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Special Relativity

Travel to the Stars

Time slows as we accelerate. Does this mean we can slow time enough to reach the stars in several years? Yes, we can. If we accelerate at 980 cm s-2, which is the average gravitational acceleration at Earth's surface, we experience a slowing of time that commences after one year of travel away from Earth.

Time Dilation During a Round Trip

In our imaginary trip, we will travel in a rocket to a destination and back with constant acceleration. We accelerate towards our destination over the first-half of the distance to our destination, and we accelerate towards Earth over the second-half of this distance, so that our spaceship reaches it highest speed half-way between Earth and our destination and is at rest at our destination. The spaceship will then make the return trip to Earth in the same way. At the end of the trip, we compare the clock carried by the spacecraft to a clock left at Earth. What we find is that substantially more time has passed for Earth than for the travelers on our spacecraft.

The equation for the time at Earth versus the time in the spaceship for a round trip is given by

t = 4 c sinh(gτ/4c)/g,

where c is the speed of light, g is the acceleration, and τ is the time for the passengers on the spacecraft, which is called the proper time of the spacecraft. These two times are given in the live figure to the left as functions of destination distance. For trips much longer than τ = 4c/g, the elapsed time on Earth increases exponentially with elapsed time on the spacecraft.

The total distance traveled in a spacecraft elapsed time of τ is given by

x = 4 c2 ( cosh(gτ/4c) - 1 )/g.

For an acceleration rate of 980 cm s-2, which is the average gravitational acceleration at Earth's surface, the constant g/c becomes 1.03 years-1, and our travelers must travel for about a year before their elapsed time is significantly less than the elapsed time at Earth. For travel times that are much less than a year, the time measured in the spacecraft is equal to the time measured at Earth, and the travel time is related to distance traveled by Newtonian mechanics: x = at2/8. For travel times as measured in the spacecraft that much longer than a year, the elapsed time measured at Earth goes to the light travel time to the destination and back; it is orders of magnitude longer than the elapsed time measured in the spacecraft. This transition from a Newtonian to a relativistic description of travel is apparent in the figure, with the time line for Earth exhibiting a change in slope the time line for the travelers flattening out at about 1 year.

This live figure shows the amount of time, given in years, that passes for a round-trip to an object at the given distance, which is given in parsecs. The travelers are accelerating at a constant rate throughout the trip. The line marked Traveler gives the elapsed time experienced by a traveler. The line marked Earth gives the elapsed time experienced on Earth. The reader can change the rate of acceleration in units of 980 cm s-2, which is the gravitational acceleration at Earth's surface; values can range from 0.25 to 50. Control of the applet from the keyboard is described in the Applet Control Guide.

Travel to the Stars and the Galaxies

The nearest star is over one parsec away from Earth; as can be seen in the figure, a round-trip to an object at 1 parsec takes 6.8 years for our traveler, but 9.65 years for those left behind on Earth. This isn't much of a time savings. The real time savings occurs for travel to objects that are 10 parsecs way, with an elapsed time of 13.9 years for our travelers, and an elapsed time of 69.0 years for those on Earth. To travel a truly galactic distance, 1 kpc, takes 31.5 years for our travelers and 6,527 years for those on Earth. In less than half a human lifetime, our travelers would pass through a large fraction of recorded human history. Travel less than twice this time, and our travels can visit galaxies that are 1 million parsecs away, farther than the Andromeda Galaxy, our companion spiral galaxy, but at the cost of jumping 6.5 million years into Earth's future, a time long enough for human evolution to occur. If our travelers go to the galaxies, they may not find on their return apes dominating humans, but they likely would find humans that are much different from themselves.

If we were in a bigger hurry, we could increase our acceleration, which would shorten the time for time dilation to commence. The time of commencement is inversely-proportional to the acceleration, so doubling our acceleration will half the time at which time dilation appears. So accelerating at twice Earth's surface gravity will produce significant time dilation in 6 months. At this acceleration, our travelers can travel to 1 kpc destinations and back in only 17.1 years, and to destinations at 1 Mpc in only 30.5 years. Elapsed times back on Earth, however, do not change significantly for distances over 10 parsecs over the elapsed times for the lower acceleration rate, because the spacecraft travel time is nearly the light-travel time, which is the minimum elapsed time that will be measured at Earth.

Realistic Applications

Our imaginary spacecraft, while not physically impossible, would be a technical colossus. Engines would have to provide continual acceleration for decades, and the amount of energy that would have to be expended would be many times the rest mass energy of the spacecraft and it occupants. Once the spacecraft experiences substantial time dilation, the surrounding starlight is blueshifted to x-ray and gamma-ray energies, with velocities counter to the spacecraft's direction of acceleration; this radiation exerts a substantial drag on the spacecraft. While these hurdles may one day be overcome, I suspect that day is many millennia way.

But while our imaginary trip is far beyond current technology, astronomers in their studies encounter matter that has undergone a similar acceleration. The radio jets of active galaxies, the gamma-ray bursts from supernovae, and the cosmic rays that strike Earth are examples of the acceleration of atoms and particles to nearly the speed of light. Nature has its accelerators that can overcome the drag of ambient radiation. Special relativity pervades the astrophysics of these accelerators and their products.

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