Cartan's method of curvature forms
Doesn't this thread belong in "Tensor Analysis & Differential Geometry"? It seems to involve a question about the mathematical theory Riemannian two-manifolds, not relativistic physics.
[EDIT: thanks to the unsung admin who moved this thread!]
(I corrected the formatting and made a small change in notation.)
Giammy85 said:
A two-dimensional Rienmannian manifold has a metric given by
<br />
ds^2= \exp(2f) \, dr^2 + r^2 \, d\theta^2<br />
where f is a function of the coordinate r.
Which can be embedded as a surface of revolution in E^3 as you probably know, at least locally (depending on f). To wit:
<br />
\left[ \begin{array}{c} h(r) \\ r \, \cos(\phi) \\ r \, \sin(\phi) \end{array} \right]<br />
leads to the line element
<br />
ds^2 = \left(1 + {h^\prime}^2 \right) \, dr^2 + r^2 \, d\phi^2<br />
from which we obtain the ODE 1 + {h^\prime}^2 = \exp(2f) which you can solve to obtain h in terms of f.
Note that the euclidean formula for the circumference of the circle r=r_0 holds good, but (for reasons easy to understand from the embedding in case of a typical function h!) the usual formula relating dr to radial distance does
not hold true, in general. In higher dimensions, BTW, in the context of gravitation physics, it is customary to refer to a radial coordinate with these properties as a
Schwarzschild radial coordinate.
Giammy85 said:
Eventually I calculated that the Ricci scalar is...
This isn't homework, is it? Or even worse, a take-home exam problem?
If so, you should have posted in the Homework Help forum, which has special rules!
The sphere is a space of constant Gaussian curvature (see the component R_{1212} of the Riemann tensor), which is an invariant property. See almost any differential geometry textbook for more information, e.g. Struik,
Lectures on Classical Differential Geometry.
Here's the slick way to compute the curvature tensor (especially efficient in higher dimensions): from the line element
<br />
ds^2 = \exp(2f) \, dr^2 + r^2 \, d\phi^2, \; 0 < r < \infty, \; -\pi < \phi < \pi<br />
(for convenience I made a slight change in notation, and where the ranges of the coordinates are the maximal permissible) we can read off the
coframe field
<br />
\sigma^1 = \exp(f) \, dr, \; \sigma^2 = r \, d\phi<br />
so that the line element is simply
<br />
ds^2 = \sigma^1 \otimes \sigma^1 + \sigma^2 \otimes \sigma^2<br />
Taking the exterior derivatives of these one forms gives
<br />
d\sigma^1 = 0, \; d\sigma^2 = dr \wedge \phi<br />
But
Cartan's first structural equations tell us that
<br />
d\sigma^j = -{\omega^j}_k \wedge \sigma^k, \;<br />
{\omega^j}_k = -{\omega^k}_j<br />
(since the exponential of anti-symmetric operator is a rotation). Plugging in the result we just obtained gives
<br />
-d\phi \wedge dr = dr \wedge d\phi <br />
= -{\omega^2}_1 \wedge \sigma^1 <br />
= -{\omega^2}_1} \wedge \exp(f) \, dr<br />
from which the (so(2) valued)
connection one-form is
<br />
{\omega^1}_2 = -\exp(-f) \, d\phi<br />
(As an easy check, the exterior product of this with \sigma^2 = r \, d\phi does indeed vanish, as it should.) Taking the exterior derivative of this one-form gives
<br />
d{\omega^1}_2 <br />
= f^\prime \, \exp(-f) \, dr \wedge d\phi <br />
= \frac{f^\prime}{r} \, \exp(-2f) \, \sigma^1 \wedge \sigma^2<br />
But
Cartan's second structural equation tells us that the (so(2) valued)
curvature-two form is
<br />
{\Omega^j}_k = d{\omega^j}_k + {\omega^j}_m \wedge {\omega^m}_k<br />
where the second term drops out in two dimensions. We can now read off the Riemann curvature tensor from
<br />
{\Omega^j}_k = {R^j}_{kmn} \, d\sigma^m \wedge d\sigma^n <br />
Finally, lowering an index gives
<br />
R_{1212} = \frac{f^\prime}{r} \, \exp(-2f)<br />
which is the only algebraically independent component in two dimensions. Note that in the coordinate cobasis dr, \; d\phi this becomes R_{r \phi r \phi} = r \, f^\prime.
This method is called the "method of Pfaffians" in Struik, and the "method of one-forms" in MTW, and was introduced by Elie Cartan c. 1905, who later championed its use in gravitation physics. In the case of surfaces embedded in E^3, it should always give the same results as the well known theory developed by Gauss c.1820 but not published until 1827-8. See Flanders,
Differential Forms with Applications to the Physical Sciences, Dover reprint, for details.
(Hmm... I see that the
current version at the time of my post of the WP article on Gauss is incorrect: Gauss presented his famous memoir to the Royal Society of Gottingen in 1827, and it was published in their journal the following year; this paper predates the publications of both Lobachevski and Bolyai and greatly generalizes their work. A English translation with commentary can be found in Karl Friedrich Gauss,
General Investigations of Curved Surfaces, Dover reprint. See also the superb comments in Spivak,
Comprehensive Introduction to Differential Geometry. Hmm... actually the WP article is pretty damn bad overall IMO, even ignoring factual errors.)
Giammy85 said:
In this case R comes to be equal to 2
I've read on wikipedia that Ricci scalar of a sphere with radius r is equal to 2/r^2
Unless stated otherwise, I always evaluate tensor indices wrt a frame field or coframe field (aka ONB of Pfaffians or one-forms or covectors, in order of increasingly modern terminology).
The sphere of radius a has Gaussian curvature R_{1212} = 1/a^2 (units of reciprocal area; components expanded wrt a coframe field). Solving
<br />
\frac{f^\prime}{r} \, \exp(-2f) = \frac{1}{a^2}<br />
gives
<br />
\sigma^1 = \frac{a \, dr}{\sqrt{a^2-r^2}}, \; 0 < r < a<br />
which agrees with line element
<br />
ds^2 = \frac{a^2 dr^2}{a^2-r^2} + r^2 \, d\phi^2, \; 0 < r < a, \; -\pi < \phi < \pi<br />
which we obtain from the obvious embedding in E^3
<br />
\left[ \begin{array}{c} \sqrt{a-r^2} \\ r \, \cos(\phi) \\ r \, \sin(\phi) \end{array} \right]<br />
of the top half of a round sphere of radius a.
A Riemannian two-manifold of non-constant curvature cannot be any of S^2, \; E^2, \; H^2, which have curvatures of form 1/a^2, 0, -1/a^2 respectively (in any chart), where a>0 is the "radius", in the first and third cases.
Struik is a great book to learn some classical surface theory from, and Flanders is a great book to learn Cartan's method (for Riemannian geometry; for Lorentzian geometry there is one small change which MTW didn't make as clear as they might have done--- I am referring to Minkowskian transpose vs. Euclidean transpose; you can figure this out from comparing standard 3x3 matrix generators for the Lie algebras so(1,2) and so(3) and asking what "skew-transpose" means!)[size=-3]<--more sloppy writing, sorry![/size][/color]