of the significance of this mathematical fact. In fact, the classical formulation of EM is mathematically equivalent to the first set above, and since this is Lorentz covariant, it does not have a privileged time variable. Rather, it has a privileged notion of non-accelerating frame. It is true that Maxwell didn't know this, however.

5. I plan to rewrite the article sometime in the next few weeks, after I have read some more math articles to get some more ideas for how to write a good math article. I can already see that it is much easier to write a new article from scratch than to try to fix a seriously flawed old one! So I am considering a "solution" which involves writing a new article on "covariance " or something like that. I am also planning to write about related topics such as the point symmetry group of a differential equation.

For the moment I have just added a citation to a good discussion of "general covariance" in a well-known and widely available gtr textbook.

--CH | Talk