Schwarzchild Solution

[Off-diagonal]

In Schwarzchild coordinates, why off-diagonal of metric tensor is vanishes?

[ajb]

I think the simple answer is because the space-time is not rotating. Terms like [math]dr d\theta[/math] mix radial and angular coordinates and more interestingly [math]dt d\phi[/math] mixes time and angular coordinates and give the possibility of CTC's.

 

For a static non-rotating spherically symmetric space-time we don't want these mixing terms when using "standard coordinates". This might not be the case in more general coordinates where it may be more difficult to interpret the coordinates.

[Off-diagonal]

What is CTC's?

[ajb]

What is CTC's?

 

Closed Time-like Curves, i.e. "Time machines"

[Norman Albers]

We enter the model saying there is spherical symmetry.

 

I'll quote from Adler, Bazin, Schiffer, Introduction to General Relativity: (p.186) "The limiting form of the line element at large distances from the origin may be expected to be Lorentzian... The reasoning which follows is based on plausibility only, in order to guess a heuristically reasonable and convenient line element. We should expect the line element to be invariant under inversion of the coordinate interval [math]dx^0[/math] (representing time); that is, [math]ds^2[/math] should be invariant under the replacement of [math]dx^0[/math] by [math]-dx^0[/math]. This dictates that we use Gaussian coordinates in which the off-diagonal elements [math]g_{0i}[/math] of the metric tensor are zero and the line element has the form [math]g_{00}(dx^0)^2+g_{ik}dx^idx^k [/math] with the [math]g_{ik}[/math] independent of the [math]x^0[/math]. This is referred to as a static metric; it is to be distinguished from a metric which is merely independent of time, or stationary. Second, if there is to be no preferred angular direction in space, the line element should be independent of a change of [math]d\theta[/math] to [math]-d\theta[/math] and a change of [math]d\phi[/math] to [math]-d\phi[/math]... so the metric tensor must be entirely diagonal for the type of solution we desire." I enjoy the mathematical clarity and honesty here.