A.1.1 Zeno’s Paradox
It is very easy to reject the conclusion of this paradox, but not so easy to refute it formally. In fact, the treatment of infinities was a tough nut for both mathematicians and scientists for many centuries, up to when modern infinitesimal Calculus, due to both Isaac Newton (1643–1727) and Gottfried Leibniz (1646–1716), provided a simple and very elegant tool for its general solution. In the case of Zeno’s paradox, however, we can offer a simple arithmetical argument to explain it. Let us assume that Achilles needs, say, 1 min to cover the 100 m. He can then run the next 10 m in 0.1 min and the following one meter in 0.01 min and so on. Hence the total time to reach the tortoise will be the sum of all those time intervals, i.e. 1 + 0.1 + 0.01 + 0.001 + …. = 1.1111…., which is a finite quantity (equal to the rational number 10/9), in spite of being the sum of an infinite number of terms.
A.1.2 Frequency/Wavelength Limits for Human Sight and Hearing
Frequency and wavelength are reciprocal quantities. The range of wavelengths of the electromagnetic spectrum visible to the human eye, called the visible spectrum, runs between 0.39 and 0.7 µ (microns) approximately, where a micron is one millionth of a metre. It is bordered to the right by the infrared and to the left by the ultraviolet. The audible frequency range for the normal human ear lies between 20 and 20,000 Hz (Hertz).
A.2.1 Left Versus Right Hemispheres of Brain
There is now accumulating evidence dispelling the idea that some people are predominantly left-brained and others right-brained, which was always a demarcation found more in the popular media than the scientific literature. See, for instance: Nielsen et al. [1].
A.2.2 Second Law of Thermodynamics
The second law of thermodynamics
received some notoriety in the 1960s when British scientist and acclaimed novelist, C.P. Snow, disparaged the lack of basic scientific knowledge among those who had received a classical education. He opined that ignoring the second law of thermodynamics was an equivalent in ignorance to never having read Shakespeare. Two satirists, Michael Flanders and Donald Swann, resolving to remedy the gap in their education, studied the law and based a song about it called: The first and Second Law, in their stage show, and in a later recording: At the Drop of Another Hat.
A.3.1 The Axioms of Euclid’s Geometry
There are five axioms that are generally recognised as the basis for ordinary (or Euclidean) geometry. Let us blow away the cobwebs that have accumulated in our heads since high school, and recap these
axioms
in Figs.
A.1,
A.2,
A.3,
A.4 and
A.5.
Fig. A.1Axiom 1: A straight line can be drawn between any two points
Fig. A.2Axiom 2: Any terminated straight line can be projected indefinitely
Fig. A.3Axiom 3: A circle with any radius can be drawn around any point
Fig. A.4Axiom 4: All right angles are equal
Fig. A.5Axiom 5: Only one line can be drawn through a point parallel to another line.
From these humble beginnings the whole of Euclidean geometry can be derived by the strict application of logic.
A.3.2 Newton’s Laws of Mechanics
Newtonian mechanics is based on three laws:
1.
An object at rest will remain at rest, and an object in motion will continue in uniform motion in a straight line, unless acted upon by an external force. This law opposes the teachings of Aristotle
who, as we saw in Chap. 2, believed that moving objects came to rest when the driving force was removed.
2.
The acceleration of an object produced by an external force is directly proportional to the magnitude of the force, in the same direction as the force, and inversely proportional to the mass of the object. The astute reader will notice that in the case where the external force is zero, this law indicates that the acceleration of the object is also zero. The object therefore remains in its original state of rest or of uniform rectilinear motion. In other words, the first law is a special case of the second, and does not need to be stated separately.
3.
Action and reaction are equal and opposite. (If you push on an object, it pushes back with an equal force.)
A.3.3 The Genius of Homer Simpson
Many of the script writers for the Simpsons are mathematicians, and some of their friends were physicists. There are two equations on the chalkboard referred to in Chap. 3. They are discussed by Simon Singh in his book: The Simpsons and their Mathematical Secrets [2].
The upper equation predicts the mass of the Higgs Boson
(see Chap. 9) to fair accuracy 14 years before the particle was discovered. This is quite an achievement for a cartoon character. The story is told in the link: [3].
The second relationship: 398712 + 436512 = 447212 is, as noted in the text, difficult to disprove by evaluating the powers explicitly. However, considering only the last two digits of each number is sufficient for this purpose, as can be seen from the following:
A number abcdefg, where a, b, c, etc. are the digits forming the number, can be split into two parts: i.e., abcdefg = abcde00 + fg, where 0 is the digit zero.
Squaring the number gives: abcde002 + 2.abcde00 x fg + fg2.
Because of the zeros, the first 2 terms contribute nothing to the last two digits of the square, and so can be disregarded.
The same is true if we multiply two different numbers together (or add them): i.e. only the product (or sum) of the last two digits of each contributes to the last two digits of the product (or sum). Using this principle it is easy to show that the last two digits of the left hand side of the above relationship are not equal to the last two digits of the right hand side, and so the relationship is invalid.
However, if one were to actually evaluate the terms in the relationship fully, one would discover that 398712 + 436512 is close to the twelfth power of 4472.0000000070576171875, which a physicist would argue is near enough to 447212.
A.3.4 Principia Mathematica
Principia Mathematica is a very intense book by Alfred North Whitehead
and Bertrand Russell
, written in 3 volumes from 1910 – 1913. Their aim was to provide a formal logical derivation for all arithmetic. The uncompromising rigour of their approach is illustrated in Fig. A.6. This excerpt shows us that at this point in their opus (page 362), Whitehead and Russell have almost, but not quite, managed to prove that 1 + 1 = 2.
Whitehead and Russell’s book is the starting point for the work of Kurt Gödel, which we discussed in Chap.
4.
Fig. A.6Extract from Whitehead and Russell’s Principia Mathematica
A.3.5 Falsifiability
The concept of falsifiability, or the notion that every physical theory must be capable, at least in principle, of being proven wrong, was introduced by the philosopher of science, Karl Popper. In Part 3, we see that there are currently diverging opinions on whether a lack of falsifiability is sufficient grounds for rejecting a physical theory. Also in Chap. 4, we see that the logician, Kurt Gödel, showed that certain propositions inside a system cannot be proved from within the system itself. In addition, undecidable statements, such as the famous paradox This sentence is false, obviously cannot be falsified.
A.3.6 Variation of the Fine Structure Constant
A very tight bound on current variations of the Fine Structure Constant
α has been set at one part in 1017 per year [4]. This result however does not preclude past variations. Research by a group at the University of New South Wales has shown a variation in α of approximately one part in 105 spanning ∼23% to 87% of the age of the universe [5]. More recent work by this group claims that the changes in α are different in different directions in the universe [6].
A.4.1 Russell’s Paradox
Russell discovered the paradox that bears his name in 1901 while using set theory to formalise the mathematics of arithmetic. A set can be considered to be a collection, e.g. the set of all dogs is a collection that contains all dogs and nothing else.
Russell explored the question of whether a set can contain itself. In the above example, this is clearly not the case; a set of dogs (i.e., a collection of dogs) is not a dog. However, if we consider a set of sets, i.e., a collection of collections, then it is possible for the set to contain itself. In fact, the set of all sets must contain itself.
Suppose now we take this process one step further, and consider the set of all sets that do not contain themselves. The paradox occurs when we ask ourselves: “Does this set, so defined, contain itself, or not?” If we say, “no, it does not contain itself”, we see from its definition as the set of all sets that do not contain themselves, that it does contain itself. Conversely, if we say that it does contain itself, we see that this is wrong, because by definition it is the set of all sets that do not contain themselves, and therefore must not contain itself.
If you are having trouble getting your head around the paradox, spare a thought for the innocent eight-year-olds of the sixties, who were expected to master set theory on their way to learning that 2 + 2 = 4.
A.4.2 Gödel’s Incompleteness Theorem
In analogy with the Liar’s Paradox
that we have already discussed, let us consider the statement: This statement is unprovable. If the statement is provable, then the proof will prove something that is false, which does not bode well for mathematics. The only alternative is that the statement is unprovable. In other words, although the statement is true, it cannot be proved, i.e., there are some statements that, although true, cannot be proved.
Of course, the above argument is only an outline of the logic that Gödel employs in his proof. His genius lies in encoding This statement is unprovable into a natural number, and then examining the implications within the rules of arithmetic. For a deeper insight into this intriguing topic, please see Gödel’s Proof by Ernest Nagel and James R. Newman, referenced in Chap. 4. A summary of Gödel’s approach is given by Natalie Wolchover [7].
A.5.1 Common Sense
The remark:
common sense
is not so common is often attributed to Voltaire: Le sens commun n'est pas si commun in the Dictionnaire philosophique portatif, 1764. However, much earlier, a Roman poet, Decimus Iunius Iuvenalis (aka Juvenal) in Book III of his Satires had written Rarus enim ferme sensus communis – Common sense is generally rare.” Much later, Mark Twain and Will Rogers added their own contributions: I’ve found that common sense ain’t so common. In any case, common sense, based as it is on our past experiences in a macroscopic world, turns out to be a poor guide in QM.
A.5.2 Black Body Radiation
A black body, when cold, is a perfect absorber of radiation, i.e. all radiation that falls on it is absorbed; none is reflected or re-emitted. When heated, a black body is the most efficient emitter of radiation allowed by the laws of physics. Normal objects, such as iron bars or lamp filaments, are less efficient radiators than black bodies.
A black body can be approximated in the laboratory by hollowing out a cavity in a metal ingot. Light falling into the cavity is repeatedly reflected around inside it before being finally absorbed. (A similar effect happens with narrow-necked bottles that are used in summer to trap flying insects.) As none of the light that enters escapes, the cavity appears black. However, if the ingot is heated, the cavity appears to be brighter than the metal around it, because the cavity is a better emitter than the surrounding metal.
A.5.3 Nature Does not Make Jumps
Natura not facit saltus or Nature does not make jumps was a truism of classical science, attributed by some to Gottfried Leibniz, co-founder of the calculus. It is the belief that nature does not allow sudden changes. In biology, it was used by Charles Darwin to support his theory of the evolution of species through small gradual changes, rather than through the sudden appearance of new species. In physics, Quantum Mechanics
, with its sudden transitions between states, violates this principle. Also in biology, the discovery of the roles of DNA and mutations in genetics shows that sudden changes, even if small, are indeed possible.
A.5.4 Laplace’s Demon
The following observation by French polymath Pierre-Simon, Marquis de Laplace (1749 – 1827), introduced what is now known as Laplace’s demon: We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes [8].
Even before the advent of QM, Laplace’s demon
had been disputed by physicists, using thermodynamical arguments. However, these refutations have themselves been criticised, and debate still continues.
A.5.5 The Monty Hall Problem
When this problem was presented in a popular magazine in the U.S., many mathematicians wrote rude letters to the editor complaining about the published solution, and how it demonstrated the deplorable lack of numeracy prevailing in the U.S. Unfortunately for them, they were wrong and the magazine solution was right.
Statisticians tackling the problem would probably use Baye’s Theorem for conditional likelihoods. However, the rest of us can use the following argument. When the contestant makes a choice for the door the car is behind, she has a 1 in 3 chance of being right, and a 2 in 3 chance of being wrong. No matter what the host does, this does not change. There is always a 2 in 3 chance that the car does not lie behind the door that the contestant has chosen. When the host, who knows where the car is, opens one of the other doors and there is no car there, the contestant knows that there is now a 2 in 3 chance that the car is behind the other unopened door. She should therefore change her selection.
If the reader still has difficulty believing this result, a simple experiment should suffice to convince them. Take three cups and a coin. Turn your back, and ask a friend to conceal a coin beneath one of them. Guess which cup hides the coin, then ask your friend to turn up one of the other two cups that does not hide the coin. Check whether your initial choice was right, or whether you would have been better off to change your selection. Repeat the experiment many times, and keep a tally of the results. You will find that you are twice as likely to win if you change your mind.
A.6.1 Origin of Relativity Limerick
A letter by A. H. Reginald Buller to the Journal of the Royal Astronomical Society of Canada in 1938 explains the origin of this much-cited limerick.
“As the author of the relativity limerick, perhaps I may be allowed to say that the limerick was made by me about fifteen years ago while sitting in the garden of my friend and former colleague, Dr. G. A. Shakespear, Lecturer on Physics at the University of Birmingham.
After conversing together on Einstein's theory, I suggested that we should each try to make a relativity limerick. At the end of about five minutes, the limericks were ready and were exchanged, but with nothing more than a trace of mutual admiration.”
A.6.2 A Land of Bliss Poem
A Thousand Reflections Quiver in These Lofty Boughs;
The Flowers of the Grotto Greet the Arriving Guest.
Near the Spring, Where then is the Herb Gatherer?
A Lone Boatman Rows on the Stream,
And His Guitar Sounds Two Notes.
The Boat Glides Lazily, the Gourd Offers Its Wine.
Shall We Ask the Boatman of Vo Lang:
“Where Are the Peach Trees of the Land of Bliss?
by George F. Schultz [9]
A.7.1 Precession of Perihelion of Mercury
The rotation of the major axis of the orbit of mercury is known, in astronomical parlance, as the precession of the perihelion of mercury, where perihelion is the term for the point of closest approach of the orbit to the sun.
The observed rate of this precession can be measured very precisely, and is found to be 5599.7 arcseconds of rotation per century, again not an effect that can be observed in the backyard. Almost all (5030 arcseconds/century) of this figure can be accounted for by Newton’s theory, and another 530 arcseconds/century is caused by perturbations to the orbit from the outer planets. This leaves a 40 arcseconds/century discrepancy, which is very well explained by General Relativity, when one includes the distortion of space–time caused by the gravitational mass of the sun.
A.7.2 Gravitational Redshift
of Light
What happens when light tries to escape from a really intense gravitational field, such as that in the neighbourhood of a black hole? In Chap. 7, we discussed the event horizon that surrounds a black hole, and saw that nothing, not even light, can exit from within this region. Another way of explaining this is to say that the gravitational redshift of light is so severe that the light is redshifted to zero frequency. Zero frequency light, of course, does not exist.
As mentioned in Chap. 7, the first reliable verification of the gravitational redshift was made in 1954 by Daniel M. Popper [10]. He utilised the line spectra (see Chap. 8) emitted by a hot gas, and measured the frequency of 42 hydrogen lines on 27 spectrograms of the white dwarf star, 40 Eridani B. He obtained a frequency shift of 0.007 percent, which was within experimental error of the value predicted by General Relativity.
So sensitive have modern experiments become that it is no longer necessary to use astronomical observations to test General Relativity. Instead, atomic clocks have been flown around the world, and rockets launched tens of kilometres into space to study the time dilation caused by gravity. Such an experiment in 1976 found General Relativity to be accurate to within 0.007% [11].
A.7.3 Gravitational Lensing
Suppose that we have a situation where a distant star or galaxy lies behind a region where the gravitational field is intense. Light passing through this field will be deflected, and as a consequence the distant object will appear to lie in a different location from its true position.
This effect, known as
gravitational lensing, is illustrated in Fig.
A.7. Light from the source S passing above the massive object G will appear to be coming from location A, while light passing below G will appear to originate at location B. Fig.
A.7 is a two-dimensional representation of a scenario that in reality is three dimensional. Light could also have passed in front of, or behind, the massive object. The net result is that light from the faraway source appears to originate from anywhere on the circumference of a circle around the source, depending on which path the light happened to take.
Fig. A.7Light from
source S is bent by the gravitational field at G, so that to an observer located at O, the light appears to come from somewhere on the circle, or Einstein Ring, joining A and B
A.7.4 Evidence of Gravitational Waves
from Binary Pulsar Systems
When stars that are 4 to 8 times the size of our sun reach the end of their lives after having consumed all of their nuclear fuel, they undergo spectacular death throes. With the pressure exerted by outflowing radiation no longer sufficient to oppose internal gravitational forces, such stars collapse under their own weight. Energy released by the collapse results in the outer layer of the star being blown away in an enormous supernova explosion. The core collapses further until all atomic structure disappears, and protons and electrons combine into neutrons,1 leaving behind what is now known as a neutron star. A pulsar is a rotating neutron star.
As we saw in Chap. 7, neutron stars may be observed as isolated objects, or in some cases, binary systems, where they rotate about another object, in which case astronomers are able to calculate their mass. As such a system loses energy, the two components of the binary system rotate more closely about each other, i.e., the radius of their orbit decreases, as they fall slowly together. Hence, the time to complete each orbit, i.e. the period of rotation, decreases as the objects move faster and faster on their paths around each other. This is the same effect that causes an ice skater to spin faster when she pulls her arms close to her body during a spin, and is a consequence of one of the most far-ranging laws in physics: the law of conservation of angular momentum.
In 1974, the discovery of the Hulse-Taylor
pulsar
, which is located 21,000 light years from earth in a binary system with another neutron star, enabled a further test of General Relativity. The period of rotation of the binary system has been measured accurately over several decades. The results are presented in Fig.
A.8.
Fig. A.8The decay of the orbital period of the binary system containing the Hulse-Taylor pulsar. The continuous line is the expected trend predicted from General Relativity. The diamonds are the measured values. Image: public domain (Image: Inductiveload—Own work, Public Domain https://commons.wikimedia.org/w/index.php?curid=9538634 (accessed 2020/7/28))
As can be seen from the figure, the period of rotation of the binary system has decayed, precisely according to the predictions of General Relativity, over thirty years of measurement. The confidence of physicists in the reality of gravitational waves therefore grew, and motivated the construction of even more sensitive apparatus in an attempt to detect these elusive ripples in space–time on their passage through the earth. As discussed in Chap. 7, the first direct detection of a gravitational wave was made on 14th September, 2015, and with improvements in the technology since that date, detections have become almost commonplace.
A.9.1 Relationship Between Range of a Force and the Mass of Its Carrier Particle
The force between two particles at a distance is visualised, in Quantum Field Theory, as arising from the exchange of carrier particles, which carry
information
between the two interacting particles. The situation is illustrated, for example, in Chap. 9, Fig. 9.5c. The emission of the carrier particle from the proton is clearly a violation of the law of conservation of mass/energy. Such violations are possible according to Heisenberg’s Uncertainty Principle, but only if they occur for a sufficiently short time. Indeed, a violation of mass/energy by an amount ΔE for a time Δt is possible if Δt is approximately Planck’s constant h divided by 4π and ΔE. During this time, the violation cannot be detected.
This process allows the temporary creation of a carrier particle of mass m, where m is obtained, according to Einstein and the one formula in this book, from ΔE = mc
2
. Hence, the larger the mass of the carrier particle, the greater is ΔE, and the shorter is the time it can exist. Such particles are said to be virtual.
The finite lifetime of the carrier particle limits its range, which is given by Δt multiplied by the velocity of light. If the particle has no mass, e.g., it is a photon, its range, and hence that of the corresponding force is infinite. As the mass of the carrier particle increases, its range decreases. Short range forces therefore have heavy carrier particles.
A.9.2 Table of Quarks
In the table below, which at first glance bears a strong resemblance to our earlier table for leptons, we summarise what is known about quarks. It should be noted that the masses of the quarks are notoriously hard to pin down [12].
Table A.1 Table of Quarks
Gen | Name | Symbol |
J
|
B
| Chargea
|
Iso-spin
|
C
|
S
|
T
|
B
’
| Anti-quark | Massb
|
|---|
1 | Up | u | 1/2 | 1/3 | 2/3 | 1/2 | 0 | 0 | 0 | 0 |
| 0.0021 |
1 | Down | d | 1/2 | 1/3 | -1/3 | -1/2 | 0 | 0 | 0 | 0 |
| 0.0051 |
2 | Charm | c | 1/2 | 1/3 | 2/3 | 0 | 1 | 0 | 0 | 0 |
| 1.36 |
2 | Strange | s | 1/2 | 1/3 | -1/3 | 0 | 0 | -1 | 0 | 0 |
| 0.098 |
3 | Top | t | 1/2 | 1/3 | 2/3 | 0 | 0 | 0 | 1 | 0 |
| 185 |
3 | Bottom | b | 1/2 | 1/3 | -1/3 | 0 | 0 | 0 | 0 | -1 |
| 4.45 |
Here “Gen” is the Generation, J is the angular momentum (spin) of the quark, B the baryon number, C the charm, S the strangeness, T the topness and B’ the bottomness. The charge is the electrical charge in units of one electron charge. It can be seen, as was mentioned earlier, that the charges of the quarks are 1/3 and 2/3 that of the electronic charge. The charges, and also the baryon numbers with their values of 1/3, give a clue that baryons are most likely constructed from three quarks. This is indeed true. The antiquarks have the opposite quantum numbers to the quarks, and are given the names: antiup, antidown, etc. Mesons are formed from a quark-antiquark pair, and are consequently bosons.
A.9.3 Grand Unified Theory (GUT) and Supersymmetry
Grand Unified Theory
(GUT)
In this theory, the three interactions we have discussed thus far (electromagnetic, weak and strong) are merged at high energy into a single force. At lower energies, the theory should result in the same predictions as the current Standard Model. However, at high energies, new particles are predicted. Unfortunately, the energies required to produce these particles are well above the capabilities of modern accelerators, and as a consequence they have not been observed.
In the absence of this type of direct evidence, support for the GUT is sought from indirect observations of physical quantities, such as proton decay and the electric dipole moments of particles, where the predictions of the Standard Model are inaccurate. Some versions of the GUT predict the existence of the magnetic monopole.
All observed magnetic fields arise from pairs of north and south magnetic poles. These are known as magnetic dipoles. A magnetic monopole would comprise an isolated north (or south) pole, in the same way that electric charges (positive or negative) can exist in isolation. Due to the lack of experimental confirmation of its predictions, there is currently no generally accepted GUT.
Supersymmetry
The distinction between fermions (particles of half odd-integer spin) and bosons (carrier particles of zero or integer spin) may seem rather arbitrary. What if each fermion had a supersymmetric partner with identical quantum numbers, but with integer spin? These supersymmetric particles (called sparticles) would comprise sleptons, squarks, neuralinos and charginos with spins differing by ½ from the normal fermions. Unfortunately, once again these new particles are so heavy that they are beyond the energy range of today’s particle colliders.
A.10.1 The Measurement of Distance in Astronomy
The measurement of distances in astronomy is fraught with difficulties. Nevertheless, it is common to see in newspapers, popular science magazines, and books such as this one, that objects have now been discovered at a distance of 10 billion light years. How are such measurements even possible?
We have already explained in Chap. 10 how nearby stars observed from earth appear to move against the background of distant stars as the earth circles the sun, and how the observed parallax
leads to the measurement of distance. The standard unit of distance in astronomy is the parsec; a star with one arc-second of observed parallax, measured when the earth is on opposite sides of the sun, is said to be at a distance of 1 parsec. (An arc-second is 1/3600 degrees.) The distance to any star in parsecs is the reciprocal of its measured parallax in arc-seconds. To convert from parsecs to units more familiar to non-astronomers (kilometres or light-years), we need to know the distance of the earth from the sun. This distance is defined as the Astronomical Unit (AU) and must be measured independently.
It is generally recognised that the first successful measurement of the Astronomical Unit was carried out in 1672 by Jean Richer and Giovanni Domenico Cassini, who measured the parallax to the planet Mars from two different locations on earth. Knowing the separation of these terrestrial locations provides a baseline length for the parallax measurement, and enables the distance from earth to Mars to be calculated. Kepler’s Laws of planetary motion give the interplanetary distances in terms of the Astronomical Unit, so working backwards, a knowledge of the earth-Mars separation can be used to estimate the value of the Astronomical Unit. The currently accepted value is: 149,597,870,691 ± 30 m. From this value for the AU, 1 parsec is found to be about 3.3 light-years.
In the early ‘90s, the Hipparcos satellite was used to take parallax measurements of nearby stars to an accuracy much greater than possible with earth-bound telescopes, thereby extending the distance measurements of these stars out to ~ 100 parsecs (~300 light-years). How can this range be further extended?
The first step involves the comparison of a star's known luminosity with its observed brightness. The latter is the brightness (or apparent magnitude) observed at the telescope. It is less than the intrinsic luminosity (or absolute magnitude) of the star because of the inverse-square attenuation with distance of the observed radiative energy. If we know how bright the star is intrinsically, we can estimate its distance using the inverse square law.
From measurements on stars in our galaxy within the 300 light-year range it was discovered that stars of similar particular types have the same absolute magnitude. These stars are dubbed
standard candles
. If stars of these types are observed outside the range where parallax measurements are observable, we can estimate their distance by assuming their absolute magnitude is the same as closer stars of the same type, observe their apparent magnitude, and use the inverse square law to compute their distances. The distance to another galaxy can be inferred from the distance to particular stars within it.
The term “standard candle” may sound strange in this context. A standard candle was originally defined as a one-sixth-pound candle of spermaceti wax, burning at the rate of 120 grains per hour, and was used when comparing the intensity of light sources. Its output was one candlepower, a unit of luminous intensity now obsolete, much to the relief of the oceans’ sperm whales. In astronomy, the term “standard candle” is used to designate a class of objects whose members have a fixed intrinsic brightness. Two examples are Cepheid Variable stars and supernovas of a particular type (Type Ia).
Supernova explosions are rare events. The last observation of a type Ia supernova explosion in our galaxy was made by Johannes Kepler, and others, on the 9th October, 1604. Earlier, in 1054 AD, Chinese astronomers had observed another famous supernova explosion, the remnants of which now make up the Crab Nebula. The extreme brightness of supernovas—whose peak light output can equal that of the entire galaxy that contains them—enables us to observe them in distant galaxies, and thereby estimate the distance to these galaxies.
A.11.1 Measurement of the Curvature of the Universe
The temperature fluctuations visible in the Cosmic Microwave Background
, and described in Sect. 11.5, can be understood by considering the plasma that existed before the Recombination Era
. The ingredients in this “soup” were electrons, atomic nuclei, photons, dark matter and various baryons, but no stable atoms, as the high temperature would have stripped the electrons from any atom that chanced to form. Occasional density fluctuations would occur, both higher and lower than the average density. The regions of high-density dark matter would tend to attract particles due to increased gravitational attraction. As plasma flowed into these regions it would become compressed.
Compressed plasma has a high internal pressure due to the electromagnetic interactions. Once the pressure had increased, it would drive the particles apart, lowering the plasma density in that region, thereby enabling the dark matter
, by means of its gravitational attraction, to pull more plasma into the region, and begin the cycle over again. What we have therefore are propagating periodic density fluctuations, which are very similar in concept to sound waves in air. As a consequence physicists call these plasma oscillations “baryon acoustic oscillations”.
The speed of these oscillations (sound waves) through the plasma is estimated to be 60% of light speed. As the age of the universe at the time of the Recombination Era was 380,000 years, the maximum distance that any oscillation could have travelled in this time is 0.6 × 380,000 light years (LY), i.e. ~ 230,000 LY. This is the largest distance over which oscillations could interfere with each other. For distances separated by more than this distance, the universe had been in existence for insufficient time for oscillations to have passed between the separated points. As a consequence this upper limit of 230,000 LY is known as the “sound horizon”.
As we saw in Sect. 11.4, a frequency analysis of the Cosmic Microwave Background (CMB) reveals the fluctuations present at the time of the Recombination Era. The oscillation with the largest spatial extent is the one corresponding to the sound horizon. However, it is not the absolute size of the sound horizon that can be measured in this way, but only its angular size, i.e. the angle it subtends in the sky. Assuming that space–time is flat, and thus that the light rays are not bent but linear, a fluctuation in the CMB the size of the sound horizon would subtend an angle of one degree when observed from earth. If space–time is not flat but curved, the angle subtended would be more or less than one degree, depending on the nature of the curvature. The measurements of the CMB obtained from the Planck observatory show that the universe is topologically flat at large scales to within 0.5%. This unexpected result is explained in Sect. 11.5.
References
1.
Nielsen JA, Zielinski BA, Ferguson MA, Lainhart JE, Anderson JS (2013) An evaluation of the left-brain versus right-brain hypothesis with resting state functional connectivity magnetic resonance imaging. Published: August 14, 2013, https://doi.org/10.1371/journal.pone.0071275. Accessed 7 July 2020
2.
Singh S (2013) The simpsons and their mathematical secrets. Bloomsbury
3.
4.
Rosenband T et al (2008) Frequency ratio of Al+ and Hg+ single-Ion optical clocks. In: Metrology at the 17th decimal place science 319(5871):1808–1812
5.
Webb JK, Murphy MT, Flambaum VV, Dzuba VA, Barrow JD, Churchill CW, Prochaska JX, Wolfe AM (2001) Further evidence for cosmological evolution of the fine structure constant. Phys Rev Lett 87:091301
6.
Webb JK., King JA, Murphy MT, Flambaum VV, Carswell RF, Bainbridge MB (2011) Indications of a spatial variation of the fine structure constant. Phys Rev Lett 107:191101
7.
8.
Laplace P-S (1951) A philosophical essay on probabilities, English Trans. by Truscott FW, Emory FL, Dover Publications, New York, p 4
9.
Schultz GF (1968) Vietnamese legends. Charles E. Tuttle Publishing Company
10.
Popper DM (1954) Red shift in the spectrum of 40 Eridani B. Astrophys J 120:316
11.
Vessot RFC et al (1980) Test of relativistic gravitation with a space-borne hydrogen maser. Phys Rev Lett 45(26):2081–2084
12.