#time-dilation

Take two identical, fully functional clocks, and while one is stationary, subject the other to motion or stronger gravity. After a predicable period of time, the stationary clock will somehow have recorded more time than its counterpart. This is time dilation -- and it is very real.

We often see time dilation occur with activities that happen in outer space or at high speeds. The people that traveled to the Moon and spacecraft that have gone to Mars have aged at a slower rate than us on Earth, while on those journeys. Also, the astronauts who inhabit the International Space Station are at a higher altitude than us on Earth and experience a weaker gravitational field. Consequently, they age faster than those on Earth. We can even measure time dilation on Earth while travelling on airplanes with super-sensitive clocks. Most of us are completely oblivious to the effect because it is negligible for our daily activities -- we either need to wear a wristwatch that can measure time in trillionths of a second, or travel at unfathomable speeds for a short time (see end of article for additional examples).

In order to be practical, my experiment can't involve space travel, super-expensive physics laboratory equipment, or large distances. Let's find out if this is achievable.

Goals

Measure 1 microsecond (1 millionth of 1 second) of time dilation in a reasonable amount of time (an hour or day).

Confine experiment to a classroom or storage facility (i.e. garage)

Don't break the bank -- spend at most $2,000, or a budget attainable by a typical science classroom.

For this experiment, we need two very sensitive clocks, far more accurate than your typical stop watch. For a couple hundred dollars, you can buy a used rubidium standard, which can measure time within 10 nanoseconds of accuracy -- more than enough for our purposes. Both clocks will be synchronized and one clock will be isolated to measure the "normal" passage of time.

Next, we can subject the experiment clock to artificial gravity and high speeds by attaching it to a rotating wheel, which gives us the benefit of a small work area. I'll assume I can find a strong-enough wheel with 1m radius to keep the calculations simple (perhaps one from a Penny Farthing bicycle), however, I can resort to using a typical racing bicycle wheel with a 29" (736mm) diameter if needed. The clock will be attached to the outermost part of the wheel and with that said, I now need to figure out how fast to spin the wheel to attain the goal.

I got some help with the required calculations from the physics community at Stack Exchange, as I suspected I used the wrong equations to model this experiment. It turns out that the equation to calculate time dilation when moving in a circle simplifies to the equation for straight-line motion:

$$T_{Moving} = T_{Stationary} \sqrt{1-\frac{v^2}{c^2}}$$ $$v^2 = c^2 (1-\frac{{T_{Moving}}^2}{{T_{Stationary}}^2})$$

It looks intimidating, but the key point for this discussion is that the value of \(v\) (speed of rotation of a 1m wheel) needs to be close to \(c\) (speed of light in a vacuum: \(c = 2.998\times10^8 m/s\)) in order for the time dilation (\(T_{Stationary}-T_{Moving}\)) to be significant. You can also say that an insignificant time dilation effect is a result of a very large gap between \(v\) and \(c\). Is 1 microsecond insignificant enough when spinning the wheel for an hour?

Unfortunately, the answer is a very definitive no. Wolfram Alpha computes that the wheel would need to spin at a whopping 7km/s!

For this sized wheel, the clock would experience almost 5 million times the gravity that you and I experience every day! Needless to say, the wheel and clock would fall apart well before reaching the required speed.

Clearly, this isn't practical. Let's vary some of the parameters to see if any speed can result in a measurable time dilation with our clock.

Time Dilation Wheel Rotation Time Wheel Diameter Wheel Rotation Speed Effective g 1µs 1h 1m 7.066 km/s ~5,000,000 1µs 1 day 1m 1.442 km/s ~212,000 1µs 7 days 1m 545.2 m/s ~30,000 1µs 28 days 1m 272.6 m/s ~7430 1µs 120 days 1m 131.7 m/s ~1735 1µs 365 days 1m 75.5 m/s ~570 25ns 1h 1m 1.117 km/s ~127,000 25ns 1 day 1m 228 m/s ~5200 25ns 7 days 1m 86.2 m/s ~743 25ns 28 days 1m 43.1 m/s ~185 25ns 120 days 1m 20.8 m/s ~43 25ns 365 days 1m 11.9 m/s ~14 25ns 28 days 0.368m 117.1 m/s ~3730 25ns 120 days 0.368m 56.5 m/s ~870 25ns 365 days 0.368m 32.3 m/s ~284

Some quick notes about the above calculations:

Wolfram Alpha's Time Dilation calculator has a maximum resolution. Once I exceeded it, I resorted to solving for \(v\) manually.

To compute \(v\) for a smaller wheel radius, I used the corresponding speed of the 1m wheel and scaled it accordingly using the rules for conservation of angular momentum.

If you're not familiar with this concept, feel free to skip the following math. Essentially, these are the same rules that make figure skaters spin faster or slower by tucking in or extending their arms.

$$L = I_{old}\omega_{old} = I_{new}\omega_{new}$$ $$\frac{m{r_{old}}^{2}v_{old}}{r_{old}} = \frac{m{r_{new}}^{2}v_{new}}{r_{new}}$$ $$v_{new} = \frac{r_{old}v_{old}}{r_{new}}$$ $$v_{new} = \frac{1}{0.368}v_{old} = 2.717v_{old}$$

I was somewhat disappointed with these results as the time needed to measure time dilation at a reasonable and safe speed is very long, generally ruling out any type of classroom demonstration. If we use the information from a Yahoo! Answers post about the limits to how fast a bicycle wheel can spin before failure (154mph = 68m/s), we quickly realize that we can't do better than waiting almost 4 months for the experiment to show a 25 nanosecond (25 billionths of a second) time dilation, which also approaches the limits of what our clock can measure. Furthermore, it is reasonable to assume that the speed limit for the wheel applies to high-performance racing wheels -- large wheels from a Penny Farthing bicycle are not built to such specifications.

Budget-wise, this all seems very doable for around $2000. As mentioned earlier, used rubidium clocks cost about $200. A single carbon-fiber performance wheel costs about $1000, and the remainder of the budget needs to be spent on an electric motor and required gears. David Cole gives information on how to design that aspect on his website, 900mpg.

Perhaps there is hope for having such an experiment on display in a science center/museum. Some people might be drawn in every four months to see the start, end, and restart of the experiment, accompanied by someone talking about relativity, angular momentum, or whatever else might be, ahem, relative to the discussion.

Further Reading

Brian Cox discusses time dilation, its derivation, and use in GPS navigation [video]

Brady Haran interviews professor Mike Merrifield on the topic of time dilation [video]

The Hafele-Keating experiment -- flying clocks around the earth [video]