Ask a Physicist: Time Dilation

This week, Amandeep from Toronto wrote in to ask:

According to Einstein’s theory of relativity time slows down as speed of the object increases. What is the rate of change of time? E.g. if time was being measured by a simple clock, can we see the hands of the clock slowing down at a certain rate as a result of increase in speed?

Great question—and a surprisingly easy one to answer! There’s a relatively simple equation that relates dilated time to regular time, and it’s something that you’ll become very familiar with if you take courses involving relativity: the Lorentz factor.

Here, v is your velocity relative to the frame of reference for “normal time”.

But the equation by itself isn’t very interesting; the exciting part is that, once it’s explained, it’s actually very easy to understand where this equation comes from, using nothing more than one of the most basic results in geometry: the Pythagorean theorem.

For our readers who are unfamiliar with the concept of “time dilation”, it’s a prediction of special relativity which says, as Amandeep’s question mentioned, that time passes slower for you the faster you’re moving. It sounds more like science fiction than science fact, that you could “travel to the future” by making a round-trip at near lightspeed, but it’s been tested repeatedly at small scales and proven real every time. (It is, however, used as a plot device in such sci-fi classics as The Left Hand of Darkness and Ender’s Game—both of which I highly recommend.)

The prediction inevitably results from the principle that the speed of light is constant to all observers. This is the cornerstone of special relativity, and it’s a slight departure from our intuitive understanding of the world. When you throw a tennis ball forward at 20 mph while in a car moving 40 mph, the ball will appear to move 60 mph to someone on the side of the road. Light, however, will always appear to move at c—2.99*108 meters per second—whether you’re in the car or on the side of the road. The only difference is that the person on the side of the road would see light from the car as being higher in energy, or blueshifted. This is a lot like how the speed of sound is constant in air: the sound from an ambulance’s siren doesn’t reach you faster if the ambulance is moving toward you, but it does seem higher in pitch.

But time dilation is another story. Imagine you’re on a spaceship passing by Earth—a futuristic one with a big bay window to let you look out at planets and stars as you whiz by—and you’ve got a mirror on both the floor and the ceiling, perfectly parallel to each other. (The classic thought-experiment used to demonstrate this idea involves mirrors on trains, but why not give it a bit of a 21st-century upgrade?)

Now let’s say you shine a light upward, toward the mirror on the ceiling, and quickly pull your hand out of the way so that some of the photons are stuck bouncing between the two mirrors. If you measured the speed of this light, you’d find it to be the standard c (with minor adjustments for the air in your ship).

But as your ship cruises by Earth at a high speed, someone on the ground looks up through a telescope and sees into your ship through the window. From their perspective, the light between those mirrors is not bouncing straight up and down, the way it is in your frame of reference—it’s also moving with the ship, tracing out a zigzag path.

From the perspective of someone on the ship (left), the light is moving straight up and down. To an observer on the ground, though (right), it covers a greater distance.
Image Credit: Sacamol, via Wikimedia Commons (CC BY-SA 4.0)

This is the crux of the issue—light is supposed to move at the same speed for all observers, but is clearly tracing out a longer path between bounces for the observer on the ground.

Puzzling over this, Einstein realized there was only one way to resolve this paradox while maintaining the constancy of the light’s speed: to warp time itself for the person on the ship, so that light takes longer to move between the mirrors. The spacefarer will still measure the light as moving up and down at c, but this is because his clock is ticking slower than the clock of the person on the ground.
So, to finally answer Amandeep’s question: how much slower? If you take a look at the diagram above, you can see why we brought up Pythagoras earlier: the path taken by light aboard the ship is the hypotenuse of a right triangle. The distance between the mirrors forms one leg of the triangle, and the distance covered by the ship in the time between bounces forms the other leg. So, if we recall the Pythagorean theorem: A2+B2=C2, the length of the hypotenuse D in the diagram above can be calculated with:
Since L and D both depend on the distance between your mirrors, finding a general equation to relate the spacefarer’s time to the ground-observer’s time means a little bit of substitution and rearrangement, so the equations only involve the time between bounces—represented by (∆t) and (∆t’) in the diagrams above for the observer on the ship and ground, respectively.
Substituting, rearranging, and simplifying this equation gets us to the equation:
…which you’ll recognize as the basis of the Lorentz factor we saw at the beginning! So if our ship is moving at 0.95c, doing the math tells us that one second of ship time (∆t) is equivalent to about 3.2 seconds of time on the ground (∆t’)! 
There’s lots more fun weirdness in special relativity—reciprocity and the Twin “Paradox” are particularly interesting—but that’s all we have time for today.
Thanks for reading, and remember to submit your questions on our “Ask a Physicist” page!
Stephen Skolnick

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