RQG: The Schwarzschild Solution

Most recent edit on 2008-05-22 03:46:55 by CharlesFrancis

Additions:
Put k = e^−λ, which will simplify calculation,

Deletions:
Put k = e^λ, which will simplify calculation,

Edited on 2007-12-21 23:56:58 by CharlesFrancis

Additions:
As might be expected, the calculation of the Schwarzschild metric is a little brutal. This included in accordance with mathematical and scientific rigour. If one wants to properly grasp the meaning of the equations of general relativity, one should follow through a calculation, even if one does not want to check every minus sign oneself!

Deletions:
As might be expected, the calculation of the Schwarzschild metric is a little brutal. This included in accordance with mathematical and scientific rigour. If one wants to properly grasp the meaning of the equations of general relativity, one should follow through a calculation, even if one does not want to check every minus sign oneself!.

Edited on 2007-12-21 23:56:32 by CharlesFrancis

Additions:
As might be expected, the calculation of the Schwarzschild metric is a little brutal. This included in accordance with mathematical and scientific rigour. If one wants to properly grasp the meaning of the equations of general relativity, one should follow through a calculation, even if one does not want to check every minus sign oneself!.

Deletions:
As might be expected, the calculation of the Schwarzschild metricis a little brutal. This included in accordance with mathematical and scientific rigour. If one wants to properly grasp the meaning of the equations of general relativity, one should follow through a calculation, even if one does not want to check every minus sign oneself!.

Edited on 2007-12-21 23:49:36 by CharlesFrancis

Additions:
The non-vanishing Christoffel symbols,

</td><tr><td><img alt="Gravitation-59" title="Christoffel symbol" src="images/gravitation/Gravitation-59.gif" align="texttop" vspace="1">, </td><td><img alt="Gravitation-60" title="Christoffel symbol" src="images/gravitation/Gravitation-60.gif" align="texttop" vspace="1">
</td><tr><td><img alt="Gravitation-61" title="Christoffel symbol" src="images/gravitation/Gravitation-61.gif" align="texttop" vspace="3"></td><td><img alt="Gravitation-62" title="Christoffel symbol" src="images/gravitation/Gravitation-62.gif" align="texttop" vspace="1">
</td><tr><td><img alt="Gravitation-63" title="Christoffel symbol" src="images/gravitation/Gravitation-63.gif" align="texttop" vspace="1"></td><td><img alt="Gravitation-64" title="Christoffel symbol" src="images/gravitation/Gravitation-64.gif" align="texttop" vspace="1"></td></table> So, the constant of integration is C = −2GM. Thus the metric outside of an isolated, spherically symmetric, non-rotating, gravitating body of mass M"" is

Deletions:
The non-vanishing Christophel symbols,

</td><tr><td><img alt="Gravitation-59" title="Christophel symbol" src="images/gravitation/Gravitation-59.gif" align="texttop" vspace="1">, </td><td><img alt="Gravitation-60" title="Christophel symbol" src="images/gravitation/Gravitation-60.gif" align="texttop" vspace="1">
</td><tr><td><img alt="Gravitation-61" title="Christophel symbol" src="images/gravitation/Gravitation-61.gif" align="texttop" vspace="3"></td><td><img alt="Gravitation-62" title="Christophel symbol" src="images/gravitation/Gravitation-62.gif" align="texttop" vspace="1">
</td><tr><td><img alt="Gravitation-63" title="Christophel symbol" src="images/gravitation/Gravitation-63.gif" align="texttop" vspace="1"></td><td><img alt="Gravitation-64" title="Christophel symbol" src="images/gravitation/Gravitation-64.gif" align="texttop" vspace="1"></td></table> So, the constant of integration is C = 2GM. Thus the metric outside of an isolated, spherically symmetric, non-rotating, gravitating body of mass M"" is

Oldest known version of this page was edited on 2007-11-28 07:53:40 by CharlesFrancis []

Page view:

The Schwarzschild Solution

As might be expected, the calculation of the Schwarzschild metricis a little brutal. This included in accordance with mathematical and scientific rigour. If one wants to properly grasp the meaning of the equations of general relativity, one should follow through a calculation, even if one does not want to check every minus sign oneself!.

In spherical coordinates, (τ, r, θ, φ), with an origin at the centre of an isolated spherically symmetric, non-rotating, gravitating body, such as a planet or a star, ignoring the gravity of other stellar objects, the metric is

Put k = e^λ, which will simplify calculation,

Using

Using prime to denote differentiation with respect to r = x¹, the non-vanishing partial derivatives of the metric are

The non-vanishing Christophel symbols,

are



,

On raising the first index,





,

The Ricci tensor is

If one calculates all the terms, one finds a diagonal matrix. However, there is only one free parameter in the metric, and it is sufficient to calculate only one component of the Ricci tensor,

Calculate the terms individually

Substitute into R₂₂,
Gravitation-85

Einstein’s field equation says this must vanish. So,

where C is a constant of integration. Thus,

In the weak field limit, Newton’s law of gravity states,

So, the constant of integration is C = 2GM. Thus the metric outside of an isolated, spherically symmetric, non-rotating, gravitating body of mass M is

This is the Schwarzschild metric.

Return

RQG : Schwarzschild

The Schwarzschild Solution