Synchronous coordinates

In general relativity, a synchronous reference system is a coordinate system in which the metric takes the form

$ds^2=-dt^2+h_{ab}dx^adx^b$,

where the Latin indices a and b are summed over the spatial directions and $h_{{ab}}$ is a spatial metric. Any metric can locally be put into this form by a coordinate transformation. This coordinate condition is called "synchronous" because the t coordinate defines proper time for all comoving observers. However, it is not uniquely defined, and therefore is not a gauge, as the spacelike hypersurface at $t=0$ can be chosen arbitrarily. Another problem with the reference system is that caustics can occur which cause the gauge choice to break down. These problems have caused some difficulties doing cosmological perturbation theory in this system, but the problems are now well understood. Synchronous coordinates are generally considered the most efficient reference system for doing calculations, and are used in many modern cosmology codes, such as CMBFAST. They are also useful for solving theoretical problems in which a spacelike hypersurface needs to be fixed, as with spacelike singularities.

See also Normal coordinates.

References

• Carroll, Sean M. (2004). Spacetime and Geometry: An Introduction to General Relativity. San Francisco: Addison-Wesley. ISBN 0-8053-8732-3.. See section 7.2.
• C.-P. Ma & E. Bertschinger (1995). "Cosmological perturbation theory in the synchronous and conformal Newtonian gauges". Astrophys. J. 455: 7–25. arXiv:astro-ph/9506072. Bibcode:1995ApJ...455....7M. doi:10.1086/176550.
• Landau, L.D. and Lifshitz, E.M. (1972). The Classical Theory of Fields. England: Elsevier Butterworth Heinemann. ISBN 0-7506-2768-9.. See section 97.