What is an orthogonal chart in the context of Gaussian Normal Coordinates?

[quantum123]

What are Gaussian Normal Coordinates?

 

[pervect]

Gaussian normal coordinates have a metric of the form

$$ds^2 = -dt^2 + g_{ij} dx^i dx^j$$

This implies that:

1) the t coordinate measures proper time of "stationary" observers, i.e. observers with constant spatial coordinates.

2) the time coordinate is orthogonal to the space coordinates (more precisely, the vector $\partial / \partial t$ is orthogonal to all of the vectors $\partial / \partial x^i$ )

These are also known as synchronous coordinates, a common example from cosmology is comvoing coordinates.

 

[Chris Hillman]

What are Gaussian Normal Coordinates?

There are various kinds of "normal coordinates" used in Riemannian geometry (positive definite metric tensors) and semi-Riemannian geometry (allows indefinite metric tensors, such as occur in Lorentzian manifolds as used in general relativity).

In Riemannian geometry, the idea is to fix a point P and to construct a chart in which metric tensor assumes the usual flat space Euclidean form at P, with the Christoffel coefficients and first partials of the metric components all vanishing at P, so that the quadratic deviation of the metric at points near P are given in terms of the components of the curvature tensor at P. It is important to understand that such charts are not unique; what is important is that near P, the metric can be approximated by the curvature components at P. See http://en.wikipedia.org/wiki/Geodesic_normal_coordinates

In general relativity, Riemann normal coordinates are often used as a technical device in various places, and Fermi normal coordinates are also useful at times. See section 3.1-3.2 of http://relativity.livingreviews.org/Articles/lrr-2004-6/title.html#articlese3.html for more about Riemann and Fermi normal coordinates.

Gaussian normal coordinates have a metric of the form

$$ds^2 = -dt^2 + g_{ij} dx^i dx^j$$

I've only seen that usage in the book by Wheeler and Cuifolini, Inertia and Gravitation. Most people would just call these synchronous coordinates.

 

[pervect]

I've only seen that usage in the book by Wheeler and Cuifolini, Inertia and Gravitation. Most people would just call these synchronous coordinates.

MTW uses it too, that's where I looked it up. (See pg 717).

 

[Chris Hillman]

Here I stand, corrected, not unlike a normal vector on a surface

MTW uses it too, that's where I looked it up. (See pg 717).

So they do! Thanks for reminding me.

 

[Thrice]

While we're on this topic, what are orthogonal curvilinear coordinates? I briefly encountered them a while ago & learned to take gradients etc, but I'm looking for a more detailed treatment.

 

[Chris Hillman]

In this context, an "orthogonal chart" probably means a chart in which the metric appears "diagonalized", e.g. the Schwarzschild chart is diagonalized but the Eddington chart is not (because of the "cross-terms" du \, dr, which signify that the coordinate vectors are "skew", i.e. non-orthogonal).

Many models, e.g. the Kerr vacuum, possesses no "diagonal" charts.