Fermi coordinates information

In the mathematical theory of Riemannian geometry, there are two uses of the term Fermi coordinates. In one use they are local coordinates that are adapted to a geodesic.^[1] In a second, more general one, they are local coordinates that are adapted to any world line, even not geodesical.^[2]

Take a future-directed timelike curve $\gamma =\gamma (\tau )$ , $\tau$ being the proper time along $\gamma$ in the spacetime $M$ . Assume that $p=\gamma (0)$ is the initial point of $\gamma$ .

Fermi coordinates adapted to $\gamma$ are constructed this way.

Consider an orthonormal basis of $TM$ with $e_{0}$ parallel to ${\dot {\gamma }}$ .

Transport the basis $\{e_{a}\}_{a=0,1,2,3}$ along $\gamma (\tau )$ making use of Fermi–Walker's transport. The basis $\{e_{a}(\tau )\}_{a=0,1,2,3}$ at each point $\gamma (\tau )$ is still orthonormal with $e_{0}(\tau )$ parallel to ${\dot {\gamma }}$ and is non-rotated (in a precise sense related to the decomposition of Lorentz transformations into pure transformations and rotations) with respect to the initial basis, this is the physical meaning of Fermi–Walker's transport.

Finally construct a coordinate system in an open tube $T$ , a neighbourhood of $\gamma$ , emitting all spacelike geodesics through $\gamma (\tau )$ with initial tangent vector $\sum _{i=1}^{3}v^{i}e_{i}(\tau )$ , for every $\tau$ .

A point $q\in T$ has coordinates $\tau (q),v^{1}(q),v^{2}(q),v^{3}(q)$ where $\sum _{i=1}^{3}v^{i}e_{i}(\tau (q))$ is the only vector whose associated geodesic reaches $q$ for the value of its parameter $s=1$ and $\tau (q)$ is the only time along $\gamma$ for that this geodesic reaching $q$ exists.

If $\gamma$ itself is a geodesic, then Fermi–Walker's transport becomes the standard parallel transport and Fermi's coordinates become standard Riemannian coordinates adapted to $\gamma$ . In this case, using these coordinates in a neighbourhood $T$ of $\gamma$ , we have $\Gamma _{bc}^{a}=0$ , all Christoffel symbols vanish exactly on $\gamma$ . This property is not valid for Fermi's coordinates however when $\gamma$ is not a geodesic. Such coordinates are called Fermi coordinates and are named after the Italian physicist Enrico Fermi. The above properties are only valid on the geodesic. The Fermi-coordinates adapted to a null geodesic is provided by Mattias Blau, Denis Frank, and Sebastian Weiss.^[3] Notice that, if all Christoffel symbols vanish near $p$ , then the manifold is flat near $p$ .

^ Manasse, F. K.; Misner, C. W. (1963). "Fermi Normal Coordinates and Some Basic Concepts in Differential Geometry". Journal of Mathematical Physics. 4 (6): 735–745. Bibcode:1963JMP.....4..735M. doi:10.1063/1.1724316.
^ Marzlin, Karl-Peter (1994). "The physical meaning of Fermi coordinates". General Relativity and Gravitation. 26 (6): 619–636. arXiv:gr-qc/9402010. Bibcode:1994GReGr..26..619M. doi:10.1007/BF02108003. S2CID 17918026.
^ Blau, Matthias; Frank, Denis; Weiss, Sebastian (2006). "Fermi coordinates and Penrose limits". Class. Quantum Grav. 23 (11): 3993–4010. arXiv:hep-th/0603109. Bibcode:2006CQGra..23.3993B. doi:10.1088/0264-9381/23/11/020. S2CID 3109453.

[1] Manasse, F. K.; Misner, C. W. (1963). "Fermi Normal Coordinates and Some Basic Concepts in Differential Geometry". Journal of Mathematical Physics. 4 (6): 735–745. Bibcode:1963JMP.....4..735M. doi:10.1063/1.1724316.

[2] Marzlin, Karl-Peter (1994). "The physical meaning of Fermi coordinates". General Relativity and Gravitation. 26 (6): 619–636. arXiv:gr-qc/9402010. Bibcode:1994GReGr..26..619M. doi:10.1007/BF02108003. S2CID 17918026.

[3] Blau, Matthias; Frank, Denis; Weiss, Sebastian (2006). "Fermi coordinates and Penrose limits". Class. Quantum Grav. 23 (11): 3993–4010. arXiv:hep-th/0603109. Bibcode:2006CQGra..23.3993B. doi:10.1088/0264-9381/23/11/020. S2CID 3109453.

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