Equation of the Day #16: Spacetime

Special relativity is over a century old, and yet due to its odd nature it still confuses people. This is not necessarily our fault – we don’t really experience the effects of special relativity on human scales, since most of us can’t claim to have ever traveled or seen someone travel appreciably near the speed of light. Light speed is just incredibly fast compared to us, traveling one foot in about 1/1,000,000,000th of a second, or about a billion feet per second. Walking speed is about 5 feet per second, so it’s hard for us to even think of traveling anywhere close to light speed; indeed, most of us cannot even fathom what a billion feet looks like. For reference, the moon is, on average, 1.26 billion feet from Earth, so the Earth-Moon distance is a decent gauge of a billion feet. This means that light takes about 1.26 seconds to travel to the Moon from Earth.

Because special relativity is so inaccessible, the other mind-bending aspects of the model seem like they’re out of a fantasy setting. Discoveries made in the 19th century revealed that the speed of light in a vacuum is measured to be the same value by any observer moving at constant velocity. As long as a measurement device is traveling at a constant speed in a fixed direction, it will detect light traveling at 299,792,458 meters per second in a vacuum. Always. So if I’m driving down the highway at 60 mph (about 100 kilometers per hour), I don’t measure the light leaving my headlights as going 60 mph slower (about 299,792,430 meters per second); I still measure 299,792,458 meters per second! This is one of the two postulates of special relativity.

  • The laws of physics are identical for all non-accelerating observers/measurement devices.
  • The speed of light in a vacuum is the same for all observers/measurement devices, regardless of the motion of the light source.

This, understandably, perplexed scientists for years, as it throws our concepts of relative velocities out the window. After all, in everyday life, if you’re traveling on the highway at 70 mph, and the car next to you is traveling at 75 mph, you see the other car as moving 5 mph relative to your seat in your car.

Let’s see if we can reconcile these two seemingly contradicting ideas and start from the assumption that the speed of light doesn’t differ in moving frames of reference, provided those reference frames are moving at constant velocity. If this assumption is false, an experiment will be able to disprove it. The thing that doesn’t change between reference frames is a speed, which by definition is a distance traveled over some interval of time. Perhaps what can change, then, are the distance traveled and the time interval over which that distance is traveled, but they do so in such a way that the overall speed, the speed of light, is invariant. This would be a way for our assumption to stay true. Can we figure out the degree to which the time and/or distance would be altered to preserve the invariant speed of light?

Let’s apply the idea of a light clock. A light clock is a set of two mirrors set a fixed distance apart between which light bounces back and forth. Each time the light hits one of the mirrors, a tick is registered on the clock. If we set the the two mirrors to be exactly 299,792,458 meters apart, then each tick will be one second. if we set the two mirrors to be 29.9792458 cm apart (about 11.8 inches), the ticks would occur every nanosecond. This clock would be easier to construct, so let’s make a clock that ticks every nanosecond; therefore, after a billion ticks, the clock’s second reading would advance by one.

In the image above, the light clock will advance each time light reflects off of mirror A or mirror B. The length L is the distance between the mirrors, which we can take to be whatever length is needed for the desired ticking interval. At rest, the clock operates normally, which should come as no surprise. The time between ticks, then, is the distance traveled from one mirror to the next divided by the speed of the light, c, or

\Delta t = L/c.

The reason for the symbol c for the speed of light is that it comes from the Latin word celer, which means fast/swift (like in the word accelerate). But what happens if the clock starts moving at a constant velocity? We’ll call the clock’s velocity v. In this moving state, the light’s motion looks different; it’s a diagonal line.

Now the light has traveled a distance that’s longer than the the distance between the two mirrors! Since the speed of light is invariant, that means the clock ticked at a different rate than when it was at rest! Fortunately, we can use trigonometry to find this distance, since the distance D, the mirror length L, and the travel distance (the bottom line segment) all form a right triangle. The travel distance is simply the speed of the mirrors, v, multiplied by the elapsed time between ticks. We’ll call this time between ticks Δt′ to distinguish it from the rest time (since we anticipate something different). The time between ticks for this clock will be

\Delta t' = D/c = \sqrt{L^2+v^2 (\Delta t')^2}/c.

You may have noticed that this equation is a bit self-referential. If we rearrange it so that all of the Δt′ terms are isolated, we end up with the relationship

\Delta t' = \dfrac{L/c}{\sqrt{1-v^2/c^2}} = \gamma \Delta t,

where I substituted in the rest frame ticking interval, and defined the Lorentz factor,

\gamma = \dfrac{1}{\sqrt{1-v^2/c^2}}.

The Lorentz factor is the quantity that tells you you’re working with special relativity. For small values of v/c, γ is approximately 1. That means for speeds that are really small compared to light speed, moving clocks don’t appear to have measurable discrepancy between their ticking rates and the ticking rate of a clock at rest. However, when v is a significant fraction of c, the Lorentz factor γ increases, becoming larger than 1 and approaching infinity as v approaches c.

What does this mean for our moving clock? It means that the time between ticks is longer for the moving clock, making the moving clock appear to run slow. This is the phenomenon of time dilation; moving clocks tick slower. Note that this has nothing to do with the clock being faulty in some way; this is just a consequence of the speed of light being the same for all observers.

Now, because the time interval between events (in this case, ticks of a clock) changes due to the invariance of the speed of light, there must be some sort of trade-off involving distance, since

\text{speed} = \dfrac{\text{distance traveled}}{\text{travel time}}.

If moving clocks run slower, that means the travel time is shorter, so for the speed of light to be unchanging for moving observers, the distance traveled of a moving object must change to keep the speed of light invariant. Think about a light clock oriented on its side, so that the travel direction is along the long side of the clock.

The speed of light is unchanged, so if the clock ticks at a slower rate, the light must traverse a shorter distance, and the factor by which the distance is shortened should be the same factor by which the travel time is dilated:

L' = \dfrac{L}{\gamma}.

This phenomenon is called length contraction. Note that the Lorentz factor is in the denominator, which ensures our distance has gotten smaller. If we wanted to, we could play similar games with the light clock to derive this result, though we would need to factor two ticks of the clock, since the sideways light clock has two time intervals associated with its ticks: a longer “forward tick” and a shorter “backward tick.” Nonetheless, the result is the same.

There are two special quantities that arise from time dilation and length contraction known as the proper time and proper length. The proper time is the time measured by a clock in its own rest frame, and the proper time interval is the ticking rate of a clock in its own rest frame. Everyone agrees on the proper time, because everyone should agree what time is measured by an object in its own rest frame. Similarly, proper length is the length of an object in its own rest frame, which all observers must also agree on. These are two more quantities that can be considered invariant, or unchanging with reference frame, in addition to the speed of light. In fact, quite a bit of special relativity consists of finding invariant quantities since they are so useful in connecting reference frames.

This give and take between space and time is what led to the idea that time and space are linked together such that they are one entity: spacetime. The thing that connects space and time together is the invariant speed of light. In a future entry, I’ll talk about visualizing spacetime and how time dilation and length contraction warp how different observers/detectors view events.