Thus spake Oz <Oz@farmeroz.port995.com>

John Baez <baez@math.removethis.ucr.andthis.edu> writes

18) Ring around a galaxy, HubbleSite News Archive, May 6, 1999,
http://hubblesite.org/newscenter/newsdesk/archive/releases/1999/16/image/a

This is surely the sort of object one would want (albeit at a better
orientation) to use to checkout galactic rotation curves.

Has anyone attempted to do this on ringed galaxies?

I recall no mention in the MOND literature on arxiv. Actually it is
quite laborious to check out rotation curves. They have looked at
different types. Low surface brightness galaxies are less dense than the
norm and the rotation curves actually continue to rise rather than level
off or fall as with bigger galaxies. They are also MONDian.
Regards
Charles Francis

On Tue, 20 Dec 2005, John Baez wrote:

David Rusin's reaction to Hoag's object was:

Cool. But what are the chances that there would be not just one but
TWO fascinating objects which have a significant plane of symmetry,
which "just happens" to be perpendicular to our line of sight?

Chance may be very very slim. Even the side view of the polar ring
shown here is not that symmetrical. Further doubt casted as in below.

He asked how many ring galaxies are known!

I checked and read there are 100 known "polar-ring galaxies". Here's
a nice one called NGC 4650:

18) Ring around a galaxy, HubbleSite News Archive, May 6, 1999,
http://hubblesite.org/newscenter/newsdesk/archive/releases/1999/16/image/a

I can imagine this thing looking like Hoag's object if we viewed
it head-on.

Maybe, the Hoag's object was a head-on view of a polar-ring galaxy, but
I have several doubts about that head-on point of view.

1.) You forgot your own words that there are galaxies between the core
(Maybe a yellow galaxy) and the ring belt. Quote,"The weirdest thing is
that inside the ring, in the upper right, you can see *another* ring
galaxy in the distance!" End quote.

If this another ring galaxy is too far behind the blue young star
ring we would not see them so clear. If they are too close to us, they
should be out of the Hubble's focal point. So it must be indeed very
close to and in-between the young blue star ring and the core.

2.) About "Ring Around a Galaxy", the vertical bluish ring is warped
and does not lie in one plane according to the accompanying commentary.
The lower clump should have more young stars. It is hardly as
symmetrical as the one we saw. The Haog's ring is more evenly bluish
and has shown a slight clockwise spiral roatation. Is it easy to get
such a higher level of plane symmetry by colliding two galaxies? I doubt it.

Look at the polar ring shown, let me emphasize again about the
commentary: "The polar ring appears to be highly distorted. No regular
spiral pattern stands out in the main part of the ring, and the
presence of young stars below the ring is warped and does not lie in
one plane" from the side view. So the head-on view should be as
symmetrical as the Haog's ring we saw.

Not so easy to explain the Haog's object.

Here's another ring galaxy, called AM 0644-741:

19) The lure of the rings, Hubblesite News Archive, April 22, 2004,
http://hubblesite.org/newscenter/newsdesk/archive/releases/2004/15/image/a

It's the result of a collision involving a galaxy that's not in this
picture. So, maybe Hoag's object is just a specially pretty case of
a galaxy collision!

Still doesn't explain "The weirdest thing is that inside the ring, in
the upper right, you can see *another* ring galaxy in the distance!"
I think it is quite hard to form a ring galaxy within another ring galaxy.

18) Ring around a galaxy, HubbleSite News Archive, May 6, 1999,
http://hubblesite.org/newscenter/newsdesk/archive/releases/1999/16/image/a

I like this photo of a jet emanating from the black hole in the
center of galaxy M87:
1) Jet in M87, http://math.ucr.edu/home/baez/m87_jet.jpg
(If anyone knows where this picture came from, please remind me!)

M87 is a giant elliptical galaxy. It's long been known as a powerful
radio source, and now we know why: there's a supermassive black hole
in the center, about 3 billion times the mass of our Sun.

The basic idea behind all this work is a "periodic table" of categorified
Frobenius algebras, which are related to topology in different dimensions.
For example, in "week174" I explained

how Frobenius algebras formalized the
idea of paint drips on a sheet of rubber. As you move your gaze down a
sheet of rubber covered with drips of point, you'll notice that drips can
merge:


\ \ / /
\ \ / /
\ \ / /
\ \ / /
\ \_/ /
\ /
| |
| |
| |
| |
| |

but also split:

| |
| |
| |
| |
| |
/ _ \
/ / \ \
/ / \ \
/ / \ \
/ / \ \
/ / \ \

Also available as http://math.ucr.edu/home/baez/week224.html

December 14, 2005
This Week's Finds in Mathematical Physics - Week 224
John Baez

This week I want to mention a couple of papers lying on the interface of
physics, topology, and higher-dimensional algebra. But first, some
astronomy pictures... and a bit about the mathematical physicist Hamilton!

I like this photo of a jet emanating from the black hole in the
center of galaxy M87:

1) Jet in M87, http://math.ucr.edu/home/baez/m87_jet.jpg
(If anyone knows where this picture came from, please remind me!)

M87 is a giant elliptical galaxy. It's long been known as a powerful
radio source, and now we know why: there's a supermassive black hole
in the center, about 3 billion times the mass of our Sun. As matter
spirals into this huge black hole, it forms an "accretion disk", and
some gets so hot that it shoots out in a jet, as envisioned here:

2) NASA, MAXIM: Micro-Arcsecond X-ray Imaging Mission,
http://maxim.gsfc.nasa.gov/docs/science/science.html

Accretion disks and jets are common at many different scales in our
universe. They're just nature's way of letting a bunch of matter fall
in under its own gravitation while losing angular momentum and energy.
We see them when dust clouds collapse to form stars, we see them when
black holes sucks in mass from companion stars, and they're probably
also responsible for slow gamma ray bursts as huge stars collapse when
they run out of fuel - see "week204" for that story.

But, among the biggest accretion disks and jets are those surrounding
supermassive black holes in the middle of galaxies. These are probably
responsible for all the "active galactic nuclei" or "quasars" that we
see. In the case of M87 the jet is enormous: 5000 light years long!
To get a sense of the scale, look at the small white specks away from the
jet in the next picture. These are globular clusters: clusters containing
between ten thousand and a million stars.

3) A jet from galaxy M87, Astronomy Picture of the Day, July 6, 2000,
http://antwrp.gsfc.nasa.gov/apod/ap000706.html

The jet in M87 is so hot that it emits not just radio waves and visible
light, but even X-rays, as seen by the Chandra X-ray telescope:

4) M87: Chandra sheds light on the knotty problem of the M87 jet,
http://chandra.harvard.edu/photo/2001/0134/

It seems the jet consists mainly of electrons moving at relativistic
speeds, focused by the magnetic field of the accretion disk. They
come in blobs called "knots". People can actually see these blobs
moving out, getting brighter and dimmer.

In fact, many galaxies have super-massive black holes at their centers
with jets like this one. The special thing about M87 is that it's fairly
nearby, hence easy to see. M87 is the biggest galaxy in the Virgo Cluster.
This is the closest galaxy cluster to us, about 50 million light years away.
That sounds pretty far, but it's only 1000 times the radius of the Milky
Way. So, even amateur astronomers - really good ones, anyway - can take
photos of M87 that show the jet. Here's a high-quality picture produced
by Robert Lupton using data from the Sloan Digital Sky Survey - you can
see the jet in light blue:

5) Robert Lupton and the Sloan Digital Sky Survey Consortium, The central
regions of M87, http://www.astro.princeton.edu/~rhl/PrettyPictures/

Backing off a bit further, let's take a look at the Virgo Cluster.
It contains over a thousand galaxies, but we can tell it's fairly new
as clusters go, since it consists of a bunch of "subclusters" that haven't
merged yet. Our galaxy, and indeed the whole Local Group to which it
belongs, is being pulled towards the Virgo Cluster and will eventually join
it. Here's a nice closeup of part of the Virgo Cluster:

6) Chris Mihos, Paul Harding, John Feldmeier and Heather Morrison,
Deep imaging of the Virgo Cluster, http://burro.astr.cwru.edu/Schmidt/Virgo/

Finally, just for fun, something unrelated - and more mysterious. It's
called "Hoag's object":

7) The Hubble Heritage Project, Hoag's Object,
http://heritage.stsci.edu/2002/21/

It's a ring-shaped galaxy full of hot young blue stars surrounding a ball
of yellower stars. Nobody knows how it formed: perhaps by a collision
of two galaxies? Such collisions are fairly common, but they don't
typically create this sort of structure. The weirdest thing is that
inside the ring, in the upper right, you can see *another* ring galaxy
in the distance!

Maybe an advanced civilization over there enjoys this form of art?
Probably not, but if it turns out to be true, you heard it here first.

Anyway... back here on Earth, in the summer of 2004, I visited Dublin for a
conference on general relativity called GR17. As recounted in "week207",
this was where Hawking admitted defeat in his famous bet with John Preskill
about information loss due to black hole evaporation. In August of this
year, Hawking finally came out with a short paper on the subject:

8) Stephen W. Hawking, Information loss in black holes, available as
hep-th/0507171.

I spent a lot of time talking to physicists, but I also wandered around
Dublin a bit. Besides listening to some great music at a pub called
Cobblestones - Kevin Rowsome plays a mean uilleann pipe! - and tracking
down some sites mentioned in James Joyce's novel "Ulysses", I went with
Tevian Dray on a pilgrimage to Brougham Bridge.

Tevian Dray is an expert on the octonions, and Brougham Bridge is where
Hamilton carved his famous formula defining the quaternions! Now there
is a plaque commemorating this event, which reads:

Here as he walked by
on the 16th of October 1843
Sir William Rowan Hamilton
in a flash of genius discovered
the fundamental formula for
quaternion multiplication

i^2 = j^2 = k^2 = ijk = -1

& cut it on a stone of this bridge

It doesn't mention that Hamilton had been racking his brain for the
entire month of October trying to solve this problem: "flashes of
genius" favor the prepared mind. But it's a nice story and a nice place.
My friend Tevian Dray took some photos, which you can see here:

9) John Baez, Dublin, http://math.ucr.edu/home/baez/dublin/

It was a bit of a challenge finding Brougham Bridge, since nobody at the main
bus station gave us correct information about which bus went there - except
the bus driver who finally took us there. So, to ease your way in case
you want to make your own pilgrimage, the above page includes directions.
And now, thanks to Dirk Schlimm, it also includes a link to a map showing
the bridge!

Speaking of Hamilton, Theron Stanford recently sent me an answer to one of
life's persistent questions: why is momentum denoted by the letter p?

Since momentum and position play fundamental roles in Hamiltonian mechanics,
and they're denoted by p and q, one wonders: could this notation be related
to Hamilton's alcoholism in later life? After all, some claim the saying
"mind your p's and q's" began as a friendly Irish warning not to imbibe too
many pints and quarts! So, maybe he used these letters in his work on
physics as a secret plea for help.

Umm... probably not. Just kidding. But in the absence of hard facts,
speculation runs rampant. So, I'm glad Stanford provided some of the former,
to squelch the latter.

He sent me this email:

While Googling various subjects, I came across the following from
your Quantum Gravity Seminar notes from 2001:

Again Oz was overcome with curiosity, so mimicking Toby's voice,
he asked, "Why do we call the momentum p?"

The Wiz glared at Toby. "Because m is already taken -- it stands
for mass! Seriously, I don't know why people call position q
and momentum p. All I know is that if you use any other letters,
people can tell you're not a physicist. So I urge you to follow
tradition on this point."

Well, I have an answer. Hamilton, the first physicist to actually
understand the importance of the concept of momentum, chose pi to
stand for momentum (actually, it's not the usual pi, but what TeX
calls varpi, a lower-case omega with a top, kinda like the top of a
lower-case tau). Jacobi changed this to p in one of his seminal
papers on the subject; he also used q in the same paper to stand for
position. In the 1800s (I want to say 1850s, though it might have
been a decade or two later) Cayley presented a paper to the Royal
Academy in which he says (and I paraphrase), "Well, it seems that p
and q are pretty well established now, so that's what I'm going to
use."

So, now the question is why Hamilton chose the letter "varpi" for
momentum. This variant of pi was fairly common in the mathematical
literature of the day, so there may be no special explanation. For
some further detective work, see:

10) Hamilton: two mysteries solved,
http://groups.google.com/group/sci.physics/browse_thread/thread/d1b7b4a998682bbb/3a868ae8218a4bca#3a868ae8218a4bca

Also see equation 12 in this paper for one of the first uses of "varpi"
to mean momentum:

11) William Rowan Hamilton, Second essay on a general method in dynamics,
ed. David R. Wilkins, available at
http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Dynamics/SecEssay.pdf

He doesn't say why he chose this letter - it may have been completely random!

Before I turn to higher-dimensional algebra, maybe this is a good time to
mention a paper related to the octonions:

12) Jakob Palmkvist, A realization of the Lie algebra associated to a
Kantor triple system, available as math.RA/0504544.

In "week193" I mentioned how 3-graded Lie algebras come from "Jordan
triple systems", and vaguely hinted that 5-graded Lie algebras come
from "Kantor triple systems". I explained how the exceptional Lie algebra
E8 gets to be 5-graded, but I didn't really say anything about Kantor
triple systems because my understanding of them was so poor. This paper
by Palmkvist explains them very clearly! And even better, he shows how the
"magic square" Lie algebras F4, E6, E7, and E8 can be systematically obtained
from the octonions, bioctonions, quateroctonions and octooctonions by means
of Kantor triple systems.

Now for some mathematical physics that touches on higher-dimensional
algebra. If you still don't get why topological field theory and
n-categories are so cool, read this thesis:

13) Bruce H. Bartlett, Categorical aspects of topological quantum field
theories, M.Sc. Thesis, Utrecht University, 2005. Available as
math.QA/0512103.

It's a great explanation of the big picture! I can't wait to see what
Bartlett does for his Ph.D..

If you're a bit deeper into this stuff, you'll enjoy this:

13) Aaron D. Lauda and Hendryk Pfeiffer, Open-closed strings: two-dimensional
extended TQFTs and Frobenius algebras, available as math.AT/0510664.

This paper gives a purely algebraic description of the topology of
open and closed strings, making precise and proving some famous guesses
due to Moore and Segal, which can be seen here:

14) Greg Moore, Lectures on branes, K-theory and RR charges,
Clay Math Institute Lecture Notes (2002), available at
http://www.physics.rutgers.edu/~gmoore/clay1/clay1.html

Lauda and Pfeiffer's paper makes heavy use of Frobenius algebras, developing
more deeply some of the themes I mentioned in "week174". In a related
piece of work, Lauda has figured out how to *categorify* the concept of a
Frobenius algebra, and has applied this to 3d topology:

15) Aaron Lauda, Frobenius algebras and ambidextrous adjunctions,
available as math.CT/0502550.

Aaron Lauda, Frobenius algebras and planar open string topological field
theories, math.QA/0508349.

The basic idea behind all this work is a "periodic table" of categorified
Frobenius algebras, which are related to topology in different dimensions.
For example, in "week174" I explained how Frobenius algebras formalized the
idea of paint drips on a sheet of rubber. As you move your gaze down a
sheet of rubber covered with drips of point, you'll notice that drips can
merge:


\ \ / /
\ \ / /
\ \ / /
\ \ / /
\ \_/ /
\ /
| |
| |
| |
| |
| |

but also split:

| |
| |
| |
| |
| |
/ _ \
/ / \ \
/ / \ \
/ / \ \
/ / \ \
/ / \ \

In addition, drips can start:

_
| |
| |
| |
| |
| |
| |
| |
| |
| |

but also end:


| |
| |
| |
| |
| |
| |
| |
| |
|_|

In a Frobenius algebra, these four pictures correspond to four operations
called "multiplication" (merging), "comultiplication" (splitting), the
"unit" (starting) and the "counit" (ending). Moreover, these operations
satisfy precisely the relations that you can prove by warping the piece
of rubber and seeing how the pictures change. For example, there's the
associative law:

\ \ / / / / \ \ \ \ / /
\ \ / / / / \ \ \ \ / /
\ \/ / / / \ \ \ \/ /
\ / / / \ \ \ /
\ \ / / \ \ / /
\ \_/ / \ \_/ /
\ / \ /
| | | |
| | | |
| | = | |
| | | |
| | | |
| | | |
| | | |
| | | |

The idea here is that if you draw the picture on the left-hand side on
a sheet of rubber, you can warp the rubber until it looks like the
right-hand side! There's also the "coassociative law", which is
just an upside-down version of the above picture. But the most
interesting laws are the "I = N" equation:

\ \ / / | | | |
\ \ / / | | | |
\ \_/ / | | | |
\ / | \ | |
| | | \ | |
| | | |\ \ | |
| | | | \ \ | |
| | | | \ \ | |
| | = | | \ \ | |
| | | | \ \ | |
| | | | \ \| |
| | | | \ |
/ _ \ | | \ |
/ / \ \ | | | |
/ / \ \ | | | |
/ / \ \ | | | |

and its mirror-image version.

So, the concept of Frobenius algebra captures the topology of regions
in the plane! Aaron Lauda makes this fact into a precise theorem in
his paper on planar open string field theories, and then generalizes it
to consider "categorified" Frobenius algebras where the above equations
are replaced by isomorphisms, which describe the *process* of warping the
sheet of rubber until the left side looks like the right. You should look
at his paper even if you don't understand the math, because it's full of
cool pictures.

Lauda and Pfeiffer's paper goes still further, by considering these paint
stripes as "open strings", not living in the plane anymore, but zipping
around in some spacetime of high dimension, where they might as well be
abstract 2-manifolds with corners. Following Moore and Segal, they also
bring "closed strings" into the game, which form a Frobenius algebra of
their own, where the multiplication looks like an upside-down pair of pants:

O O
\ \ / /
\ \ / /
\ /
| |
| |
| |
| |
| |
O

These topological closed strings are the subject of Joachim Kock's
book mentioned in "week202"; they correspond to *commutative*
Frobenius algebras. The fun new stuff comes from letting the open
strings and closed strings interact.

You can read more about Lauda and Pfeiffer's work at Urs Schreiber's
blog:

15) Urs Schreiber, Lauda and Pfeiffer on open-closed topological strings,
http://golem.ph.utexas.edu/string/archives/000680.html

In fact, I recommend Schreiber's blog quite generally to anyone interested
in higher categories and/or the math of string theory!

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

Quote of the Week:

Here's how you do it:
First you're obtuse,
Then you intuit,
Then you deduce!
- Garrison Keillor

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

Consider that the mass of the entire
observable universe, if it could be contained in a ball with the density
of an atomic nucleus, would be in a ball with a radius only a little
larger than the orbit of Mars.

Oz
Now, what would be the s/c radius of a black hole with the mass of the
observable universe.

About the same as the "radius of the observable universe". This has
been discussed here before.

John Baez <baez@math.removethis.ucr.andthis.edu> writes

18) Ring around a galaxy, HubbleSite News Archive, May 6, 1999,
http://hubblesite.org/newscenter/newsdesk/archive/releases/1999/16/image/a

This is surely the sort of object one would want (albeit at a better
orientation) to use to checkout galactic rotation curves.

Has anyone attempted to do this on ringed galaxies?

John Baez <baez@math.removethis.ucr.andthis.edu> writes

18) Ring around a galaxy, HubbleSite News Archive, May 6, 1999,
http://hubblesite.org/newscenter/newsdesk/archive/releases/1999/16/image/a

This is surely the sort of object one would want (albeit at a better
orientation) to use to checkout galactic rotation curves.

Has anyone attempted to do this on ringed galaxies?