Time Dilation and the Theory of Relativity

With our totally unrestricted guidelines on Blog 4, I would like to make a bit of a deviation from the content of our course and discuss time dilation and the theory of relativity!

In the movie Interstellar, a planet is so close to a black hole that the immense force of gravity of the black hole causes one hour of time on the planet to constitute seven years on Earth. Source

When I first learned about time dilation, which refers to the difference in elapsed time as measured by different clocks due to different relative velocities of the clocks or different gravities acting on the clocks, I couldn’t wrap my brain around it. Time dilation hinges on the idea that time is not a universal constant, rather, the speed of light is constant. This is extremely difficult to understand on Earth–we have no firsthand experience with anything other than the perceived constant, slow march of time steadily ahead. I could not understand how the speed of a photon exiting a flashlight held by a person at rest could possibly be the same as for a photon exiting a flashlight held by a person riding a bike–obviously speeds are additive!

In fact they are not–not where relativistic speeds are concerned. The speed of light is a universal speed limit, no object can move faster. What finally helped me understand this was a YouTube video, which explains a couple of Albert Einstein‘s thought experiments. I would like to explain it here!

While riding the train home from his job as a clerk, Einstein looked at a clocktower behind him, receding into the distance. He could see the second hand ticking as time passed and he moved further away. He thought–what if this train was moving at the speed of light? The clock hand would appear to be still–photons must travel from an object to your eye for you to be able to see it, and if you are moving away from the photon as fast as it is moving toward you, it will never reach your eye. The clock hand would appear frozen in time, even though the clock would still appear to tick away from the perspective of someone standing next to the clock. (Note: technically time can’t be “frozen”–since no object can actually travel at the speed of light, we must settle for very close to the speed of light, so the clock hand will still move very slowly.)

How can this be possible? Isaac Newton‘s laws of motion state that velocities are never absolute, but must be described in relation to something else. For example, a car may travel 60 mph with respect to a person standing on the side of the road, but it will appear to travel 40 mph with respect to a person in a car traveling 20 mph in the same direction. It will appear to travel 80 mph with respect to a person in a car traveling 20 mph in the opposite direction. However, James Clark Maxwell found experimentally that the speed of light is fixed, regardless of who is observing it.

Newton’s laws describe that velocity is relative to some observer. Source

These two ideas are, on the surface, contradicting. How can the speed of light be constant, if speeds must be measured relative to some other object?

Einstein’s solution was to make a small adjustment to Newton’s laws, while still upholding the constancy of the speed of light. He proposed that time must slow down for objects traveling at great speeds, in order to keep the speed of light constant. This was called time dilation–time does not move steadily forward, but can stretch and contract with varying velocity of motion. It is important to note that Newton’s laws still work for non-relativistic situations, which is why we haven’t completely thrown them out!

To accommodate time dilation, Einstein further developed the concept of spacetime–the idea that time and space are not separate entities, but are inextricably intertwined into one entity. Gravity causes distortions of spacetime–this can be imagined as 3D dips in a 2D “fabric” of spacetime, meant to represent the actual 4D spacetime. Smaller objects orbit around larger ones since they are caught in the curved dip of spacetime around the larger object.

We know that the force of gravity on an object increases with decreasing distance from another object–most noticeably if a small object is approaching a larger one. This difference in forces represents a difference in accelerations–imagine a person falling faster as they approach the Earth. Now, consider what we just learned–that the faster you move through space, the slower you move through time. A clock in high-Earth orbit around will tick faster than the clock on your desk, since the gravitational forces, and thus accelerations, on the two are different.

We can look at another thought experiment to understand why this is so. Imagine a person falling from high-Earth orbit down to the ground, carrying a photon clock–a theoretical clock for which it takes one second for a photon to bounce between two reflective surfaces. Another observer stands on the ground. What will they observe as the person falls from space?

The falling person will see the light from their own clock traveling in a straight line back and forth, much like when you throw a ball upward and catch it while traveling in a car–it doesn’t move behind your head. On the other hand, the observer will see the light traveling in diagonal lines, much like if an observer outside the car were to see you throwing the ball up and down. The net movement of the ball, or the photon, would appear to be a zig-zag.

What does this mean for our clock? Light is observed by the person on Earth to travel a greater distance–the diagonal lines are longer than the straight ones. Since the speed of light is constant, and the confines of the start and the end of the event are the same, this must mean that time has gotten shorter. The duration of a second is not constant–it is proportional to the velocity of the object in motion.

In order to accommodate light traveling a longer distance at the same speed, the time between two set start and end events must change. Source

This has been experimentally proven–time records of clocks on spaceships that have spent a decent amount of time in space are different from time records of the same “event” as measured by clocks on Earth. Time passes at different speeds, and thus the total amount of time elapsed is different.

We have explored here time dilation and the theory of relativity–the idea that if the speed of light is constant, which has been repeatedly proven to be true, then time must not be constant in order to compensate for light traveling different distances based on motion and gravity. Thank you all for sticking around, and if you never understood Interstellar, I hope this explanation helps!