Diagonalization of metric matrix in general relativity

[exponent137]

1. Is its possible diagonalization of metric matrix (g_{uv}) in general relativity?

2. If we include imaginary numbers, can this help?

 

[pervect]

It's possible to diagonalize the metric in a small region, by the proper choice of coordinates. It's not possible to diagonalize it everywhere, because space-time is not flat. It's rather like asking if it's possible to make an accurate map of the Earth's surface that correctly represents distances "to scale" without distortion, covering the entire surface. This is possible on a globe, but not possible on a flat sheet of paper.

I'm not sure where complex numbers come into the problem, perhaps you're asking some question other than the one I thought you were asking.

 

[Bill_K]

It's possible to diagonalize the metric in a small region, by the proper choice of coordinates. It's not possible to diagonalize it everywhere, because space-time is not flat. It's rather like asking if it's possible to make an accurate map of the Earth's surface that correctly represents distances "to scale" without distortion, covering the entire surface. This is possible on a globe, but not possible on a flat sheet of paper.

Flat is not the issue. The metric on the Earth's surface in the usual spherical polar coordinates *is* diagonal. The Schwarzschld metric is diagonal too.

 

[pervect]

Oooops. My remarks were mistakenly off-topic (about making the metric unity, not diagonal).

Offhand, I'm not sure what it takes to make a general metric diagonal - though the only non-diagonal metric I can think of offhand is the Kerr metric.

 

[Ben Niehoff]

Every quadratic form (and hence every line element) can be written as a sum of N squares, where some of the squares have a minus sign in front, according to the signature of the quadratic form. So yes, the metric can always be diagonalized. However, the basis in which it diagonalizes is not always a coordinate basis.

Actually, what I've just described is the process of finding the orthonormal frames.

 

[jfy4]

Say we are given a particular matrix that has the following form

<br /> G_{\alpha\beta}=<br /> \begin{pmatrix}<br /> 0 &amp; 0 &amp; 0 &amp; 0 \\<br /> 0 &amp; G_{11} &amp; G_{12} &amp; G_{13} \\<br /> 0 &amp; G_{21} &amp; G_{22} &amp; G_{23} \\<br /> 0 &amp; G_{31} &amp; G_{32} &amp; G_{33}<br /> \end{pmatrix}<br />

Is it possible to diagonalize this matrix using only a general algorithm, or must the components be known explicitly?

 

[DrGreg]

Say we are given a particular matrix that has the following form

<br /> G_{\alpha\beta}=<br /> \begin{pmatrix}<br /> 0 &amp; 0 &amp; 0 &amp; 0 \\<br /> 0 &amp; G_{11} &amp; G_{12} &amp; G_{13} \\<br /> 0 &amp; G_{21} &amp; G_{22} &amp; G_{23} \\<br /> 0 &amp; G_{31} &amp; G_{32} &amp; G_{33}<br /> \end{pmatrix}<br />

Is it possible to diagonalize this matrix using only a general algorithm, or must the components be known explicitly?

I used to know how to do this, 30 years ago. Look up "Sylvester's Law of Inertia".

 

[Ben Niehoff]

Say we are given a particular matrix that has the following form

<br /> G_{\alpha\beta}=<br /> \begin{pmatrix}<br /> 0 &amp; 0 &amp; 0 &amp; 0 \\<br /> 0 &amp; G_{11} &amp; G_{12} &amp; G_{13} \\<br /> 0 &amp; G_{21} &amp; G_{22} &amp; G_{23} \\<br /> 0 &amp; G_{31} &amp; G_{32} &amp; G_{33}<br /> \end{pmatrix}<br />

Is it possible to diagonalize this matrix using only a general algorithm, or must the components be known explicitly?

Of course you can, using the same method you would use to diagonalize any 3x3 matrix.

 

[exponent137]

So, because diagonalization is always possible, space-time of Minkowski can be generalized to general relativity. The question is only, if such changed space-time axes have any meaning?

 

[tom.stoer]

1. Is its possible diagonalization of metric matrix (g_{uv}) in general relativity?

2. If we include imaginary numbers, can this help?

By introducing the tetrad (4-bein) formalism you do something like that; the tetrad is a map between the manfold and its tangent space; on the tangent space you have a trivial (+---) metric. The tetrads can be used as dynamical variables instead of the metric. They have real components, so there is no need for complex numbers.