最近在學(xué)?Do Carmo 的Differential Geometry 和 Carroll 的 Spacetime and Geometry（雖然后者是基于manifold講的，但是學(xué)校教的還是古典微分幾何），記錄一下學(xué)習(xí)過(guò)程 ovo

1. Definition of geodesics

A curve on a surface is said to be a geodesic if the geodesic curvature ??at every point of . Thus, a geodesic curves only because of the curving of the surface – the extent of internal curving is zero. Here,?.

Free particles move along paths of “shortest possible distance,” or geodesics. In other words, particles try their best to move on straight lines, but in a curved spacetime there might not be any straight lines.

Light is always travelling along geodesics (null geodesics, with??)

2. Velocity and accleration vectors

Suppose there exists a diffeomorphism map for a regular surface S, and a regular curve ?that on S , then consider the tangent vector(or velocity vector) is given by:

? ? ? ? ? ? ? ? ? ? ? ? ? ?

then the accleration vector is given by:

? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

also by chain rule?we have:

? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ??

and?

? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ??

therefore,??the?accleration vector by chain rule above is exactly in the form:

The formular above is very complicated, therefore we apply?Einstein notation to simplify.Here, donate??, obtain:

? ? ? ? ? ? ? ? ? ? ? ? ? ?

the first term????indicates the linear combination of natural basis??and??in tangential space?.Then we consider the second term?.

Actually, the form of??indicates the change of basis along the trajectory?. To express this change in canonical basis?, we have the following in?Einstein notation :

? ? ? ? ? ? ? ??

Here, since there are 4 directions for change and each direction has 2 basis,

thus there are 8 different?.The??is called?Christoffel symbol,?and the??for the normal component is the second fundamental form of the surface.(Revision for the second fundamental form:?)

Christoffel symbols describe how the direction of basis deviates from the straight line. The Christoffel symbols however, indeed describe the chage of grid of the coordinate on the surface.

3.Calculation of?Christoffel symbols

3.1. Review of the first fundamental form

First, let's review the first fundamental form of regular surfaces. The 2 tangential vectors at the point??on the surface are??and?, and a trajectory???.Also review the tangent vector of the trajectory at??is?.

the first fundamental form is the self inner product of the tangent vector, that is:

In tensor form, the first fundamental form is given by?, which is actually the metric tensor of the surface. In??, the metric tensor is a?? symmetric matrix, with:

? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

its inverse, in?Einstein notation, is expressed as?, therefore?

? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

by calculation of inverse matrix, it is trivial to obtain:

? ? ? ? ? ? ? ? ? ? ? ??

3.2. Calculate Christoffel symbols

Remind that the acceleration vector is:

? ? ? ? ? ? ? ?

and noticing the geometric relationship, the normal vector??is perpendicular to the tangent space?, therefore we can calculate the?Christoffel symbols by doing inner product, that is:

? ? ? ? ? ? ? ? ???

therefore by mutipulying the inverse, we have:

? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

or

? ? ? ? ? ? ? ? ? ?? ? ??

In detail, in an orthonormal coordinate,where?, we have the following:

4.Geodesics

With the arrangement above, we have the accleration vector below:

? ? ? ? ? ? ??

By the definition of geodesics,?，and because??

we have:??

???

because?

? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ???

therefore??if and only if

? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

which is the differential form of geodesics.

Expand Einstein notation, we have the following :

With orthonormal basis, the geodesics behaves:

5.Null geodesics

Back to the beginning, consider the null geodesics, in which massless particles are travelling along,like photon. Suppose a path along the??,then the metric is:

? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ??

With?parametrization ? , we have:

? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ??

donate???? ,the null geodesics is :

? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

With parametrization of propertime??, we have:

? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

where?

? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ??