Saturday, July 18, 2015

The fallacy of 'billions of billions', or: Why popular arguments that aliens must exist are bogus.

The universe is an awfully big place.  Granted, most of it is empty space.  But within that empty space, there are trillions of stars.  Maybe more.  At least some of those stars have planets around them, and some of those planets are the kind that could give rise to life.  That's still hundreds of billions (at least) of planets that can support life out there.  And so, the popular argument goes, even if the odds of life arising on another planet are very small, there are so many planets that it is bound to happen.  Thus, there almost certainly exist extraterrestrial life forms.  It isn't a matter of if, but of when we find them.

Here's a video of Dr. Carl Sagan presenting a more sophisticated version of this (with actual numbers) to estimate the number of inhabited planets in our galaxy.  Or try this worksheet on the Drake Equation on the BBC website.

It's a common argument.  And it sounds pretty convincing.  If you keep trying over and over, even though something is unlikely, eventually you will succeed.

It's common and convincing, but it's also fallacious.  Here's the problem: How many times do we have to try before we're guaranteed to succeed?

The mathematical answer is infinitely many times.

But that's to guarantee we succeed, with 100% probability.  So a better question might be: what happens to the probability of success as we keep trying?

Let the probability of a success be very low, set to $10^{-X}$, where $X$ is some large number.  This makes $10^{-X}$ a very small number.  Then let the number trials be $10^{Y}$, where $Y$ is some large number.  This makes $10^{Y}$ a very large number.  Now we define a quantity $P_0$, which is the probability of never succeeding, even after $10^Y$ trials.  (If you can't see my math, check your browser's plugin settings)


Assuming whether we succeed or not on a given trial is a simple coin flip with probability $10^{-X}$ of success, then the probability of failure in a single trial is $(1-10^{-X})$.  The probability of never succeeding after $10^{Y}$ trials is just the product
$$P_0 = \left(1-\frac{1}{10^{X}}\right)^{10^Y}.$$

We have said that $X$ is large.  Maybe you remember from algebra learning the formula for continuously compounded interest, where you ended up with an exponential, like so:
$$e^{x} = \lim_{n\rightarrow \infty} \left(1 + \frac{x}{n}\right)^n.$$

Well, in our case, if $X$ is large, then $10^{X}$  is really large, and
$$\left(1 + \frac{-1}{10^{X}}\right)^{10^X} \approx e^{-1} \approx 0.36788$$

If we re-write our expression for $P_0$, then, we find
$$P_0 = \left(1-\frac{1}{10^{X}}\right)^{10^{X + Y-X}} = \left[\left(1+\frac{-1}{10^{X}} \right)^{10^X}\right]^{10^{Y-X}} \approx = \left[e^{-1}\right]^{10^{Y-X}} = e^{-10^{Y-X}}.$$

Now, $e^{-1} \approx 0.36788$ is less than one, so squaring it or tripling it will make it even smaller.  However, taking the square root of it will make it larger.  The resolution comes down to: how does $X$ compare to $Y$?

Consider a simple case, where $X=Y$.  Then $10^{Y-X} = 10^0 = 1.$  So $P_0=e^{-1} = 0.36788.$  That is, there is only about a 37% chance of there being no successes, or in other words, there is a 63% chance of a success happening at some point.  It's not a guarantee, but it's more likely than not.

Now suppose that $Y = X+1$.  This means that we do ten times as many trials as our inverse probability; if the probability is a 1/10, do 100 trials, if the probability is 1/2, do 20 trials, etc.  Then $10^{Y-X} = 10^{1}= 10$, so $P = e^{-10} = 0.0000454$.  That is, the probability of success is 99.995%.  As we increase $Y$, this probability gets even closer to 100%.  Success is all-but guaranteed.

However, now suppose that $Y = X-1$.  This means that we only do a tenth as many as the inverse probability; if the probability is 1/10, do 1 trial.  If the probability is 1/20, do 2 trials, etc.  Then $10^{Y-X} = 10^{-1} = 0.1$, so $P_0 = e^{-0.1} = 0.9048.$  That is, the probability of success is down to a measly 9.516%.  As we increase $X$, this number gets even closer to 0%.

As we can see, our confidence of success depends drastically on the value of $Y-X$.  Even slight differences here can mean huge changes in the probability of success, $P_{\geq1} = 1-P_0$.

Simple graph showing the steep rise from 0 to 1 in the probability of success.


What this comes down to is whether $X$ is greater than or less than $Y$.  Put differently, does the probability of a single success compare to the number of trials?  Or put in terms of aliens, is the number of planets out there that can give rise to life close to the inverse of the probability of life actually arising?

And the answer is: no one knows!

We do not know how many planets there are.  If we estimate this as $N_p = 10^Y$, then $Y$ might be off by 2 or 3 in either direction.  There might be a thousand times as many as we think now, or there might be only a hundredth of our current guess.  As we just saw, for a fixed $X$, changing $Y$ even by 1 can drastically affect our confidence of extraterrestrial life existing.

Way more crucially, we have no idea how likely it is for life to occur on a planet that can give rise to life.  Think about this.  We have only ever observed life arising on a planet once.  This means that we don't have a very good definition of a planet where life can arise (see above), but it also means that we have a single data point upon which to base a probability.  If I were a pollster, and I went out on the street and asked a single person who they were voting for, and from that concluded that 54\% of voters supported the candidate, you would rightly question my methodology.  If we estimate this probability of life arising on a planet as $p_L = 10^{-X}$, then we don't even know what $X$ is.

Since we do not know what $X$ is, then we don't know what $P_{\geq 1}$ is.  This is a simple model, but it makes its point: Even small differences between $Y$ and $X$ can lead to very different predictions.

Consider again what it would mean for $Y=X-1$.  Take $10^Y$ to be the total number of planets what ever exist or will exist in our universe's lifetime.  Take $10^{-X}$ to be the probability of life ever arising on any given planet in our universe other than our own.  Then if $Y = X-1$, as above, we have $P_{\geq1} \approx$ 10% as the probability of extraterrestrial life ever arising in this universe.  This means that we'd need roughly 10 other universes just like our own before we can be back at roughly 63% probability of life arising again.

The popular statement that the universe is so big that there must be life in it somewhere is a false one.  The universe is quite big, but the probability of life arising can also be so small as to negate this bigness, and we have no way to know if this is the case or not.

The universe can still be as big as it is, and yet still not be big enough for life to arise anywhere else within it.

Saturday, March 21, 2015

Faeries in Phase Space

A few weeks ago, Professor Ben Zuckerman spoke at my school. He is one of the editors of the popular book "Extraterrestrials: Where Are They?" which explores in part the Fermi paradox: Why haven't extra-terrestrials tried to contact us yet? While he gave two lectures, I only attended one, where he went over many of the ideas in his book, explaining the rarity of technological life and the improbability of us ever making "contact".

I went to the colloquium talk rather interested. In honesty, I kind of misunderstood the intention of the talk (I thought he was going to be arguing against the existence of extraterrestrial life), but I was not disappointed. There were a lot of interesting ideas brought up about how to make contact or about what sorts of development projects we should pursue. And while I think some of them were really bad, they were thought-provoking. (Dr. Zuckerman of course recognizes the flaws of these, saying they are just stage 1 prototype designs).

The talk was very well done and I won't touch on it too much. What I really wanted to address was an audience question asked by one of the professors at my university. First let me provide some background.

Friday, November 7, 2014

How To Read the Voynich Manuscript

In case you aren't familiar with it the Voynich Manuscript (pictured at left) is currently one of the bigger linguistic mysteries out there .  It is a set of some 240 hand-illuminated pages, bound in codex form, making what appears to be a reference work on such topics as herbalism, biology, and astronomy.  Many of the illustrations are of plants and flowers that do not actually exist or cannot be precisely identified.  Most puzzling is the text, which is written in an unknown and undecipherable script that bears no relation to any known language or script.  You can see high-quality scans of the book here, courtesy of the Yale Library.

It is believed that the manuscript is a pharmacopoiea, as it bears some similarities to other such works.  However, much of it is puzzling, and incomprehensible.  Some scholars have proposed the manuscript to be a fake, one of a number of herbals made in the Middle Ages by alchemists and charlatans to impress simple people with the possessor's supposed knowledge.  The text is gibberish, mere squiggles on a page, meant to look like writing and yet containing no message.  That's one proposal.

Yet, the script looks intentional.  The same letters are repeated, and even specific ligatures are discernible.  The letters are repeated in such a way that shows consistency, as though the author were writing in an actual script, and not merely scribbling.

There are all kinds of hypothesis about how and why the manuscript was authored.  The most plausible is probably that the text is an invented script meant to write an East Asian tonal language.  Other theories are that it is a secret script or language invented by the author to hide his writing, or that the script is a code, containing information in some secondary feature of the words.

Those are the best theories.

But I want to propose a crazy theory, and a way to test it.

Monday, September 1, 2014

And Ye Shall Likewise


In days past and in lands forgotten - certainly far before either you or I were born - there in that far-away land stood a great kingdom.  And that kingdom was ruled, of course, by the King, a man of great honor and of great love for his people.  This King was a wise monarch, who dealt with his subjects fairly; and so there was peace in the land, prosperity in the markets, and the people there felt safe to leave their doors unlocked at night.

While the castle slept, the King would disguise himself as a pauper and sneak through passages unknown to his guards to the town beyond the castle walls.  He would take with him a wallet of gold coins out of his treasury, and so equipped he would wander the streets of his capital looking for those in need so that, in secret, he might give them comfort.  Orphans, widows, beggars; he would visit each in turn, under cover of night, and when he left they would discover the King's gift.

And it happened one night, as the King was about this business, that his attendants in court learned of their King's absence.  They learned that the King had left the castle, and left it empty, and that now it was theirs.  They learned that the King was outside of its walls and unable to stop them, unable to enforce his reign.  They learned that they could do whatever they wanted.

Now they could be kings and queens.

Monday, June 30, 2014

The Extent to Which I Could Change My Mind


It pays sometimes to ask yourself the question, what would change my mind?  To what extent could my mind be changed?  What alternatives would I consider?

Asking this kind of question is an important part of basic mental hygiene.  After all, if you never consider the possibility of being wrong, then for all you know you are wrong.  If you never consider the possibility of truth in other systems, then for all you know they are true.

This is a personal post, where I'm going to talk about my religious beliefs.  That's not what I normally do on this blog, but it's my blog and I'll do what I want.  I should mention, this post was inspired in part by a good friend of mine, Nathaniel Givens, in his post here on Times and Seasons.

Monday, June 23, 2014

On the Berenstein Bears Switcheroo


Two years ago, I wrote a post about one of the icons of my childhood, the Berenstein Bears.  Except, as I learned, they aren't called the Berenstein Bears.  As it turns out, they're the Berenstain Bears.

BerenstAin.  With an "A".

My mind was blown.  I had very distinct memories of the bears.  I grew up reading their books and watching them on TV in school, and remember how it used to be spelled.  I tried to figure out when the name had changed.

Wednesday, April 30, 2014

Time Travel Doesn't Work That Way


Tales of characters finding themselves hundreds of years in the past or the future are as old as the human imagination.  In recent years, since the codification of special relativity, this idea has taken a slightly more scientific bent, time travel now being the domain of crazed scientists, as opposed to fairy tricksters.

There are basically two ways to talk about time travel.  One is the literary, magical kind of time travel in stories like Rip Van Winkle, Harry Potter, Back to the Future, or Terminator; this is how time travel is almost always treated in popular culture.  The other is the scientific, relativistic approach, found in stories such as... well, none of them that I know of, because it's too difficult for liberal arts majors to understand, and doesn't make for as fun of a story.

compliments SMBC
I don't necessarily have a problem with the convention of literary time travel.  I mean, it's a story.  Even though I've written before about how some of the ideas that appear in fictional time travel don't make sense, such as changing the past or alternative time lines, the point of a story is to entertain, and not necessarily educate about the forefront of theoretical physics.  Plus, I frequently read stories about dragons and wizards and magic swords.

Nevertheless, in this post I want to offer an alternative explanation, of how time travel would work physically within the framework of General Relativity.

Special Relativity came first, as a theory that united space and time in to a single 4-dimensional entity called spacetime.  The "special" here means that it was later realized to be a special case of a more general theory -- the General Theory of Relativity.   More on that later.  The "relativity" is the more intriguing part.  This name was given to the theory because of how it treated distances in space and intervals in time.  Namely, that they could be mixed together, depending on the speed of the observer.  A scientist riding a spaceship and a scientist standing on an asteroid will get two different measurements of the length of the asteroid.  If the scientist on the asteroid drops a rock, he and the spaceship scientist will measure two different amounts of time it takes the rock to fall.

The lynch pin in Special Relativity is the experimental observation that the speed of light is independent of how fast you move relative to the light source.  What does this mean?

Suppose you're in a giant warehouse.  You're standing still.  In the center of the warehouse is one of those machines that shoots tennis balls.  Using some sort of experimental apparatus, you can measure the speed of the tennis balls.  Say you measure them to have a speed of 20 mph.  Then you start walking towards the machine, at a speed of 2mph.  Your measurement apparatus is not very smart, and doesn't know that you're moving.  In fact, as far as it can tell, all that happened is the tennis balls sped up; now instead of moving at 20 mph, they move at 20 + 2 = 22 mph.  By moving towards something, speeds add!  If you walked away from the  machine at the same speed, then your apparatus would measure 20 - 2 = 18 mph.  The speeds subtract!  This makes sense, intuitively.  When you're driving, the passing cars seem to move a lot faster than the trees on the side of the road.  The important key here is, the speed you measure for the objects depends upon your own speed.

This is called the principle of Galilean Relativity, after Galileo Galilei.  The speeds of objects add relative to other objects.

Light does not work that way.

In the center of the warehouse, there is also a light bulb.  Using a different apparatus, you can measure the speed of light.  Standing still, you measure the speed of light to be 670,616,629 mph.  Light moves pretty fast, as it happens.  You start running at the light bulb, but you still measure the speed to be 670,6616,629 mph.  Maybe you're not running fast enough for the apparatus to notice the change?  So you get in a car and rev the engine up to 200 mph at the light bulb, but still measure 670,616,629 mph.  You get on a high speed, magnetically-hovering bullet train and travel 10,000 mph toward the light bulb, and you still measure 670,616,629 mph.  It never goes up.  Dejected, you turn the train around, and to your continued frustration, even when going 10,000 mph the other way the speed of light from the light bulb is still 670,616,629 mph.

The speed you measure for light does not depend upon your own speed.

When scientists first discovered this, there was a lot of debate trying to figure it out.  A lot of it had to do with "ether", a fictitious fluid through which photons were supposed to move, and a dragging of the ether relative to moving bodies that affected the speed of the interior light beams.  It was confusing stuff.  Then Albert Einstein had an insight.  At first glance, if you didn't know it, it was one of the most ridiculous things you could propose, but it turned out to be real genius.

Einstein decided that the speed of light wasn't the speed of an actual object, but some kind of fundamental constant that was always the same in every frame of reference.  He called this constant c, the speed of light.  But since speed has units of distance per time, this gave c as a fundamental scaling factor between distances and time intervals.  The result was a unification of the two in to a single geometric entity called spacetime.

It was important that spacetime was geometric.  This meant that the same concepts that are normally used in discussion of space -- such as distances, directions, axes, speeds -- could be applied at a higher level to talk about movement in this new four dimensional spacetime.  Objects were given what is called a 4-velocity, measuring their rate of movement in spacetime.  The 4-velocity of every object (at least all those you've ever seen) has a constant magnitude of c.  Now, mind you, that's the 4-velocity, not the normal velocity.  When you see an object "standing still", it is actually moving forward in time at the speed of light.  When the object "speeds up" or "slows down" (from  3D point of view), it is actually only changing direction in spacetime; now it is moving partially forward in time and partially forward in space.

Make sure to keep these words straight: "speed" is an object's rate of movement in space; "velocity" is an object's rate of movement and the direction of its movement; "four-speed" is an object's rate of movement in spacetime, and is always c; "four-velocity" is an object's movement and the direction of its movement in space and in time; the magnitude of the four-velocity is the four-speed.

The car turns from N to NW.
It's speed stays the same,
but its velocity changes direction
Consider a car driving North at 100 mph.  It then comes to a bend at the road, where the road now moves true North-West, exactly midway between North and West.  The driver, looking at the speedometer, measures his speed to still be 100 mph.  However, the car is not moving North as quickly as it once was, as part of its movement is now directed to the West; its speed to the North is in fact now only 70 mph (approximately).  Contrariwise, the car used to not be moving Westward at all; but, since changing direction, the car is now moving to the West with a definite speed.

This is a similar situation to an object in spacetime.  When it is "stationary", the direction of its movement is strictly temporal, with four-speed c; just like the car going North at 100 mph, which we do not perceive as moving West.  When the object accelerates in space, it actually only changes direction in spacetime, keeping the same four-speed of c; much like our driver turning the car to go partially West, but still moving 100 mph.

You can see how this would put limits on the spatial velocity of objects.  Because the four-speeds of objects in spacetime are always c, and because "acceleration" only changes direction in spacetime, it means the largest speed that you could ever give to a real object is c.  Nothing can move faster than the speed of light.

The car is moving West a quickly as it can
Think of the car now turning to point true West, but still moving 100 mph.  There is no amount of turning that can make the driver appear to be moving faster westward; the only option would be to increase the speed of the car from 100 mph to say 200 mph.  It's the same thing in spacetime; once you're moving with speed c, no amount of turning (i.e. spatial acceleration) can increase your four-speed; you'd need to increase your 4-velocity itself, and that is impossible.

Now, this is not a true analogy.  If it were, then scientists would have discovered spacetime a long time ago.  The difference is a geometric one, because the geometry of spacetime is not quite the same as the geometry of space.  In our car example, it is possible to turn the car so that it only moves West, and not North.  In spacetime, it is not possible to turn an object so that it only moves in space and not in time.

To see this, remember that nothing moves faster than light.  But light definitely has spatial movement; otherwise, the universe would be pitch black.  Light also definitely moves forward in time; otherwise, lightbulbs wouldn't do anything and there'd be no way to light up a dark room.

This then suggests a method of visualizing spacetime.  We make the following diagram, and draw what is called a light cone.  A point on this diagram is called an "event"; just as normal events like birthday parties or weddings have locations and times, so too do spacetime events.  These tell you when and where.  At the center of the diagram, where the time and space axes cross, we are going to have a physicist stand.  The physicist has a stop watch and a lightbulb.  At the same time he hits the stopwatch and turns on the lightbulb, and watches as light shoots out from the bulb.  It moves with speed c, and gets farther away from the physicist with each passing second.  The lines that light follows in the spacetime diagram make the light cone.
How else would you know he's a physicist,
without mad scientist hair and a lab coat?
Notice that the light beams move at the speed of light, so that c = ∆x/∆t, or ∆x = c∆t, at all points in the light's trajectory.  If we consider the length ∆s of the hypotenuse of the triangle made by light's movement, based on the Pythagorean Theorem, we might be tempted to say
$$\Delta s^2 = \Delta x^2 + c^2\Delta t^2.$$
However, what makes spacetime different from normal geometry is that the time coordinate has a negative sign, meaning the Pythagorean Theorem in spacetime looks like this
$$\Delta s^2 = \Delta x^2 - c^2\Delta t^2.$$
The negative sign is important, and is what gives spacetime most of its interesting properties.

If you are standing still, then the path you follow in spacetime is a straight line parallel to the time axis; i.e., you only move forward in time.  Your four-velocity points straight up.  If you begin walking, then you rotate your four-velocity, so that it points more along the x axis [see note below about reference frames].  Moving faster, you go a larger distance in a similar amount of time, and so your four-velocity points further along the x axis.  The limit is when you are moving with a speed of c; this is as fast as you can ever go, but here we see your four-velocity only makes a 45 degree angle with the space and time axes.

This puts limits not only on our ability of spatial movement, but also on our ability to exploit the connection between space and time.  If time is just a direction, after all, why not just walk to the past, or to the future, as surely as we walk North or West?

As  you can see in the diagram, the process of, say, instantaneous teleportation is available; just move only along the x axis as far as you like without moving on the t axis.  However, doing that would put you outside of the light cone, which requires moving faster than the speed of light, which is impossible.

Also, the diagram makes the process of time travel available.  All it requires is "turning around" completely in spacetime, so that your four-velocity points in the opposite direction.  However, this is also impossible, for at some point in the process of turning around, you must rotate your four-velocity through 45 degrees, which makes your speed faster than light, and is thus forbidden.  So even though special relativity illustrates how time travel would work, special relativity also forbids time travel to work.

The problem is that our four-velocity has to stay inside of the light cone.  This is a fundamental fact of nature; we can never move outside of a light cone.  A lesser man would give up at this point, but not us.  Rather than admit defeat, we will simply change the light cones, and we will do this with general relativity.

In special relativity, the path we would like to follow requires us to leave the light cones.

 But what if we just tipped them all over, in just such a way that our backwards-in-time path was always inside the cones?

General relativity introduces the possibility of spacetime curvature; most importantly, the possibility of space-time mixing.  When space and time are mixed in very extreme cases, a spatial direction like North becomes equivalent to a time direction; by walking North, you actually move forward or backward in time.

These sorts of spacetimes are easy to construct, mathematically.  Just add a term that mixes space and time in to the equations in general relativity.  Physically, however, these spacetimes are impossible to construct.

Of the proposal for time travel metrics, most of them succumb to the principle of garbage in/garbage out.  The "garbage out" in this case would be the possibility of time travel to the past.  The "garbage in" is the distribution of mass that you'd have to make in order to curve space in the right way.

For instance, the simplest metric allowing for time travel is van Stockum space, which was shown by Frank Tipler to allow for time travel.  However, van Stockum space describes an infinitely long rotating cylinder.  It is impossible to construct an infinitely long cylinder (obviously).

Another example are wormholes, connecting two temporally separated regions of spacetime.  These require negative mass, and there does not exist negative mass.

The Kerr spacetime, which describes space in the vicinity of a rotating massive object, allows for time travel.  However, time travel only occurs for the blackhole case, and only inside of the event horizon; a time traveler from the future could never reach the outside world and have any impact on the past.  These are just a few examples, but every other proposal has failed on similar grounds.

Time travel to the past, so far as well know, is physically prohibited.  Nonetheless, Einstein's relativity explains to us how it would work if there were such a thing.  Let us point out some facets.

Firstly, space and time are united in to a single geometric entity called spacetime.  While the geometry of spacetime bears subtle differences to the geometry of space, it is still a geometric entity.

A reasonable way to visualize world lines
Imagine, if you like, a tall transparent rectangle sitting on a table on your desk, and imagine embedded in this rectangle little lines of red, blue, or green, running from top to bottom in the rectangle, and sometimes moving laterally from side to side.  These lines are like the positions of objects inside of the universe.  The process of experiencing time is the process of running your finger from the bottom of the rectangle to the top.   The technical word for them, in terms of a spacetime diagram, is world lines.

The disturbing part of this picture is that the top of the rectangle is always there.  Even when your finger is half-way to the top, the positions of all the world lines are already there.  They don't need you to catch up to exist.  The future is written, even if we'll never know what it is until we get there.

If you're having trouble understanding why this implies world lines are fixed, hold a pen in your hand.  Look at the bottom of the pen.  Is the top of the pen still there?  It's the same thing here, but now "the top" is "the future".  You may say that you can change the top of the pen, maybe by squeezing the bottom, but when you change it, you are changing it in time.  If you think maybe the world lines might change, then keep in mind, time is a direction in this picture; the world lines definitely do change in that direction, which is moving from the bottom to the top of the rectangle.  There is no other "outside time" for them to change with respect to.

This is disturbing to humans because we like to imagine ourselves as having free wills (and are we free to imagine otherwise?), but there simply isn't any other way to look at things once you've acknowledged spacetime as valid... and there isn't really any way to understand the universe without spacetime being valid.

You shouldn't think of this view as denying you anything, like freewill or boundless possibilities.  If you really had them at all, then they must be compatible with this view; if they aren't, then you never had them to begin with, so you really haven't lost anything.  I'm not saying there isn't freewill; I'm saying if there is, then it's compatible with this view.

The green world line goes back in time
Any affect it has on the other lines, it had before it 'left' 
Not only is the future written.  The past is written as well.  This is where the physical picture of time travel contradicts that of contemporary popular culture most strongly.  Anyone traveling to the past must do so within spacetime - that is, within the rectangle of our visualization.  When they get there, they find themselves not in some new spacetime, but in the exact same spacetime, but now at a point farther to the bottom of the time axis.  Once more, the future is written; the world lines at the top of the rectangle go where they go, and we're just tracing them.  This means that, while we can certainly impact the past, we cannot alter the past.  Whatever a time traveler does in the past, the future must already know that he did it.  Step on all the butterflies you want; you already know how bad the future will be.

To answer all possible time travel paradoxes, again, turn to our visualization of the transparent rectangle with the world lines embedded as colored lines inside of it.  Lines can only exist that run from the bottom to the top of the cylinder (or, perhaps, make closed loops within it).  Lines that do are possible.  Lines that don't are impossible.  This resolves nearly every possible paradox.

Another important point to note is that the act of traveling backwards in time is not an "instantaneous" thing.  What would that even mean, anyway?  Rather, traveling backward in time is literally traveling, backward in time.  You have to rotate your four-velocity to point backward, and then move in spacetime.

This also does not back-trace your previous world line, putting you as yourself in high school.  Technically, from an outside perspective (the world lines are embedded in a rectangle sitting on your desk), you still are yourself in high school; if you're not experiencing yourself in high school right now, then blame entropy.

Rather, moving backwards in time traces a new world line that moves from top-to-bottom, and then at some point turns around and moves bottom-to-top again.  During this top-to-bottom process, it is possible to interact with other objects.  Someone traveling backwards in time is perfectly visible to anyone around them.  As you travel back in time, you will see a second copy of yourself walking backward towards you and finally merging with you.  This occurs when you switch from backward to forward traveling.  You will then see a second copy of yourself split off from you.  That copy will be doing all the things you did while time traveling, only in reverse order.

It would probably be a pretty horrifying experience, really.

How long is Red's beard when Blue enters?
One fortunate fact is that you cannot move backwards in time without also moving in some direction.  This is from the frame dragging effect mentioned earlier.  If it were otherwise (if you could travel backwards in time and stand still) then your conscious mind at two separate ages -- perhaps separated by years of experienced time -- would exist at the exact same time and in a brain in the exact same location.  Whose thoughts would you think?  If you grew a beard in your time traveling backwards, then how long would your beard appear to a non-backwards-traveling observer in the same room?  As long as when you left, or as long as when you arrive?  Or somewhere in between?  Fortunately, it is only possible to rotate four-velocities to point backwards by moving through a frame dragging spacetime, so these sorts of considerations are mostly meaningless.

This sort of thinking though -- world lines and four-velocities -- is how time travel "actually" works.  While time travel is impossible (currently, anyway), the picture from general relativity tells us that the only way for movement backwards in time to work is if it respects the geometric picture of spacetime.  The geometric picture of spacetime would make such things as changing the past, separate world lines, instantaneous "jumps", or stationary machines all impossible, and would instead imply a single, fixed, and deterministic sequence of events that may include reverse time travel, but is not changed by reverse time travel.

There are some fascinating stories to be told about the physical picture of time travel, to be sure.  Sadly, however, it is normally the literary picture that we see in films and books.  Hopefully this article has help clear up the differences, and explained the faults of the popular view.

NOTE:  In this post, I do not introduce reference frames, which is a point against me.  I intended to explain the geometry of spacetime, without dealing with the pesky issue of how things change depending on who is watching, so I left out that additional complication.  When I talk about four-velocities (as above), I am assuming all of this is measured by a stationary observer.  Just know that the simple picture I painted here is complicated by the fact that most observables in relativity change depending on who is doing the observation.