How Big is Big and How Small is Small
Page 13
If the strong force is only a hundred times stronger than the electromagnetic force, why are the energies a million times different? This is because the constituents of an atom are almost 100,000 times farther apart than the constituents of a nucleus. The electromagnetic force drops off dramatically with this extra distance.
We can also use this technique to compare the strength of the electromagnetic force to gravity. This time we will use the simplest and most common atom to compare: hydrogen. The hydrogen atom is made of an electron and a proton. Electrons and protons are bound together primarily because they have a charge and therefore an electromagnetic attraction. They also have mass, and therefore exert a gravitational attraction. The electromagnetic attraction is 1036 times stronger than the gravitational attraction. In fact if you were to consult a number of references that compare forces, this is the ratio which is most often cited. This ratio is independent of separation distance, unlike our previous comparison, since the electromagnetic force and gravity tail off in exactly the same way at large distances. But the ratio is not without caveats.
The electromagnetic repulsion between two protons is only about 1033 times greater than the gravitational attraction. However, in the exotic and unstable positronium atom—made up of an electron and a positron (anti-electron)—the ratio is about 1039. What is happening here is that in all three cases the electromagnetic attraction or repulsion is exactly the same, but gravity depends upon the mass of the particle involved. The reason why I have offered the positronium and repelling proton case is that it reminds us that the commonly quoted ratio of 1036 is not an absolute of nature. It is a function of the system we picked to use when comparing the forces. The comparison of forces may be instructive, but it is not absolute.
So here we are at the end of our discussion of energy. We were trying to find some large guiding principle such that we could order small energy objects at one end of our scale and large energy objects at the other end. We then saw a separation between chemical fuels and nuclear fuels, but energy related to gravity did not fit. We also saw that chemical and nuclear forces are not fundamental, but rather forces derived from the residues of the electromagnetic and strong forces. There are trends here, but still no deep enlightenment.
There is one force I have not discussed much: the weak force. This is responsible for radioactive decays. Like the strong force it is short ranged. Also, as the name implies, it is very weak: about 1–7 orders of magnitude weaker than the electromagnetic force. The gravitational force may be trivial in strength, but it does have a long reach, all the way across the universe. The strong force might be short ranged, but it is powerful. The weak force is both short ranged and feeble. It also does not contribute to the shape of anything in the universe and so it is tempting to neglect it. But the weak force lies behind radioactive decay and if there was never a radioactive decay much of the energy tied up in the nuclear force would never be released. The weak force is what unbalances the systems of quarks, neutrons and protons and sets the pace for the release of nuclear energy. It is not only the reason that stars burn, but why they burn over eons and not instantly. It can be argued that the universe has a history of 1017 seconds and is still dynamic and still unfolding because of this force.
8
Fleeting Moments of Time
I am sitting on the grass of Salisbury Plain staring at Stonehenge and thinking about time. When you are sitting inside of earthworks that are 5000 years old, looking at stones that were dragged here as much as 4500 years ago, it is hard to not think about time. It is tempting to think about this stone ring as a hole in the fabric of the universe, a portal into another age, an era before the machines of today. Can I see a bit of the Neolithic world? Or perhaps it is the other way around and it is a bit of the Neolithic world protruding into our own age? It may be tantalizing to think about Stonehenge in those terms, even picturesque and poetic, but it really is not quite right.
Those blue stones were raised up on end about 4500 years ago and they have passed through all 4500 intervening years to get to the present. They are not in the Neolithic world any longer and they bear the scars of age to prove it. I am sitting on the plains of Salisbury in the twenty-first century on a beautiful sunny August day. The forests of the Mesolithic and Neolithic have vanished and instead there are sheep grazing on the downs. I am in the midst of a crowd of a hundred people speaking a dozen languages and coming from six continents. The stones themselves are also travelers, much older than a mere 4500 years. Geologists tell us that the blue stones of Stonehenge came from the Preseli Hills in Pembrokeshire in the southwest corner of Wales. They were formed in the Ordovician era, about 450–500 million years ago. So how old is what I am looking at? These stones and earthworks were arranged here between four and five thousand years ago. But it was in a volcanic eruption 100,000 times older than Stonehenge that the blue stones were formed. And the atoms and quarks within the stones? They date back to a time shortly after the big bang, 13–14 billion years ago.
The formation of the stony Welsh mountains, the building of Stonehenge, this day when I sit in the sunshine on the grass of Salisbury Plain are all “events.” It is time that separates events that take place in the same location. I may be at Stonehenge, but I am not witnessing John Aubrey performing his seventeenth century survey, or the Battle of the Beanfield of 1985. I am not bumping into the Neolithic engineer who paused at this spot to consider the stone’s alignment. For we are separated by 4500 years.
***
Over two chapters I will describe the many scales of time: in this chapter short time and in the next chapter long time. We will look at the types of events that are measured in years or seconds, in eons, or in the “shakes of a lamb’s tail.” We will also look at how we measure time and how we would recognize a good clock. Also how we can date the universe, measure the lifetime of an exotic, fleeting particle, or measure the speed of a nerve pulse.
***
Time seems like such a simple thing as it flows by us, or as we pass through it, from the past into the future. We like to think of time as a line, and even chart history on a timeline, with the Romans a long way off to the left, the Middle Ages closer and the present right in front of us. And off to the right? That is where the question mark goes; that is the unknown future. So we can think of time as just a line, like latitude or longitude, that we travel along. In fact, in relativity, we are taught to calculate the separation of events by combining latitude, longitude and elevation (x, y, z) with time (t) in an equation that looks like the Pythagorean theorem for triangles. However, time is not just like distance in a different direction. We are always compelled to travel along it. We cannot sit temporally still and stay in the same moment, and we most certainly cannot move backwards. That makes time a unique type of dimension. We must follow time’s arrow in one direction, towards the future.
For millennia, people have written, discussed, and argued about what time really is and the nature of this thing called time’s arrow? Is it a necessary feature of the universe? Is it a consequence of entropy and the second law of thermodynamics? Or is it the other way around? The philosophy of what time is is beyond the scope of this book. Instead I will pose two simpler questions that we can answer. Why do we know time exists? And how do we measure it?
We know time exists because we see change. That may seem self-evident, but not to Parmenides of Elea. Parmenides (520–450 BC) could not understand how things could move, because in his view the universe was packed tight. He rejected the idea of empty space and is attributed with saying “Nature abhors a vacuum.” In his view the world is tightly packed and so you have no space to move into. Therefore motion is an illusion and time is unreal. His most famous student, Zeno, left behind a number of motion paradoxes to puzzle us. I will, however, take the more pragmatic view that motion is not an illusion, for if it was, trying to do anything, such as write a book, would be pointless.
***
We know time exists because we see change, whi
ch also gives us a handle on how to approach the second question: how do we measure time? To measure time we look at how something is changing. For example, I could mark time by the growth of a tree or the march of a glacier. Since the last time I visited these mountains it has been 25 tree-millimeters or 5 glacial-meters. But these are not the best candidates for a clock because on my next return I may find only 22 tree-millimeters but 8 glacial-meters of time, meaning these two timepieces are out of synch. What I want for a good clock is something with consistent motion or constant change.
Humans have used the Sun, the Moon and the stars to mark time since before our collective memory. The day, as marked by the passing of the Sun, is ingrained in our daily routine. We rise and set with the Sun. But we needed to subdivide the day into smaller chunks of time—into hours—and the most natural tool for doing this is the sundial. The shadow cast by a stick, or gnomon, sweeps across the ground as the day progresses. But the rate of that sweep is not uniform. It creeps at mid-day and rushes when the Sun is on the horizon. The difficulty arises from the fact that the Sun follows a circular path—an arc—across the sky whereas the shadow is confined to a flat plane. Not only is there this arc-versus-plane mismatch, but the arc is tilted at an angle that depends upon your latitude on the globe as well as the season.
Through good design you can solve part of this problem. The sundial face need not be a flat plane. If it too is an arc then the shadow will march an equal angle and distance for every hour. Also you can tip the dial to compensate for your longitude and in the process create a beautiful, useful and accurate time piece … almost. The most obvious problem is what do you do the other half of the time: at night?
Determining the hours through the night can be done with something called an astrolabe. This essentially measures the positions of stars and converts that information to hours, much like a sundial measures the position of the Sun. An astrolabe needs to be a bit more sophisticated than a sundial because the night sky changes throughout the year. For example, Orion is a winter constellation whereas Scorpio is most visible in the summer sky. However, since the march of the constellations across the sky is very regular an astrolabe has a disk that you turn to the date, and then you sight one of many stars that are visible and read off the hour.
Cloudy days call for a different solution, but humans have been resourceful and have solved this problem with clocks based on dripping water, burning candles and flowing sand. In all these cases we knew that we had a good and regular clock if it exhibited a constant motion or rate of burn, especially when compared to a sundial or astrolabe. A good clock is one that agrees with the Sun.
As our pendulum clocks became more and more accurate a problem in our very concept of a day became more and more apparent. With a pendulum clock we could compare the hours of a sundial and astrolabe and find that they did not agree; that is, the Sun and the stars disagreed on the length of a day. To understand this problem we need to step back and think about what we mean by a day. A day is the amount of time it takes for the Sun to pass due south of you (if you are in the northern hemisphere) until the next time it passes due south of you. Now let me build my best pendulum clock and adjust the length of the pendulum such that it ticks once a second. A day contains 24 hours, each hour has 60 minutes and each minute has 60 seconds, so there are 86,400 seconds in a day. Now in the middle of July I meticulously adjust my pendulum such that from noon to noon my pendulum swings pack and forth 86,400 times. I can even cross check that with the stars. I can count the ticks from the time when Altair, a star in the constellation Aquila, crosses my north–south meridian one night until it does so again the next night. I count 86,164 seconds. This is okay because stars mark out sidereal time and the Sun solar time and they shift by about four minutes a day because we are going around the Sun. Four minutes is 24 hours divided by 365 days. This is the same reason that different constellations show up in the summer and winter sky.
To prove that our clock is a good one we will repeat our measurement at the beginning of the New Year. Altair is not visible at this time of year, but we can use Betelgeuse in the constellation of Orion. The time for Betelgeuse to transit the meridian from one night to the next is 86,164 seconds. The sidereal day is the same as when I measured it in July. But my measurement of the Sun from noon to noon is about 86,370 seconds! The day is half a minute shorter than we expected.
Sundials gain time from February to May and again from August to November. They lose a little bit of time from May to August and a lot of time from November to February; as much as half a minute a day. The reason for this is that the axis of the Earth is tilted compared to the plane of our orbit and the fact that our orbit is not circular; it is an ellipse with our motion speeding up as we approach perihelion (January 3), the point where we are closest to the Sun. That means that this time creep is very regular and we can adjust for it. The adjustment is called the equation of time. Sometimes it is combined with the north–south seasonal changes to give a figure-of-eight figure called the analemma (see Figure 8.1).
These adjustments are calculable, but they leave one with a dissatisfied feeling. Whenever we have a new technique for measuring time we find that it is better than previous techniques, but it never seems to be perfect. Further corrections and adjustments always seems to linger. We are left wondering if we always will need to tweak and fine-tune our clocks.
The stars really are a good way of marking time. They travel across the sky at a very regular pace because the Earth is rotating at a very regular rate. Our home planet is a ball spinning in frictionless space and, by Newton’s laws and the conservation of angular momentum, we will keep spinning at the same rate far into the future. The same stars will cross the meridian every 23 hours, 56 minutes and 4.0916 seconds … almost … if it were not for the tides.
Figure 8.1 The analemma, as viewed from New York City. The location of the Sun at 12:00 noon changes through the year: north–south due to the season, east–west due to the shape of our orbit. The dotted vertical line is due south. It is not centered since New York City is not in the middle of its time zone.
The Earth’s rotation rate is actually slowing down and the speed at which the Moon orbits is speeding up. It is not changing much. Our days will be a second longer 40,000 years from now and if our Sun were to burn long enough the Moon’s orbit and our rotation would eventually synchronize. A lunar month would equal a solar day. This is because of tides. We think of tides as a seaside phenomena, but the Moon pulls on the waters mid-ocean as well. In fact the Moon also pulls on the rocks in our planet, flexing and distorting our crust twice a day, each and every day. This flexing of rocks from the tidal effect works like friction, slowing down our rotation and every century adding 2.3 ms to each day. But tides are not the only thing affecting the Earth’s rotation rate.
The Earth is changing its shape and has been since the end of the ice age. With glaciers retreating, continents, especially near the poles, have undergone rebound and are rising up. This changes the rate of our rotation in the same way a spinning ice skater can change their spin rate by pulling in their arms. This speeds our day up by −0.6 ms per day per century. Tidal and post-glacial rebound combine to give a slowing of 1.7 ms per day per century. That does not sound like much, but it adds up. The year 1900, which was taken as the standard year, is now a century ago, which means our days are now 1.9 ms longer than in 1900, and by 2100 they will be 3.4 ms longer. Presently that 1.9 ms adds up to a whole second after about 526 days at which point we need to add a leap second to our clocks. The rate of correction will only increase.
In addition there are other things going on that have less predictable effects. For example the Indonesian earthquake in 2004, which led to a tsunami, shifted enough mass under the Indian Ocean to effect the Earth’s rotation rate by 2.68 μ s. When the winds pick up for El Niño, the Earth slows down. When ocean levels change and their mass moves, these changes are measurable, but have somewhat erratic effects on the length of day. This is why
the adding of leap seconds to our clocks is sporadic. It starts to make one think that the rotation rate of the Earth might not be the best way, the ultimate way, of measuring time.
We need to go back to the question of what does it mean to say that we have a good clock? It means that when this clock marks out a day or a second it is the same amount of time as the next day or second that it marks. But that does not tell us how we know. The way we know that a clock design is good is to build a bunch of independent clocks, start them together and see how they vary after a while. If we have ten clocks of one design and after a day they vary by an hour, that is not a good design. However if after a year they very by a tenth of second, that is a much better timepiece.
Through the centuries, horology, the study of the measurement of time, has devised a number of techniques for building clocks. These include the chronometers used to solve the longitude problem of navigation. Every generation created more and more accurate clocks. Today the gold standard of horology is a type of atomic clock called the cesium-fountain clock. The title of the best clock is a coveted prize and is constantly shifting between various national standards institutions and observatories throughout the world. In fact, we can only determine how good a clock is by comparing it to other accurate clocks. In the US, there are cesium-fountain clocks at the Naval Observatory and the National Institute of Standards and Technology. In Germany there is one at the Physikalisch-Technische Bundesanstalt. In the UK the National Physical Laboratory hosts NPL-CsF2, which at this time is the best clock on Earth, accurate to 2.3 × 10−16 or 10−11 seconds a day. Another way of think about that is if you had two clocks like NPL-CsF2 and set them going, after about 100 million years they would differ by no more than about a second. If you had started these clocks at the time of the Big Bang, they would now differ by no more than two minutes.