sharp@kpnoa.UUCP (05/21/84)
<Does your chewing gum lose its flavour on the bedpost overnight ?>
**Warning** All of this article is based on general relativity.
Those of you who dislike that theory can argue with me privately.
1) Time machines
First, just what exactly IS a time machine. We are all familiar with the
standard science fiction device, which transports matter (usually people) into
the future and/or past of their particular location. One feature not usually
addressed in stories is how the machine manages to maintain a planet-based
coordinate system. Thus, in addition to the rotation of the Earth and its
orbit about the sun, the sun is moving around the centre of the galaxy, and the
galaxy is moving relative to other galaxies. In order to make a time-machine
work in the fictional sense, these motions are very relevant: what is the
SPATIAL reference frame for the "time travel" ?
This is particularly problematic for general relativity. The reason for
calling it relativity is because the laws of physics are the same in all frames
of reference, and GR is a way to compare the descriptions made from the point of
view of any frame to those made from a different frame (note that this is NOT
the popular interpretation that "All things are relative": GR enables one person
to interpret his observations relative to another person's observations).
(The reason that GR is a theory of gravity AS WELL is that observers can be
accelerated with respect to each other, and acceleration is locally
indistinguishable from gravity: hence, relative descriptions include gravity.)
Physical systems in GR are usually described by means of a space-time diagram.
In such diagrams, motion in time is included by making one of the axes a time
axis. (This is well explained in all simple texts on GR.) Consider a simple
diagram having one vertical time axis and one horizontal space axis. A particle
which is stationary is a vertical line: constant spatial position, but
continuous in time. A particle with constant velocity is a tilted line: spatial
position moves uniformly with time. Usually, the speed of light is set to 1 and
both time and space are measured in the same units (for the curious, one nano-
second of time is almost exactly a foot of length). This way, photons which
travel at the speed of light are represented by lines tilted at 45 degrees.
No material object can travel faster than this, so no lines are tilted at more
than 45 degrees. This is all true in the absence of gravity. Now, if we put
in a gravitational field, this can be viewed as a bending of the space-time (the
most famous part of GR - we live in a warped space-time !) In a simple one-time
one-space diagram, we can imagine the field making the tilted lines representing
uniform motion tilt over further (or less far), even to the extent of making
45 degree lines, which represent light rays, point straight up. At this level,
a photon is moving "at the speed of light", but it is not making any progress !!
Its spatial coordinate (along the horizontal axis) is not changing with time.
This is what happens with the space-time diagram for a simple black hole.
From this example, it is easy to imagine much more complex gravitational fields
which have the effect of causing one of our lines to curve around and join up
with itself: fields bend lines in space-time diagrams, so we just bend one
into a closed loop. This is a CLOSED TIME-LIKE CURVE, and it is the mathematical
structure in GR which equates to a time machine (see, I got there in the end!!)
Now let's pause and think what this means. A closed curve. By moving forward
along this curve, we come back to EXACTLY THE SAME POINT IN BOTH SPACE AND TIME.
This is exactly what we want, slightly modified so we don't "materialise" inside
ourself (it is easy to show in GR that if closed curves exist, then there exist
curves arbitrarily near to closed). So, if we can find a circumstance in GR that
admits of closed timelike curves(CTC), the second question is, can we use them ?
To put it another way, we have to be able to come in from some other curve until
we are very close to the CTC, follow it around until we reach the time we want,
and then move away from the CTC until we reach the place we want.
2) Tipler machines
This is quite popular with a lot of people. What Tipler showed, in a 1974
article (Physical Review D, vol. 9, p2203), is the existence of CTC
around a special configuration of matter, to wit, an infinitely long rotating
cylinder. The rotation is VERY rapid, but the surface is still travelling at
less than the speed of light, and the strains are not necessarily inconsistent
with the known strength of materials. One problem is the length, since it is
not possible to construct an infinite cylinder, but it seems that a very long
one (compared to its radius) would be adequate (although this was not proven).
Another problem is that the solution is a stationary one - i.e. in the jargon
of GR, a solution which has existed for all time and which will continue so to
exist. It is not clear that constructing such a cylinder would necessarily
give rise to the same gravitational system, although the expectation is that
it would just be a question of waiting long enough for it to settle down.
In a later paper (Physical Review Letters, vol. 37, p879, 1976) Tipler argued
that it is not possible for creatures to manipulate matter in such a way as
to create CTC without also creating what is called a naked singularity.
This is a whole new can of worms: basically, a naked singularity is a place
where classical, non-quantized GR predicts a point of infinite density.
Such a point has no known properties, since no theory can cope with infinity.
The current feeling is in two parts: proper quantisation will remove the
"infinity" to a fuzzed-out "very big" (or something similar, like the debate
over the infinite self-energy of the electron), and naked singularities cannot
be created or observed (just because there are mathematical solutions to the
equations of GR does not mean that such solutions are physically realisable).
If there is enough interest, I may expand on Tipler machines & NS later.
3) Black holes as time machines
No way. This comment is a little blunt, so ...
There are no CTC for black hole solutions to the equations of GR. There are
naked singularity solutions closely related to the black hole solutions which
contain CTC (and are examples of Tipler's second theorem mentioned above),
although much work has been devoted to showing that these CTC are not accessible
in the sense I mentioned above, for using as time machines.
The idea of using BH for time machines, which can be found in various places
(particularly Kaufmann's popular works, which are not recommended) comes from
a curious property of the "throats" of BH. The very simplest BH, the
non-rotating, uncharged Schwarzschild hole, cannot be used as a "passage",
since anything which falls into it is dragged inevitably into the central
crushing singularity. However, the solutions which have rotation or charge
(Kerr, Reissner-Nordstrom, or, for both, Kerr-Newman) WILL allow a particle
to enter the BH at a speed below that of light, and to leave it again by an exit
different from the surface through which it entered. OK, I hear you crying
about not getting out of BH. Let's look at this again. The solutions which
represent BH (and there's a lot of work that shows that the mathematical
solutions are the same as the end-point of real physical collapse) go down to
a radius of zero, covering a whole exterior region (our Universe, if you will),
an event horizon (the surface of the BH), and an interior region. Now, this
is an incomplete manifold (I'm sorry, I'm going to have to use some technical
terms), which is a mathematical way of saying there's more in there somewhere !
So, a technique known as analytic continuation is used to extend the manifold
across the Killing horizon (i.e. what is this "more" that's in there somewhere?)
and create a bigger space-time. When you've done all this mathematical
jiggery-pokery, you find that NEXT to the interior solution you had before is
another interior solution, joined by a horizon to a whole new exterior solution
- a whole new Universe, in fact (a conjecture much abused by Disney's BH movie).
In fact, under certain conditions, you can get an infinite progression of such
"interior-exterior" or "black hole-Universe" sections, all joined together.
But, what does all this mathematical extension of a nice simple BH mean ?
Now we get into a little philosophy - i.e. interpretation of mathematics in a
way that can never be compared to reality. (what's that symbol ? :-) )
There are two simple possibilities, which is to say that the "other Universe"
sections are either whole new Universes (as mentioned, Disney abused this, as
did others, noticeably a certain very poor TV series), or that they are to be
identified with our own Universe section (the bit we first thought of).
If you identify them with our own Universe, then the structure does not tell
you either where or when the horizon should be placed - i.e. where the BH is.
The structure is such that if you pass into the BH in our Universe, it is
possible (with good rockets !) to avoid the singularity and re-emerge through
the next horizon into the next Universe. You therefore have two possible
travel plans, whereby you pass to an unknown time and place in either another
Universe or the one you started in. Since you have no control over your exit
point, this is not a very useful form of time-machine. Of course, you could
always dive into another BH, going through to yet another Universe (you can't
go back, so that you really can't get out of the BH you went into), and keep
this up in the hope of ending up where you want to be - if you live long enough.
However, there's yet another problem with this idea. (And you thought us
relativists didn't work for a living :-) ) It turns out, and this really is
complicated mathematics, that these "tunnels" are unstable to perturbations.
This means that even the slightest disturbance will cause the structure to
collapse into something which hasn't been investigated fully (yet), closing
off the passage into other Universe sections. But, how is it possible to go
into the BH without disturbing the tunnel ? After all, even the slightest
effect - such as your radio waves, or your gravitational influence - will
close the door. There is no "back door" to a black hole.
This is now a whopping article, but I hope that at least some of this will
be clear to at least some of you (although I won't pretend that it's all clear
to me). The Tipler discussion is brief, and might be expanded. The rest is
probably open to some expansion, but not much.
Any queries and comments should be sent to me (I don't know your best path,
but I'm kpno!sharp), and I may post summaries and/or further explanations.
Nigel Sharp