Geodesic time travel in Gödel’s universe
Godel Meets Einstein has 18 ratings and 0 reviews. What happens when the country's greatest logician meets the century's greatest physicist? In the case. The easiest type of time travel universe looks like a trick that is stipulated into existence. Einstein showed us through the Einstein universe that we can curve space back must conform to in order that distant future and distant past mesh when they meet. . The Goedel universe is a solution of Einstein's equations with the. Although Einstein was a physicist and Gödel a mathematician, they shared When the two men met, in , word of young Gödel's genius had yet to and succeeded in coaxing a new and flamboyant universe from the alembic of its symbols. After all, space and time are fused in ordinary life as well.
These differences were inevitably reflected in the nature of the friendship. In a letter written to the biographer Carl Seelig, Einstein's secretary remarked on the "awed hush" that greeted Einstein whenever he appeared at seminars or conferences. Not even the sharp-tongued Wolfgang Pauli, a fellow Nobel Prize winner in physics, could bring himself to treat the great man as if he were mortal.
In letters to his mother, he appeared to take pleasure in affirming that through his friendship with Einstein, he was basking in reflected glory. Although the content of their conversations has been lost, we can imagine at least one topic they must have discussed on those long evening walks. He did so by providing an exact solution to the heart of the theory—a field equation that allows one to calculate the force of a gravitational field—and his analysis reflects the distinctive characteristics of all his work.
It is original and logically coherent, the argument set out simply but with complete and convincing authority. A sense of superb taste prevails throughout. There is no show.
Godel Meets Einstein: Time Travel in the Godel Universe by Palle Yourgrau
And it is odd. It is distinctly odd. The leading idea of general relativity—the fusion of space and time—is not hard to grasp. After all, space and time are fused in ordinary life as well.
We locate an event the assassination of JFK, for example both in terms of where it took place Dallas, Texas and when it took place roughly 1: Three numbers suffice to mark the space of Dallas, Texas, on a three-dimensional map: The place is pinpointed as an event in space-time if another number, the time, is added. And if an event can be defined by four numbers, then a series of events can be defined by a series of such numbers, trailing one another like elephants marching trunk to tail.
In general relativity such series are called world lines. General relativity then forges a far-flung connection between the geometry of space and time and the behavior of objects in motion within space and time. Imagine a marble placed on a mattress. Given a tap, the marble will move in a straight line.
But place a bowling ball on the mattress, too, and the marble, given precisely the same tap, will roll down the sagging surface, its path changing from a straight to a curved line.
The bowling ball's weight deforms the medium of the mattress, and the deformed medium influences the marble's movement. Replace the bowling ball and marble with planets, stars, or wheeling galaxies, and the mattress with space-time itself, and a homely metaphor is transformed into the leading principle of a great physical theory.
In a universe with no massive objects, there is no deformation of space and time, and the shortest route between two points is a straight line. When matter makes its fateful appearance, the shortest routes will curve.
The first and most celebrated confirmation of this theory came inwhen astronomers established that the mass of the sun causes a beam of light to curve, as Einstein had predicted. Imagine a group of observers scattered carelessly throughout the cosmos. Each is able to organize the events of his life into a linear order—a world line of the kind just described.
Time Travel Universes
Each is convinced that his life consists of a series of nows, moving moments passing from the past to the present to the future. Special relativity urges a contrary claim. The observers scattered throughout space and time are all convinced their sense of now is universal.
Now is, after all, now, is it not? Time passes at a different rate depending on how fast a person is moving: While one hour passes on Earth, only a few seconds might pass on a rocket ship hurtling away from Earth at nearly the speed of light. 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.
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: 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.
John Dawson, Gödel Meets Einstein: Time Travel in the Gödel Universe by Palle Yourgrau - PhilPapers
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: 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.
Einstein and Gödel
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.Visualization of the Gödel universe
That meshing with the future pieces can only happen if the time traveler fails in the assassination attempt. The "99" Puzzle Here's a simple example in a different arena of how these sorts of global constraints can work. It is the arithmetic puzzle, " Is there some combination of additions and subtractions that will get you to 99?
Locally, there is no obstacle to getting to 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. Globally, however, there is a constraint that necessarily defeats your attempts to arrive at 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 This global constraint assures your failure to solve the puzzle. 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: Now pick some time slice. What configurations of the particle are possible? A One Particle Time Travel Universe 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.
- Godel Meets Einstein: Time Travel in the Godel Universe
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!
Wormhole Time Travel Universes The cylinder universe described above is the simplest way of creating a time travel universe by attaching one part of spacetime to another.
A fancier way of achieving a similar end is with a "wormhole. Then a traveler can enter the first region and reappear earlier in time at the second. The figure shows the idea. The sorts of time travel issues that arise in this wormohole universe are pretty much the same as those that arise in the cylinder universe. We next exhibit that such time-travelling possibility do not exist in his model universe.
This is done in the most straightforward way possible, framing the discussion as to serve as a simple example for students of General Relativity. You may imagine such inertia compass as a set of gyrocompasses fixed to every galaxy in GU and such that all such galaxies rotate in unison about its prescribed parallel-transported normals, so indicating that the entire GU rotates rigidly in the opposite sense.
Therefore, GU is homogeneous but cannot be isotropic, a feature that prevents the definition of a unique time valid for the whole universe 2. For example, a universe in which matter rigidly rotates 4 means that GU is homogeneous but not isotropic. Moreover, it is an example of a cosmology exhibiting properties associated with the rotation of the universe as a whole.