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Solar neutrinos and Gravitational differences of a chemical nature
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Johannes Swartling
science forum beginner


Joined: 11 Mar 2006
Posts: 2

PostPosted: Sat Mar 11, 2006 8:45 pm    Post subject: Re: Optics Researchers See the Light Reply with quote

"Andrew" <andrew_zi@nospam.yahoo.com> wrote in message
news:c9085c$qjm$1@netlx020.civ.utwente.nl...
Quote:


Lehigh's Jean Toulouse and Iavor Veltchev are studying a phenomenon that
few
scientists in the world have been able to achieve.

Two physicists at Lehigh have produced a rainbow of visible and invisible
colors by focusing laser light in a specially designed optical fiber that
confines light in a glass core whose diameter is 40 times smaller than
that
of a human hair.

Full story: http://www.physorg.com/news110.html

This is a very interesting phenomenon called "supercontinuum generation in
nonlinear fibers", which is very difficult to achieve!

Supercontinuum generation in non-linear fibers is routine in laser labs
everywhere and has been for a few years. A pretty strange press release for
something which is not really new.

Johannes
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Lostgallifreyan
science forum beginner


Joined: 12 Mar 2006
Posts: 11

PostPosted: Sun Mar 12, 2006 6:59 am    Post subject: Re: Optics Researchers See the Light Reply with quote

"Johannes Swartling" <johannes.swartling@home.se> wrote in
news:c9250b$57f$1@newsserver.cilea.it:

Quote:
Supercontinuum generation in non-linear fibers is routine in laser
labs everywhere and has been for a few years. A pretty strange press
release for something which is not really new.

Johannes


There is so much I don't know... Smile But unless I'm missing something even
in what I thought I knew, this is odd. Even at a few thousand dollars per
meter of special fibre, that alone wouldn't stop there being fully tunable
diode-based lasers. If this supercontinuum generation were routine, I'd
have thought these lasers would be a seriously desireable and much talked
about icon, a holy grail for light show makers, for one thing, yet I've
never heard of that. Can their output be coherent, if it's in a wide
spectrum? And even if not, can the source be tuned and coupled to a fibre
or other means of making a tiny bright source for collimating?
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j.m.1491@gmx.net
science forum beginner


Joined: 13 Mar 2006
Posts: 1

PostPosted: Mon Mar 13, 2006 1:32 am    Post subject: Re: Optics Researchers See the Light Reply with quote

Lostgallifreyan wrote:
Quote:
"Johannes Swartling" <johannes.swartling@home.se> wrote in
news:c9250b$57f$1@newsserver.cilea.it:

Supercontinuum generation in non-linear fibers is routine in laser
labs everywhere and has been for a few years. A pretty strange press
release for something which is not really new.

Johannes


There is so much I don't know... Smile But unless I'm missing something even
in what I thought I knew, this is odd. Even at a few thousand dollars per
meter of special fibre, that alone wouldn't stop there being fully tunable
diode-based lasers.

Supercontinuum has about the same tuneability as a light bulb. If you
add a monochromator you can select a wavelength, but the source itself
is, well, a continuum. Very wide spectrum, but white.

About diodes: Most diodes will not give you short enough pulses with
high enough peak powers to create a supercontinuum.

Quote:
If this supercontinuum generation were routine,

http://www.google.ca/search?hl=en&q=supercontinuum+generation
http://www.google.ca/search?hl=en&q=supercontinuum+comb
http://www.google.ca/search?q=supercontinuum+optical+frequency+metrology
http://www.google.ca/search?hl=en&q=supercontinuum+spectroscopy

and so on...

Quote:
I'd
have thought these lasers would be a seriously desireable and much talked
about icon, a holy grail for light show makers,

A continuum in visible wavelength range is white. What light shows like
to have are a few base colors, modulated separately.

Quote:
for one thing, yet I've
never heard of that. Can their output be coherent, if it's in a wide
spectrum?

Usually it's created with ultrashort pulses. In a first approximation
it's coherent as long as the pulse lasts, i.e. there is a fixed phase
relation between the individual frequencies, as in an ordinary
femtosecond pulse e.g. created by a Ti:Sapphire laser. If you overlap
multiple pulses you may not find any coherence, or extremly short
coherence time. If you look in detail at one pulse, you can have the
situation that the pulse splits up in multiple pulses, each with a
different center wavelength. Then it's getting problematic to define
what you mean with coherence.

Quote:
And even if not, can the source be tuned and coupled to a fibre
or other means of making a tiny bright source for collimating?

The way to create this continuum creation works is by using nonlinear
optical effects. Doing that in fibers keeps the power needed down
because you can maintain a high intensity over a long propagation
distance (until GVD kills you). Photonic crystal fibers are nice because
you can engineer the GVD and make very small modes.

So, yes, it's possible to make this tiny source.

HTH,
j.m.
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Uncle Al
science forum Guru


Joined: 24 Mar 2005
Posts: 1226

PostPosted: Sat May 13, 2006 1:07 pm    Post subject: Re: This Week's Finds in Mathematical Physics (Week 231) Reply with quote

John Baez wrote:
Quote:

Also available at http://math.ucr.edu/home/baez/week231.html

May 9, 2006
This Week's Finds in Mathematical Physics (Week 230)
John Baez
[snip]


Quote:
Hmm. Do those quantities mean as little to you as they do to me?
A "pascal" is a unit of pressure, or force per area, equal to one
newton per square meter. An "atmosphere" is another unit of pressure,
basically the average air pressure at sea level here on Earth. This
has the annoying value of 101,325 pascals. Personally I have some
trouble getting a feel for how much pressure this is, since a newton
per square meter isn't much, but 101,325 of them sounds like a lot.
So for me, being an American, it's helpful to know that an atmosphere
equals 2116 pounds per square foot. If you're a metric sort of person,
that's about the weight of 1 kilogram pushing down on each square
centimeter. That's a lot of pressure we're under! No wonder we feel
stressed sometimes.

Illustrating the deep sourcing of Intelligent Design, the IDiots and
we note that the average weight of the average apple is one newton.

Quote:
(Yes, I know a kilogram is not a unit of weight. I mean the weight
corresponding to a mass of a kilogram in the Earth's gravitational field
at sea level. Sheesh!)

[snip]

Quote:
Cool Light gas gun, Wikipedia, http://en.wikipedia.org/wiki/Light_Gas_Gun

It's not called a "light" gas gun because it's wimpy - in fact they're
huge, and everyone evacuates the lab when they run the one at NASA!
It's called that because the speed of the projectile is limited only
by the speed of sound in the gas, which is higher for a light gas like
helium - or even better, hydrogen. Even better, that is, you don't
mind exploding gunpowder near highly flammable hydrogen! But, as you
can imagine, people who do this stuff are precisely the sort who don't
mind. You may enjoy reading how folks at Lawrence Livermore National
Laboratory used a light gas gun to compress hydrogen to pressures of
up to 200 gigapascals, enough to convert it into a metal:

9) Robert C. Cauble, Putting more pressure on hydrogen,
http://www.llnl.gov/str/Cauble.html

To my mind, a most interesting seriously big squishie would be the
zeta-pinch facility at Sandia. Adding the delight of a huge aligned
magnetic field to the study of compressed light magnetic nuclei is the
right thing to do.

http://zpinch.sandia.gov/
Not the gizmo, just its power supply.
http://physicsweb.org/articles/news/7/4/7/1/z-machine
another view

Quote:
13) P. M. Celliers et al, Electronic conduction in shock-compressed
water, Plasmas 11 (2004), L41-L48.

They also mention that "a single datum at 1.4 terapascals from an
underground nuclear experiment has never been repeated." Some people
just don't know when to stop in the quest for higher pressures.

Consider the planned popping of 700 tons of ANFO to model a
nuclear-based underground bunker buster the US is not Officially
developing. Given the physics of long rod penetration, the depth of
the presumed targets, and the (radiative) political inconvenience of a
megatonne shallow depth detonation, one extrapolates the merry
weaponeers are aiming for a nuclear (or nukular, all things
considered) Monroe effect.

<http://www.globalsecurity.org/military/systems/munitions/bullets2-shaped-charge.htm>

[snip]

Quote:
-----------------------------------------------------------------------
Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics, as well as some of my research papers, can be
obtained at

http://math.ucr.edu/home/baez/

For a table of contents of all the issues of This Week's Finds, try

http://math.ucr.edu/home/baez/twfcontents.html

A simple jumping-off point to the old issues is available at

http://math.ucr.edu/home/baez/twfshort.html

If you just want the latest issue, go to

http://math.ucr.edu/home/baez/this.week.html

--
Uncle Al
http://www.mazepath.com/uncleal/
(Toxic URL! Unsafe for children and most mammals)
http://www.mazepath.com/uncleal/qz3.pdf
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Richard Saam
science forum Guru Wannabe


Joined: 20 May 2005
Posts: 137

PostPosted: Sun May 14, 2006 8:26 pm    Post subject: Re: This Week's Finds in Mathematical Physics (Week 231) Reply with quote

Uncle Al wrote:
Quote:
John Baez wrote:


[snip]


8) Light gas gun, Wikipedia, http://en.wikipedia.org/wiki/Light_Gas_Gun

It's not called a "light" gas gun because it's wimpy - in fact they're
huge, and everyone evacuates the lab when they run the one at NASA!
It's called that because the speed of the projectile is limited only
by the speed of sound in the gas, which is higher for a light gas like
helium - or even better, hydrogen. Even better, that is, you don't
mind exploding gunpowder near highly flammable hydrogen! But, as you
can imagine, people who do this stuff are precisely the sort who don't
mind. You may enjoy reading how folks at Lawrence Livermore National
Laboratory used a light gas gun to compress hydrogen to pressures of
up to 200 gigapascals, enough to convert it into a metal:

9) Robert C. Cauble, Putting more pressure on hydrogen,
http://www.llnl.gov/str/Cauble.html


To my mind, a most interesting seriously big squishie would be the
zeta-pinch facility at Sandia. Adding the delight of a huge aligned
magnetic field to the study of compressed light magnetic nuclei is the
right thing to do.

http://zpinch.sandia.gov/
Not the gizmo, just its power supply.
http://physicsweb.org/articles/news/7/4/7/1/z-machine
another view


13) P. M. Celliers et al, Electronic conduction in shock-compressed
water, Plasmas 11 (2004), L41-L48.

They also mention that "a single datum at 1.4 terapascals from an
underground nuclear experiment has never been repeated." Some people
just don't know when to stop in the quest for higher pressures.


"Insanity Insanity Insanity ..."
spoken by British Corpsman
In the context of
'Bridge over the River Kwai'
Colonel Saito (Sessue Hayakawa)
Colonel Nicholson (Alec Guinness)

Do a Google Search on

Critical optical volume energy (COVE)

A pittance spent
to test structural elegance
resulting in benign energy
rather than bruit force
with massive destructive implications.

Richard
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baez@galaxy.ucr.edu
science forum addict


Joined: 21 Oct 2005
Posts: 53

PostPosted: Wed May 17, 2006 2:32 am    Post subject: Re: This Week's Finds in Mathematical Physics (Week 229) Reply with quote

In article <1145101189.273980.185650@i40g2000cwc.googlegroups.com>,
Squark <top.squark@gmail.com> almost wrote:

Quote:
Hello John and everyone!

Hello! Long time no see! How are you doing? I showed up at
the Perimeter Institute yesterday, and I should be getting a talk
ready:

http://math.ucr.edu/home/baez/quantum_spacetime

but I'm goofing off.

Quote:
There's a simple topological interpretation of the element of the
rational projective line associated to a rational tangle. I don't know
how to use this to prove the theorem, and I don't know a reference for
it (maybe it is in one of the references you cited). Anyway, regard a
rational tangle as a two-component curve C in the 3-ball B^3 whose four
boundary points are on the 2-sphere S^2. Consider the double branched
cover of B^3 along C.

What is "_the_ double branched cover"? Is there a way to choose a
canonical one, or is there only one in this case, for some reason?

Good point. I hope there's a specially nice one.

To pick a branched cover of B^3 along C, it's necessary and sufficient
to pick a homomorphism from the fundamental group of B^3 - C to Z/2.
This says whether or not the two sheets switch places as we walk around
C following some loop in B^3 - C.

Quote:
In the case of a sphere with 4 points removed it should be easy to
check.

Yes.

Quote:
The fundamental group has 4 generators - a, b, c, d (loops around each
of the points) and one relation abc = d (since we're on a sphere). Hence,
it is freely generated by a, b, c (say).

[I changed your presentation slightly here, for my own convenience.]

Right, the fundamental group of the four-punctured sphere is
the free group on 3 generators, F_3. I believe the "specially nice"
homomorphism

f: F_3 -> Z/2

is the one that sends each generator to -1, where I'm thinking of
multiplicatively:

Z/2 = {1, -1}

One reason this homomorphism is especially nice is that it also sends
d = abc to -1.

So, if you walk around ANY of the four punctures, the two sheets switch!

This is just what you want for the Riemann surface of an elliptic integral,
as someone else pointed out in another post: there are four branch points
each like the branch of point of sqrt(z). It's also the most symmetrical,
beautiful thing one can image.

Now let's see if and how this branched cover extends to a branched
cover of the ball B^3 with C (two arcs) removed. The fundamental group
of B^3 - C is the free group on two generators, say X and Y.

The inclusion of the 4-punctured sphere in B^3 - C gives a homomorphism

g: F_3 -> F_2

as follows

a |-> X
b |-> X^{-1}
c |-> Y
d |-> Y^{-1}

So, to extend our branched cover, we need to write our homomorphism

f: F_3 -> Z/2

as

f = hg

for some homomorphism

h: F_2 -> Z/2

The obvious nice thing to try for h is

X |-> -1
Y |-> -1

It works, and it's unique!
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Gerard Westendorp
science forum beginner


Joined: 30 Apr 2005
Posts: 48

PostPosted: Fri May 19, 2006 11:46 am    Post subject: Re: This Week's Finds in Mathematical Physics (Week 231) Reply with quote

Squark wrote:

Quote:
Hello John and everyone!

John Baez wrote:

While we tend to take it for granted, water is a very strange chemical:

5) Martin Chaplin, Forty-one anomalies of water,
http://www.lsbu.ac.uk/water/anmlies.html


I heard this claim many time by now and I begin to wonder about it.
Most of the explanations of water's "strange" properties (like high
melting point, boling point, dielectric coefficient, viscosity etc.) I
heard
are based on hydrogen bonds. However, why is this phenomenon so
unique to water? For instance, why wouldn't hydrofluoric acid (HF)
exhibit the same properties? Fluorine is even more electronegative than
oxygen, so the molecule is bound to be very polar. Nevertheless, the
boiling point is merely 20C and the melting point is as low as -83C
(according to Wikipedia). So, there must be more to water than polarity
and hydrogen. Multipole moments possibly??

There is a nice explanation of this in the Wikipedia article on
'hydrogen bond'.

Around each atom, you draw 4 bars, which represents 4 pairs of
electrons, except for hydrogen, which only needs one bar. eg:

H
|
water: |O|
|
_ H
HF: |F|
|
H
H
|
Amonia: |N-H

|
H

Next, a hydrogen bond is a bond between an H and a 'lone pair', eg:

H
| _
|O|...H-O-H
| -
H

Now (I just learned this from Wikipedia), water has the highest amount
of possible H-bonds per molecule, that is, if it binds to itself. HF
would have too many lone pairs, amonia too many H.

For the rest, the Wikepedia is a lot better than this post, except
perhaps drawing out the molecular diagrams explicitly, like I did in
ASCII, is helpful, at least it was for me.

Gerard
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John Baez
science forum Guru Wannabe


Joined: 01 May 2005
Posts: 220

PostPosted: Sun May 28, 2006 10:29 pm    Post subject: Re: This Week's Finds in Mathematical Physics (Week 232) Reply with quote

In article <1148433330.385151.129800@j33g2000cwa.googlegroups.com>,
Greg Egan <gregegan@netspace.net.au> wrote:

Quote:
John Baez wrote:

The really cool part is the relation between the Lie algebra element p
and the group element exp(p). Originally we thought of p as momentum -
but there's a sense in which exp(p) is the momentum that really counts!

Would it be correct to assume that the ordinary tangent vector p still
transforms in the usual way?

Hi! Yes, it would.

Quote:
In other words, suppose I'm living in a
2+1 dimensional universe, and there's a point particle with rest mass m
and hence energy-momentum vector in its rest frame of p=m e_0. If I
cross its world line with a certain relative velocity, there's an
element g of SO(2,1) which tells me how to map the particle's tangent
space to my own. Would I measure the particle's energy-momentum to be
p'=gp? (e.g. if I used the particle to do work in my own rest frame)
Would there still be no upper bound on the total energy, i.e. by making
our relative velocity close enough to c, I could measure the particle's
kinetic energy to be as high as I wished?

To understand this, it's good to think of the momenta as
elements of the Lie algebra so(2,1) - it's crucial to the
game.

Then, if you have momentum p, and I zip past you, so you
appear transformed by some element g of the Lorentz group
SO(2,1), I'll see your momentum as

p' = g p g^{-1}

This is just another way of writing the usual formula for
Lorentz transforms in 3d Minkowski space. No new physics
so far, just a clever mathematical formalism.

But when we turn on gravity, letting Newton's constant k
be nonzero, we should instead think of momentum as group-valued, via

h = exp(kp)

and similarly

h' = exp(kp')

Different choices of p now map to the same choice of h.
In particular, a particle of a certain large mass - the
Planck mass- will turn out to act just like a particle
of zero mass!

So, if we agree to work with h instead of p, we are now
doing new physics. This is even more obvious when we decide
to multiply momenta instead of adding them, since multiplication
in SO(2,1) is noncommutative!

But, if we transform our group-valued momentum in the correct
way:

h' = ghg^{-1}

this will be completely compatible with our previous transformation
law for vector-valued momentum!

Quote:
I guess I'm trying to clarify whether the usual Lorentz transformation
of the tangent space has somehow been completely invalidated for
extreme boosts, or whether it's just a matter of there being a second
definition of "momentum" (defined in terms of the Hamiltonian) which
transforms differently and is the appropriate thing to consider in
gravitational contexts.

Good question! Amazingly, the usual Lorentz transformations still work
EXACTLY - even though the rule for adding momentum is new (now it's
multiplication in the group). We're just taking exp(kp) instead of
p as the "physical" aspect of momentum.

This effectively puts an upper limit on mass, since as
we keep increasing the mass of a particle, eventually it "loops
around" SO(2,1) and act exactly like a particle of zero mass.

But, it doesn't exactly put an upper bound on energy-momentum,
since SO(2,1) is noncompact. Of course energy and momentum don't
take real values anymore, so one must be a bit careful with this
"upper bound" talk.

Quote:
In other words, does the cut-off mass apply only to the deficit angle,
and do boosts still allow me to measure (by non-gravitational means)
arbitrarily large energies (at least in the classical theory)?

There's some sense in which the answer to this question is "yes",
but one must be careful: what coordinate on SO(2,1) shall we call
"energy".

Maybe you can figure out some more intuitive way to say what's going on.
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Aaron Bergman
science forum addict


Joined: 24 Mar 2005
Posts: 94

PostPosted: Mon May 29, 2006 7:43 pm    Post subject: Re: This Week's Finds in Mathematical Physics (Week 232) Reply with quote

In article <e5crae$f77$1@glue.ucr.edu>,
baez@math.removethis.ucr.andthis.edu (John Baez) wrote:
Quote:

Maybe you can figure out some more intuitive way to say what's going on.

I think I'm very confused here. Can you define for me what 'momentum'
and 'mass' mean in these contexts?

Aaron
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John Baez
science forum Guru Wannabe


Joined: 01 May 2005
Posts: 220

PostPosted: Tue May 30, 2006 6:07 pm    Post subject: Re: This Week's Finds in Mathematical Physics (Week 232) Reply with quote

In article <abergman-D08EC7.20583628052006@geraldo.cc.utexas.edu>,
Aaron Bergman <abergman@physics.utexas.edu> wrote:

Quote:
In article <e5crae$f77$1@glue.ucr.edu>,
baez@math.removethis.ucr.andthis.edu (John Baez) wrote:

Maybe you can figure out some more intuitive way to say what's going on.

I think I'm very confused here. Can you define for me what 'momentum'
and 'mass' mean in these contexts?

We're studying point particles in 2+1 gravity.

So, "energy-momentum" and "mass" are properties of a solution of
the vacuum Einstein equations that's singular along a curve - the
worldline of the particle.

It turns out that the "energy-momentum" is something we can detect
by parallel transporting a little arrow around the worldline of the
particle. Parallel transporting it around a loop, it comes back
rotated and/or Lorentz transformed, so we get an element g of the Lorentz
group SO(2,1).

If one calculates for a while, one sees that g is a function of
the traditionally defined energy-momentum of the particle, say p:

g = exp(kp)

where k is Newton's constant times something like 4pi.

Of course, the trick here is to think of p as living not in R^3 but
in the isomorphic vector space so(2,1) - the Lie algebra of SO(2,1).
That lets us exponentiate it!

However, operationally, all we have access to is g, not p.

Next, we see that the traditionally defined mass of the particle
goes into determining the *conjugacy class* of our element g.
This famous picture:

http://math.ucr.edu/home/baez/loopbraid/lightcone.jpg

which looks like a picture of mass hyperboloids in R^3, is actually
a picture of adjoint orbits in so(2,1) - which look exactly like
*conjugacy classes* in SO(2,1) near the identity.

So, after thinking a while, we see that in the context of 2+1
gravity, the really sensible sort of "energy-momentum" for
point particles takes values in SO(2,1) - and "mass" means
"conjugacy class".

The fun starts when we realize that the correct way to "add"
energy-momenta, e.g. in particle collisions, is to *multiply*
elements of SO(2,1). And it gets even more fun when we realize
that the statistics of these point particles are anyonic - even
classically. Everything I've said so far is classical, but it
all has a precise quantum version as well.

For more, try these:

http://math.ucr.edu/home/baez/week232.html
http://arxiv.org/abs/gr-qc/0603085

I know this stuff sounds a bit wacko, but that's precisely why
it's interesting - it sounds wacko at first, but it's not.
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John Baez
science forum Guru Wannabe


Joined: 01 May 2005
Posts: 220

PostPosted: Tue May 30, 2006 10:08 pm    Post subject: Re: This Week's Finds in Mathematical Physics (Week 232) Reply with quote

In article <1148885617.080909.91450@i39g2000cwa.googlegroups.com>,
Greg Egan <gregegan@netspace.net.au> wrote:

Quote:
Thanks for the reply! I've been reading "2+1 Gravity and Double
Special Relativity" by Friedel, Kowalski-Glikman and Smolin, and
"Quantum Mechanics of a Point Particle in 2+1 Dimensional Gravity" by
Matschull and Welling, trying to understand what's going on here, but
I'm still confused.

Yeah, they're confusing. I never understood this stuff until
I read the introduction to this paper:

http://arxiv.org/abs/gr-qc/0603085

so I recommend that you start there. But maybe you did....

Quote:
I always thought that the whole idea of DSR was to find a new
representation of the Lorentz group in which the statement "this system
has the Planck energy" takes on an invariant meaning -- but not in the
trivial sense that rest mass is already an invariant for any particle!

Heh. Don't worry, I don't think it's *that* trivial.

First of all, some history.

"Doubly special relativity" is a general name for attempts to
modify the way quantities transform so that not only the speed
of light but also the Planck energy is invariant under changes
of reference frame.

At first you might think this is impossible. In fact, there turn
out to be *lots* of ways to achieve it.

In 4 dimensions, the most interesting way seems to be replacing
the Poincare group by a quantum group, the "kappa-deformed Poincare
group", where kappa is a parameter that optimists believe might be
related to Newton's constant (i.e., gravity).

In 3 dimensions the most interesting way comes from simply working
with general relativity and noticing that energy-momenta of point
particles act differently when you turn on gravity! I like this
because you don't have to fight to get it to happen: it just does.
Also, the math is nice.

Quote:
Smolin et al. seem to be insisting that the symmetries that apply to
2+1 gravity "preserve an energy scale", but I can't really figure out
exactly what this means, or even whether it's supposed to be true in
the classical theory as well as the quantum version.

Well, I can't talk about doubly special relativity in general because
there are lots of versions and I can never keep them all straight.
So, I will only talk about 2+1 gravity. Here it doesn't require
quantum mechanics to see the interesting effects. So, let's set
hbar = 0 and work with classical GR.

Quote:
One thing I'd like to do is get a handle on collision theory in
classical 2+1 gravity, since it's collision theory that helps make
sense of the definition of energy-momentum in Newtonian physics and SR.

Right. The key thing is that if we parallel transport a tangent
vector around a loop enclosing the worldline of a point particle,
it gets Lorentz transformed, so we get an element of SO(2,1). This
doesn't change when we do any basepoint-preserving homotopy to the
loop. But, if we move the basepoint of the loop, it gets conjugated.

So, we have this fairly robust concept of "energy-momentum" taking
values in SO(2,1). You can go ahead and relate it to the traditional
so(2,1)-valued notion, but I don't feel like doing that here.

Instead, what I want to say is that if we let two particles collide,
and merge into one, their SO(2,1)-valued energy-momenta multiply *if*
we let the new loop for the new particle be the obvious "product" of
the two loops for the original particles.

This is easy to see if you know how parallel transport for flat
connections works. It doesn't require any scribbling, just drawing
pictures.

Quote:
Using the recipe in Matschull and Welling for mapping R^3 to
so(2,1)~sl(2), we have:

e_t = | 0 1 |
| -1 0 |

e_x = | 0 1 |
| 1 0 |

e_y = | 1 0 |
| 0 -1 |

Yikes! I've never needed these explicit formulas to understand this
subject. They could be helpful, but I will just skip over them
and focus on your words.

Quote:
Suppose we have a collision between two particles of equal mass, m,
moving along the x-axis with opposing velocities, +/-v.

This is where things start to become extremely strange. How do we know
in which order to multiply these two group elements?

The key is to understand the meaning of these group elements: they
are holonomies around loops that go around the particles. It helps
to draw a worldline diagram like I did in "week232":

p p'
\ /
\ /
\ /
|
|
|
p"

If you pick a loop going around the worldline of particle p and
the worldline of particle p', you can multiply these loops - in
some order that your picture will tell you!!! - and do a homotopy
to get a loop going around particle p".

If you change which loops you pick, the correct order of multiplication
will change.

This freaked me out too until I realized this. In particular,
it's a good exercise to find the convention that gives this rule:

exp(kp") = exp(kp) exp(kp')

and the convention that gives this rule:

exp(kp") = exp(kp') exp(kp)

Quote:
Now, to zeroth order in k this gives the right result:

To zeroth order in k, multiplication in the group matches
addition in the Lie algebra, so yes, everything works - and it
works regardless of these "ordering" issues. The fun starts when
we look at the order-k corrections, which involve commutators
in the group. As I mentioned in "week232":

exp(kp") = exp(kp) exp(kp')

gives

p" = p + p' + (k/2) [p,p'] + terms of order k^2 and higher

thanks to Baker-Campbell-Hausdorff.

Maybe this will also help:

SO(2,1) has a circle in it, the SO(2), which describes rotations.
It also has a plane's worth of boosts. Its topology is S^1 x R^2.

When we use this space to describe energy-momenta, it's the S^1 that
corresponds to "energy", since its the energy of a particle at rest
that determines the "deficit angle" in the conical singularity in
the spacetime metric. This angle can be anything from 0 to 2pi,
but 2pi is the same as zero. So, energies are S^1-valued.

Similarly, the plane R^2 is what corresponds to "momentum". Unlike
energy, the x and y components of momentum can be unbounded.

To make this more precise, one might try to find "preferred coordinates"
on SO(2,1) which one could call "energy", "x-momentum" and "y-momentum".
Energy should be S^1-valued, while x- and y-momentum should be R-valued.

Probably some lovers of coordinates have already figured out a way
to do this. To see if their answer is "right" one needs to decide
what counts as "preferred" coordinates. Clearly they should match
the usual coordinates on so(2,1) = R^3 in the limit of small
energy-momenta - i.e., near the identity of the group SO(2,1).

But what else? You might want the coordinates to transform nicely
under rotations, but note that there's no "preferred" SO(2) in SO(2,1) -
there's an "obvious" choice, but this consists of rotations in a
specific rest frame, which is presumably no better than any other.

Hmm. I do like the exponential map

p |-> exp(kp)

from so(2,1) to SO(2,1). This is perfectly canonical, hence
"preferred" in every sense. It's not 1-1 - energy "wraps around" -
but I guess you could just work with a quotient space of so(2,1).
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Aaron Bergman
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Joined: 24 Mar 2005
Posts: 94

PostPosted: Fri Jun 02, 2006 12:38 pm    Post subject: Re: This Week's Finds in Mathematical Physics (Week 232) Reply with quote

In article <e5ht2m$p6n$1@glue.ucr.edu>,
baez@math.removethis.ucr.andthis.edu (John Baez) wrote:

Quote:
In article <abergman-D08EC7.20583628052006@geraldo.cc.utexas.edu>,
Aaron Bergman <abergman@physics.utexas.edu> wrote:

In article <e5crae$f77$1@glue.ucr.edu>,
baez@math.removethis.ucr.andthis.edu (John Baez) wrote:

Maybe you can figure out some more intuitive way to say what's going on.

I think I'm very confused here. Can you define for me what 'momentum'
and 'mass' mean in these contexts?

We're studying point particles in 2+1 gravity.

So, "energy-momentum" and "mass" are properties of a solution of
the vacuum Einstein equations that's singular along a curve - the
worldline of the particle.

It turns out that the "energy-momentum" is something we can detect
by parallel transporting a little arrow around the worldline of the
particle. Parallel transporting it around a loop, it comes back
rotated and/or Lorentz transformed, so we get an element g of the Lorentz
group SO(2,1).

If one calculates for a while, one sees that g is a function of
the traditionally defined energy-momentum of the particle, say p:

g = exp(kp)

where k is Newton's constant times something like 4pi.

I'm sorry, but I'm still confused. Should I think of this 'traditionally
defined energy-momentum' as a sort of ADM-like thing in 2+1D?

I'm sure you can define all the things as you say, but I'm not quite
sure that this is physically significant. Does this just translate to a
holonomy in the related CS theory? Anyways, the structure group of the
tangent bundle still reduces to SO(2,1), so I don't understand how this
relates to what I normally think of as the (local) Lorentz symmetry in
GR.

Aaron
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Greg Egan
science forum addict


Joined: 01 May 2005
Posts: 75

PostPosted: Sat Jun 03, 2006 8:56 pm    Post subject: Re: This Week's Finds in Mathematical Physics (Week 232) Reply with quote

Daryl McCullough wrote:
[snip]
Quote:
I'm wondering if there is an even more elementary introduction
to gravity in n+1 spacetime for various values of n. What I'm
specifically wondering is this: Does General Relativity in various
numbers of dimensions have a non-relativistic "Newtonian" limit
in the same way it does in 3+1 spacetime?

My intuition (which might be completely wrong) is that gravity
in 2+1 spacetime is like gravity in 3+1 spacetime if we impose
translational symmetry in one spatial direction. If we start
off with a Newtonian inverse-square law for point-masses in 3+1,
and then impose translational symmetry in the z-direction, then
instead of point-sources we have line-sources. In Newtonian gravity,
the gravitational acceleration due to a line source is

g = -2G lambda/r

where lambda is the mass per unit length.

Is there a "Newtonian" limit of 2+1 GR? If so, is
my speculation right, that instead of inverse-square
law, there is a 1/r law of attraction?

2+1 GR is both simpler and weirder than a 1/r law of attraction.

The vacuum Einstein equations require that the Ricci curvature tensor
is zero, but in three dimensions it turns out that this also implies
that the whole Riemann curvature tensor is zero. In other words,
spacetime is completely flat, and so apart from topological
differences, regions of vacuum are, locally, just like Minkowski
spacetime.

In the vacuum around a point particle the correct vacuum solution turns
out to be one you get by cutting a wedge out of spacetime with an edge
that runs along the world line of the particle, and then identifying
points on the opposite planes of the wedge. As a result of this, there
is an angle of less than 2pi in the space around the world line.

Because spacetime is flat, there is nothing that truly resembles a
Newtonian acceleration. Two point masses at rest wrt each other will
remain at rest forever! However, there is an effect a bit like
gravitational lensing: if you send a swarm of initially parallel test
particles towards a massive object, their trajectories will veer
together.

I guess it might be possible to "work backwards" and look for a
tensor-based theory that reduces to a 1/r Newtonian limit ... Maxwell's
equation in flat 2+1 spacetime would be one such theory, but that's
cheating because particles don't follow geodesics.

One big obstacle to finding a curvature-based 2+1 theory with a 1/r
Newtonian limit is the following: in 3+1 GR, the fact that the Ricci
curvature tensor is zero is connected to the fact that the second rate
of change with time of the volume of a small ball of free-falling dust,
in a vacuum, is zero. This result also holds in the 3+1 Newtonian
theory, with a 1/r^2 acceleration. But in a 2+1 Newtonian theory with
1/r acceleration, the second rate of change with time of the *area* of
a small disk of free-falling dust will also be zero! This in turn
makes it hard to see how the tensor-based theory to which it is an
approximation could fail to have a Ricci curvature tensor of zero. And
in three dimensions, that means totally flat spacetime, and no
Newtonian attraction at all.
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Greg Egan
science forum addict


Joined: 01 May 2005
Posts: 75

PostPosted: Sat Jun 03, 2006 8:56 pm    Post subject: Re: This Week's Finds in Mathematical Physics (Week 232) Reply with quote

I wrote:

Quote:
Now this is where I get confused again. Shouldn't I be able to get from
the group-valued momentum to some *globally* applicable description of the
outgoing particle's world line? Obviously its precise momentum (whether
vector or group-valued) will depend on the chosen frame of reference ...
but surely it's an invariant question to ask whether the outgoing world
line lies in the same plane as the two incoming world lines -- and if not,
on which side?

Sorry, I was being naive about the symmetries of the situation. There
should be a kind of dihedral symmetry in which the outgoing particle's
world line *is* coplanar with the world lines of the two incoming
particles, but the angular deficits mean that when we parallel
transport the tangent vectors of the three world lines to a common
point on either side of that plane, we shouldn't expect those vectors
to be coplanar.

It's like the situation where you slice a cone into two halves,
vertically, straight through the tip. When you flatten out the two
pieces, the edges won't be colinear. And if instead of doing any
cutting and flattening you'd simply drawn two lines from the tip of the
cone to infinity that together divided the surface symmetrically in
two, then if you parallel-transported tangent vectors to those lines to
some common point, the tangent vectors would not be colinear.
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tessel@um.bot
science forum addict


Joined: 20 Sep 2005
Posts: 60

PostPosted: Mon Jun 05, 2006 9:01 am    Post subject: Re: This Week's Finds in Mathematical Physics (Week 233) Reply with quote

On Tue, 30 May 2006, John Baez wrote:

Quote:
And, R^3 minus the trefoil knot is secretly the same as SL(2,R)/SL(2,Z)!

This is actually incredibly interesting for me- what is a reference for
this? (I couldn't find it in either cited paper, and Gannon gives no
source).

As usual, I gave all the references I know. I too find this fact
incredibly interesting. I first heard of it from Chris Hillman:

http://www.lns.cornell.edu/spr/2002-04/msg0040885.html

Maybe ask Graeme Segal (Math, University of Cambridge) for a citation?
This is mentioned (without further explanation) on p. 59 of "Lie Groups"
by Segal, which is comprises third of the inspiring little book

Roger Carter, Graeme Segal, Ian MacDonald
Lectures on Lie Algebras and Lie Groups
London Mathematical Society Student Texts 32
Cambridge University Press, 1995

Great stuff, BTW!

"T. Essel"
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