4
$\begingroup$

I teach myself mathematics, but those days I wanted to learn about General relativity (not to pursue in it but only to have some background), perhaps because I am very curious to learn why exactly We needed such curved space-time to interpret Gravity ... But I faced some problems when I began reading about it because my knowledge of physics is very little (It felt also that I know no maths at all !! that I felt a little frustrated ). The problem that I know all what was required, but in the text-book I am reading they were working with the tools "I know" in strange way that I am not understanding fully. For example in Newtonian gravitation, the author derived the formula $$ \nabla \cdot \frac{\mathbf{x}}{|\mathbf{x}|^3} = 4\pi \delta(\mathbf{x}) $$ , and used it to derive the divergence of gravitation at some point $\mathbf{r}$ as following; $$ \nabla\cdot\mathbf{g}(\mathbf{r}) = -4\pi G\int \rho(\mathbf{s})\ \delta(\mathbf{r}-\mathbf{s})\ d^3\mathbf{s} = -4\pi Gp(r) $$, the last equality is one of those problems facing me, how I can derive that and I am assuming that I use the ordinary jordan measure used in definition of reimanian integral, I can understand it using the "convention" that $\delta$ is sum sort of concentrating mass at $0$ formalizing that using measure which the book didn't do, It only defined $\delta$ to be that function of all of $\mathbb{R}^3$, for which $$\int_V f(\mathbf{x}) \delta( \mathbf{x}) \space{} \mathrm{d}\mathbf{x} = f(0) $$ whenever $0$ lies in $V$, and the integral is zero otherwise, which can't make sense in the ordinary reimanian integral, so how must I understand it ??

For the formula $$ \nabla \cdot \frac{\mathbf{x}}{|\mathbf{x}|^3} = 4\pi \delta(\mathbf{x}) $$, it can't make sense to me except that it is some type of convention not a formula, though it was *derived# in the textbook through the divergence theorem where $$\int_{V} \nabla \cdot \frac{\mathbb{x}}{|\mathbf{x}|^3} \mathrm{d} \mathbf{x} = \int_{\partial V} d \Theta = 4 \pi $$, where $V$ is some body "3-manifold with boundary which is compact" which contains the origin, $d \Theta$ is the solid angle form. But the divergence theorem can't be used like this because of the underlying field is not differentiable, moreover the left integral must be zero which doesn't happen here, except if there is some variant of divergence theorem for other measure spaces where some point has concentrated mass, or it is some type of convention where we want the divergence Th. to hold.

So what exactly I wanted to know, how must I understand these things exactly and does physics learners are caring about the formalism as those of mathematics?

And why the text-book doesn't differentiate between theorems, definitions and conventions ?

Finally, what do you advice me to have a good background in General relativity, and what are its problems which are purely mathematical like equations which arise, structures and others ? Thanks

$\endgroup$
  • $\begingroup$ It might be wise to study (a) differential geometry, and (b) partial differential equations. Differential geometry for the obvious reasons. But (b) because classical field theory, for the most part, boils down to PDEs [or, depending on your outlook, some "abstract nonsense" involving sections of fiber bundles ;)]. $\endgroup$ – Alex Nelson Feb 2 '14 at 0:05
3
$\begingroup$

You can understand everything classically if you're careful. Then you can just remember what the shorthand notation means, and allow yourself to arrive at the correct answer a little more quickly. For example, just remember that $$ V(x)=\int_{V}\rho(x')\frac{1}{|x-x'|}\,d^{3}x' $$ is a solution of Poisson's equation $$ \nabla^{2}V(x)=-4\pi\rho(x). $$ So $\vec{E}=-\nabla V$ is a solution of $\nabla\cdot \vec{E}=4\pi\rho$ (sorry for the definition that omits $\epsilon_{0}$.) You need some smoothness on the function $\rho$ to say this, but not much. This formula can be verified, just not as trivially as a delta function suggests. The delta function becomes a way of writing such an equation in order to make the formula intuitive.

You correctly noted that the formalism is lacking in proper rigor. Classically, one is not allowed to interchange differentiation and integration in such a case; however, if you allow the abstraction of a delta function, then you can write the above in a memorable and suggestive way as $$ \nabla^{2}V(x)=\int_{V}\rho(x')\nabla^{2}_{x}\frac{1}{|x-x'|}dV(x') \\ =-\int_{V}\rho(x')4\pi\delta(x-x')d^{3}x'=-4\pi\rho(x). $$ There's nothing wrong in doing this if you're aware that correct logic backs it up, and that a few minimal conditions do need to be verified. There exists considerable abstraction that will allow you to rigorously define such things, and have them work the way you'd like. However, I suggest that you remember that there are classical methods to make sense of such formalism; just remember that, even if a step or two in between is not classically correct, the final result can be correct under fairly general conditions.

I recommend that you learn about the Helmholtz Theorem. It will simplify a lot what you're trying to do, and will leave you more confident about what you're doing: http://en.wikipedia.org/wiki/Helmholtz_decomposition .

It is also instructive to see how professional Physicists work with delta functions:
Reference: J.D. Jackson, Classical Electrodynamics. Look for delta function in the index.

$\endgroup$
  • $\begingroup$ For my sorry I haven't learnt about Poisson equation, but it is ok as one can check that this is a solution by some limiting argument isolating the singularity at 0 ... But what is the standard formalization for delta's ? Thank you $\endgroup$ – Mahmoud Abdulrazek Feb 2 '14 at 14:46

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.