Time and Distance

Notes from "Chapter 5: Time and Distance", from The Feynman Lectures on Physics: Volume I.

5.1 Motion

The development of the physical sciences has depended to a large extent on making quantitative observations. Galileo is recognized as the founder of this method. He experimented with rolling balls down an incline, and measured how far they went in how long a time. Since there were no modern clocks, he used his pulse to approximate time.

5.2 Time

What really matters is not how we define time, but how we measure it. One way of measuring time is to use something which happens over and over again in a regular fashion—something which is periodic. For example, a day. We can also engineer something that’s periodic, e.g. the time it takes a fixed amount of sand to flow through an hourglass.

While it doesn’t prove that either a day or an hourglass is actually periodic, the comparison and consistency of the two gave early scientists confidence that they are.

5.3 Short times

Galileo noticed that a pendulum swings back and forth in equal intervals of time so long as the size of the swing is kept small. If we use a mechanical device to count the swings—and to keep them going—we have the pendulum clock of our ancestors. If our pendulum oscillates 3600 times in one hour (and if there are 24 hours in a day), we call each period of the pendulum one “second.”

We can also measure time using a different technique, by observing the distance an object has traveled if the velocity of that object is known. With this techniques, we’ve been able to infer the duration of faster physical events—say the time it takes light to cross the nucleus of a hydrogen atom.

5.4 Long times 

We’ve been able to count days and years for a while, since someone has been around to count them. But how do we know the age of objects that came before us?

One of the most successful methodologies is using radioactive material as a clock. We find that the radioactivity of a particular sample of material decreases by the same fraction for successive equal increases in its age.

We know, for example, that the carbon dioxide in the air contains a certain small fraction of the radioactive carbon isotope $C^{14}$ (replenished continuously by the action of cosmic rays). If we measure the total carbon content of an object, we know that a certain fraction of that amount was originally the radioactive $C^{14}$; we know, therefore, the starting amount A to use in the formula above. Carbon-14 has a half-life of 5000 years. By careful measurements we can measure the amount left after 20 half-lives or so and can therefore “date” organic objects which grew as long as 100,000 years ago.

Uranium, for example, has a half-life of $10^9$ years. Uranium disintegrates into lead as it decays. We’ve been able to use this method to infer that certain rocks are several billion years old. Using averages of uranium and lead found throughout the planet, we’ve been able to estimate that the Earth itself is about 4.5 billion years old.

Meteorites that have hit the Earth have found to be about the same age, which is an encouraging sign that this methodology is correct (implying that these meteorites are left over material in space from when the Earth formed).

We now believe that the universe began about 14 billion years ago. We do not know what happened before then, or if anything could have happened (does time have any meaning without the periodic measurements we’ve established?)

5.5 Units and standards of time

There is a story of a Swiss boy who wanted all of the clocks in his town to ring noon at the same time. So he went around trying to convince everyone of the value of this. Everyone thought it was a marvelous idea so long as all of the other clocks rang noon when his did!

For a long time the rotational period of the earth has been taken as the basic standard of time. However, it has been found that the rotation of the earth is not exactly periodic, when measured in terms of the best clocks. These “best” clocks are those which we have reason to believe are accurate because they agree with each other. We now believe that some days are longer than others, and on the average the period of the earth becomes a little longer as the centuries pass.

We’ve recently gained experience with some natural oscillators which we now believe would provide a more constant time reference than the rotation of the Earth. These are based on natural phenomenon available to everyone (aliens?). These are the so-called “atomic clocks.” Their basic internal period is that of an atomic vibration which is very insensitive to the temperature or any other external effects. These clocks keep time to an accuracy of one part in $10^9$ or better.

5.6 Large distances

The classical way of measuring distance would be to use a meter stick. Historically, this was defined as 1/10,000,00 of the circumference of Earth through Paris to the equator. However, by 1960, the meter was redefined as equal to 1,650,763.73 wavelengths of the orange-red line in the spectrum of the krypton-86 atom in a vacuum.

But what about for larger distances. Say, between two mountain tops? For that, we’ve been able to use triangulation to calculate the distance geometrically.

We’ve also been able to measure distances using radar, such as the distance from Earth to Venus. This is an inferred measurement, which depends on assumptions about the speed of light (and therefore, radar waves). From the amount of time it takes to receive a reflection, we infer the distance.

5.7 Short distances

We cannot “see” objects smaller than the wavelength of visible light (about $5 \times 10^{-7}$ meter), but we are able to photograph things on a smaller scale using an electron microscope.

By indirect measurements—by a kind of triangulation on a microscopic scale—we can continue to measure to smaller and smaller scales. First, from an observation of the way light of short wavelength (x-radiation) is reflected from a pattern of marks of known separation, we determine the wavelength of the light vibrations. Then, from the pattern of the scattering of the same light from a crystal, we can determine the relative location of the atoms in the crystal, obtaining results which agree with the atomic spacings also determined by chemical means. We find in this way that atoms have a diameter of about 10−10  meter.

Atoms are about $10^{-10}$ of a meter, nuclei are substantially smaller, and about $10^{-15}$ meter. This unit, $10^{-15}$, is known as a fermi, in honor of Enrico Fermi.

A note on relativity: measurements of distance and of time give results which depend on the observer. Two observers moving with respect to each other will not measure the same distances and times when measuring what appear to be the same things.