Derive Lanczos Equation: Step-by-Step Guide

[mersecske]

Where can I found the derivation of the Lanczos equation

 

[arkajad]

Go to arxiv.org
Select find-all, choose author Lanczos

 

[mersecske]

I think this is a bad idea.
Lanczos' papers is not freely available,
because he wrote his papers in the 20'

 

[arkajad]

I think this is a bad idea.
Lanczos' papers is not freely available,
because he wrote his papers in the 20'

First try it - then tell me whether it was a bad idea or a good one.

 

[mersecske]

I didnt found useful stuff on arxiv

 

[arkajad]

I didnt found useful stuff on arxiv

I found. By Cornelius Lanczos (reprints from 1929 etc)

1) On the covariant formulation of Dirac's equation
2) Dirac's wave mechanical theory of the electron and its field theoretical interpretation
3) The tensor analytical relationships of Dirac's equation
4) The relations of the homogeneous Maxwell's equations to the theory of functions
5) The conservation laws in the field theoretical representation of Dirac's theory

Some of the above are quoted in "Lanczos's equation to replace Dirac's equation?"
which you also find on arxiv?

It isn't useful for someone who wants to know about Lanczos's equation? Then you have a peculiar taste.

 

[mersecske]

I am interested about the following Lanczos equation:

$$K^{+}_{ab} - K^{-}_{ab} = 8\pi\left(S_{ab} - \frac{1}{2}h_{ab}S\right)$$

where K is the intrinsic curvature, S is the energy-momentum tensor on the boundary surface, h is the induced metric on the surface.

 

[arkajad]

Then perhaps this will help you: "[URL equivalence of Darmois-Israel and distributional
method for thin shells in general relativity[/URL]

 

[mersecske]

I have this paper, but the equation (22) is just put there and no real derivation. How can you perform the integral on the left hand side?

 

[George Jones]

Take a look at section 3.7 (particularly the development leading to equation 3.7.11) in Eric Poisson's notes,

http://www.physics.uoguelph.ca/poisson/research/agr.pdf.

Better yet, see if your library has a copy of the excellent book, A Relativist's Toolkit: The Mathematics of Black Hole Mechanics, into which the notes evolved.

 

[mersecske]

In spherically symmetric case the dynamics of a thin shell is described by the above Lanczos equation and the radial conservation equation which gives us the sigma(r) function, where sigma is the surface energy density, and r is the radius. For dust shells sigma=const, and the Lanczos equation discribes the dynamics alone.

However in some literature other equations are used also. For example in vacuum:

$$S^{ab}K_{(ab)}=0$$

I think this is not an independent equation.
What is the intuitive meaning of this equation, and what is the minimal derivation of this equation?