###### By Dan Hooper, Ph.D., The University of Chicago

## Kurt Gödel, one of the premier most mathematicians and logicians of the twentieth century, thought his cosmological solution to the gravitational field equations of general relativity exposed a logical inconsistency with the theory itself. However, Einstein was skeptical of Gödel’s work and wasn’t convinced that Gödel’s cosmological solution had any real physical significance or implications for the nature of time. Was Einstein correct in doing so?

Interestingly, Einstein did publicly endorse and promote Gödel’s essay on time. He hailed it as, ‘an important contribution to the general theory of relativity’ and ‘especially to the concept of time’.

Learn more about the problems with time travel.

### Not All Time Travel Is Illogical

Gödel hoped that the logical inconsistency he had found in the theory of general relativity would force us to revisit, and perhaps radically revise, our concept of time.

Einstein and Gödel did agree about the questions raised by Gödel’s essay. In particular, they both recognized and agreed that the existence of closed timelike curves would make it impossible for one to distinguish the past from the future. They also agreed that the lack of a well-defined direction of causality in such a system could lead to paradoxes, and oftentimes to illogical nonsense.

This, however, doesn’t mean that all kinds of time travel are problematic. In some cases, they’re not. Some kinds of time travel are entirely logically self-consistent.

Let’s take the example involving a pair of twins. One twin travels through space at nearly the speed of light and returns to Earth. To the moving twin, only a few years of time passed during the trip. But on Earth—and to the stationary twin—a full century of time and history had played out.

By moving at nearly the speed of light, the moving twin traveled not only through space, but also through time, and into the future. There are no logical problems with this. In principle, if you can move fast enough, you can move arbitrarily far into the future. A thousand years. A million years. A billion years. Or more.

Learn more about the search for a unified field theory.

### The Grandfather Paradox

The serious logical problems start to appear with backward time travel. The most famous illustration of these kinds of problems is known as the ‘grandfather paradox’.

Imagine that you follow a closed timelike curve to a point in the past. At this point, you encounter and kill your own grandfather while he’s still a child. As a consequence of these actions, your grandfather never grows up. He never meets your grandmother, and he never has any children or grandchildren. This means you’re never born. Therefore, you never exist, and that means that you never travel backward through time to kill your grandfather.

So, since he was never killed, your grandfather survives to meet your grandmother, and they do have children and grandchildren together. So once again, you do exist. And then you do travel through time to kill your grandfather.

Backward time travel makes it impossible for there to be a self-consistent timeline.

This is a transcript from the video series

What Einstein Got Wrong. Watch it now, on The Great Courses Plus.

The grandfather paradox has been a staple of science fiction since the 1930s. In addition to producing some very entertaining storytelling, it also serves to illustrate the logical hazards that can come with unrestricted time travel.

Any system in which it is possible to change the past suffers from these kinds of problems, which means that any system containing closed timelike curves seems sure to lead to paradoxical nonsense.

You should also keep in mind that this conclusion doesn’t only apply to people or other living things that might travel through time. An electron, for example, might travel through a closed timelike curve only to prevent it from ever coming into existence. On fairly general grounds, the existence of closed timelike curves seems to break the very logical self-consistency of a universe.

For these and other reasons, Einstein doubted that Gödel’s result could have any real physical meaning or other physical implications. He didn’t doubt that Gödel’s math was correct because it was perfect. However, Einstein didn’t think that all mathematically valid solutions to the equations of general relativity were necessarily physically valid solutions.

Learn more about Einstein’s rejection of black holes.

### Einstein Prematurely Dismissed Many Valid Solutions

There are a number of cases where Einstein applied his intuition in order to decide whether or not a given solution to his field equations was physically meaningful.

Take black holes, for example. Black holes are described by a valid solution to the field equations, first identified in 1915. Einstein had refused to believe that objects like these could ever form. Only well after Einstein died, did astronomers begin to discover real black holes in our universe?

We now know that very massive stars inevitably become black holes, and supermassive black holes occupy the centers of most galaxies. However, throughout his entire life, Einstein refused to believe that such objects were, in fact, physically possible.

Another example would be the expansion of the universe. Until Edwin Hubble proved him wrong, Einstein rejected the possibility that the universe might be expanding or contracting. Instead, his intuition led him to insist that the universe must be static.

When we take examples such as these into account, we see that Einstein had a far from the perfect record when it came to deciding which solutions to his field equations were physically real and which were not.

This leads us back to the question at hand. Was Einstein correct to reject the physical significance of Gödel’s universe? And what, if anything, does Gödel’s work tell us about general relativity? Or about the nature of time itself?

At the time, it was probably impossible to know whether Einstein was right to disregard any physical significance of Gödel’s cosmological solution. Throughout his life, Einstein often played the role of the skeptic, and this was no exception. Gödel’s essay on time had identified what seemed to be a very surprising aspect of general relativity, but when faced with these surprising consequences, Einstein’s instincts told him that this result probably didn’t really matter.

Einstein did have some good reasons to support this choice. Gödel’s solution to the field equations doesn’t describe the universe that we actually live in. Gödel’s universe isn’t expanding, while ours definitely is. Also, there’s no evidence that our universe is rotating the way that Gödel’s is. So, the universe that we live in is clearly very different from the one described by Gödel’s solution.

So, someone with Einstein’s outlook could simultaneously acknowledge that Gödel’s universe seems to have some logical inconsistencies, but at the same time, he could argue that our universe—the real universe—doesn’t necessarily suffer from any of these problems. Maybe our universe doesn’t contain any closed timelike curves.

If there are no closed timelike curves in our universe, then maybe there aren’t any underlying problems with time either—at least not with the way that time seems to exist in our world.

However, in my opinion, this is where Einstein went wrong. Or at least where he seems to have reached these conclusions prematurely.

When Gödel pointed out that closed timelike curves could exist, even in a hypothetical universe, this gave us a deep and good reason to worry about the self-consistency of general relativity itself. Even though we don’t actually live in a universe like the one described by Gödel’s solution, it’s not clear that our universe is entirely safe from the kinds of logical inconsistencies that could be associated with the existence of closed timelike curves.

### Common Questions about Kurt Gödel’s Universe

**Q: Is it possible to travel back in time?**

Time travel is not possible, as of now. However, mathematically, if one can move at the speed of light or faster, one can move arbitrarily far into the future. In the case of backward time travel, serious logical problems start to appear, even in a mathematical system.

**Q: Who created the grandfather paradox?**

The grandfather paradox is the most famous example of the types of problems that present themselves with backward time travel. This form of a time paradox has been written in science fiction stories as early as the 1930s. So, we don’t know who was the first to create the grandfather paradox, but one of the earliest examples can be found in a letter to the American science fiction magazine Amazing Stories.

**Q: How do you make a closed timelike curve?**

A closed timelike curve is essentially a path through space and time that makes it possible for someone to be present for some event, and then to travel through space only to later encounter the same event again. Here the same event isn’t a recurrence of the original event. It is the original event. The observer has simply followed a path through their universe that has taken them from the future into the past.

**Q: What would happen if a time paradox occurred?**

In a mathematical sense, certain forms of time travel are entirely logically self-consistent. However, in the case of backward time travel, a number of time paradoxes appear. One of the most famous illustrations of such a time paradox is the grandfather paradox. The result of such a time paradox is that there’s no possibility of a self-consistent timeline.