## General Relativity Tutorial - Torsion

#### John Baez

Oz asks: "Why do we assume [the connection] in GR is torsion free?
A short couple of non-sarcastic information-rich understandable
sentences would suffice. Enough to place it in a footnote of the brain."
Non-sarcastic, eh? You must think I'm really mean and nasty, when
actually I am the sweetest, nicest guy you could ever meet.

Relatively few people understand why in GR we assume the connection ---
the gadget we use to do parallel translation --- is torsion-free. It's
often presented as a technical assumption not worth trying to
understand.
But here's a nice way understanding what torsion-free-ness means (in
conjunction with our other assumptions: that parallel translation be
linear and metric-compatible).

Take a tangent vector v at P. Parallel translate it along a very short
curve from P to Q, a curve of length epsilon. We get a new tangent
vector w at Q. Now let two particles free-fall with velocities v and w
starting at the points P and Q. They trace out two geodesics. Let me
try to draw this:

| |
| |
| |
^v ^w
| |
P------Q

Remember, this is a picture in spaceTIME. Here I've drawn what it might
look like in flat Minkowski spacetime, where the geodesics are boring
old straight lines, and I've drawn everything very rectilinearly, since
ASCII is so bad at drawing curves. Okay. Now, let's call our two
geodesics C(t) and D(t), respectively. Here we use as the parameter t
the proper time: the time ticked out by stopwatches falling along the
geodesics. (We set the stopwatches to zero at the points P and Q,
respectively.)
Now we ask: what's the time derivative of the distance between C(t) and
D(t)? Note this "distance" makes sense because C(t) and D(t) are really
close, so we can define the distance between them to be the arclength
along the shortest geodesic between them.

If, no matter how we choose P and Q and v, the time derivative of the
distance between C(t) and D(t) at t = 0 is ZERO, up to terms
proportional to epsilon^2, then the torsion is zero! And conversely!
(One can derive this from the definition of torsion, assuming our recipe
for parallel transport is metric preserving.)

If v got "rotated" a bit when we dragged it over to Q, and things looked
like this:

| /
| /
| /
^v ^w
| /
P------Q

then the time derivative of the distance would not be zero (it'd be
proportional to epsilon). In this case the torsion would not be zero.
In GR we assume this kind of "rotation" effect doesn't happen. In some
other theories of gravity there is torsion. But there's no experimental
evidence for torsion, so most people stick with GR.