Fermi coordinates

In the mathematical theory of Riemannian geometry, Fermi coordinates are local coordinates that are adapted to a geodesic.[1]

More formally, suppose M is an n-dimensional Riemannian manifold, ${\displaystyle \gamma }$ is a geodesic on ${\displaystyle M}$, and ${\displaystyle p}$ is a point on ${\displaystyle \gamma }$. Then there exists local coordinates ${\displaystyle (t,x^{2},\ldots ,x^{n})}$ around ${\displaystyle p}$ such that:

• For small t, ${\displaystyle (t,0,\ldots ,0)}$ represents the geodesic near ${\displaystyle p}$,
• On ${\displaystyle \gamma }$, the metric tensor is the Euclidean metric and all its first derivatives vanish,
• On ${\displaystyle \gamma }$, all Christoffel symbols vanish.

Such coordinates are called Fermi coordinates and are named after the Italian physicist Enrico Fermi. The above properties are only valid on the geodesic. For example, if all Christoffel symbols vanish near ${\displaystyle p}$, then the manifold is flat near ${\displaystyle p}$.

See also

• Proper reference frame (flat spacetime)#Proper coordinates or Fermi coordinates
• Geodesic normal coordinates
• Christoffel symbols
• Isothermal coordinates

References

1. Manasse and Misner , Fermi Normal Coordinates and Some Basic Concepts in Differential Geometry. Journal of Mathematical Physics 4:6, 1963.