differential geometry

[nibot]

Blah. Covariant derivative. Parallel transport. What's the story?

"a geodesic is a curve whose parametrization, when viewed from within the surface, appears to have zero acceleration" (i.e., all of the acceleration is normal to the surface)

The phrase "objects not experiencing external forces follow geodesics" is more of a tautology than I thought.

If gravity is actually a warping of space-time — so that there's not actually any 'force' of gravity, but rather falling objects travel in "straight lines" along geodesics in space time — why is there still the occasional mention of graviton messenger particles for the gravitational force?

sleepy time.

Comments

[calbruin.livejournal.com]

I remember reading about such things in Gravitation (http://www.amazon.com/exec/obidos/tg/detail/-/0716703440/qid=1081793808/sr=1-5/ref=sr_1_5/103-4920087-4747065?v=glance&s=books). I remember it taking me a long time, since I was studying on my own outside of any classroom guidance, to understand what it was describing. In effect, parallel transport is merely "transporting" picking up and moving the vector to another place on a curve. What annoyed me was when I wanted to learn more about "warped" curves, most other texts focused upon the local space-time being flat so that ordinary 1-forms may be applied. I did not care about the baby "flat" region sceniro, I wanted to learn about the hard core problems investigating curved surfaces.

hmmm..

[ucbfumbler.livejournal.com]

I thought the idea of a "gaviton" is like the "current flow" (ie. to tell where something is going, not the actual carrier). Hence, the term "messenger." Basically, a way of quantifying effects.

[travisgarrett.livejournal.com]

Yeah, gravitons and Classical GR are somewhat disjoint. In GR you can assume weak feilds (chap 18 in Gravitation, or MTW as we call it), i.e. guv=nuv+huv where g is the metric (u,v indices) n is the metric for flat space - Minkowskian - and h is a pertubation on n, with |h|<<|n|. You can then get various things out of h, like Newtonian gravity (laplacian psi = rho...) and then post-newtonian effects (precession of Mercury's orbit, gravitational redshifting and time dilation - and hopefully frame dragging will be detected shortly with Gravity probe B (http://einstein.stanford.edu/)). Using a 'Lorentz' gauge, you can also get box(h)=T, i.e. the wave equation. So in the weak field limit you can think of h as a separate field on top of a flat background, and you quantize just h (creation and annhilation operators for quanta of h with certain momentum), and this spin 2 field is the graviton, all of this following the standard prescription of quantum field theory for QED and so on. Note in string theory also you have this spin-2 h field on top of a fixed background metric. So String theory will not be the final theory, we'll probably also need to unite it with quantum loop gravity for the strong field limit, although this is probably premature since we can't even extract the minkowski spacetime limit from quantum loop gravity yet! Not to mention classical GR needs it's strong field limit tested also, which hopefully will be done with LIGO (http://www.ligo.caltech.edu/) and LISA (http://lisa.jpl.nasa.gov/), and it may well not hold up. Lots of people are also trying to modify GR due to dark matter and dark energy - mostly dark energy I think because gravitational lensing makes it fairly clear there are smooth galactic scale lumps of matter out there.