Learn About Tetrads: Definition, Meaning & Problems Solved

[m4r35n357]

Well, I got the Schwarzschild, Gullstrand-Painleve and Doran metrics "deconstructed" very easily. For some reason the Boyer-Lindquist metric seems unusually resistant to such analysis. I first tried mutating the Doran metric without success, the tried to factorize the metric into four squares by hand (again unsuccessfully).
Then I stumbled across this paper: http://casa.colorado.edu/~ajsh/phys5770_08/grtetrad.pdf . In section 5.22 it gives the factorization and the inverse vierbein just in precisely the form I was looking for! Unfortunately it does not seem to give the correct metric when applied to the Lorentz frame.
After much experimentation I went back to the ctensor documentation and found there is a way to do the calculation without relying explicitly on matrix multiplications, which I show below:

kill(all)$
if get('ctensor,'version)=false then load(ctensor)$
cframe_flag:true$
verbose:true$
("Covariant frame metric");
lfg: matrix([-1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]);
("Inverse frame matrix to metric");
("Minkowski");
fri: matrix([1, 0, 0, 0], [0, 1, 0, 0], [0, 0, r, 0], [0, 0, 0, r * sin(%theta)]);
cmetric()$
lg;
("Schwarzschild");
fri: matrix([sqrt(1 - %beta^2), 0, 0, 0], [0, 1 / sqrt(1 - %beta^2), 0, 0], [0, 0, r, 0], [0, 0, 0, r * sin(%theta)]);
cmetric()$
lg;
("Gullstrand-Painleve");
fri: matrix([1, 0, 0, 0], [%beta, 1, 0, 0], [0, 0, r, 0], [0, 0, 0, r * sin(%theta)]);
cmetric()$
lg;
("Boyer-Lindquist");
fri: matrix([R*sqrt(1-%beta^2)/%rho,0,0,-R*sqrt(1-%beta^2)/%rho*a*sin(%theta)^2],[0,%rho/(R*sqrt(1-%beta^2)),0,0],[0,0,%rho,0],[-a*sin(%theta)/%rho,0,0,R^2*sin(%theta)/%rho]);
cmetric()$
lg;
("Doran");
fri: matrix([1, 0, 0, 0], [%beta * R / %rho, %rho / R, 0, - %beta * R / %rho * a * sin(%theta)^2], [0, 0, %rho, 0], [0, 0, 0, R * sin(%theta)]);
cmetric()$
lg;

Unfortunately, the BL metric still looks wrong to me (in the g_{tt}{}^{} and g_{\phi\phi}{}^{} components). I tend to trust Hamilton's work ;) so I'm a bit puzzled about this. Can anyone spot my error(s)?

 

[Mentz114]

("Boyer-Lindquist");
fri: matrix([R*sqrt(1-%beta^2)/%rho,0,0,-R*sqrt(1-%beta^2)/%rho*a*sin(%theta)^2],[0,%rho/(R*sqrt(1-%beta^2)),0,0],[0,0,%rho,0],[-a*sin(%theta)/%rho,0,0,R^2*sin(%theta)/%rho]);
cmetric()$
lg;

Unfortunately, the BL metric still looks wrong to me (in the g_{tt}{}^{} and g_{\phi\phi}{}^{} components). I tend to trust Hamilton's work ;) so I'm a bit puzzled about this. Can anyone spot my error(s)?

Your BL tetrad is correct. You probably need one more trigsimp. The acceleration of the observer in the static frame has one component in the r direction given by
$$\frac{m}{r\,\sqrt{{r}^{2}-2\,m\,r+{a}^{2}}}-\frac{{a}^{2}}{{r}^{2}\,\sqrt{{r}^{2}-2\,m\,r+{a}^{2}}}$$
You can see the gravitational and rotational accelerations. If a=0 the result is the same as for the Schwarzschild stationary frame. This is exactly what is expected.

Also I subtracted the result of cmetric() from the Kerr-BL metric from Visser and the result is zero ( after a trigsimp).

 

[m4r35n357]

Ah, thanks for the confirmation, now I see how it all comes together, unfortunately Maxima doesn't have the information to fix it, so I'll need to recast variables a bit. My problems are caused by trying to make BL look more like Doran than it otherwise would, but then so does Hamilton, so I don't feel bad!

 

[m4r35n357]

The acceleration of the observer in the static frame has one component in the r direction given by
$$\frac{m}{r\,\sqrt{{r}^{2}-2\,m\,r+{a}^{2}}}-\frac{{a}^{2}}{{r}^{2}\,\sqrt{{r}^{2}-2\,m\,r+{a}^{2}}}$$
You can see the gravitational and rotational accelerations. If a=0 the result is the same as for the Schwarzschild stationary frame. This is exactly what is expected.

OK, you've got me! Thought I'd take a little while to try to understand this, but after some head-scratching I'm not sure where it comes from or what it's telling me. Would you mind elaborating as I can't work out what acceleration you are referring to or how you are deriving it?

 

[Mentz114]

Would you mind elaborating as I can't work out what acceleration you are referring to or how you are deriving it?

The proper acceleration of a 4-velocity u^μ is given by the covariant derivative of u_μ projected onto u^μ, i.e.
$$\frac{d^2x^\mu}{d\tau^2}=\nabla_\nu u_\mu u^\nu$$
If the vector is a geodesic then the acceleration is zero, otherwise the frame is experiencing a force.

That is rather informal description. The subject is covered in textbooks and the oft-quoted Wiki article is a decent introduction ( which you've probably read ).
http://en.wikipedia.org/wiki/Frame_fields_in_general_relativity

I'll PM you with the gory details of calculating this with Maxima.

[edit]
The acceleration I gave in my previous post is only valid in the equatorial plane $\theta=\pi/2$. Also, if $mr=a^2$ it's a geodesic, and is the path of a circular orbit where the centripetal force cancels the gravitational field. Isn't that a nice way to find a geodesic !

 

[m4r35n357]

Ah, that explains it; I haven't done any actual covariant derivative calculations before, just read the theory and gone straight on to the geodesic equations ;)
Yes I've seen that wikipedia article but I find it very confusing, with inconsistent notation and lots of missing steps . . . pity none of my books go into tetrads in any depth.
All I know about tetrads is what I've learned in this forum and what I've tried myself. So having found out what tetrads are, I am only now in a position to attempt to use them, I look forward to the gory details ;)