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Relativity is mind-bending. Rightly so, Einstein and his equations represent some of the most amazing advances in modern physics. But fundamentally, many of these counter-intuitive space-time properties can be derived with simple thought experiments and some algebra. Of course, discovering that time moves differently for different observers at different velocities takes a flash of insight that few of us have, but at least we can proceed from there.
The fact that the speed of light is the same in all reference frames has the consequence that moving clocks run slow. This stunning fact applies to time itself; astronauts orbiting the earth at speed biologically age more slowly than the rest of us. So, if two events occur at the same place, such as the ticks of a clock, a moving observer will measure the time between the events to be longer.
We can start deriving how time differs for stationary and moving observers by considering a light clock. This imaginary clock would work by ticking every time a light pulse (a photon moving at the speed of light) reflected back to the lower mirror as shown below:
In the diagram, “h” represents the vertical distance between the mirrors. The stationary observer here observes a particular time, but what about a moving observer? The diagram below shows the change:
Here we have moving mirrors with the photon moving at the same speed and hitting the same place on the mirrors (so that it would be identical to the first diagram if the mirrors were not moving). All the parameters here are the same, expect for the movement, and relativity states that then the time will change. Let’s derive that change for ourselves, no Einstein required. I have outlined each step in the necessary tedium below:
There you have it! With some simple algebra and a clever thought experiment, we have derived ourselves how time changes as you move closer and closer to the speed of light.
Let’s give it a quick test run. Suppose that we wish to get to the nearest star beyond our sun, Proxima Centauri.
Proxima Centauri (center inset) as seen by 2MASS
Proxima Centauri is 4.24 light years away from Earth. The speed of light is around 186,282 miles per second (299,792,458 meters per second). This means that the star is nearly 25,000,000,000,000 miles away (38,000,000,000,000 kilometers).
To get there, we could use one of the fastest spacecraft ever built:
The figure always cited here is the heliocentric speed of Voyager 1, some 17.05 kilometers per second, which is faster than any of our outward bound spacecraft…
At this speed, it would take us over 70,000 years to get there! But what if we could move fast enough to make our equation that we derived above matter? What if we could move at 30% the speed of light? How long would that take us?
At 30% the speed of light, our trip would shorten to only 14 years. But time dilation then becomes a factor:
This means that in our ship we would be .65 years or around 8 months younger than people back on Earth when we reached the star! And this effect get exponentially larger as we get closer to the speed of light, as seen in the graph below:
For example, at 99% the speed of light, for our 14 year journey people on Earth would age 85 years more than us!
By the time we returned from our journey, because of time dilation, everyone we knew would have long since died. This is the peril of long-distance space travel.
And all this from a mirror-clock and some algebra.
Science-Based Life 
I’ve always wondered. Why is it the spacecraft that ages less? Isn’t the planet moving just as fast away from the spacecraft as the other way around? Is it the gravity well that makes the planet the origin point, or the universe in general, or what?
The “origin point” in your example would be a stationary observer on the planet. Suppose a stationary observer watches a spacecraft take 20 years to reach some other planet. To him, the spacecraft does in fact take 20 years to travel the distance. However, to an observer in the spacecraft, time actually expands (though he experiences the flow of time normally) when traveling close enough to the speed of light, enough that he might only experience a few months.
I thought the whole point of relativity is that light moves the same speed regardless of which point you observe from.
Actually, my question would be better posed as…two asteroids lie outside a solar system. One person on each. They both move away from each other at a percentage of light speed. Will one observer age faster than another?
Sorry for being dense, but this has actually bothered me for more than two decades.
quote: “Let’s derive that change for ourselves, no Einstein required.”
The fact that you’re assuming a constant speed of light in ALL inertial frames is precisely using Einstein (it’s mentioned explicitly in his theory of special relativity). In fact, the speed of light being constant in all inertial frames is actually _more_ counter-intuitive than time dilation for moving objects.
Trying to explain time dilation while assuming constant speed of light in all inertial frames is like trying to explain Heisenberg’s Uncertainty Principle while assuming Bell’s inequality to be true. Kinda defeats the purpose.
I think the confusion is thinking of the relative speed (v) rather than the acceleration. It’s true since there is no absolute reference point in the Universe you can’t officially say the spacecraft is moving and the earth is not, you can say that the spacecraft has had acceleration acted on it, it’s the acceleration that dilates time. Also described by Einstein is that gravity is the same as acceleration, so being in a strong gravity field also slows time for that person.
If you analyze absolute motion present within an absolute Space-Time continuum, you end up finding a full understanding of relativity and produce all of the SR equations in almost no time at all. It’s a breeze, just as seen here…