Astronomy 162: Professor Barbara Ryden

# TIME TRAVEL

``There was a young lady named Bright
Whose speed was far faster than light;
She set out one day
In a relative way
And returned home the previous night.''
- Arthur Buller

## Key Concepts

• Time travel into the future is permitted by the laws of General Relativity.
• Time travel into the past is possible if wormholes exist.
• Travel into the past can result in paradoxes.

#### (1) Time travel into the future is permitted by the laws of General Relativity.

In one sense, time travel into the future is trivial; we are all traveling into the future. What would be interesting would be to travel into the future at a different rate from the people around you. In order, for instance, to attend the opening ceremonies of the Olympics in the year 3000, you would want to age by only a small amount while the rest of the world aged by 997 years. In short, time travel into the future requires making your clock run at a slower rate than the clocks around you. General relativity permits this to happen; according to the laws of general relativity, an accelerated clock runs at a slower rate than a clock which has a constant velocity. To travel into the future, accelerate yourself.
The Adventures of Waldo and Wanda,
the Amazing Time-Traveling Warthogs!!

To illustrate how travel into the future might work, consider a pair of warthogs named Waldo and Wanda. Waldo will be accelerated while Wanda stays home on Earth, and we will see how the passage of time differs for Waldo and Wanda.

Waldo boards a rocket ship and accelerates away from Earth at 10 g's (10 times the gravitational acceleration at the Earth's surface). He maintains this acceleration for six months, as measured by the clock aboard his spacecraft. Then, to bring his ship to a halt, he accelerates toward Earth at 10 g's for another 6 months. Waldo is now at rest relative to the Earth, 18 light years from home. Waldo immediately starts back to Earth. He accelerates toward Earth at 10 g's for 6 months, and then, to bring his ship to a halt, he accelerates away from Earth at 10 g's for six months. Waldo's spacecraft is now at rest on the Earth's surface.

By Waldo's clock, which has been continuously accelerated, 2 years have passed.
By Wanda's clock, which has not been accelerated, 42 years have passed.

Waldo has traveled into the future; while two years have passed for him, 42 years have passed on Earth. (Notice that this method of time travel, although permitted by the laws of physics, is not technologically feasible for our culture. Accelerating a spacecraft for 6 months at 10 g's brings it close to the speed of light, and requires a gargantuan amount of energy.)

#### (2) Time travel into the past is possible if ``wormholes'' exist.

Acceleration makes clocks run more slowly, but can't make them run backward. As a clock is accelerated to speeds closer and closer to the speed of light, it ticks more and more slowly - for it to run backward, it would have to travel faster than light.

Does the impossibility of faster-than-light travel, then, imply that time travel into the past is impossible? Well, there is one loophole that we can take advantage of: time travel into the past is permitted by the laws of general relativity if you have a wormhole present. A wormhole is a tunnel of space-time which can connect two widely separated points. We know that massive objects cause `dimples' in space-time. There are other types of curvature permitted by the laws of General Relativity. A wormhole is an example of a curved surface permitted by General Relativity.

Suppose there existed a wormhole with one end near Earth and the other end near the star Vega. We would then have two alternate routes to take between here and Vega; we could either take a path through ordinary space or we could go through the wormhole. The incredibly useful thing about wormholes is that their length can be shorter than the distance between their ends in ordinary space. The distance between Earth and Vega is 26 light-years in ordinary space; however, a wormhole between them might only be a kilometer long (for instance).

Wormholes permit ``faster-than-light'' travel. If the Earth and Vega were connected by a short wormhole, we could travel to Vega in less than a day, rather than having to laboriously follow the 26 light-year long path in ordinary space.

Since wormholes permit faster-than-light travel, they also permit travel into the past. To illustrate how a wormhole can be turned into a time machine, let us call on Waldo and Wanda, the Amazing Time-Traveling Warthogs.

Waldo takes the same accelerated voyage that he took in section (1) of the notes. This time, though, Waldo and Wanda have provided themselves with a wormhole, one meter long. Waldo takes one end of the wormhole with him on the spaceship, and Wanda keeps the other end at home on Earth. During the voyage, Waldo and Wanda exchanges messages through the wormhole. Since the wormhole's length remains at one meter, Waldo and Wanda agree, when they talk through the wormhole, that neither of them is being accelerated relative to the other. Thus, when Waldo and Wanda exchange messages, they agree on what time it is.

Suppose that Waldo leaves Earth on February 14, 2003. He returns to Earth on February 14, 2005 according to his own clock. Just before he exits the hatch of his spaceship, he asks through the wormhole, ``Wanda, what day is it?'' Wanda replies, ``February 14, 2005''. After exiting the hatch, he asks a passing warthog, ``What day is it?'' The passerby replies, ``February 14, 2045''. The wormhole is now a time machine. Waldo's end, which has been accelerated, is now located in the year 2045 (Earth time). Wanda's end, which has not been accelerated, is located back in the year 2005 (Earth time). If Waldo decides he doesn't like the year 2045, he can crawl through the wormhole to the year 2005. Conversely, if Wanda in 2005 decides she wants a sneak peek at the future, she can crawl through the wormhole and take a look at life in the year 2045.

Note that if Waldo has the foresight to supply himself with back issues of the New York Times for the years 2005-2045 before crawling through the wormhole to 2005, he can make himself into a very wealthy warthog indeed. He will have advanced knowledge of which stocks will go up, which sports teams will win, and which numbers will win the lottery.

#### (3) Travel into the past can result in paradoxes.

Although the laws of general relativity allow travel into the past (if wormholes exist), traveling into the past can result in some nasty paradoxes. The most famous of these paradoxes, familiar to all readers of science fiction, is the ``Grandmother Paradox''. To illustrate the Grandmother Paradox, let us use the time machine that Waldo and Wanda have set up, allowing travel between the years 2045 and 2005. Wanda's grandson, in the year 2045, decides to murder his grandmother Wanda. He dives through the wormhole to the year 2005 and fatally gores Wanda. What makes the murder develop into a paradox is that in the year 2005, Wanda has not yet had her first litter of piglets. Therefore, by murdering Wanda, her grandson has ensured that she will have no descendants. In particular, she will not have a grandson who will travel back in time to murder her.

Wanda is killed by her grandson.
THEREFORE, Wanda has no grandchildren.
THEREFORE, Wanda is not murdered by her grandson.
THEREFORE, Wanda has grandchildren.
THEREFORE, Wanda is killed by her grandson.
THEREFORE, Wanda has no grandchildren.
THEREFORE, Wanda is not murdered by her grandson.
THEREFORE, Wanda has grandchildren. (ad infinitum....)

You get the picture. We are now stuck with the nonsensical, self-contradictory picture of Wanda both having and not having grandchildren and of Wanda both being and not being murdered. How can we escape this paradox?

One proposed solution is the ``Alternate Histories Hypothesis''. This hypothesis states that when a time traveler goes into the past and changes the course of history, the universe splits into a pair of separate universes with different histories. For instance, when Wanda is fatally gored by her time-traveling grandson, the universe splits into the two universes described below:

Universe One:
In 2005, a crazed warthog bursts out of a wormhole and kills Wanda.
Wanda leaves no descendants.
Universe Two:
Wanda is not gored in 2005. Wanda lives a long, happy life and leaves many descendants.
In 2045, Wanda's grandson dives through a wormhole, and is never heard from again.

The reason why Wanda's grandson is never heard from again in Universe Two is that he has traveled over to Universe One. Once Wanda's grandson changes history by killing his grandmother, he is stuck in Universe One. Even if he tries to escape the scene of the crime by diving through the wormhole back to the future, the future he escapes to is the future of Universe One, not the future of Universe Two which he came from.

More prosaically, Steven Hawking and other physicists have speculated that although the laws of general relativity permit the construction of time machines from wormholes, the laws of quantum mechanics will FORBID the construction of time machines, thus making the universe safe from paradoxes.

Prof. Barbara Ryden (ryden@astronomy.ohio-state.edu)

Updated: 2003 Feb 14