Fear of Physics (24 page)

Since Dirac’s proposal, many direct and indirect tests have been performed to check to see whether not only the strength of gravity but also the strength of the other forces in nature has been changing with time. Very stringent astrophysical and terrestrial limits have in fact been set on the variation of fundamental constants, and no definitive evidence of variation has been observed. While one group has claimed evidence, based on observations of the spectrum of light from distant sources, for a possible variation of 1 part in 100,000 in the strength of electromagnetism over the past 10 billion years, at present there is no independent confirmation of this claim, which many physicists argue could easily be due to complicated atomic interactions that must be taken into account. On other fronts, observations of the abundance of light elements created in the big bang compared to theoretical predictions
obtained by utilizing today’s fundamental constants in the calculations are sufficiently good, for example, to imply that the strength of gravity could not have changed by more than about 20 percent in the 10 billion or so years since the universe was only one second old! Thus, as far as we can tell, gravity doesn’t change with time.

Nevertheless, even if the formulation of microphysical laws
were
tied in some way to the macroscopic state of the universe, we would expect the underlying physical principles that tie the two together to remain fixed. In this case, it would always be possible to generalize our definition of energy so that it remains conserved. We are always free to generalize what we mean by energy as new physical principles arise on ever larger, or smaller, scales. But that
something,
which we can then call energy, remains conserved, as long as these principles do not change with time.

We have already had a number of occasions to revise our concept of energy. The most striking example involved Einstein’s special and general theories of relativity. These, I remind you, imply that different observers may make different, but equally valid, measurements of fundamental quantities. As a result, these measurements must be considered relative to a specific observer rather than as absolute markers.

Now, when trying to understand the universe as a whole, or indeed any system where gravitational effects become strong, we must utilize a generalization of energy that is consistent with the curvature of space-time. However, if we consider the dynamics of the universe on scales that are small compared to the size of the visible universe, the effects of curvature become small. In this case, the appropriate definition of energy reduces to its traditional form. This in turn allows an example of how powerful the
conservation of energy can be, even on cosmic scales. It is conservation of energy that determines the fate of the universe, as I now describe.

 

 

It is said that “what goes up must come down.” Of course, like many an old saying, this one is incorrect. We know from our experience with spacecraft that it is possible to shoot something up off the Earth’s surface so that it doesn’t come back down. In fact, a universal velocity (the same for all objects) is required in order to escape the Earth’s gravitational pull. (If this were not the case, the Apollo missions to the moon would have been much more difficult. The design of the spacecraft would have had to take explicitly into account how much each astronaut weighed, for example.) It is the conservation of energy that is responsible for the existence of a universal escape velocity.

We can define the energy of any object moving in the Earth’s gravitational pull in terms of two parts. The first part depends upon the velocity of the object. The faster it is traveling, the more of this energy of motion—called
kinetic
energy, from the Greek word for motion—it has. Objects at rest have zero kinetic energy. The second part of the energy an object can have in a gravitational field is called
potential
energy. If a grand piano is hanging from a rope fifteen stories high, we know that it has a potential to do great damage. The higher something is, the more potential energy it has, because the greater might be the consequences should it fall.

The potential energy of well-separated objects is considered to be a negative quantity. This is merely a convention, but the logic behind it is this. An object at rest located infinitely far away from the Earth or any other massive body is defined to have zero total
gravitational energy. Since the kinetic energy of such an object is zero, its potential energy must be zero as well. But since the potential energy of objects decreases as they get closer and closer to each other—as the piano does when it gets closer to the ground—this energy must get more negative as objects are brought together.

If we stick with this convention, then the two parts of the gravitational energy of any object in motion near the Earth’s surface have opposite signs—one positive, one negative. We can then ask whether their sum is greater than or less than zero. This is the crucial issue. For
if energy is conserved,
then an object that starts out with negative total gravitational energy will never be able to escape without returning. Once it gets infinitely far away, even if it slows down to a halt, it would then have zero total energy, as I described above. This is, of course, greater than any negative value, and if the total energy starts out negative it can never become positive, or even zero, unless you add energy somehow. The velocity where the initial (positive) kinetic energy is exactly equal in magnitude to the (negative) potential energy, so that the total initial energy is zero, is the escape velocity. Such an object can, in principle, escape without returning. Since both forms of energy depend in the same way upon the mass of an object, the escape velocity is independent of mass. From the surface of the Earth, for example, the escape velocity is about 10 kilometers per second, or about 20,000 miles per hour.

If the universe is
isotropic
—the same everywhere—then whether or not the universe will expand forever is equivalent to whether or not an average set of well-separated galaxies will continue to move apart from each other indefinitely. And, barring exotic new sources of energy, such as the energy of empty space, which we shall consider in a bit, this is identical to the question
of whether or not a ball thrown up from the Earth will come down. In this case if the relative velocity of the galaxies, due to the background expansion, is large enough to overcome the negative potential energy due to their mutual attraction, they will continue to move apart. If their kinetic energy exactly balances their potential energy, the total gravitational energy will be zero. The galaxies will then continue to move apart forever, but will slow down over time, never quite stopping until they (or what remains of them) are infinitely far apart from one another. If you remember my characterization of a flat universe in which we think we live, this is how I described it. Thus, if we do live in a flat universe today, the total (gravitational) energy of the entire universe is zero. This is a particularly special value, and one of the reasons a flat universe is so fascinating.

I gave a proviso that the arguments above were based on the reasonable assumption that there are no exotic new sources of energy in the universe, such as an energy of empty space. However, we have seen that this reasonable assumption is, in fact, incorrect. One forgoes it, in one sense this changes everything, because then all bets are off: an open universe can collapse, and a closed universe can expand forever, depending upon the sign and magnitude of this extra energy component. But in a more basic sense it does not, for a proper energy accounting will still determine the future of the universe.

Consider, for example, the universe we appear to live in, which is flat, meaning it has zero total gravitational energy, and yet the bulk of this energy appears to reside in empty space. In this case the observable universe will expand forever, with the rate of expansion continuing to increase instead of decreasing as described earlier for a flat universe. What has changed? Well, it turns out that because the energy of empty space is gravitationally
repulsive, the total energy contained in any expanding region does not remain constant. Instead, the universe does work on this region as it expands. This additional work means that additional energy keeps getting pumped into the expanding universe allowing it to speed up even though the total gravitational energy remains zero.

In the end, whether or not the universe will end with a bang or a whimper will be determined by energy, exotic or otherwise. The answer to one of the most profound questions of human existence—How will the universe end?—can be determined by measuring the expansion velocity of a given set of galaxies, measuring their total mass, and finally determining the abundance and nature of a possible “dark energy.” We will not know what currency will govern the energy accounting until we figure out the nature of the mysterious dark energy currently making up 70 percent of the energy of the universe, but ultimately the presently undetermined fate of the universe will be a simple matter of energy bookkeeping.

Another symmetry of nature goes hand in hand with time-translation invariance. Just as the laws of nature do not depend upon
when
you measure them, then they should not depend on
where
you measure them. As I have explained to some students’ horror, if this were not so we would need an introductory physics class not just at every university but in every building!

The consequence of this symmetry in nature is the existence of a conserved quantity called
momentum,
which most of you are familiar with as inertia—the fact that things that have started moving tend to continue to move and things that are standing
still stay that way. Conservation of momentum is the principle behind Galileo’s observation that objects will continue to move at a constant velocity unless acted upon by some external force. Descartes called momentum the “quantity of motion,” and suggested that it was fixed in the universe at the beginning, “given by God.” We now understand that this assertion that momentum must be conserved is true precisely because the laws of physics do not change from place to place.

But this understanding was not always so clear. In fact, there was a time in the 1930s when it seemed that conservation of momentum, at the elementary-particle level, might have to be dispensed with. Here’s why. Momentum conservation says that if a system is at rest and suddenly breaks apart into several pieces—such as when a bomb explodes—all the pieces cannot go flying off in the same direction. This is certainly intuitively clear, but momentum conservation makes it explicit by requiring that if the initial momentum is zero, as it is for a system at rest, it must remain zero as long as no external force is acting on the system. The only way that the momentum can be zero afterward is if, for every piece flying off in one direction, there are pieces flying off in the opposite direction. This is because momentum, unlike energy, is a directional quantity akin to velocity. Thus, nonzero momentum carried by one particle can be canceled only by a nonzero momentum in the opposite direction.

One of the elementary particles that make up the nucleus of atoms, the
neutron,
is unstable when it is isolated and will decay in about 10 minutes. It decays into a proton and an electron, which can both be detected because they carry (equal and opposite) electric charge. Thus, one might observe the following tracks in a detector after a neutron, initially at rest, decays:

But momentum conservation tells us that, just as for a bomb, the proton and electron should not both go flying off to the right. One of them should go off to the left. Instead, at least some of the time, the configuration in this diagram was observed. The question thus arose: Does momentum conservation not apply for elementary particles? After all, no one at that time understood the nature of the force that was responsible for neutron decay in the first place. But the thought of giving up this conservation law (and also the conservation of energy, which was also apparently violated in these observations) was so unpalatable that Wolfgang Pauli, one of the most eminent theoretical physicists of the time, suggested another possibility. He proposed that an undetectable particle might be produced in neutron decay along with the proton and electron. This could be the case if the particle were electronically neutral, so that it could not be detected in standard charged-particle detectors. It would also have to be very light, because the sum of the masses of the proton plus the electron is almost equal to the mass of the neutron. For this reason, Pauli’s Italian colleague Fermi called this particle a
neutrino,
Italian for “little neutron.” It is the very particle I spoke of earlier in connection with the nuclear reactions that power the sun. If the neutrino
were produced in neutron decay, its direction could be such that its momentum would cancel that of the other two particles: