Time and Relativity
Do you and your friends experience the same time? Is time travel possible?
Time and Length are Not Absolute
In 1905, physicist Albert Einstein published a paper in the journal Annalen der Physik titled “On the Electrodynamics of Moving Bodies.” In his paper, Einstein states that a rigid sphere in motion along the x-axis relative to an equally rigid but stationary sphere will have its x-dimension appear shortened to the motionless sphere by the ratio:
In less mathematical terms, the greater the relative velocity of a moving body, the more the body is shortened along its direction of travel as observed by the relatively stationary body. When the relative velocity of the moving body reaches c, the shortening along its axis of motion is so much that it becomes infinitely small, or it effectively becomes a two-dimensional object with the value of the length of its axis of motion reaching zero at the speed of light when viewed by a stationary observer.
When v=c:
This phenomenon is known as Lorentz-FitzGerald Contraction, after the physicists George FitzGerald and Hendrik Lorentz who first postulated the effect in 1889 and 1892 respectively. During non-relativistic velocities, which are not a significant portion of c, the effects of the phenomenon are extremely small, which is the reason length contraction is not perceived in day-to-day experience. The motions experienced by humans on Earth are so low compared to the speed of light that the effects can be ignored for most applications.
Furthermore, Einstein also postulated that time dilates in a similar manner to length for bodies undergoing relative motion. In his paper, he uses the example of two clocks with different positions on Earth, and therefore different motions relative to each other.
“We conclude that a balance-clock at the equator must go more slowly, by a very small amount, than a precisely similar clock situated at one of the poles under otherwise identical conditions.” — Albert Einstein (1905)
Time Dilation Formula
The time periods experienced by two observers, one stationary and the other in relative motion, can be calculated with the formula:
Time Dilation on Earth
How does time dilation affect us during everyday activities such as walking down the street or flying in an airplane?
Taking an average of reported average walking speeds from multiple popular sources gives us a walking speed of 3.4mph, or about 0.002km/s, for the average adult (source 1, source 2, source 3)*. Velocity and speed are not the same thing. Speed is a scalar quantity, which only has magnitude, while velocity is a vector quantity, which has both magnitude and direction. But for the purposes of these calculations, speed can be used in place of the velocity term since the direction of travel is unimportant. What is important is the relative difference in speed between the two bodies (the stationary observer and the moving person).
[*] Note: Peer-reviewed information could not be located for average walking speeds so an average was taken from three averages reported by popular (non-peer-reviewed) sources instead.
The difference in experienced time between a stationary observer and a person walking by at 0.002km/s can be calculated with the following equation by using one second for the time experienced by the moving person, 0.002km/s for the relative velocity, and 299,792.5km/s for the speed of light:
So, for every 1.00000000000000002 (one and two hundred-quadrillionths) seconds that tick by on the watch of a person sitting on the couch, the watch of a person walking by would read only one second precisely (assuming they had watches precise enough to make the measurements). Although an incredibly small difference, the person walking by has technically experienced slightly less time relative to the person sitting on the couch due to their motion. What if we speed things up a bit?
The Boeing Next-Generation 737 aircraft, which was deemed the most-produced commercial jet airplane of all time by Guinness World Records in 2018, has a cruising speed of 590mph, or roughly 0.3km/s. Inputting 0.3km/s into the formula gives us the following equation and approximate value for T:
This means that for every one second experienced by someone in an airplane, a stationary observer would experience roughly 1.0000000000005 (one and five ten-trillionths) seconds. The difference in gravitational acceleration due to altitude is ignored in this calculation for the sake of simplicity, and the reader can assume that this difference will be ignored for the remainder of the calculations. The time dilation experienced in an airplane is greater than the time dilation experienced while walking, but it’s not a very noticeable difference.
During its final flight, NASA’s X-34A aircraft set the record for the fastest air-breathing aircraft at 6,800mph, or about 3.04km/s. The X-34A aircraft is unmanned, but if it did have a pilot, said pilot would only measure one second passing for every 1.00000000005 (one and five hundred-billionths) seconds measured by a relatively stationary observer.
It turns out that regular speeds obtainable on Earth aren’t going to cut it in terms of perceivable differences in the passage of time. But what about speeds obtainable outside of Earth?
The Fastest Man-Made Objects
On July 8, 2011, the last space shuttle, Atlantis, launched from the Kennedy launch complex in Florida. In order to remain in low-Earth orbit and meet up with its target, the International Space Station, space shuttle Atlantis needed to reach speeds of 17,500mph (7.8km/s) or about 2.6 times that of the X-34A aircraft. At this speed, the astronauts onboard Atlantis would only experience one second for every 1.0000000003 (one and three ten-billionths) seconds experienced by a stationary observer due to their relative motion.
17,500mph is fast, but still only creates a difference in experienced time of three ten-billionths of a second. What about the absolute fastest man-made object ever?
NASA’s Parker Solar Probe launched on August 12, 2018 and is currently attempting to orbit within four million miles (6,437,376km) of the Sun’s surface for the study of solar winds. According to NASA, the current record speed for the probe at the time of the writing of this publication is 531,670kph (147.7km/s). That’s about 18.9 times faster than the space shuttles.
A super precise watch attached to the Parker Solar Probe would read exactly one second for every 1.0000001 (one and one ten-millionth) seconds measured by the watch of a relatively stationary observer. As we have seen from perhaps too many examples, any significant time dilation is going to require intensely high speeds. Although it’s the fastest thing humanity has ever created, the Parker Solar Probe is embarrassingly slow when compared to the speed of light (about 0.05% the speed of light to be more accurate).
Percentages of Light Speed
We’ve seen how little time dilates when experiencing speeds that are a tiny fraction of a percent of the speed of light (0.05%), but how does it compare when more significant percentages of light speed are achieved?
A watch traveling at 1% the speed of light (2,997.9km/s) would experience one second for every 1.00005 (one and five hundred-thousandths) seconds experienced by a relatively stationary watch.
The difference between a stationary watch and a watch moving relative to it at 10% the speed of light (29,979.5km/s) would be approximately 0.005 (five thousandths) of a second.
At 25% light speed (74,948.1km/s) the difference between the two watches is approximately 0.03 (three hundredths) of a second.
At 50% light speed (149,896.3km/s) the difference is approximately 0.2 (two tenths) of a second.
What speed would you have to reach so that the watch you’re carrying reads half the time as your relatively stationary friend? For this calculation, we can input two seconds for the observer time, and one second for the elapsed time at the moving body and solve for the unknown velocity.
For you to measure ½ a second for every one second measured by your stationary friend due to your relative motion, you’d need to travel at approximately 259,627.9km/s, or about 86.6% the speed of light.
At 99% the speed of light (296,794.6km/s) the difference in time measured between your clock and your friend’s would be about 6.08 seconds, with your friend measuring approximately 7.08 seconds for every one second measured by your watch.
If we input values into a graph using the time dilation formula, we can see that time dilation is not linear, but follows a curve. The closer you get to the speed of light,the more dilation occurs for the same increase in speed. In mathematical terms, as v approaches c, T becomes infinite.
At the speed of light, the equation breaks down similarly to the first undefined equation presented in this paper.
A person moving at the speed of light would experience no time relative to a stationary observer, just like how a rigid body moving at light speed has its length along its axis of motion shortened to zero. Thankfully, moving at light speed is impossible for anything with mass because it would require an infinite amount of energy. As Einstein wrote in his 1905 paper after the equation for the energy of motion of an electron (W) under the action of an electrostatic force:
“When v=c, W becomes infinite. Velocities greater than that of light have-as in our previous results-no possibility of existence.” — Albert Einstein (1905)
Time Travel
Time dilation could be used to travel to the future…sort of. Everyone on Earth is currently moving to the future at very close to one second per second. There is some small variation among people dependent on their velocities relative to each other as we spoke about earlier, as well as their position within Earth’s gravitational field (altitude) since gravity is an acceleration toward an object’s center of mass via the warping of spacetime. These differences are small compared to the rest of this thought exercise so they can be ignored. But as established earlier, if you moved with a speed of 86.6% the speed of light relative to the people on Earth, you would experience one second for every two seconds experienced by those on the planet.
Let’s say you made a round trip that started on Earth and ended up back on Earth in a spacecraft. During this trip your spacecraft had a constant speed of 86.6% the speed of light, and the trip took you twenty years from your point of view. Everyone that remained on Earth would have, from their point of view, experienced forty years of time. When you got back, everyone you knew would have aged twice as much as you did during your trip. Your friend that started out the same age as you would now be older than you. In effect, you would be traveling to the future in half the time, or twice as fast as normal. If you increased your relative speed more toward that of light, then this time travel effect would be even more pronounced. It’s not quite the time travel featured in sci-fi movies and books, but it’s a very real phenomenon. Satellites, such as those used for GPS, must take time dilation into account for accurate triangulation due to their relative velocity.
Conclusion
Everyone experiences time slightly differently based on how fast they’re moving relative to others. Although this effect is so small that it’s not perceived in everyday life for most people, it can be manipulated to perform a certain type of forward time travel under extreme circumstances.
This article is dedicated to David Andrew Taylor.
