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.
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:
Code:
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
<br />
\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)<br />To interpret the above, you need to assign
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
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.
Last edited:
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Mentz114
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You can also do this with the free program Maxima
Code:
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
OK, so this has bugged me for a while about the equivalence principle and the black hole information paradox.
If black holes "evaporate" via Hawking radiation, then they cannot exist forever.
So, from my external perspective, watching the person fall in, they slow down, freeze, and redshift to "nothing," but never cross the event horizon.
Does the equivalence principle say my perspective is valid? If it does, is it possible that that person really never crossed the event horizon?
The...
ASSUMPTIONS
1. Two identical clocks A and B in the same inertial frame are stationary relative to each other a fixed distance L apart. Time passes at the same rate for both.
2. Both clocks are able to send/receive light signals and to write/read the send/receive times into signals.
3. The speed of light is anisotropic.
METHOD
1. At time t[A1] and time t[B1], clock A sends a light signal to clock B. The clock B time is unknown to A.
2. Clock B receives the signal from A at time t[B2] and...
From $$0 = \delta(g^{\alpha\mu}g_{\mu\nu}) = g^{\alpha\mu} \delta g_{\mu\nu} + g_{\mu\nu} \delta g^{\alpha\mu}$$ we have $$g^{\alpha\mu} \delta g_{\mu\nu} = -g_{\mu\nu} \delta g^{\alpha\mu} \,\, . $$ Multiply both sides by ##g_{\alpha\beta}## to get $$\delta g_{\beta\nu} = -g_{\alpha\beta} g_{\mu\nu} \delta g^{\alpha\mu} \qquad(*)$$ (This is Dirac's eq. (26.9) in "GTR".) On the other hand, the variation ##\delta g^{\alpha\mu} = \bar{g}^{\alpha\mu} - g^{\alpha\mu}## should be a tensor...
