Differential Geometry

Metric

Definitions

Denote the basis in use as ˆeμ, then the metric can be written as

gμν=ˆeμˆeν

if the basis satisfies

Inversed metric

gμλgλν=δνμ=gνμ

How to calculate the metric

Let’s check the definition of metric again.

If we choose a basis ˆeμ, then a vector (at one certain point) in this coordinate system is

xa=xμˆeμ

Then we can construct the expression of metric of this point under this coordinate system,

gμν=ˆeμˆeν

For example, in spherical coordinate system,

(6)x=rsinθcosϕˆex+rsinθsinϕˆey+rcosθˆez

Now we have to find the basis under spherical coordinate system. Assume the basis is ˆer,ˆeθ,ˆeϕ. Choose some scale factors hr=1,hθ=r,hϕ=rsinθ. Then the basis is

ˆer=xhrr=ˆexsinθcosϕ+ˆeysinθsinϕ+ˆezcosθ,

etc. Then collect the terms in formula (6) is we get x=rˆer, this is incomplete. So we check the derivative.

dx=ˆex(drsinθcosϕ+rcosθcosϕdθrsinθsinϕdϕ)ˆey(drsinθsinϕ+rcosθsinϕdθ+rsinθcosϕdϕ)ˆez(drcosθrsinθdθ)=dr(ˆexsinθcosϕ+ˆeysinθsinϕˆezcosθ)dθ(ˆexcosθcosϕ+ˆeycosθsinϕˆezsinθ)rdϕ(ˆexsinϕ+ˆeycosϕ)rsinθ=ˆerdr+ˆeθrdθ+ˆeϕrsinθdϕ

Once we reach here, the component (er,eθ,eϕ) of the point under the spherical coordinates system basis (ˆer,ˆeθ,ˆeϕ) at this point are clear, i.e.,

dx=ˆerdr+ˆeθrdθ+ˆeϕrsinθdϕ=erdr+eθdθ+eϕdϕ

In this way, the metric tensor for spherical coordinates is

gμν=(eμeν)=(1000r2000r2sin2θ)

Connection

First class connection can be calculated

Γμμνλ=ˆeμˆeμ,λ

Second class connection isfootnote{Kevin E. Cahill}

[μν,ι]=gιμΓμμνλ

Gradient, Curl, Divergence, etc

Gradient

Tbbc;a=aTbbc=Tbbc,a+ΓbadTddcΓdacTbbd

Curl

For an anti-symmetric tensor, aμν=aνμ

Curlμντ(aμν)aμν;τ+aντ;μ+aτμ;ν=aμν,τ+aντ,μ+aτμ,ν

Divergence

divν(aμν)aμνμν;ν=aμνxν+Γμντaτν+Γνντaμτ=1gxν(gaμν)+Γμνλaνλ

For an anti-symmetric tensor

div(aμν)=1gxν(gaμν)

Annotation Using the relation g=gμνAμν, Aμν is the algebraic complement, we can prove the following two equalities.

Γμμν=νlng
Vμμ;μ=1gxμ(gVμ)

In some simple case, all the three kind of operation can be demonstrated by different applications of the del operator, which ˆxx+ˆyy+ˆzz.

  • Gradient, f, in which f is a scalar.
  • Divergence, v
  • Curl, ×v
  • Laplacian, Δ2