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.

[nibot.livejournal.com]

Thanks for the informative response. I'll have to learn a bit more field theory and GR before I can parse it fully, though. (-: Exciting stuff!

How do you like UNC so far? I gather that you're a second year student there? re: your research summary, it sounds like you have your work cut out for you.. numerical GR, would you say? I'm curious about this bit:

I just noticed something interesting. For a certain amount of mass, if you compress it enough it will form a black hole with radius R=2MG/c^2, where M is the mass, G is the gravitational constant, and c is the speed of light. For the sun this radius is about 1.5 kilometers. For all the mass in the visible universe, the black hole radius is (drumroll), the current radius of the universe. More on this latter...

We're living in a black hole?! (-:

I heard someone say that the incompatibility between GR and QM arises because the quantum fluctuations of the vacuum at the planck scale would case miniscule black-holes to arise everywhere. My own personal theory is that this is actually what happens — that there really is an infinite lattice of black holes, and that this wouldn't really mess up physics because of the Bloch theorem. You see, this is what my condensed matter course is doing — warping my mind into that of a crackpot! hehe.

[travisgarrett.livejournal.com]

There is a theory going around - the holographic principle - that the amount of information that can be stored in a region goes as the surface area of the region (let's just say a sphere) and NOT the volume. For instance this meshes with black hole thermodynamics - the temperature is related to the mass (inversely - a black hole's Hawking radiation has an average wavelength the size of the black hole, so a solar mass black hole would emit 1.5 kilometer radio waves - i.e. it would be much colder than the current microwave background) and thus to the surface area of the event horizon. Another derivation is given by UNC's own Jack Ng and Hank Van Dam - gr-qc/0403057 (http://www.arxiv.org/abs/gr-qc/0403057) - check it out, it's an easy read. Thus my instinct when presented with something like this is to try and break it - and since the volume grows much faster than the surface area it seems like you should take a giant volume and then cram as much matter (and thus info) inside it as possible. The problem is that a black hole's radius grows that much quicker - the event horizon is linear in mass: R=2M (G=c=1). Thus very large black holes have very low average densities - indeed as I found (trivial really), current intergalactic densities give a black hole radius of about the radius of the visible universe. So to break the volume/area thing you actually want to go to small volumes or else you'll form black holes immediately. Let's see, the earth has a mass of about 10^25 kg, which is 10^52 protons, and a Schwarzchild radius of about 1cm, and Lplanck = 10^-33 cm, so there are 10^66 Planck lengths squared on an Earth sized black holes - no good. Hold on, matter at nuclear densities also won't work no matter how small, but taken at schwarzchild radius density... OK 1 kg of neutrons is ~10^27 particles, with a RBH~10^-27 meter = 10^7*Lplanck, so there are ~ 10^15 possible bits on the horizon according to the holographic principle, but 1 bit/neutron would bit 10^27 bits! You can beat it at tiny volumes! I'm going to have to look at this more, but the densities here are much higher than nuclear density - not that this would stop a theorist! Thanks for getting me back onto this Tobin! I'll see if the details pan out... For one thing it would be a quark-gluon plasma, not neutrons - but I don't think the binding energy mass would increase much - only gets big when you try to separate them. Ah, but they are fermions - degeneracy pressure would push the temperature and energy way up - have to check it out.

Oh yeah, and we're not in a black hole because there must be matter beyond our visible horizon at 13.7 billion light years - can tell that because from measuring the CMB radiation we know space is very flat. If it was a vacuum outside, our visible universe would collapse to a singularity in time T=pi/2*(3/(8*pi*G*rho))^1/2, rho~10*10^30*10^11*10^11/(10^10*10^16)^3=10^-25kg/m^3, so T=2*10^17 seconds, which, I'll be damned, is about the age of the universe. But then the metric for a collapsing ball of dust is the same as the Freidmann-Robertson-Walker metric, so maybe I shouldn't be surprised. Huh. So the evolution of the universe forward from the big bang really is very similar to a black hole collapse. Lee Smolin has the idea that collapsing black holes seed new baby universes - then if the physical constants can change at each singularity the system will evolve towards universes that have physical constants most favorable to the formation of lots of new black holes - these will dominate the counting. That's looking even clearer to me now. I can't wait to talk to Lee on Monday, he's coming to give a colloquia. I need to really go back and look at Martin Bojowald's singularity evolution code...

[travisgarrett.livejournal.com]

This has been great Tobin - you should come to UNC! I like it a lot at least. Yeah, my official project is working on computer simulations of black hole mergers - should be the first test of strong gravity when LISA and Ligo work - I've got a summer grant to work on LISA motivations, so I'm really happy. And there's lots of cool stuff here - in addition to my theory stuff we're getting strong in cosmology with new hires, the carbon nanotube stuff is really cool (I really hope this space elevator thing pans out), we share a mid-energy particle accelerator at TUNL, the biophysics, especially the neuron stuff is fascinating - I almost thought about switching to that - and if you're considering astronomy we're about to get our own telescope in Chile (17% viewing time, not too shabby) which is really cool. Anyways, good to talk to you, see ya later...

[nibot.livejournal.com]

LISA sounds awesome.. it's almost too bad it's a JPL project, 'cause it would be so much fun to work on.

I should check out this neuron stuff. Hadn't heard about that. Man, it's getting down to the wire.