HPS 0410 | Einstein for Everyone |

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John
D. Norton

Department of History and Philosophy of Science

University of Pittsburgh

Before we turn to pursue the spacetime that best resembles our own in the next chapter, it is interesting to review how Einstein's theory allows us describe universes in which time travel is possible.

The easiest type of time travel universe looks like a trick that is stipulated into existence. However there is nothing illicit about it. And its great simplicity enables us to refine our intuitions about just how time travel can arise.

Einstein showed us through the Einstein universe that we can curve space back onto itself and thus produce a closed space.

That construction proved a little complicated for Einstein since there are three dimensions of space that need to be accommodated. If we want to do it in the time direction, it is much easier. There is only one dimension of time. The simplest case arises if we wrap up the time direction of a Minkowski spacetime. As before, if we consider only one dimension of space, we recover a cylinder. The spacetime is on the surface of the cylinder. For the new case the cylinder is wrapped up in the timelike direction.

You might wonder if a trick like this is really allowed by Einstein's gravitational field equations. It is. Recall that Einstein's gravitational field equations merely fix how each little patch of spacetime must look. A solution is admissible if each patch connects properly with those next to it. That will happen in this spacetime. In any not too big piece, this cylinder universe is exactly the same as a Minkowski spacetime; each piece connects with the one next to it just as they do in a Minkowski spacetime. That is all that is needed for the spacetime to count as a solution of Einstein's gravitational field equations. |

The timelike curve on the spacetime represents the life of a traveler who stays at one point in space, but passes through time merely by being. Eventually that worldline will wrap all the way around the spacetime and reconnect. At that point, the traveler will meet his or her former self.

The traditional "grandfather" paradox of time travel arises if the latest stage of the traveler (now imagined to be the grandson of the original traveler) were to kill the original one (the grandfather). A contradiction would ensue. With the assassination complete, there would no traveler to pass through time and commit it. So the assassination happening entails that it doesn't happen.

We have already discussed the difficulties raised by such a paradox in the discussion of tachyons earlier. The difficulty is that we would visit a logical contradiction on our physical theory and that is unsustainable in any cogent physical theory.

Time traveler goes back in time. | Time traveler kills grandfather | Time traveler does not go back in time. | |||

Time traveler does not go back in time. | Time traveler does not kill grandfather | Time traveler goes back in time. |

So we have a contradiction:

Time traveler goes back in time. | if
and only if |
Time traveler does not go back in time. |

Contradictions may be fine in science fiction or when we are writing a screen play for a time travel movie. However contradictions are a mortal threat to a physical theory. From time to time, we will work with an inconsistent theory. We shall see that this happened in the early years of quantum theory. However this tolerance has to be temporary. For an inconsistent theory cannot reliably tell you what is the case. It assures you that some assertion is both true and false at the same time.

The possibility of such paradoxes has led some to conclude that time travel universes are logical impossibilities. That is too hasty. There is an obvious loophole in the paradox. If the assassination attempt fails, then there is no contradiction.

So that is what must happen in a time travel universe. The grandson's bullet must miss; or the gun misfire; or the grandfather ducks; or who knows what. For if the assassination attempt didn't fail, there would be no assassin to attempt it.

To put it another way, when the grandson arrives to assassinate his grandfather, the failure of the assassination has already happened in the grandson's past. It has already happened and so cannot be undone by whatever the grandson may try to do.

That resolution is, as far as I know, admissible. Many find it objectionable since there seems to be no reason in the physics itself that forces the failure of the assassination attempt. What if the grandson takes all due care, aims carefully with a new gun, and so on? How can we be so sure that the attempt will fail.

We can. The intuitions that tell us it will not fail are honed in a type of universe that is quite different from a time travel universe. In the ordinary time travel free universes, such as we presume we inhabit, local constraints prevail. If the gun misfired, for example, it was because something in the state of the gun immediately prior to to the assassination attempt intervened. Perhaps the grandson passed through a rain shower and a component of the gun began to rust.

In a time travel universe, in addition to these sorts of local constraints, we have a new type: global constraints. These are extra constraints that all processes must conform to in order that distant future and distant past mesh when they meet. These global constraints do not arise in time travel free universes. They are what assures us that the assassination attempt must fail.

We can get an idea of how they work from the jigsaw puzzle analogy for solving Einstein's equations.

First consider a universe without time travel. We
start with a row of pieces that represents space in the present
instant. Then we add successive rows that correspond with space in
successive future times. The pieces we add are constrained
only by the local requirement that each piece mesh with
those immediately before and after it in time; and those around it
in space. These pieces specify the local condition of spacetime: the curvature of its geometry and its matter content. So one piece will have a grandfather; and the successive pieces fitted to it will have the grandfather engendering children, aging and so on. There will also be pieces in which assassins shoot their victims. |

Now take the case of a time travel universe. All
these constraints apply. But, in addition, as we keep adding the
successive rows, we will eventually end up going all the way round
the space and then the new and powerful constraint will come into
force. The last row we add has to be so perfectly built that it
meshes with the past edge of the first row we put in place. That is
a global constraint. It means that in our planning of which pieces
to lay down, we had to worry about the local meshing of the pieces;
and, in addition, we had to select pieces now so that eventually the
final meshing of last and first row
would work out. If that last piece that we add contains the time traveler attempting to assassinate his grandfather, then that piece must mesh with the past. Those past pieces would contain the time traveler loading his gun and walking down the street to the house of his grandfather. It must also mesh with the future pieces. Those future pieces will contain a live grandfather who survives, for those pieces will in turn develop forward to give us the time traveler. That meshing with the future pieces can only happen if the time traveler fails in the assassination attempt. |

Here's a simple example in a different arena of how these
sorts of global constraints can work. It is the arithmetic
puzzle, "99." In the puzzle, you are to start at zero and may add
or subtract any number you like between 0 and 10, as many times as you
like, provided that the numbers that you are adding or subtracting are
always *even* numbers. Is there some combination of additions and
subtractions that will get you to 99?

Locally, there is no obstacle to getting to
99. If you could somehow get your sum to 97 or to 95, you could complete
the task by adding 2 or 4. Just looking locally at the numbers around 99
reveals no problems.

..., 87, 95, 91, 97, 99.

Globally, however, there is a constraint that
necessarily defeats your attempts to arrive at 99. Since you start with
zero and may add or subtract even numbers only, your sum must always be an
even number. So you can never get to 97 or 95 or any other number that is
an even number removed from 99. This global constraint assures your
failure to solve the puzzle.

0, 8, 4, 12, 16, 24, 22, 30, ...

That is how things would work also for the time-traveling would-be assassin. The assassin can get to states very close to the successful assassination--the "99" of the puzzle. However those close states would always be the even states, "94," "96", "98," ... They look close to the assassination sought, but physical laws preclude them developing into the successful assassination.

Another simple illustration shows just how powerful these global restriction on a spacetime can be. Consider just about the simplest possible time travel universe: a universe empty of all matter excepting just one mass. Now pick some time slice. What configurations of the particle are possible?

In an ordinary time travel free universe, at some initial moment of time, we can have the worldline of the mass with any initial velocity.

If the spacetime is a time travel, cylinder universe, we are strangely restricted in the possibilities for this time slice. We could choose a mass at rest. That corresponds to the case of a single worldline that eventually wraps back onto itself. But if we have the mass initially moving, then we must also stipulate that clone masses be distributed in space at uniform intervals. These will be the repeated returns of the single mass as it travels all the way round spacetime and back to the present.

The global constraint says that if we have a moving mass here and now, we must also have a moving mass there and now; and there and now; and so on. That sort of constraint would be incomprehensible in a universe without time travel. What reason of physics, we would exclaim, requires it--just as we ask, what reason of physics requires the grandson's assassination attempt to fail!

There is something that looks just a little fishy about the way time travel is arrived at in the cylinder universe. It does not seem to arise from the physics of the spacetime. It comes from a stipulation on our part that the future wrap back onto past. Einstein's theory seems only to get involved in so far as it raises no objection. There is nothing wrong with this way of introducing time travel, of course.

It is nice to know, however, that time travel also can arise more naturally. The Goedel universe is one such example. This solution to the Einstein equations was arrived at by the famous logician, Kurt Goedel, in the 1940s, when he was a colleague of Einstein's at the Institute for Advanced Study in Princeton, and published in 1949.

The Goedel universe is a solution of Einstein's equations
with the cosmological term. Its signature property is that it contains closed
timelike worldlines. As a result, it is hard to
pick out a single timelike direction globally in the spacetime. Rather, we
can get a feel for its spacetime properties by taking just a single two
dimensional slice of it. It will become clear that this is *not* a
spacelike slice.

If we consider some observer in the middle of this slice, the observer will find all the matter in a great cosmic rotation around them. (For this reason, the Goedel universe cannot be ours. We don't see such rotation.) The reason for the rotation lies with the structure of spacetime itself. As we consider positions in the slice further away from the observer, the light cones start to tip over. So if we consider a large enough chunk of the slice, we can find a timelike curve that loops back onto itself. It forms a closed timelike curve, the hallmark of universes that admit time travel.

The timelike curve is not a geodesic; it represents the trajectory of an accelerating spaceship. To achieve time travel, the spaceship would need to accelerate quite considerably. Most interestingly, the Goedel universe uses no stipulations about past wrapping back onto the future to achieve the possibility of time travel.

There are other universes that admit time travel. Often rotation is involved. Spacetime around an infinitely long, very dense, rapidly rotating tube of matter admits closed timelike curves, for example. Some of the most fascinating of the time travel universes are those in which one part of spacetime is connected to another by a wormhole. That is just a tunnel of spacetime that provides an alternative route from one part of spacetime to another.

- How time travel can arise in relativistic cosmologies.
- Cylinder Universe
- Goedel Universe
- How the "kill your grandfather paradox" arises.
- How it is resolved in time travel universes.
- The difference between global and local constraints in spacetimes.

Copyright John D. Norton. March 2001; January 2007; February 16, October 15, 27, 2008. March 6, 2013, March 4, 2015. March 15, 2017.