Re: questions about the Christoffel symbol
- From: "Puppet_Sock" <puppet_sock@xxxxxxxxxxx>
- Date: 18 Dec 2006 14:30:16 -0800
GRseminar@xxxxxxxxx wrote:
Dear all,
I'm reading a physics book about general relativity(book I) and a
mathematic book about geometry(book II). In book II, the author
introduces the "formula of Gauss":
X_{ij} = \Gamma^r_{ij} X_r + L_{ij} U -------( eq I )
where X(u,v) is a 2-surface, \Gamma^r_{ij} is the Christoffel symbol,
U is the surface normal vector and X_{ij} means the derivative of X_i
with respect to coordinate j (where X_i is the derivative of X w.r.t.
coordinate i, one takes as the basis vector on a regular surface)
However, in the text of GR(book I), the Chrisoffel symbol is defined
as the expansion coefficients of the derivative of basis vectors with
respect to a coordinate, i.e.
e_\alpha_,\beta = \Gamma^\mu_{\alpha\beta} e_\mu -------(eq II)
where the comma of the left hand side stands for the partial
derivative.
My wonder is, in (eq I), there is a term "L_{ij} U" which accounts for
the component of X_{ij} normal to the surface, however, in the book II
about relativity, there is no term about "the component outside the
space?" Why is the derivative of the basis vectors can be expanded by
the original old basis?! In the case of 2-surface, this is obviously no
Well, it's a little difficult to tell, not having the texts. But I
*think* I've
understood what is going on.
The math book would seem to be dealing with a 2-D surface embeded
in a 3-D manifold. In such a case you can construct the 3-D connection
and get the result you've called (eq I). But note that this is a 3-D
statement, a result in three dimensions.
It is also possible to construct the 2-D version of this. That is,
suppose
you were a triangle... Well, suppose you get the book _Flatland_
and read about Mr. R. Square. That is, you were only aware of two
dimensions of the surface, and did everything in the surface. The
question then is, could you detect the nature of your surface by
measurements in the surface?
This is what is happening in the GR case of your (eq II). That is a
4-D equation about properties intrinsic to the 4-space that is being
worked with. You don't, in the usual approach, worry about trying
to embed the manifold in GR in a larger space.
Depending on exactly what is going on, you could try to write down
such things as the connection and metric on the surface of a sphere.
Or a coordinate patch on a sphere since you will find that you get
some "weirdness" if you try to cover a sphere with a single set of
coordinates. (Think of the North pole in standard polar coords.)
So you work out the connection of a 2 sphere by observing what
happens in measurements entirely within the surface of the sphere,
and never use that it is embeded in a larger space.
Socks
.
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