Finding Components of the Christoffel Symbol

[MidnightR]

I'm finding it hard to understand this, does anyone know where I can find worked examples of how to find the components of the christoffel symbols of a metric? Please don't give me one to try, I really need a worked example.

Thanks :S

 

[pervect]

Well, the easy way is to let the computer do it for you, i.e. if you have grtensor (which requires maple), we can compute the Christoffel symbols for the Schwarzschild metric as shown. This will give you a number of worked examples to check that you're doing it right:

qload(schw);
                       Default spacetime = schw                       For the schw spacetime:                             Coordinates                                x(up)                         a
                       x   = [r, theta, phi, t]                             Line element             2
     2    d r       2         2     2           2       2
   ds  = ------- + r   d theta   + r  sin(theta)   d phi
             2 m
         1 - ---
              r

           /     2 m\     2
         + |-1 + ---|  d t
           \      r /          The Schwarzschild metric in curvature coordinates

> grcalc(CC(up,dn,dn));

                          CPU Time  = 0.047

> grdisplay(_);

                       For the schw spacetime:                             CC(up,dn,dn)                        r                 m
                      CC   [r r] = - -----------
                                     (r - 2 m) r                         theta
                       CC       [theta r] = 1/r                           phi
                         CC     [phi r] = 1/r                         t               m
                       CC   [t r] = -----------
                                    (r - 2 m) r                         theta
                       CC       [r theta] = 1/r                      r
                    CC   [theta theta] = -r + 2 m                     phi                cos(theta)
                   CC     [phi theta] = ----------
                                        sin(theta)                           phi
                         CC     [r phi] = 1/r                     phi                cos(theta)
                   CC     [theta phi] = ----------
                                        sin(theta)                 r                                   2
               CC   [phi phi] = -(r - 2 m) sin(theta)               theta
             CC       [phi phi] = -sin(theta) cos(theta)                         t               m
                       CC   [r t] = -----------
                                    (r - 2 m) r                         r          (r - 2 m) m
                       CC   [t t] = -----------
                                         3
                                        r

But you can go back to the definition

$$\Gamma^{\rho}{}_{\mu\nu} = \frac{1}{2} \sum_{\sigma} g^{\rho\sigma} \left( \frac{\partial g_{\nu\sigma}} {\partial x^{\mu}} + \frac{\partial g_{\mu\sigma}} {\partial x^{\nu}} - \frac{\partial g_{\mu\nu}} {\partial x^{\sigma}} \right)$$To interpret the above, you need to assign

$x^0 = t \, , \, x^1 = r \, , \, x^2 = \theta \, , \, x^3 = \phi$

So if we want $\Gamma^{r}{}_{rr} = \Gamma^{1}{}_{11}$ in our example of the Schwarzschild metric, we see that the only non-zero term is for $\sigma = 1$ which gives

$$\frac{1}{2} g^{11} \left( \frac{\partial g_{11}}{\partial r} + \frac{\partial g_{11}}{\partial r} - \frac{\partial g_{11}}{\partial r} \right)$$

and
$$g_{11} = \frac{1}{1-2m/r}$$

We also know
$$g^{11} = 1-2m/r$$

because $g^{\mu\nu}$ is just the inverse of $g_{\mu\nu}$ , ie. $g^{\mu\nu} g_{\mu\nu} = 1$ with the usual summation convention, and g is diagonal.

 

[Mentz114]

You can also do this with the free program Maxima

load(ctensor);
/* set some flags */
cframe_flag: false;
ratchristof: true;
ratriemann : true;
ratfac : true;
ctrgsimp: true;
/* define the dimension */
dim: 4;
/* list the coordinates */
ct_coords: [t,r,theta,phi];
/* set up the metric */
/* assign to lg a matrix of zeros  ':' means 'assign' */
lg:zeromatrix(4,4);
/* now add the Schwarzschild coefficients */
lg[1,1]:-c^2*(1-2*m/r);
lg[2,2]:1/(1-2*m/r);
lg[3,3]:r^2;
lg[4,4]:r^2*sin(theta)^2;
/* make the inverse matrix */
ug:invert(lg);
/* get Christoffels */
christof(mcs);

The last Christoffel symbols I ever worked out by hand were for the (nonsense) metric

$$ds^2=-dt^2+exp(2ax)dx^2+dy^2+dz^2$$

(the only non-zero CS is $\Gamma^x_{xx}=a$)

and that was enough.