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John Baez
Works at Centre for Quantum Technologies
Attended Massachusetts Institute of Technology
Lives in Riverside, California
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John Baez

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An infinite corridor of universes

Einstein's equations for gravity have some amazing solutions.  Some describe things we see:  the Big Bang and black holes.  Others don't - like white holes, wormholes, and the infinite corridor of universes shown here. 

As far as we know, all real-world black holes were formed at some moment in time by collapsing matter.   But it's easier to find solutions of Einstein's equations that describe an eternal black hole whose shape doesn't change with time.   

A rotating eternal black hole is called a Kerr black hole, because this solution of Einstein's equation was first found by Roy Kerr in 1963.   However, he just found part of the solution - not the whole picture here!

You see, when you solve Einstein's equations, you get a world obeying the rules of general relativity.  But sometimes, if you're not careful, somebody else can find a bigger world that contains yours!    It's like you drew a map of the world but you forgot there was anything south of the equator.   A solution is called maximally extended if you can't make it any bigger. 

This picture shows the maximally extended Kerr solution.  It's a Penrose diagram, so moving up the page takes you forward in time, while moving to the right or left edge of the page takes you away from the black hole.  Light moves along diagonal lines.

It's a single world, but it has portions called Universe, Parallel Universe, Antiverse, and Parallel Antiverse.   Each of these is roughly like our universe, but with no Big Bang.  Each lasts forever: time is not drawn to scale.

Each universe, and each parallel universe, has a black hole in it - and also a white hole!   Each antiverse, and each parallel antiverse, has a black hole with negative mass, and also a white hole with negative mass.

Only a few of these universes and antiverses are shown here.   But there are infinitely many.  The pattern repeats forever as you continue to go up or down the picture - that is, forwards or backwards in time.  

There's also an infinite repeating sequence of black holes and white holes.    And there's more - you can see singularities drawn as wiggly lines.   But let's not worry about those yet.  There's too much to take in at once.

Let's just follow the blue curve as it goes up the page.   This describes a path you could take through space and time.

You could shoot out of a white hole at the very bottom of the picture and wind up in our universe. 

Then you could jump into the black hole. 

If you dodge the singularities, you could wind up in a new white hole!

And at this point, you have a choice.  Swerve right and you go into a new universe. Swerve left and you go into a new parallel universe.  They're different - but there's no big difference.  In this picture, you choose to enter the new universe. 

And so on!

It would be great fun if our universe were part of a grand infinite corridor of universes like this.   As far as we know, it's not.   I suspect the real universe will be even more amazing.  However, we will need much better science and technology to discover what's out there.  Right now most of us are stuck here on Earth, and we need to learn to live here.  That's a tough challenge too.

My picture is from Andrew Hamilton's wonderful website:

http://jilawww.colorado.edu/~ajsh/insidebh/penrose.html

I would like to tell you more about the Kerr black hole - but if I don't get around to it, also check out David Madore's page:

http://www.madore.org/~david/math/kerr.html

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+Steve Wenner Electric charge creates electromagnetic fields, and electromagnetic fields carry energy, momentum and force, and these things are sources of gravity, just like matter is. And on the other side of the coin, the geometry of spacetime affects how electromagnetic fields propagate. So they very much interact in the dance.
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Poncelet's Porism

If you can fit a triangle snugly between two circles, you can always slide the triangle around.  The triangle may have to change shape, but it stays snug!   All 3 corners keep touching the outside circle, and all 3 sides keep touching the inside circle.

That's really cool.  But even better, it also works for polygons with more than 3 sides!

This amazing fact is called Poncelet's Porism

A porism is like a theorem, but much cooler.  Poncelet was a French engineer and mathematician who wrote a famous book on 'projective geometry' in 1822. 

What's a porism, really? 

Well, Euclid is famous for his Elements, but he also wrote a more advanced book called Porisms.  Unfortunately that book is lost.  I hear that someone checked it out from the library of Alexandria and never returned it.   By now the overdue fee exceeds the annual GDP of Greece, so we'll never see that book again... and we'll never know exactly what Euclid meant by 'porism'.

Wikipedia starts by saying:

A porism is a mathematical proposition or corollary. In particular, the term porism has been used to refer to a direct result of a proof, analogous to how a corollary refers to a direct result of a theorem. In modern usage, a porism is a relation that holds for an infinite range of values but only if a certain condition is assumed, for example Steiner's porism.  [...]  Note that a proposition may not have been proven, so a porism may not be a theorem, or for that matter, it may not be true.

In short: nobody knows what a porism is, but people are willing to make stuff up.

Pappus of Alexandria managed to write down a few of Euclid's porisms around 400 AD, before the book got lost.  They are quite advanced facts about geometry.  Poncelet was inspired by Pappus, so when he proved his cool result, maybe he wanted to call it a porism too.  I don't know.

A slick modern proof of Poncelet's porism uses 'elliptic curves'.  Check out David Speyer's explanation:

https://sbseminar.wordpress.com/2007/07/16/poncelets-porism/

In case you're not a mathematician, beware!  An 'elliptic curve' is jargon for a surface shaped like a doughnut.  We just call them 'elliptic curves' to keep people like you confused.

For some truly amazing connections between Poncelet's Porism and other math problems, see this paper by J. L. King:

https://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/King609-628.pdf

An elementary proof of Poncelet's Porism is here:

http://user.math.uzh.ch/halbeisen/publications/pdf/poncelet.pdf

In math, 'elementary' means that we don't use fancy concepts.  It doesn't mean 'easy'. 

When I retire, I want to quit proving theorems, and prove a porism.

#geometry  
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+John Baez OK, having googled it again, maybe "famously" is a bit of an overstatement.

At the page http://math.colorado.edu/~jonathan.wise/etc/moduli/index.html Jonathan Wise says

Michael Artin supposedly said that to understand stacks you really only need to understand the moduli space of triangles.

Similarly this lecture by Kai Behrend: http://www.math.ubc.ca/~pooya/files/sms-Behrend.pdf has the quote

I like triangles anyway! Mike Artin himself is reputed to saying [sic] that if you study the stack of triangles, you can understand everything about stacks.

By the way, the first linked page has some nice videos of loops in the moduli space of triangles, showing why it is really a (n Artin) stack. Wise is one of those amazing people who gives me the impression that working with stacks is the most natural thing in the world.
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Understanding black holes

This is a diagram of a Schwarzschild black hole - a non-rotating, uncharged black hole that has been around forever. 

Real-world black holes are different.  They aren't eternal - they were formed by collapsing matter.  They're also rotating.  But the Schwarzschild black hole is simple: you can write down a formula for it.  So this is the one to start with, when you're studying black holes.

This is a Penrose diagram.  It shows time as going up, and just one dimension of space going across.  The key to Penrose diagrams is that light moves along diagonal lines.  In these diagrams the speed of light is 1.   So it moves one inch across for each inch it moves up - that is, forwards in time.

The whole universe outside the black hole is squashed to a diamond. The singularity is the wiggly line at top.   The blue curve is the trajectory of a cat falling into the black hole.  Since it's moving slower than light, this curve must move more up than across.  So, once it crosses the diagonal line called the horizon, it is doomed to hit the singularity. 

Indeed, anyone in the region called "Black Hole" will hit the singularity.   Notice: when you're in this region, the singularity is not in front of you!  It's in your future.  Trying to avoid it is like trying to avoid tomorrow.

But what is the diagonal line called the antihorizon?   If you start in our universe, there's no way to reach the antihorizon without going faster than light.     But we can imagine things crossing it from the other direction: entering from the left  and coming in  to our universe! 

The point is that while this picture of the Schwarzschild black hole is perfectly fine, we can imagine extending it and putting it inside a larger picture.   We say it's not maximally extended

The larger picture, the maximally extended one, describes a very strange world, where things can enter our universe through the antihorizon.   But that's another story, which deserves another picture.

If we stick with the diagram here, nothing can come out of the antihorizon, so it will look black.  In fact, to anyone in the "Universe" region, it will look like a black sphere.  And that's why a Schwarzschild black hole looks like a black sphere from outside!

The weird part is that this black sphere you see, the antihorizon, is different than the sphere you can fall into, namely the horizon.

If this seem confusing, join the club.   I think I finally understand it, but nobody ever told me this - at least, not in plain English - so it took me a long time.

What could be behind the antihorizon?   If you want to peek, try Andrew Hamilton's page on Penrose diagrams, where I got this picture:

http://jila.colorado.edu/~ajsh/insidebh/penrose.html
 
I wish that Wikipedia had a really nice Penrose diagram like this!  It's very important.  They have some more complicated ones, but the most basic important ones are not drawn very nicely.  You need to think about Penrose diagrams to understand black holes and the Big Bang!

Still, their article is worth reading:

https://en.wikipedia.org/wiki/Penrose_diagram

For more on the Schwarzschild black hole, read this:

https://en.wikipedia.org/wiki/Schwarzschild_metric

#physics  
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+James Salsman - I didn't notice. I think I know his work, but I forget why.
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Exploring black holes - with cats!

There should be a series of videos exploring black holes with cats. 

So far all we have is this gif made by +Dragana Biocanin.   A cat can orbit just above the photon sphere of a non-rotating black hole, moving at almost the speed of light.   It's impossible for a cat to orbit below the photon sphere.   As long as it's outside the event horizon it can accelerate upwards and escape the black hole's gravitational pull.   But if it crosses the event horizon, it's doomed! 

The event horizon is an imaginary surface in spacetime that's defined by this property: once a cat crosses this surface, it can't come back without going faster than light!   This property involves events in the future, so there's no guaranteed way for the cat to tell when it's crossing an event horizon.

For example, if two supermassive black holes were shooting toward our Solar System right now and collided in an hour, forming a black hole that swallowed the Earth, at some moment your cat would cross the event horizon.  That's the moment when, no matter how hard it tried, it could no longer escape.  But this moment could be happening right now, and your cat might not notice!   No alarm bells ring at this moment.

What happens inside the event horizon?

For a non-rotating black hole formed by the collapse of matter, the answer is pretty well understood - except at the 'singularity', where the laws of physics we know break down.   

Your cat will fall in, getting stretched ever thinner.    For a hypothetical non-rotating black hole with the mass of our Sun, once it crosses the event horizon it will hit the singularity in about 10 microseconds.  That's not much time!

In fact, all known black holes are heavier than our Sun.   If you double the mass of the black hole, you double the amount of time it takes to hit the singularity, and so on.  So, for a non-rotating black hole 100,000 times the mass of our Sun, it takes 1 second to hit the singularity after  crossing the horizon.

The biggest known black holes are about 30 billion times the mass of our Sun.  For a non-rotating black hole this big, it would take three and a half days for your cat to hit the singularity after it crosses the horizon!   You might want to send it in with some cat food.

But there's a catch.  Real-world black holes are always rotating!  This makes them much more complicated.  For starters, frame-dragging tends to pull you along with the black hole's rotation.

We began to see that yesterday when I showed you +Leo Stein's website about how photons orbit a black hole.  There's not just one photon sphere - there's a bunch!   

There's also a region called the ergosphere where frame-dragging becomes so strong that your cat can't stand still.   And Penrose discovered something interesting about this.

You can send a cat into the ergosphere with rockets strapped to its back.  When it shoots back out, it can carry angular momentum and energy out of the black hole!   It's a bit like how we use Jupiter to fling satellites to Pluto - except we're using the rotation rather than the motion of the black hole!  

So, we can in theory "mine" a rotating black hole, removing energy from it until it's not rotating.

Beneath the ergosphere lies the horizon.  Inside the horizon of a rotating black hole, things get even weirder.  More on that later, I hope.  But probably not with cats.

For now, try this:

https://en.wikipedia.org/wiki/Ergosphere

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thank you +John Baez will do.
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Just because someone's on crutches doesn't mean they're handicapped

Nomads kick ass.   James Dator explains:

The World Nomad Games concluded on Friday in what can only be described as the greatest week-long sporting event on the planet. The games, intended to showcase ethnic sports of Central Asia, featured things you have never heard of, athletes you’ll never learn about and sports that sound absolutely terrifying.

There were 16 sports with medals up for grabs. These are the ones that are the absolute wildest.

Cirit

This Turkish equestrian sport involves teams of riders chasing each other and throwing javelins at each other while on horseback. Yes, seriously.

Er Enish

It’s wrestling, except you’re on a horse. You win by pulling your opponent off their horse.

Kok-boru

There’s no delicate way to explain Kok-boru. It’s horseback basketball using a goat carcass. You win by tossing the dead goat into your opponent’s well. It comes from a tradition of beating up wolves that attacked your herd of sheep and throwing a dead wolf to your friends who went wolf hunting with you.

Mas-wrestling

In this form a wrestling, athletes fight over a stick. Each wrestler is given part of the stick to hold and are seated facing each other with their feet on a plank. Whoever gets the stick wins.

Salbuurun

A three-step hunting sport involving animals.  Competitions are held in the following disciplines:

1. Burkut saluu - hunting with golden eagles. Composition of the team - 6 people: 1 leader and 5 berkutchi (hunter with eagles).

2. Dalba oynotuu - falcon flying to the lure. Composition of the team - 6 people: 1 leader and 5 Kushchu (falconer).

3. Taigan jarysh - dog racing among breeds of greyhound. Composition of the team - 6 people: 1 leader and 5 owners of dogs.

Traditional Archery

This has to be the biggest misnomer of the World Nomad Games. They say “traditional,” but really they mean on horseback and also this.

(The picture of this woman here.  Who is she?)

James Dator explains more games here:

http://www.sbnation.com/2016/9/12/12888720/world-nomad-games-burning-horseriders-dead-goat-basketball-eagle-hunting-wow

There's more here:

https://www.washingtonpost.com/news/early-lead/wp/2016/09/14/first-ever-u-s-team-plays-rugby-on-horses-with-a-decapitated-goat-at-world-nomad-games/

Not for the squeamish!  However, excellent pictures of hunters with their eagles, horse riders, etc.
 
The World Nomad Games, Kyrgyzstan. 
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Just above the photon sphere

This gif shows what it's like to orbit a non-rotating black hole just above its photon sphere.

That's the imaginary sphere where you'd need to move at the speed of light  to maintain a circular orbit.    At the photon sphere, the horizon of the black hole looks like a perfectly straight line!

But since you can't move at the speed of light, this gif shows you orbiting slightly above the photon sphere, a bit slower than light. 

We cannot go to such a place - not yet, anyway.  The gravity would rip us to shreds if we tried.   But thanks to physics, we can figure out what it would be like to be there!   And that is a wonderful thing.

The red stuff drawn on the black hole is just to help you imagine your motion.  You would not really see that stuff. 

The light above the black hole is starlight - bent and discolored by your rapid motion and the gravitational field of the black hole.

This gif was made by Andrew Hamilton, an expert on black holes at the University of Colorado.  You can see a lot more explanations and movies on his webpage:

http://jila.colorado.edu/~ajsh/insidebh/schw.html

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Cool thanks. 
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Solar wind

This is the solar wind, the stream of particles coming from the Sun.  It was photographed by STEREO.  That's the Solar Terrestrial Relations Observatory, a pair of satellites we put into orbit around the Sun at the same distance as the Earth, back in 2006.  One  is ahead of the Earth, one is behind.  Together, they can make stereo movies of the Sun!

One interesting thing is that there's no sharp boundary between the 'outer atmosphere' of the Sun, called the corona, and the solar wind.  It's all just hot gas, after all!   STEREO has been studying how this gas leaves the corona and forms the solar wind.  This picture is a computer-enhanced movie of that process, taken near the Sun's edge.

What's the solar wind made of?   When you take hydrogen and helium and heat them up so much that the electrons get knocked off, you get a mix of electrons, hydrogen nuclei (protons), and helium nuclei (made of two protons and two neutrons).   So that's all it is.

The Sun's corona is very hot: about a million degrees Celsius.  That's hotter than the visible surface of the Sun!  Why does it get so hot?  When I last checked, this was still a bit mysterious.   But it has something to do with the Sun's powerful magnetic fields. 

When they're this hot, some electrons are moving fast enough to break free of the Sun's gravity.   Its escape velocity is 600 kilometers per second.  The protons and helium nuclei, being heavier but having the same average energy, move slower.  So, few of these reach escape velocity.

But with the negatively charged electrons leaving while the positively charged protons and helium nuclei stay behind, this means the corona builds up a positive charge!   So the electric field starts to push the protons and helium nuclei away, and some of them - the faster-moving ones - get thrown out too.  

Indeed, enough of these positively charged particles have to leave the Sun to balance out the electrons, or the Sun's electric charge would keep getting bigger.   It would eventually shoot out huge lightning bolts!  The solar wind deals with this problem in a less dramatic way - but sometimes it gets pretty dramatic.  Check out this proton storm:

http://www.spaceweather.com/images2014/08jan14/x1s2_anim.gif

When storms like this happen, the US government sends out warnings like this:

Space Weather Message Code: WATA50
Serial Number: 48
Issue Time: 2014 Jan 08 1214 UTC
WATCH: Geomagnetic Storm Category G3 Predicted
Highest Storm Level Predicted by Day:
Jan 08: None (Below G1) Jan 09: G3 (Strong) Jan 10: G3 (Strong)
THIS SUPERSEDES ANY/ALL PRIOR WATCHES IN EFFECT
Potential Impacts: Area of impact primarily poleward of 50 degrees geomagnetic latitude.
Induced Currents – Power system voltage irregularities possible, false alarms may be triggered on some protection devices.
Spacecraft – Systems may experience surface charging; increased drag on low Earth-orbit satellites and orientation problems may occur.
Navigation – Intermittent satellite navigation (GPS) problems, including loss-of-lock and increased range error may occur.
Radio – HF (high frequency) radio may be intermittent.
Aurora – Aurora may be seen as low as Pennsylvania to Iowa to Oregon.

The solar wind is really complicated, and I've just scratched the surface.  I love learning about stuff like this, surfing the web as I lie in bed sipping coffee in the morning.  Posting about it just helps organize my thoughts - when you try to explain something, you come up with more questions about it.

For more on space weather, visit this fun site:

http://www.spaceweather.com/

You can see space weather reports put out by the National Oceanic and Atmospheric Administration here:

http://www.swpc.noaa.gov/products/alerts-watches-and-warnings

For more on the solar wind, see:

https://en.wikipedia.org/wiki/Solar_wind

For more on STEREO, see:

https://en.wikipedia.org/wiki/STEREO

#physics   #astronomy  
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legendary "music of the spheres."
that is beautiful,
Thank you for clarifying.
+Garry Gust 
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Life on the Infinite Farm

This is a great book about infinity - for kids.   For example, there's a cow named Gracie with infinitely many legs.  She likes new shoes, but she wants to keep wearing all her old shoes.  What does she do?

Life on the Infinite Farm is by Richard Evan Schwartz, and it's free here:

https://www.math.brown.edu/~res/farm.pdf

Later it will be published on paper by the American Mathematical Society.  I really like turning the pages when I'm reading a book to a child.  Is that old-fashioned?  What do modern parents think?

Gracie's tale is just a retelling of the first Hilbert Hotel story.  There's a hotel with infinitely many rooms.  Unfortunately they're all full.  A guest walks in.  What do you do? 

You move the guest in room 1 to room 2, the guest in room 2 to room 3, and so on.  Now there's a room available!

The Hilbert Hotel stories were introduced by the great mathematician David Hilbert in a 1924 lecture, and popularized by George Gamow in his classic One Two Three... Infinity.   That book made a huge impression on me as a child: one of my first times I tasted the delights of mathematics.    

But that book is not good for children just learning to read.  Life on the Infinite Farm is.  And there's nothing that smells like "education" in this book.  It's just fun.

You can read more Hilbert Hotel stories here:

https://en.wikipedia.org/wiki/Hilbert's_paradox_of_the_Grand_Hotel

But it's probably more fun to read Gamow's One Two Three... Infinity.   He was an excellent astrophysicist who in 1942 figured out how the first elements were created - the theory of Big Bang nucleosynthesis.    He was also a coauthor of the famous Alpher-Bethe-Gamow paper on this topic, also known as the αβγ paper.    Alpher was a grad student of Gamow, and they added the famous nuclear physicist Hans Bethe as a coauthor just for fun - since 'Bethe' is pronounced like the Greek letter 'beta':

It seemed unfair to the Greek alphabet to have the article signed by Alpher and Gamow only, and so the name of Dr. Hans A. Bethe was inserted in preparing the manuscript for print. Dr. Bethe, who received a copy of the manuscript, did not object, and, as a matter of fact, was quite helpful in subsequent discussions. There was, however, a rumor that later, when the alpha, beta, gamma theory went temporarily on the rocks, Dr. Bethe seriously considered changing his name to Zacharias.

Gamow also had a real knack for explaining things in fun ways, with the help of charming pictures.   I don't do many advertisements for commercial products, but I will for this!  You can get his book for as little as $2.98 plus shipping:

https://www.amazon.com/One-Two-Three-Infinity-Speculations/dp/0486256642/

You should have read it by the time you were a teenager - but if you didn't, maybe it's not too late.

For more about Gamow, see:

https://en.wikipedia.org/wiki/αβγ_paper

#bigness
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+John Baez there is a poet hiding within you, so you forgive me easily.
Every mistake is a learning experiece, not that i wish for too many learning experiences.
but looked up Zaniness.. that was cool
..
comedy, comicality, comicalness, drollery, drollness, farcicality, funniness, humor, humorousness, jocoseness, jocosity, jocularity, ludicrousness, ridiculousness, wit, wittiness.
2. Foolish behavior:
absurdity, folly, foolery, foolishness, idiocy, insanity, lunacy, madness, nonsense, preposterousness, senselessness, silliness, tomfoolery.
I confirm you are the Poet, also.
and of course i fit all those descriptions.
I promise to be better behaved in the future.
I like your presence in my life. There is something about Math and Physics that always evaded me, so i keep pursuing those.
Your posts are tantalizing, and a dance of sorts ranging from dolphins to black Saturn, and a proton glistening like a blue diamond between the teeth of v. calipers, unafraid.
The last image i cannot get rid of, from my mind.
A very sweet day to you!
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Black hole versus white hole

Last time I showed you a Schwarzschild black hole... but not the whole hole.  

Besides the horizon, which is the imaginary surface that light can only go in, that picture had a mysterious "antihorizon", where light can only come out.    When you look at this black hole, what you actually see is the antihorizon.    The simplest thing is to assume no light is coming out of the antihorizon.  Then the black hole will look black.

But I didn't say what was behind the antihorizon!

In a real-world black hole there's no antihorizon, so all this is just for fun.  And even in the Schwarzschild black hole, you can never actually cross the antihorizon - unless you can go faster than light.  So there's no real need to say what's behind the antihorizon.    And we can just decree that no light comes out of it.

But inquiring minds want to know...  what could be behind the antihorizon?

This picture shows the answer.  This is the maximally extended Schwarzschild black hole - the biggest universe we can imagine, that contains this sort of black hole.

It's really weird.

It contains not only a black hole but also a white hole.  The wiggly lines are singularities.  Matter and light can only fall into the black hole from our universe... passing through the horizon and hitting the singularity at the top of the picture.   And they can only fall out of the white hole into our universe... shooting out of the singularity at the bottom of the picture and passing through the antihorizon.

If that weren't weird enough, there's also a parallel universe, just like ours.  

Someone from our universe and someone from the parallel universe can jump into the black hole, meet, say hi, then hit the singularity and die.    Fun!

But we can never go from our universe to the parallel universe.  :-(

Why not?   Remember, the only allowed paths for people going slower than light are paths that go more up the page than across the page - like the blue path in the picture.  To get from our universe to the parallel universe, a path would need to go more across than up.

If you could go faster than light for just a very short time, you could get from our universe to the parallel universe by zipping through the point in the very middle of the picture, where the horizon and antihorizon meet. 

Puzzle 1.  Suppose the parallel universe has stars in it more or less like ours.  You can't see it from our universe - but you could see it if you jumped into the black hole!  What would it look like?

Puzzle 2.   How would my story change if the "arrow of time" in the parallel universe pointed the other way from ours?  In other words, what if the future for them was at the bottom of the picture, rather than the top?

I should emphasize that we're playing games here, but they're games with rules.  We're not talking about the real world, but the math of this stuff is well-understood, so you can't just make stuff up.  Or you can, but it might be wrong.  These puzzles have right and wrong answers!

Unfortunately I haven't really explained things well enough, so you may need to guess  the answers instead of just figure them out.  For more info, try Andrew Hamilton's page, from which I took this picture:

http://jila.colorado.edu/~ajsh/insidebh/penrose.html

And for more, try this:

https://en.wikipedia.org/wiki/Schwarzschild_metric

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+David Chudzicki - No.  First, the concept of "closed timelike curve" has nothing to do with the arrow of time (the distinction between future and past).   The arrow of time lets us distinguish between "future-pointing" and "past-pointing" timelike vectors, but a closed timelike curve is simply a loop whose tangent vector is always timelike.

Second, this spacetime doesn't have any closed timelike curves.
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The mystical hexagram theorem

The picture explains this amazing result, which was discovered by Pascal in 1639, when he was only sixteen.

Take six points on an ellipse, called A,B,C,D,E,F.  Connect each point to the next by a line.

The red lines intersect in a point G.
The yellow lines intersect in a point H.
The blue lines intersect in a point K. 

And then the cool part:

The points G, H and K lie on a line!

I'm teaching a course on 'algebraic groups' starting on Thursday, so I need to review a bit of the history of projective geometry.   This result of Pascal, called the Hexagrammum Mysticum Theorem, was the first exciting theorem about projective geometry after the old work of Pappus.  So I'll mention it in my course!   But I don't really understand why it's true.  Do you know a nice explanation?

I'll start by reading this:

https://en.wikipedia.org/wiki/Pascal%27s_theorem

If you can manage to enable Java applets on your device - a task made ever harder by those worried for our safety - you should check out this:

http://www.cut-the-knot.org/Curriculum/Geometry/Pascal.shtml

You can move six points around a circle and see how things change.

#geometry
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+David Roberts - that's a nice one.  Maybe it is invariant under projective transformations, despite appearances, because if such a transform squashes one circle to an ellipse it does the same to all the rest, giving similar ellipses, so one could formulate it as "given an ellipse, draw 6 similar ellipses tangent to it and touching each other..."    To see if it's a theorem of complex projective geometry one would want to look for an analogue involving hyperbolae.
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Light moves around a rotating black hole

This gif by +Leo Stein shows a photon orbiting a black hole.  Since the black hole is rotating, the photon traces out a complicated path.  You can play around with the options here:

 https://duetosymmetry.com/tool/kerr-circular-photon-orbits/

If a black hole is not  rotating, light can only orbit it on circles that lie on a special sphere: the photon sphere

But if the black hole is rotating, photon orbits are more complicated!  They always lie on some sphere or other — but now there's a range of spheres of different radii  on which photons can move! 

The cool part is how a rotating massive object — a black hole, the Sun or even the Earth — warps spacetime in a way that tends to drag objects along with its rotation.  This is called frame-dragging.  

Frame-dragging was one of the last experimental predictions of general relativity to be verified, using a satellite called Gravity Probe B.    Frame-dragging was supposed to make a gyroscope precess a bit more.   This experiment was really hard.  It suffered massive delays and cost overruns.   When it was finally done, the results were not as conclusive as we'd like.   I believe in frame-dragging mainly because everything else about general relativity works great, and it's hard to make up a theory that differs in just this one prediction.

It's pretty bizarre that instead of following orbits that move in and out from the black hole - like ellipses, or something - photons can move only in orbits of constant radius, with a range of different possible radii  being allowed.   Leo Stein explains:

After you study the radial equation, you learn that the only bound photon trajectories — that is, orbits! — are those for which r=const in Boyer-Lindquist coordinates. This is why these photon orbits are sometimes called “circular” or “spherical.”

In the end, you see that for each angular momentum parameter a for the black hole, there is a one-parameter family of trajectories given by the radius r, which must be between the two limits

r₁(a) ≤ r ≤ r₂(a)

The innermost photon orbit is a prograde circle lying in the equatorial plane, and the outermost orbit is a retrograde circle lying in the equatorial plane.

Prograde means that this orbit goes around the same way the black hole is rotating; retrograde means it's moving in the opposite direction.

These orbits are all unstable.  Push the photon slightly inward and it will fall into the black hole.   Push it outward just a bit and it will fly away.  So, this stuff is mainly interesting for the math.  You won't actually find a lot of light orbiting a black hole.

For more of the math, see Leo Stein's website.  It's great!  But the most fun part is using some sliders to play with photon orbits.

For more on frame-dragging, see:

https://en.wikipedia.org/wiki/Frame-dragging

#physics  
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Noah Friedman's profile photoWillie Wong's profile photoMatt McIrvin's profile photoJohn Baez's profile photo
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+Matt McIrvin - that sounds vaguely familiar, but the Wikipedia article on frame-dragging doesn't give experimental evidence for it other than GPB:

https://en.wikipedia.org/wiki/Frame-dragging#Experimental_tests_of_frame-dragging

https://en.wikipedia.org/wiki/Frame-dragging#Astronomical_evidence

In the second section it says some effects could  give evidence.  I guess I need to look elsewhere...
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An even bigger particle accelerator?

This is Chen-Ning Yang.  He helped create Yang-Mills theory - the wonderful theory that describes all the forces in nature except gravity.    He helped find the Yang-Baxter equations, which describe what particles do when they move around on a thin sheet of matter, tracing out braids. 

He's one of China's top particle physicists... and he's come out against  building a new, bigger particle accelerator!   This is a big deal, because only China has the will to pay for the next machine.

In 2012, two months after the Large Hadron Collider (near Geneva) found the Higgs boson, a Chinese institute called for a bigger machine: the Circular Electron Positron Collider or CEPC.

This machine would be a ring 80 kilometers around.  It would collide electrons and positrons at an energy of 250 GeV, about twice what you need to make a Higgs.   It could make lots of Higgs bosons and study their properties.  It might find something new, too!  Of course that would be the hope.

It would cost $6 billion, and the plan was that China would pay for 70% of it.  Nobody knows who would pay for the rest.

On 4 September, Yang, in an article posted on the social media platform WeChat, says that China should not build a supercollider now. He is concerned about the huge cost and says the money would be better spent on pressing societal needs. In addition, he does not believe the science justifies the cost: The LHC confirmed the existence of the Higgs boson, he notes, but it has not discovered new particles or inconsistencies in the standard model of particle physics. The prospect of an even bigger collider succeeding where the LHC has failed is “a guess on top of a guess,” he writes. Yang argues that high-energy physicists should eschew big accelerator projects for now and start blazing trails in new experimental and theoretical approaches.

That same day, the director of the institute that wants to build the machine posted a rebuttal on WeChat.   I can't read it, because it's in Chinese:

http://chuansong.me/n/680325551051

It will be interesting to see how this plays out.  Personally I think we as a species need to focus on global warming and the Anthropocene: the way we're transforming the Earth. 

In the last 25 years, 10% of the world's remaining wilderness has disappeared.   Temperatures are rising at an ever-increasing rate.   If we keep it up, we'll melt Greenland and the Antarctic, eventually flooding all coastal cities.  Even now, weather patterns are changing, with big heat waves, floods and droughts becoming more common. 

Surviving the Anthropocene will require new math, new physics, new chemistry, new biology, new computer science, and new technology of many kinds.  Most of all, it will require new attitudes - new politics and economics.

One thing that won't  be required is new elementary particles.  I love fundamental physics.   But finding new particles can wait.  They'll still be here in a century or two.  Our civilization, and the natural world we love, may not.

What really matters here is not the money.  $6 billion is not much in the grand scheme of things.  For example, this fall California is voting on a $9 billion bond measure for its schools.    What really matters is where scientists put their energy

The quote is from Science magazine:

http://www.sciencemag.org/news/2016/09/debate-signals-cloudy-outlook-chinese-supercollider

The Earth's disappearing wilderness:

https://www.theguardian.com/environment/2016/sep/08/humans-have-destroyed-a-tenth-of-earths-wilderness-in-25-years-study

xkcd has a great graph of the Earth's temperature - check it out!  You'll learn a lot and have fun:

http://xkcd.com/1732/

#physics  
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I'm a mathematical physicist.
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  • Centre for Quantum Technologies
    Visiting Researcher, 2011 - present
  • U.C. Riverside
    Professor, 1989 - present
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I teach at U. C. Riverside and work on mathematical physics — which I interpret broadly as ‘math that could be of interest in physics, and physics that could be of interest in math’. I’ve spent a lot of time on quantum gravity and n-categories, but now I'm starting to work on more practical things, too.

Why? I keep realizing more and more that our little planet is in deep trouble! The deep secrets of math and physics are endlessly engrossing — but they can wait, and other things can’t.
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  • Massachusetts Institute of Technology
    Mathematics, 1982 - 1986
  • Princeton University
    Mathematics, 1979 - 1982
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