Notes for Nineteenth Lecture

Lecture 20, April 21, Richard Gott

• Read Chapter 3 of Prof. Gott's book.
• There will be the last observing session of the semester this evening, starting at 9 PM.
• Homework 6 is due on the last day of class, May 1.
```We're talking about time travel to the past today.  A space-time
diagram shows both space and time; think about it as giving a full
history of the path in space and time of whatever objects you are
looking at; a story-board, if you will.  Time travel means looping
back in time; the first requirement is whether you can represent the
whole thing in a space-time diagram in a self-consistent way, with one
past, one future, and perhaps some interesting twists in the middle.

Self consistent time travel stories have some interesting features. We
introduced the concept of a "Djinn" - a particle whose world line is
like a hula hoop - a circle with no ends (the name was introduced by
Igor Novikov).  An example was given with the 1980s movie "Somewhere
in Time", (in which a pocket watch that was given to the protagonist
by an old lady is brought back in time and given to the younger
version of this lady, who in turn grows old to give it to the
protagonist - thus this pocket watch, which has never been near a
watch factory, is a "djinn").  In special relativity, you're always
going forward in time.  But in general relativity, space-time can be
curved, so you can visit the past, at least in principle.

One way one might conceivably make a djinn particle is to use a
strange property of quantum mechanics, whereby a particle and its
corresponding anti-particle are created spontaneously out of the
vacuum.  This is possible for simple objects such as electrons; for
much more complex objects (such as pocket watches), it is *much* less
likely.

One of the standard questions when thinking about time travel to the past is,
"Can't you go back into the past and change the present"?  For example, if you
could go back in time and kill your grandmother as a young girl, then you
wouldn't exist.  There are two possible responses to this:
a).  The story *has* to be consistent; each event in space-time has only one
version.  In this case, it would be possible to travel back in time, but not
b).  Quantum mechanics interprets essentially every event in terms of
probabilities.  For example, if you have a Hydrogen atom with its electron in
the n=2 excited energy level, quantum mechanics gives a probability for that
electron to decay to its ground state at each particular moment in time.  One
way to interpret these probabilities is to say that each moment, the universe
splits into two parallel universes, one in which the electron decays to n=1,
one in which it doesn't.  So one possibility for time travel is that when you
go back in time, you get two parallel universes: one in which you kill your
grandmother, and one in which you don't.   This idea is called the
"Many Worlds Theory".

In special relativity, you can certainly visit the *future*.  If you
travel in a rocket ship at high speed, your time goes slower than for
your twin that stayed on Earth, and so you can move to the future much
faster than "normal."  Similarly, if you go to the vicinity of a black
hole, your time goes slower than the person far away.

You might think that if you could go faster than the speed of light,
you would see time reverse direction.   But as we've seen, there is
no way we can go faster than the speed of light.  So that won't
work.

However, in general relativity, the trick we will try to use is to use
curved-space time to take a short cut between two points to go faster
than light.  If we're clever, we can use this to travel back in time.

We'll use a cosmic string to give us curved space-time.  Think of a
cosmic string as a 1-dimensional analog to a black hole; essentially
infinitely thin, but stretched out.  A black hole is a point, of zero
dimensions.  A cosmic string is a long, stretched out, very massive
object, with a gravitational field which is not spherically symmetric
around a point (like a black hole), but cylindrically symmetric around
a line.  Cosmic strings have no ends, and therefore are either
infinite in extent, or are closed loops.  They are predicted to exist
by various particle physics theories, created in the Big Bang, but
unlike black holes, there is no observational evidence that they
actually do exist. For the sake of argument, we'll assume they do.

The space-time near a cosmic string is like that of a pizza, with one
slice taken out, and the edges next to the missing slice associated
with one another.

This distortion can give you gravitational lensing.  This means that
the light from an object beyond a cosmic string will be bent by the
gravitational influence of the string, so that it can come to us by
two (or more) different paths.  We therefore see two (or more) images
of the object.  Gravitational lensing can be caused by more ordinary
objects, like galaxies, and indeed, a number of distant quasars and
galaxies are observed to be gravitationally lensed (you see two or
more images of the same object, separated by a few arcsec).  In one
famous case, studied in detail by folks here at Princeton, one sees
directly that the two light paths take different amounts of time:
quasars are known to vary in brightness, and one sees the two images
varying out-of-synch by 417 days! One light beam travels on a shorter
path, that is it has beaten the other one by 417 days while traveling
the 8.9 billion light years to us.

Similarly, the two light paths around the cosmic strings have slightly
different lengths.  So one of the paths is indeed a short cut. So you
can beat a light beam from one point to another if you take the short
cut, and therefore "travel faster than light", one of the requirements
of time travel, as we saw above.

Prof. Gott invented a time machine in 1991 (at least on paper).  Take
two cosmic strings, moving transversely with respect to each other at
very close to the speed of light.  As they whiz past each other, zip
around the two of them in a circle in your spaceship, again at close
to the speed of light.  You can show that you can come back to your
starting place sooner than you left, and therefore meet yourself
before you left.  A time machine!  You have to get these strings
moving at high speed, so as to use all the tricks from special
relativity to play with the notions of simultaneity.  (And no, you are
not responsible for the details of this time machine).

One way to proceed would be to make a loop of cosmic string.  Like a
rubber band, these things have a lot of internal tension, and so tend
to collapse under tension.  As they do, the two opposite ends will
pass each other at high speed, approximating the situation we want
above.  People think, however, that in this circumstance, you could
indeed travel a bit back in time, but the whole thing would end up
collapsing into a black hole, leaving you trapped inside.  Not very
practical.  Really understanding this may require us to have a full
understanding of General Relativity on microscopic scales (i.e., where
quantum mechanical effects are also important); we're not there yet.

To make this work in practice, there are a few practical difficulties.
First, you have to find some cosmic strings; a problem, because they
haven't been discovered yet.  These have to be particularly massive
strings, with a mass of at least 4 x 10^{16} tons for every inch of
length.  You have to get them each moving at 0.9999992 times the speed
of light.  The energy required to do this is roughly the rest-mass
energy (i.e., m c^2) of an entire galaxy.  So we're not about to apply
to NASA to build one of these things quite yet!  But it is interesting
to demonstrate that the laws of physics allow time travel in
principle.

A time machine like this doesn't allow us to connect *all* events in
spacetime.  Crudely speaking, you can't travel back to a time before
the time machine itself actually exists.  So if you put your wormhole
in place next year, you can use it (in principle) to travel back from
the year 2009 to the year 2008, but not back to, say, 1963 with
warnings about Kennedy's assassination.  This is why we don't see lots
of time-traveling tourists from the future; the time machine hasn't
been invented yet.

Another approach to time travel: a wormhole was proposed by Kip
Thorne; think of a black hole that opens up on "the other side", to
another region of space.  Another kind of shortcut.

I can make a time machine out of a wormhole, which I move back and
forth at close to the speed of light.  Imagine a wormhole connecting
us and Alpha Centauri.  You step inside and move the mouth of the
wormhole on this end at high speed (close to the speed of light).
Special relativity says your time passes much slower than those of us
here on Earth.  Then step through the wormhole to Alpha Centauri and
high-tail it back to Earth on a rocket travelling close to the speed
of light.  You can in principle get back here before you left.  Again,
wormholes are not yet known to exist, and so this remains speculative
in the extreme.

```
Notes for Twenty-first Lecture