Large Number Hypothesis P.
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Large Number Hypothesis
Prof. Tai Kai NG
Department of Physics, Hong Kong University of Science and Technology
Universal Constants
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 3

10
8 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., F

GM
1
M
2
/ r
2 . 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
Boltzmann constant k c

3 .
00

10
8 ms

1
Gravitational constant
Planck constant h

G

6 .
67

10
6 .
63

10

34
Js

11
Nm
2 kg

2

1 .
38

10

23
JK

1
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.
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Large Number Hypothesis P.
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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.
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Large Number Hypothesis P.
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Large Number Hypothesis (LNH)
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 ~ 10
39
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 10 39 . 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 10 39 , 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.
Value of G
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.
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Large Number Hypothesis P.
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Fine Structure Constant
Results from spectral analysis of quasars (quasar is the short from for quasi-stellar object
– 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.
Amount of Substance in the Universe
The Amount of Particles in the universe is estimated to be about 10
78
, 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.
Models of the Universe
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.
Two Different Time Evolutions
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
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Large Number Hypothesis P.
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[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]
________________________
[About the author: Professor Tai Kai NG received his BSc degree from the University of
Hong Kong in 1981. He completed his PhD degree at Northwestern University in 1987.
He joined the Physics Department of HKUST in 1991. In 1993, Professor Ng received an
Honorable Mention in the 1993 Achievement in Asia Award of the Overseas Chinese
Physics Association, for his contribution to the understanding of the non-equilibrium mesoscopic Kondo effect. He is a Fellow of the American Physical Society since 2000.
He is awarded the Croucher Senior Research Fellowship for 2002-2003. Prof. Ng also participates actively in Hong Kong's secondary and primary school science education. He is a (Hong Kong) representative in the joint Hong Kong-Mainland expert working groups on Science Education.]
Keywords:
Universal constants, Planck constant, Boltzmann constant, Photon, Quantum theory,
Quanta, Large number hypothesis, Electromagnetic waves, Big bang, Gravitational constant, spectral analysis, quasars, cosmological (macroscopic) and microscopic processes, Coulomb force, atomic unit.
Related Topics in the syllabus:
Newton's law of gravitation, Electromagnetism, wave, frequency, electric charge, electromagnetic fields (electric field and magnetic field), gravitational force, mass, velocity, radiation, temperature
Extensions (from the syllabus):
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Bring out ideas: The meaning and importance of universal constants in nature; the consequence of the change of these constants have on nature; the significance of the order of 10 39 and how it can be constructed; recent measurements of G.
Warm-up discussion:
1.
What is universal constant?
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.
2.
What is universal constant usually related to?
3.
What is the corresponding physical principle behind concerning the following universal constant?
(a) speed of light
(b) universal gravitational constant
(c) Boltzman Constant, and
(d) Plank Constant.
4.
Apart from the four constants mentioned above, can you tell more
Universal constants that you might have come across in your study?
5.
What is the main difference between Quantum theory and
Classical ideas of energy propagation?
6.
What is the theoretical basis of LNH?
7.
How much particles in the universe is estimated to have?
Points for further discussion:
1.
Why the investigation of those universal constant is so important regarding to the universe evolution?
2.
Other than those value in the paragraph, please try to identify/discuss with your teacher, which of the following is/are universal constant?
(i) gravitational acceleration (g = )
(ii) boiling point of water at the sea level (100 at 1 atm)
(iii) electronic charge (
(iv) speed of sound (330
)
at 25 )
3.
Why is the speed of light always quoted in vacuum?
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4.
What is the different interpretation of G in Newton’s Law of gravitation and Einstein’s general relativity?
5.
What may happen if G is not a constant?
6.
What effect does 0.6% increase in the value of G have on you?
7.
In terms of space exploration and research, give your opinion(s) why those universal constants is so important in our understanding of the universe. If they are really changed with time?
Related web sites: o Interview with Paul Adrian Maurice Dirac - F. David Peat
An interview with P.A.M. Dirac, who is one of the greatest physicists in
20th century, his contributions in physics include the quantum theory of matter and radiation, the prediction of the spin of the electron, and the existence of the positron as well as an attempt to form a marriage between quantum theory and the theory of relativity. He was awarded the Nobel
Prize in 1933. This interview also includes “large number hypothesis”, which has occupied him in recent years. http://www.fdavidpeat.com/interviews/dirac.htm o The Controversy over Newton's Gravitational Constant
The value of the fundamental constant G has interested physicists for over
300 years. However, the fractional uncertainty in G is thousands of times larger than those of other important fundamental constants, such as
Planck's constant or the charge on the electron. As a result, the mass of the
Earth is known far less precisely than, for instance, its diameter. http://www.npl.washington.edu/eotwash/gconst.html o Paul Adrien Maurice Dirac
A brief history of P.A.M. Dirac---one of the greatest physicists in 20th century. http://www-groups.dcs.standrews.ac.uk/~history/Mathematicians/Dirac.html
o BBC NEWS - Science-Nature Einstein proved right on gravity
The speed of gravity has been measured for the first time, revealing that it does indeed travel at the speed of light. It means that Einstein's General
Theory of Relativity has passed yet another test with flying colours. http://news.bbc.co.uk/2/hi/science/nature/2639043.stm o New Scientist - “Black hole theory suggests light is slowing”
One of Einstein's most dearly held concepts - that the speed of light is constant - is looking a little fragile. Physicists in Australia claim there is good reason to think the speed of light has slowed over time… http://www.newscientist.com/news/news.jsp?id=ns99992650
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