From: baez@galaxy.ucr.edu (John Baez) Subject: Re: Operational Meaning of Parallel Transport Date: 30 Jan 2000 22:55:06 -0800 Newsgroups: sci.physics.research Summary: [missing] In article <86scbj$1rbl@edrn.newsguy.com>, Daryl McCullough wrote: >Oh, I guess I should get the terminology right, as well: > >(0) Is "torsion" the term used to describe the difference >between the actual connection and the one derivable from >the metric? No! And since understanding this issue is a necessary prerequisite for thinking about your other questions, let's straighten this out before tackling the others. In general relativity we start with a metric and derive from it a connection called the Levi-Civita connection. The Levi-Civita connection is uniquely determined by 2 properties: 1) its torsion is zero, 2) it is compatible with the metric. Condition 1) doesn't refer to the metric. Condition 2) does. There are infinitely many connections satisfying condition 1) alone. And given a metric, there are infinitely many connections satisfying condition 2) alone. But given a metric, there is exactly one connection satisfying both conditions 1) and 2)! And this is the connection we use in general relativity: the Levi-Civita connection. To understand general relativity, we need to understand what's so great about the Levi-Civita connection. The best way to do that is to understand the physical significance of conditions 1) and 2). So: what do these conditions mean? Heuristically, a connection is something that tells you how to parallel translate vectors. We say a connection is "metric-compatible" if parallel translating a vector never changes its length. Of course, the notion of "length" depends on the metric. But this notion is simple and straightforward. The tricky part is understanding the notion of "torsion". Crudely put, a connection has zero torsion if when you parallel translate a vector it never rotates unnecessarily. The hard thing is to see that you can formulate this condition in a coordinate-independent and metric-independent manner. There's a nice heuristic definition of torsion in terms of rockets. Take two rockets right next to each other and pointing the same way, and parallel translate one a bit to one side. Fire them off at the same speed and measure their relative velocity before they travel very far. If the torsion is zero, this relative velocity must be zero to first order. In other words, the only way the relative velocity could be nonzero to first order is if the rocket you parallel translated *rotated* a bit in the process! For pictures and a few more details see this: http://math.ucr.edu/home/baez/gr/torsion.html Once you see the role of "metric-compatibility" and "vanishing torsion" in general relativity, you can investigate the consequences of dropping these assumptions. Weyl had a theory in which the connection was not metric-compatible, but this was shot down by Einstein, who noted that if atoms could get bigger upon parallel translation, spectral lines of stars would not be so sharply defined. (See "The Dawning of Gauge Theory" for reprints of Weyl's original paper, which has Einstein's rebuttal appended at the end.) These days, more people are interested in theories where the connection has nonzero torsion. I don't understand these theories, but I'd like to, because apparently these theories are deeply related to particles with nonzero spin: spin couples to torsion, or something like that. In fact, Ashtekar's "new variables" for general relativity naturally include torsion when you have particles with nonzero spin around, and there's a guy who claimed to derive a value for the Immirzi parameter - which determines the fundamental unit of area - using this fact. This is the sort of thing I'm supposed to understand, so it bugs me that I don't understand it very well. ============================================================================== From: Roland Cruesemann Subject: Re: Operational Meaning of Parallel Transport Date: Thu, 3 Feb 2000 05:07:02 GMT Newsgroups: sci.physics.research John Baez schrieb: > Crudely put, a connection has zero torsion if when you parallel > translate a vector it never rotates unnecessarily. The hard > thing is to see that you can formulate this condition in a > coordinate-independent and metric-independent manner. > > There's a nice heuristic definition of torsion in terms > of rockets. Take two rockets right next to each other and > pointing the same way, and parallel translate one a bit to > one side. Fire them off at the same speed and measure their > relative velocity before they travel very far. If the > torsion is zero, this relative velocity must be zero to first > order. In other words, the only way the relative velocity could > be nonzero to first order is if the rocket you parallel translated > *rotated* a bit in the process! There is also a simple geometrical characterization of torsion, which should somehow be related to your heuristic definition. Let v, w be two "infinitesimal" vectors in the tangent space V_p at point p. One can regard the vectors v, w as infinitesimal displacements to the neighboring points q, r of p. Now we parallel transport v along p-r and w along p-q. In the case of non-vanishing torision the resulting parallelogram does not close. ============================================================================== From: baez@galaxy.ucr.edu (John Baez) Subject: Re: Operational Meaning of Parallel Transport Date: 3 Feb 2000 00:01:43 -0800 Newsgroups: sci.physics.research In article <3896D5BA.56CBD4CB@trustcenter.de>, Roland Cruesemann wrote: > John Baez schrieb: >> There's a nice heuristic definition of torsion in terms >> of rockets. Take two rockets right next to each other and >> pointing the same way, and parallel translate one a bit to >> one side. Fire them off at the same speed and measure their >> relative velocity before they travel very far. If the >> torsion is zero, this relative velocity must be zero to first >> order. In other words, the only way the relative velocity could >> be nonzero to first order is if the rocket you parallel translated >> *rotated* a bit in the process! >There is also a simple geometrical characterization of torsion, which >should somehow be related to your heuristic definition. Let v, w be two >"infinitesimal" vectors in the tangent space V_p at point p. One can regard >the vectors v, w as infinitesimal displacements to the neighboring points >q, r of p. Now we parallel transport v along p-r and w along p-q. In the >case of non-vanishing torsion the resulting parallelogram does not close. Right, this is really the same as my heuristic definition. Let v be the velocity vector of the first rocket and let w be the vector along which we displace the second rocket. In the case of nonvanishing torsion, the "parallelogram" you describe does not close - which means that the rockets start moving away from each other the moment you fire them off! Hmm, will anyone understand what I just said? The only way I know to really understand it is to draw some pictures and do some calculations. I have some pictures on my general relativity tutorial website - check out http://math.ucr.edu/home/baez/gr/outline2.html and look under "torsion". Does anyone lurking out there happen to be an expert on torsion theories of gravity? (I.e., theories of gravity where the connection has torsion.) This area has always seemed like a bit of a quagmire to me, but there might be something really interesting about it. ============================================================================== From: dragon@itp.uni-hannover.de (Norbert Dragon) Subject: Re: Operational Meaning of Parallel Transport Date: 3 Feb 2000 11:15:39 -0600 Newsgroups: sci.physics.research * John Baez baez@galaxy.ucr.edu writes > Does anyone lurking out there happen to be an expert on torsion > theories of gravity? (I.e., theories of gravity where the connection > has torsion.) This area has always seemed like a bit of a quagmire > to me, but there might be something really interesting about it. You are expert enough to consider the following arguments: 1) From the invariance of the action under general coordinate transformations it follows that the energy-momentum tensor is covariantly conserved where the covariant derivative is torsionfree and metric compatible, i.e. the connection is given by the Christoffel symbol. 2) From the covariant conservation of the energy momentum tensor it follows (shown e.g. in Papapetrou "Lectures on General Relativity") that in spacetime a tube of nonvanishing energy and momentum, the worldline of a test particle, is concentrated around a geodesic corresponding to the covariant derivative -- i.e. test particles follow the geodesics defined by the Christoffel symbols. 3) The geometric optics approximation to Maxwell's equation show that lightrays trace out geodesics of the torsionfree, metriccompatible parallel transport. I conclude: The field equations of gtr imply that neither test particles nor light care for torsion. But I am no expert. -- Norbert Dragon dragon@itp.uni-hannover.de http://www.itp.uni-hannover.de/~dragon Aberglaube bringt Unglück.