Blogging Bone II
Unfortunately, Carl's excellent PF thread on Schwarzschild orbits has been locked by the moderators after some negative reports from GR experts. A number of particle physicists have expressed some appreciation for Carl's work, but of course they couldn't possibly know anything. Don't forget to look at Carl's new gravity applet.
On a positive note, Louise tells us that her paper about $GM = t c^3$ has finally been accepted for publication by a major journal. It will be nice to see it in print at last. Maybe she will talk about it at GRG18 in Sydney in July. I wish Carl could make this conference as well. Is there anybody that would be willing to buy him an airline ticket?
On a positive note, Louise tells us that her paper about $GM = t c^3$ has finally been accepted for publication by a major journal. It will be nice to see it in print at last. Maybe she will talk about it at GRG18 in Sydney in July. I wish Carl could make this conference as well. Is there anybody that would be willing to buy him an airline ticket?
posted by Kea | 10:26 AM
6 Comments:
There is something I wanted to say about the Painleve metric that didn't make it into that "why" file I typed up.
The Painleve metric mixes dr and dt, that is, it is not diagonal. The (Cartesian) metric looks like:
-dt^2 + (dx +\sqrt(2/r^3)x dt)^2
plus similar for y and z. If you square the terms, you end up with the usual stuff for a Newtonian potential, plus the Minkowski metric, plus some stuff that looks like:
\sqrt(8/r^3) x dx dt,
and similar in y and z. These terms scale as 1/sqrt(r), which is kind of strange.
This resonates a little with the square root that shows up in the Koide equation for the masses. Another place it shows up, one we haven't discussed much since dark matter seems to have killed it, is in the MOND version of gravity.
So my speculation is that when you find the quantum theory of a particle that produces an acceleration equal to the Painleve metric for strong gravitational fields, you will find that in the weak field it tends towards the MOND value:
\sqrt(a a_0).
Carl, MOND is alive and well, despite the consensus. The term 'Dark Matter' doesn't mean very much.
Okay I'm looking at TeVes right now. They have a paper, astro-ph/0403694, which computes the post-Newtonian corrections to the Schwarzschild metric. I suggest that we replace that calculation with the Painleve metric and see what happens.
I forgot to mention that yet another cool thing about the Painleve metric is that it defines an arrow of time. Namely, there are two Painleve solutions, the one above, plus one with the sqrt(2) replaced by -sqrt(2).
The other Painleve corresponds to a white hole. The event horizon is reversed so things get pushed out of it. This is the white hole Painleve solution.
We only see black holes, which means that gravity defines an arrow of time (if you believe that the Painleve metric is the true one, as indicated by the geometric algebra geometry theorists).
Also, if you write the Painleve metric and look at it in powers of r, at large r you end up with Minkowski space, plus a 1/r potential, plus that sqrt(2/r) mixed term.
For large r, the Minkowski terms stay constant. The 1/r term drops off faster than the 1/sqrt(r) dr dt cross term. So if you get far enough away, the dr dt cross term dominates.
If Minkowski relativity were perfect, the dr dt cross term would give no acceleration (it just redefines time as a function of radius). So maybe MOND is not a correction to Newtonian gravity, but instead should be seen as a correction to Minkowski relativity.
So maybe MOND is not a correction to Newtonian gravity, but instead should be seen as a correction to Minkowski relativity.
Well, if the TeVeS people were thinking of gravity in flat space, I think that's what they'd be doing. OK, something cool for to work on. Um, that would add galactic rotation curves to the experimental picture.
It all gets back to the question of what should be at the foundations of physics, simple and beautiful symmetries, or simple and beautiful equations of motion.
Almost all bets have been on simple symmetries, but those bets have not paid off, and newer theories are less and less simple and less and less beautiful.
So my reason for writing Painleve as equations of motion (instead of using the symmetry of Christoffel symbols) is to look for an approximation that arises from a simple and beautiful equation of motion.
Still working on it. Oh, and I updated the gravity simulator so that it automatically goes through a loop of demonstrations of facts about gravity. I know that few people are patient enough to set up the initial conditions. If there's some cool fact about gravity you'd like me to type in, I've got room for many more simulations.
Thanks, Carl. All in all, you've given me more work than I can handle! Some things will have to wait until I get a chance to teach courses on the subject - if I ever do - and I'm certainly not expecting that I will. Actually, I loved Euler-Lagrange when I was a kid. It was one of my favourite undergrad topics.
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