cf. metric.typ
[geodesic]
Geodesics as possible "shortest length paths". Action
ODE initial value . The definition of geodesics does not depend on coordinate selection
is the metric-volume-form restricted to a 1 dimension path. is the induced quadratic-form
The Lagrange equation is
For unit length parameter of the path, , the equation becomes
product-rule expansion , where . Transposing terms and using , the equation becomes
Or written as
where is [metric-connection] alias [Levi-Civita-connection] alias [Christoffel-symbols]
metric-connection is not a tensor. The transformation of metric-connection [connection-transformations]
by the transformation of metric in the definition of
[geodesic-exponential] โฆ
[geodesic-coordinate]
Geodesic or with gives coordinate
It is a local diffeomorphism by at , by
- The solution of the ODE analytically depends on the initial values
In geodesic coordinates, the geodesic equation is . Proof The geodesic is
In geodesic coordinates at point , the connection at point is zero,
Proof
ODE
Initial value and
Substitute the solution of the ODE into the ODE to get
Thus, at point , for all directions , ==> at
[Taylor-expansion-of-metric-in-geodesic-coordinate]
In geodesic coordinates, the Taylor expansion of the metric is
-
The zeroth-order term is the standard metric
-
The first-order term is zero i.e. the first derivative is zero
Combining the two,
Proof
In geodesic coordinates at point
-
0th. For geodesic coordinates, at point , , which copies the orthonormal basis of the origin in geodesic coordinates to the coordinate-frame of the tangent space at point
-
1st.
[differenial-of-metric-vs-connection] Prop Proof Directly substitute the definition of into the formula
Then use to get at point
In coordinates
Can also be written as
For the inverse matrix, there is a similar one
[differenial-of-metric-inverse-vs-connection] Prop
Proof Using and
In coordinates
Can also be written as