# May 18, 2006 {#week232}

I'm at the Perimeter Institute now. It's great to see how it's
developed since I first saw their [new building](PI1.jpg) back in 2004
(see ["Week 208"](#week208) for the story).

There's now a busy schedule of seminars and weekly colloquia, with
string theorists and loop quantum gravity people coexisting happily.
Their program of Superstring Quartets features some really hot bands,
like the Julliard and Emerson --- unfortunately not playing while I'm
here. The Black Hole Bistro serves elegant lunches and dinners, there
are at least two espresso machines on each floor, and my friend Eugenia
Cheng will be happy to hear that they still have a piano available
(after 6 pm).

But don't get the impression that it's overly sophisticated: there are
also a couple of guys constantly playing [foosball](https://math.ucr.edu/home/baez/foosball.jpg) in the
Feynman Lounge.

Since I'm here, I should talk about quantum gravity --- so I will. But
first, let's have the astronomy picture of the week.

This week it comes, not from outer space, but beneath the surface of the
South Pole:
$$\href{http://icecube.wisc.edu/gallery/detector_concepts/ceren_hires}{\includegraphics[max width=0.65\linewidth]{../images/ice_cube.jpg}}$$

1) Steve Yunck / NSF, "Cerenkov light passing through the IceCube neutrino detector", `http://icecube.wisc.edu/gallery/detector_concepts/ceren_hires`

This is an artist's impression of a huge neutrino observatory called
"IceCube". (Maybe they left out the space here so the rap star of that
name doesn't sue them for trademark infringement, or go down there and
shoot them.)

IceCube is being built in the beautifully clear 18,000-year old ice deep
beneath the Amundsen-Scott South Pole Station. When a high-energy
neutrino hits a water molecule, sometimes the collision produces a muon
zipping faster than the speed of light in ice. This in turn produces
something like a sonic boom, but with light instead of sound. It's
called "Cerenkov radiation", and it's the blue light in the picture.
This will be detected by an array of 5000 photomultiplier tubes --- those
gadgets hanging on electrical cables.

One thing the artist's impression doesn't show is that IceCube is
amazingly large. The whole array is a cubic kilometer in size! It will
encompass the already existing AMANDA detector, itself 10,000 meters
tall, shown as a yellow cylinder here with a neutrino zipping through:
$$\href{http://icecube.wisc.edu/gallery/detector_concepts/icecubeencomp_300}{\includegraphics[max width=0.65\linewidth]{../images/ice_cube_AMANDA.jpg}}$$

2) Darwin Rianto / NSF, "Comparison of AMANDA and IceCube", `http://icecube.wisc.edu/gallery/detector_concepts/icecubeencomp_300`

Even the very top of IceCube is 1.4 kilometers beneath the snowy
Antarctic surface, to minimize the effect of stray cosmic rays. The
station on top looks like this --- not very cozy, I'd say:
$$\href{http://icecube.wisc.edu/gallery/antarctica/PC140287_300}{\includegraphics[max width=0.65\linewidth]{../images/south_pole_station.jpg}}$$

3) Robert G. Stokstad / NSF, "South Pole Station", `http://icecube.wisc.edu/gallery/antarctica/PC140287_300`

I heard about IceCube from Adrian Burd, one of the old-timers who used
to post a lot on `sci.physics`, a former cosmologist turned oceanographer
who recently visited Antarctica as part of an NSF-run field course. He
ran into some people working on IceCube. It sounds like an interesting
community down there! You can read about it in their newspaper, the
Antarctic Sun. For example:

4) "Ice Cube turns up the heat", _The Antarctic Sun_, January 29, 2006, `http://antarcticsun.usap.gov/2005-2006/contentHandler.cfm?id=959`

For more on IceCube and Amanda, these are fun to read:

5) Francis Halzen, "Ice fishing for neutrinos", `http://icecube.berkeley.edu/amanda/ice-fishing.html`

6) Katie Yurkiewicz, "Extreme neutrinos", _Symmetry_, volume **1** issue 1, November 2004, `http://symmetrymagazine.org/cms/?pid=1000014`

For some of AMANDA's results, including a map of the sky as seen in
neutrinos, try this:

7) M. Ackermann et al, "Search for extraterrestrial point sources of high energy neutrinos with AMANDA-II using data collected in 2000--2002", available as [`astro-ph/0412347`](https://arxiv.org/abs/astro-ph/0412347).

For much more, try these:

8) AMANDA II Project, `http://amanda.uci.edu/`

9) Welcome to IceCube, `http://icecube.wisc.edu/`

And now, on to gravity.

You may have heard of the gravitational 3-body problem. Well, Richard
Montgomery (famous from ["Week 181"](#week181)) recently pointed
out this movie of the $60$-body problem:

10) Davide L. Ferrario, "Periodic orbits for the $60$-body problem", `http://www.matapp.unimib.it/~ferrario/mov/index.html`

60 equal masses do a complicated dance while always preserving
icosahedral symmetry! First 12 groups of 5 swing past each other, then
20 groups of 3. If you want to know how he found these solutions, read
this:

11) Davide L. Ferrario and S. Terracini, "On the existence of collisionless equivariant minimizers for the classical $n$-body problem". _Invent. Math._ **155** (2004), 305--362.

It's quite math-intensive --- though just what you'd expect if you know
this sort of thing: they use the $G$-equivariant topology of loop spaces,
where $G$ is the symmetry group in question (here the icosahedral group),
to prove the existence of action-minimizing loops with given symmetry
properties.

Next, I'd like to say a little about point particles in 3d quantum
gravity, and some recent work with Alissa Crans, Derek Wise and
Alejandro Perez on string-like defects in 4d topological gravity:

12) John Baez, Derek Wise and Alissa Crans, "Exotic statistics for strings in 4d $BF$ theory", available as [`gr-qc/0603085`](https://arxiv.org/abs/gr-qc/0603085).

13) John Baez and Alejandro Perez, "Quantization of strings and branes coupled to $BF$ theory", available as [`gr-qc/0605087`](https://arxiv.org/abs/gr-qc/0605087).

(Jeffrey Morton is also involved in this project, a bit more on the
$n$-category side of things, but that aspect is top secret for now.)

In ["Week 222"](#week222) I listed a bunch of cool papers on 3d
quantum gravity, but I didn't really explain them. What we're trying
to do now is generalize this work to higher dimensions. But first, let
me start by explaining the wonders of 3d quantum gravity.

The main wonder is that we actually understand it! The classical version
of general relativity is exactly solvable when spacetime has dimension
3, and so is the quantum version. Most of the wonders I want to discuss
are already visible in the classical theory, where they are easier to
understand, so I'll focus on the classical case.

A nice formulation of general relativity in 3 dimensions uses a
"Lorentz connection" $A$ and a "triad field" $e$. This is a gauge theory
where the gauge group is $\mathrm{SO}(2,1)$, the Lorentz group for 3d spacetime. If
we're feeling lowbrow we can think of both $A$ and $e$ as $\mathfrak{so}(2,1)$-valued
$1$-forms on the $3$-manifold $M$ that describes spacetime. The action for
this theory is:
$$\int_M \operatorname{tr}(e\wedge F)$$
where $F$ is the curvature of $A$. If you work out the equations of motion
one of them says that $F = 0$, so our connection $A$ is flat. The other,
$d_A e = 0$, says $A$ is basically just the Levi-Civita connection.

This is exactly what we want, because in the absence of matter, general
relativity in 3 dimensions says spacetime is *flat*.

A fellow named Phillipp de Sousa Gerbert came up with an interesting way
to couple point particles to this formulation of quantum gravity:

14) Phillipp de Sousa Gerbert, "On spin and (quantum) gravity in 2+1 dimensions", _Nuclear Physics_ **B346** (1990), 440--472.

He actually did it for particles with spin, but I'll just do the
spin-zero case.

The idea is to fix a $1$-dimensional submanifold $W$ in our $3$-manifold $M$ and
think of it as the worldlines of some particles. Put $\mathfrak{so}(2,1)$-valued
functions $p$ and $q$ on these worldlines --- think of these as giving the
particles' momentum and position as a function of time.

Huh? Well, normally we think of position and momentum as vectors. In
special relativity, "position" means "position in spacetime", and
"momentum" means "energy-momentum". We can think of both of these as
vectors in Minkowksi spacetime. But in 3 dimensions, Minkowski spacetime
is naturally identified with the Lorentz Lie algebra $\mathfrak{so}(2,1)$. So, it
makes sense to think of $q$ and $p$ as elements of $\mathfrak{so}(2,1)$ which vary from
point to point along the particle's worldline.

To couple our point particles to gravity, we then add a term to the
action like this:
$$S = \int_M \operatorname{tr}(e\wedge F) - \int_W \operatorname{tr}((e + d_A q)\wedge p)$$
Now if you vary the $e$ field you get a field equation saying that
$$F = p \delta_W$$
Here $\delta_W$ is like the Dirac delta function of the worldline $W$; it's a
distributional $2$-form defined by requiring that
$$\int_W X = \int_M (X\wedge \delta_W)$$
for any smooth $1$-form $X$ on $W$. This sort of "distributional differential
form" is also called a "current", and you can read about them in the
classic tome by Choquet-Bruhat et al. But the main point is that the
field equation
$$F = p \delta_W$$
says our connection on spacetime is flat except along the worldlines of
our particles, where the curvature is a kind of "$\delta$ function". This is
nice, because that's what we expect in 3d gravity: if you have a
particle, spacetime will be flat everywhere except right at the
particle, where it will have a singularity like the tip of a cone.

A cone, you see, is intrinsically flat except at its tip: that's why
you can curl paper into a cone without crinkling it!

So, our spacetime is flat except along the particles' worldlines, and
there it's like a cone. The "deficit angle" of this cone --- the angle
of the slice you'd need to cut out to curl some paper into this cone -
is specified by the particle's momentum $p$.

Since delta functions are a bit scary, it's actually better to work
with an "integrated" form of the equation
$$F = -p \delta_W$$
The integrated form says that if we parallel transport a little tangent
vector around a little loop circling our particle's worldline, it gets
rotated and/or Lorentz transformed by the element
$$\exp(p)$$
in $\mathrm{SO}(2,1)$. This will be a rotation if the particle's momentum $p$ is
timelike, as it is for normal particles. Again, that's just as it
should be: if you parallel transport a little arrow around a massive
particle in 3d gravity, it gets rotated!

If $p$ is timelike, our particle is a tachyon and $\exp(p)$ is a Lorentz
boost. And so on... we get the usual classification of particles
corresponding to various choices of $p$:
$$\includegraphics[max width=0.65\linewidth]{../images/lightcone.jpg}$$

There are other equations of motion, obtained by varying other fields,
but all I want to note is the one you get by varying $q$:
$$d_A  p = 0$$
This says that the momentum $p$ is covariantly constant along the
particles' worldlines. So, momentum is conserved!

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!

First, $\exp(p)$ is what we actually detect by parallel transporting a
little arrow around our particle.

Second, suppose we let two particles collide and form a new one:
$$
  \begin{tikzpicture}
    \node[label=above:{$p$}] at (0,0) {};
    \node[label=above:{$p'$}] at (1,0) {};
    \node[label=below:{$p''$}] at (0.5,-2) {};
    \draw[thick] (0,0) to (0.5,-1) to (0.5,-2);
    \draw[thick] (1,0) to (0.5,-1);
  \end{tikzpicture}
$$
Now our worldlines don't form a submanifold anymore, but if we keep our
wits about us, we can see that everything still makes sense, and we get
momentum conservation in this form:
$$\exp(p'') = \exp(p) \exp(p')$$
since little loops going around the two incoming particles can fuse to
form a loop going around the outgoing particle. Note that we're getting
conservation of the *group-valued* momentum, not the Lie-algebra-valued
momentum --- we don't have
$$p'' = p + p'$$
So, conservation of energy-momentum is getting modified by gravitational
effects! This goes by the name of "doubly special relativity":

15) Laurent Freidel, Jerzy Kowalski-Glikman and Lee Smolin, "2+1 gravity and doubly special relativity", _Phys. Rev._ **D69** (2004) 044001. Also available as [`hep-th/0307085`](https://arxiv.org/abs/hep-th/0307085).

This effect is a bit less shocking if we put the units back in. I've
secretly been setting $4\pi G = 1$, where $G$ is Newton's gravitational
constant. If we put that constant back in --- let's call it $k$ instead of
$4\pi G$ --- we get
$$\exp(kp'') = \exp(kp) \exp(kp')$$
or if you expand things out:
$$p'' = p + p' + \frac{k}{2}[p,p'] + \mbox{terms of order $k^2$ and higher}$$
So, as long as the momenta are small compared to the Planck mass, the
usual law of conservation of momentum
$$p'' = p + p'$$
*almost* holds! But, for large momenta this law breaks down --- we must
think of momentum as group-valued if we want it to be conserved!

I think this is incredibly cool: as we turn on gravity, the usual
"flat" momentum space curls up into a group, and we need to *multiply*
momenta in this group, instead of *add* them in the Lie algebra. We can
think of this group has having a "radius" of $1/k$, so it's really big
and almost flat when the strength of gravity is small. In this limit,
multiplication in the group reduces to addition in the Lie algebra.

I should point out that this effect is purely classical! It's still
there when we quantize the theory, but it only depends on the
gravitational constant, not Planck's constant. Indeed, in 3d quantum
gravity, we can build a unit of mass using just $G$ and $c$: we don't need
$\hbar$. This unit is the mass that curls space into an infinitely skinny
cone! It would be a bit misleading to call it "Planck mass", but it's
the maximum possible mass. Any mass bigger than this acts like a
*negative* mass. That's because the corresponding group-valued momenta
"wrap around" in the group $\mathrm{SO}(2,1)$.

We also get another cool effect --- exotic statistics. In the absence of
gravitational or quantum effects, when you switch two particles, you
just switch their momenta:
$$(p, p') \mapsto (p', p)$$
But in 3d gravity, you can think of this process of switching particles
as a braid:
$$
  \begin{tikzpicture}
    \begin{knot}[clip width=5]
      \strand[thick] (0,0) to (1,-2);
      \strand[thick] (1,0) to (0,-2);
    \end{knot}
  \end{tikzpicture}
$$
and if you work out what happens to their group-valued momenta, say
$$
  \begin{aligned}
    g &= \exp(kp)
  \\g' &= \exp(kp')
  \end{aligned}
$$
it turns out that one momentum gets conjugated by the other:
$$(g, g') \mapsto (gg'g^{-1}, g)$$
To see this, remember that we get these group elements by doing parallel
transport around loops that circle our particles. When we move our
particles, the loops get dragged along, like this:
$$\includegraphics[max width=0.65\linewidth]{../images/particles_braiding.jpg}$$
Note that the left-hand red loop moves until it looks just like the
right one did initially, but the right-hand one gets wrapped around the
left one. If you ponder this carefully, and you know some math, you can
see it yields this:
$$(g, g') \mapsto (gg'g^{-1}, g)$$
So, the process of braiding two particles around each other has a
nontrivial effect on their momenta. In particular, if you braid two
particles around other twice they don't wind up in their original
state!

Thus, our particles are neither bosons nor fermions, but "nonabelian
anyons" --- the process of switching them is governed not by the
permutation group, but by the braid group. But again, if you expand
things out in powers of $k$ you'll see this effect is only noticeable for
large momenta:
$$(p, p') \mapsto (p' + k[p,p'] + \mbox{higher order terms}, p)$$
Summarizing, we see quantum gravity is lots of fun in 3 dimensions:
it's easy to introduce point particles, and they have group-valued
momentum, which gives rise to doubly special relativity and braid group
statistics.

Now, what happens when we go from 3 dimensions to 4 dimensions?

Well, we can write down the same sort of theory:
$$S = \int_M \operatorname{tr}(B\wedge F) - \int_W \operatorname{tr}((B + d_A q)\wedge p)$$
The only visible difference is that what I'd been calling "$e$" is now
called "$B$", so you can see why folks call this "$BF$ theory".

But more importantly, now $M$ is an $4$-dimensional spacetime and $W$ is an
2-dimensional "worldsheet". $A$ is again a Lorentz connection, which we
can think of as an $\mathfrak{so}(3,1)$-valued $1$-form. $B$ is an $\mathfrak{so}(3,1)$-valued $2$-form.
$p$ is an $\mathfrak{so}(3,1)$-valued function on the worldsheet $W$. $q$ is an
$\mathfrak{so}(3,1)$-valued $1$-form on $W$.

So, only a few numbers have changed... so everything works very
similarly! The big difference is that instead of spacetime having a
conical singularity along the worldline of a *particle*, now it's
singular along the worldsheet of a *string*. When I call it a
"string", I'm not trying to say it behaves like the ones they think
about in string theory --- at least superficially, it's a different sort
of theory, a purely topological theory. But, we've got these closed
loops that move around, split and join, and trace out surfaces in
spacetime.

They can also braid around each other in topologically nontrivial ways,
as shown in this "movie":
$$\includegraphics[max width=0.65\linewidth]{../images/loopbraid.jpg}$$
(By the way, all the math pictures this week were drawn by Derek for our
paper.)

So, we get exotic statistics as before, but now they are governed not by
the braid group but by the "loop braid group'', which keeps track of
all the ways we can move a bunch of circles around in 3d space. Let's
take our spacetime M to be $\mathbb{R}^4$, to keep things simple. Then our circles
can move around in $\mathbb{R}^3$... and there are two basic ways we can switch
two of them: move them around each other, or pass one *through* the
other, like this:
$$\includegraphics[max width=0.65\linewidth]{../images/loop_switch.jpg}$$
If we just move them around each other, they might as well have been
point particles: we get a copy of the permutation group, and all we see
are ordinary statistics. But when we consider all the ways of passing
them through each other, we get a copy of the braid group!

When we allow ourselves both motions, we get a group called the "loop
braid group'' or "braid permutation group'' --- and one thing Alissa
Derek and I did was to get a presentation of this group. This is an
example of a "motion group'': just as the motion group of point
particles in the plane is the braid group, and motion group of point
particles in $\mathbb{R}^3$ is the permutation group, the motion group of strings
in $\mathbb{R}^3$ is the loop braid group.

As before, our strings have group-valued momenta: we can get an element
of the Lorentz group $\mathrm{SO}(3,1)$ by parallel transporting a little tangent
vector around a string. And, we can see how different ways of switching
our strings affect the momenta. When we move two strings around each
other, their momenta switch in the usual way:
$$(g, g') \mapsto (g', g)$$
but when we move one through the other, one momentum gets conjugated by
the other:
$$(g, g') \mapsto (gg'g^{-1}, g)$$
So, we have exotic statistics, but you can only notice them if you can
pass one string through another!

In the paper with Alejandro, we go further and begin the project of
quantizing these funny strings, using ideas from loop quantum gravity.
Loop quantum gravity has its share of problems, but it works perfectly
well for 3d quantum gravity, and matches the spin foam picture of this
theory. People have sort of believed this for a long time, but Alejandro
demonstrated this quite carefully in a recent paper with Karim Noui:

15) Karim Noui and Alejandro Perez, "Dynamics of loop quantum gravity and spin foam models in three dimensions", to appear in the proceedings of the _Third International Symposium on Quantum Theory and Symmetries (QTS3)_, available as [`gr-qc/0402112`](https://arxiv.org/abs/gr-qc/0402112).

The reason everything works so nicely is that the equations of motion
say the connection is flat. Since the same is true in $BF$ theory in
higher dimensions, we expect that the loop quantization and spin foam
quantization of the theory I'm talking about now should also work well.

We find that we get a Hilbert space with a basis of "string spin
networks", meaning spin networks that can have loose ends on the
stringy defects.

So, there's some weird blend of loop quantum gravity and strings going
on here --- but I don't really understand the relation to ordinary string
theory, if any. It's possible that I can get a topological string
theory (some sort of well-defined mathematical gadget) which describes
these stringy defects, and that would be quite interesting.

But, I spoke about this today at the Perimeter Institute, and Malcolm
Perry said that instead of "strings" I should call these guys
$(n-2)$-branes, because the connection has conical singularities on them,
"which is what one would expect for any respectable $(n-2)$-brane".

I will talk to him more about this and try to pick his, umm, branes. In
fact I took my very first GR course from him, back when he was a postdoc
at Princeton and I was a measly undergraduate. I was too scared to ask
him many questions then. I'm a bit less scared now, but I've still got
a lot to learn. Tomorrow he's giving a talk about this:

17) David S. Berman, Malcolm J. Perry, "M-theory and the string genus expansion", _Phys. Lett._ **B635** (2006) 131--135. Also available as [`hep-th/0601141`](http://arxiv.org/abs/hep-th/0601141).

------------------------------------------------------------------------

**Addenda:** Here's an email from Greg Egan, and my reply:

> 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? 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 $p=me_0$. If I
> cross its world line with a certain relative velocity, there's an
> element $g$ of $\mathrm{SO}(2,1)$ 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?
>
> 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.
>
> 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)?

I replied:

> Greg Egan wrote:
>
> > 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.
>
> > 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 $\mathrm{SO}(2,1)$ 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 $\mathfrak{so}(2,1)$ --- 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 $\mathrm{SO}(2,1)$, 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 $\mathrm{SO}(2,1)$ 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!
>
> > 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"
> $\mathrm{SO}(2,1)$ and act exactly like a particle of zero mass.
>
> But, it doesn't exactly put an upper bound on energy-momentum, since
> $\mathrm{SO}(2,1)$ 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.
>
> > 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 energy-momenta can be arbitrarily large.
> That's because the space of energy-momenta, namely $\mathrm{SO}(2,1)$, is
> noncompact. Maybe you can figure out some more intuitive way to
> express this.

------------------------------------------------------------------------

> *I was sitting in a chair in the patent office in Bern when all of a
sudden a thought occurred to me. If a person falls freely, he will not
feel his own weight.* 
> 
> --- Albert Einstein