# Get Lorentzian Spherically Symmetric Metric to Sylvester Form

• tut_einstein
In summary, the conversation is about determining the transformation that brings a spherically symmetric spacetime metric in spherical coordinates to the Sylvester normal form, with just 1 or -1 on its main diagonal and all other elements equal to zero. This transformation can be calculated from the matrix of eigenvectors and results in a non-holonomic basis. An example of this is seen in Schwarzschild spacetime.
tut_einstein
Hi,

I'm trying to determine the exact transformation that brings a spherically symmetric spacetime metric in spherical coordinates to the Sylvester normal form (that is, with just 1 or -1 on its main diagonal, with all other elements equal to zero.) Assuming that the metric has Lorentzian signature, does anyone know how to determine the exact transformation that achieves this?

Thanks.

This is essentially nothing more than the orthogonal diagonalisation of a symmetric matrix that you probably did loads of times when you first learned about matrices. The coordinate transformation can be calculated from the matrix of eigenvectors. This gives a diagonal matrix, which should have one negative and the rest positive entries. Then you just have to rescale the coordinates to make the entries -1 and +1.

henry_m said:
Then you just have to rescale the coordinates to make the entries -1 and +1.

tut_einstein, this, in general, results in a non-holonomic basis. i.e., one that is not induced by any coordinate system. As a specific example, consider Schwarzschild spacetime,

## 1. What is the significance of transforming a Lorentzian spherically symmetric metric to Sylvester form?

Transforming a Lorentzian spherically symmetric metric to Sylvester form allows for a simpler representation of the metric, making it easier to analyze and solve equations in curved spacetime. It also helps to identify the curvature properties of the metric.

## 2. How is the transformation from Lorentzian spherically symmetric metric to Sylvester form performed?

The transformation is achieved by using a set of coordinate transformations and tensor manipulations. This process is known as the Sylvester algorithm and involves transforming the metric tensor to a diagonal form, with only three non-zero elements.

## 3. What are the advantages of using Sylvester form for Lorentzian spherically symmetric metrics?

Using Sylvester form allows for easier calculations and interpretations of the metric, as well as a clearer understanding of the underlying geometry. It also simplifies the process of solving the Einstein field equations in curved spacetime.

## 4. Can any Lorentzian spherically symmetric metric be transformed to Sylvester form?

Yes, any Lorentzian spherically symmetric metric can be transformed to Sylvester form, as long as the necessary coordinate transformations and tensor manipulations are performed correctly.

## 5. What are some applications of transforming a Lorentzian spherically symmetric metric to Sylvester form?

One application is in general relativity and cosmology, where the Sylvester form can be used to analyze the properties of spacetime and make predictions about the behavior of objects in curved spacetime. It is also helpful in understanding the gravitational effects of massive objects, such as black holes.

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