Prof. Tai Kai NG
Department of Physics, Hong Kong University of Science and Technology

In physics, we encountered many numbers that we believe do not change with time. These numbers are usually called constants or more appropriate, universal constants. The most familiar example is the speed of light, usually denoted by a small letter c, and has a value of about metres per second, or 300,000 kilometres per second. Another familiar example is the gravitational constant G, derived from the proportionality of the force between two masses and the square of their distance apart, i.e., . G is known as the Newton's law of gravitation.

There are about four so-called universal constants of nature: c, the speed of light, G, the Gravitational constant, h, the Planck constant and k, the Boltzmann constant. Some people believe there are more but to keep within the scope of this essay, only these four will be discussed. The values of these four constants appear unchanged over time and valid throughout the entire universe, i.e., it doesn't matter whether you live on Earth or in another galaxy, and they all have the same values.

Speed of light in vacuum
Gravitational constant
Planck constant
Boltzmann constant

Light is an electromagnetic wave. In the absence of matter, the speed of light is a constant. When light travels through a medium, e.g., glass, its speed is reduced according to the density of the glass. In space, however, the density of matter between stars is very low, so that speed of light is essentially the same as in vacuum.

(The change of speed of light in air and glass)

Newton's law of universal gravitation states that any two objects attract each other with a force proportional to the product of their masses divided by the square of their separation. The constant of proportionality, called G, is what is known as the Gravitation constant. The value of G was first measured in 1798 by Cavendish and co-workers. The value obtained was accurate to about 1%. At the turn of century, Einstein's theory of General Relativity reinterprets the meaning of G in terms of the curvature of space-time around the objects, i.e., G is proportional to the curvature of space-time when a unit mass of material is present. In this theory, the gravitational attraction between two objects is a result of the curvature of space-time around them.

Before the advent of modern physics, electromagnetic radiation was thought to be continuous wave. The concept of discrete radiation was first devised by Planck. His theory of quantized energy, which we now called quantum theory, proposed that radiations are emitted or absorbed in discrete units called quanta. The energy of these quanta is proportional to the frequency of the radiation. The constant of proportionality is what is known as the Planck constant. The value of the Planck constant was first measured in 1916 by the American physicist Robert Millikan.

The ideal gas law says that the product of the pressure and volume is proportional to the number of molecules times the temperature. The constant of proportionality is known as the Boltzmann constant k. The constant relates the average kinetic energy of a molecule to its absolute temperature.

The Large number hypothesis (LNH) was first conceived and coined by the famous physicist Paul Dirac, one of the pioneers of modern physics. What Dirac realised was that several very large dimensionless numbers of the order ~ are constructible from certain physical parameters of nature. For example, the ratio of the Coulomb force (the force between electromagnetic charges) to the gravitational force (the force between masses in the universe) for a proton-electron pair is of the order of . As a coincidence, these two forces are both inversely proportional to the square of the distance between their two charges or masses. Since we now believe that stars and galaxies are moving away from us at a monstrous velocity proportional to their distance from us (known as the Hubble's law), and the fact that everything started from the so-called "Big Bang", i.e., the universe started from an explosion at the single point known as the singularity to the physicists, the age of the universe can be determined by measuring how far the distant stars have moved and the velocity at which they are moving. In terms of atomic units, the age of the universe turns out to be of order of , same as the ratio between the gravitational and the coulomb forces. An example for contrast, the ratio of the Planck constant and the electric charge is about 137.

Dirac believes these large dimensionless numbers are somewhat related to each other, e.g., a relationship exists between cosmological and microscopic processes. If we assume the above two numbers are connected, then the forever changing age of the universe suggests that the ratio of the gravitational and the coulomb forces is not a constant, but increases proportionally with the age of the universe.

The more recent measurements of the universal gravitational constant G have created some controversies over its value. Several research teams around the world have come up with a value of G which differs from the previous accepted value by 0.1% to 0.7%. Two possibilities exist: (1) we underestimated the uncertainty in our measurement of G; (2) G actually varies with time. The variation in G may be small but its implication in physics is overwhelming. For instance, Einstein's theory of relativity requires G to be a constant. So one either modifies the theory to incorporate the idea of varying G with time or abandon it altogether.

Results from spectral analysis of quasars (quasar is the short from for quasi-stellar object ¡V quasars are the probably the cores of most distant visible galaxies in the universe) indicates that the fine structure constant which is a combination of electric charge, Planck constant and speed of light and measures the strength of interactions between charged particles and electromagnetic fields, may be changing with time. If the fine structure constant varies with time then at least one other constant - the electric charge, the Planck constant or the speed of light - must also vary with time.

The Amount of Particles in the universe is estimated to be about , which is equal to the square of the age of the universe. What this entails is that the amount of particles in the universe may not be conserved but may increase with the square of the age of the universe. In other words, new matter may be continually being created.

In modern cosmology, there exist three different models of the universe: the open, the closed and the stationary model. In the open model, the universe is forever expanding and there will be no end of the universe. In the closed model, the theory predicts that at some point in the future, the universe will collapse into a point, i.e., a big crunch, denoting the end of the universe. In the stationary model, the universe neither expands nor contracts; the universe will just remain as it is. From modern observation, we know that the universe is expanding, therefore the stationary model can be ruled out. In the closed model, the universe will expand to a maximum size at some point in time and then starts to collapse to a point. If we relate this maximum size to the age of the universe, then the idea of contracting or collapsing universe is also ruled out. What remains is the open universe.

Dirac's work on LNH led him to conclude that there are two different time evolutions in nature that are closely related to one another: one that governs the motions of galaxies, planets, and the evolution of the Universe; and one that governs the periodic vibrations of atoms, the atomic time. The former appears in the Einstein's theory of gravitation, or the general theory of relativity, and the latter appears in the theory of atoms, or the quantum theory. Whether this is true is still an open problem for you to solve.

[This essay is an extract of one of the Appendices written by one of us (CWC) for Prof Tai Kai NG's Lecture Notes on Physical Phenomena for Everyday Life]