XVI(D): Everything in the universe is made of math including you and I.
[Contd. A Journey to the Wonderland of Math.by Ajay Kumar Chaudhuri.]
"When you are courting a nice girl an hour seems like a second. When you sit on a red-hot cinder a second seems like an hour. That`s relativity."----------Albert Einstein.
Next let us explore the contributions of Einstein’s theory
of relativity in our real world modern life where math has great role behind
the scenes.
Albert Einstein (1879 – 1955) is regarded as the greatest
Scientist of the 20th century for discovery of his famous “theory of
Relativity” which brought the Nobel Prize for him in 1921.
But many people are curious to know, what is the utility and
practicality of relativity? What purpose does it serve for us? Most of them
think of it as an abstract and highly mathematical theory understood by a few
but unfathomed for most of them and has no consequence for everyday life. But
this is in fact, far from the truth.
Yes, the theory of relativity is of course mathematical, but
not bizarre. This theory gave the world the equation E=mc2, which
expresses the relationship of energy to mass. Here E is the energy possessed by
a body of mass m and c is the speed of light. In other words, this equation
tells that mass and energy are synonymous. As the speed of light has been
calculated to be about 300000 km per second, so a mass1 gm of an ordinary
object, like a piece of stone, possesses surprisingly an enormous amount of
energy.
It is the most famous, simplest and innocent looking
equation in the world but was used with dangerous consequences as the key
component to the building of the bomb in U.S.A’s president Franklin D.
Roosevelt’s Manhattan Project. This atomic bomb was used against Japan as a
measure to end World War-II, This horrible incident in the history of human
race occurred on the fateful day 6th of August, 1945 when the
dreaded atom bomb, nick-named, “Little Boy” was dropped on Hiroshima and just
after three days another bomb, named “Fat Man” was dropped on Nagasaki, comparatively
two small cities of Japan. Those two bombs caused deaths of thousands and
thousands of people, made a large number of people permanently disabled and
left a trail of devastation including dangerous after effect of nuclear weapon
for the people of future generation of Japan. But Einstein was no way
responsible for misuse of his discovery to cause such mass-killing and
devastation.
Is Einstein’s theory of relativity far from common people’s
reach? It is contrary to the reality if only the salient features of the theory
are concerned without going in to minute details and profound implications in various
fields of physics and astrophysics.
Curiosity is our inborn companion which we inherit from the
very moment of our birth. We want to know anything and everything around us. When
we cast our look at the overhead boundless sky we are struck with wonder and
utter bewilderment. In fact, these thoughts lead us to the door step of the
vast universe full of mysteries, wonders, hazards and boundless beauties. But
how can we quench this thirst for knowledge? We need not be great physicists or
mathematicians, only our inquisitiveness will do. For this, as common people,
we must know some rudiments of modern physics.
The two pillars of modern physics and perhaps the most
revolutionary theories in the history of physics are theory of quantum
mechanics and the theory of relativity. At present physicists have two separate
rule books explaining how nature works.
The theory of quantum mechanics handles the mechanism of the
smallest, like properties of elementary particles those constitute a matter.
It is established conclusively that this vast universe is
governed by four fundamental forces: Firstly, the force of gravitation which
acts between any two objects lying at whatever distance may be. Secondly, the
electromagnetic forces which act between two electrically charged particles and
is much more many times stronger than the gravitational force. Thirdly, the
highly powerful strong nuclear force which acts in the core or nucleus of an
atom within an extremely short range. Fourthly, the weak nuclear force which
plays a greater role in things falling apart or decaying such as decay of
uranium, radium by radioactive radiation. So, the first three forces hold
things together while the fourth separates them. Interestingly, Einstein tried
hard for last 30 years of life to unify the last three forces, excluding
gravity, to unify and establish the Grand Unification Theory (G.U.T.). But
unfortunately he was not successful.
Quantum physics very skillfully explain the properties and
the applications of those last three forces in different fields of physics and their
applications to our everyday life which we have already seen.
On the other hand, Einstein’s theory of relativity deals
with the problems related to the biggest. It beautifully accounts for gravity
and all of the things it dominates such as orbiting planets, colliding galaxies,
the dynamics of the expanding Universe etc. That’s big.
The theory of relativity consists of two parts. Einstein
formulated the first part – “Special Theory of Relativity” or “Special
Relativity” in 1905. Then after 10 years’ labour he published the Second or
final part – “General Theory of Relativity” or “General Relativity” in 1915. As
a mark of recognition of his outstanding theory, he was honoured with the Nobel
Prize in 1921.
The theory essentially explains the behavior of objects in
space and time moving with uniform or non-uniform motion. The special
relativity includes only the special case (hence the name) where motion is
uniform. The motion it explains only if you are travelling in a straight line
at a constant speed (called ‘inertial frame). As soon as you accelerate or
curve or do anything that changes the nature of motion in any way, then the
special relativity ceases to apply. That’s where Einstein’s general relativity
comes in, because it explains the general case of any sort of motion.
The special theory of relativity based on two postulates:
Nothing is absolute (except speed of light) in this universe
which means that all physical quantities like mass of a body, velocity,
momentum, acceleration etc. of an object or length of time are always measured
relative to something or a reference
frame. Also the laws of physics are the same everywhere, that is, the physical
laws have the same mathematical form everywhere.
Secondly, the speed of light is the same, no matter who
measures it or how fast the person is moving who is measuring it. In other
words, only the speed of light is absolute (unchangeable) in this universe.
Derivable from these postulates of special relativity, the startling
conclusions are:
There can be no motion of anything at a speed greater than
that of light in vacuum.
Mass of a body in motion increases as its velocity increases
Mass and energy are equivalent which means we may get energy
from mass and vice-versa. We have already seen the dangerous consequences of
the equation E=mc2, relating mass and energy.
Time depends on the relative motion of an observer measuring
time. So, time is not absolute or fixed which is contrary to our age-old
previous notion.
So far as general Theory of Relativity is concerned, in a
nutshell, its central premise is that massive objects cause distortion or
curvature of space-time which is determined by the distribution of matter and
energy within it, is felt as ‘gravity’.
It will be very interesting to know that this vast universe
was born about 13.8 billion years from now by a mysterious explosion, famously
called ‘Big Bang’ of a particle smaller than a grain of sand, coined as ‘Cosmic
egg’. At that time there was nothing, absolutely nothing. The whole of the
universe was lying within the cosmic egg as an embryo.
The Big Bang generated unimaginably enormous heat energy
during explosion which we are unable to determine. According to Einstein’s
theory of relatively, it is ascribed as ‘Singularity’, which means a point
which contains a huge mass in infinitely small space where density and gravity
become infinite and space-time curves infinitely where the laws of physics as we
know them cease to operate. All other forms of energies like light, electrical
etc., matter of all kinds and everything of today’s universe, known and unknown
were subsequently formed by the passage of millions and billions of years of
time.
To get an idea of Einstein’s general theory of relativity,
one of the towering achievements of the 20th century physics, at
least in a nut shell, a well-informed person should know mainly the entities
like, “Space”, “time”, “Space-time curvature” and “gravity”. It will help to
grasp how this theory finds its applications to unravel many great mysteries of
this universe and its usage in our everyday life. Now let us first consider,
what is meant by “Space”. It is a term that can refer to various contexts in science,
math and communications.
In astronomy and cosmology, space the vast 3-dimensional
(having length, breadth and height as each dimension) expanse which is free or
unoccupied. Space is usually thought to begin at the lowest altitude at which
Satellites can maintain orbit for a reasonable time without falling into
atmosphere. This is approximately 160 km
above the surface of the Earth. Astronomers may speak of interplanetary space
(the space between planets in our Solar System), interstellar (space between
stars in galaxies) or intergalactic space (space between galaxies in the
universe). Some scientists believe that space extends infinitely far in all
directions while the others believe that space is finite but unbounded.
It will be very appropriate to mention an interesting fact
that one of the greatest scientist like, Einstein believed that this universe
along with all heavenly bodies are fixed and eternal. But in 1929 an American
astronomer Edwin Hubble proved from his observations of sky for long 10 years
through Hooker’s telescope, the then largest 100 inches telescope in diameter,
set up on the hills of Mount Wilson near California, that this universe is
expanding with tremendous speed.
In Einstein’s theory of relativity there was seed of the idea
that this universe will either expand or contract. But Einstein himself failed
to understand this truth which was concealed in the equation related to the
very theory he propounded. So, sometimes a mathematician may be wrong but his
math is always right.
However, after Hubble’s this amazing discovery, Einstein met
him to confess with regret for his idea that this universe is static is ‘the
biggest blunder of (his) life’.
Another important component of Einstein’s theory of
relativity is “time” which is something that we deal with every day and
something that everyone thinks he understood. But, in fact, time is curious and
slippery concept which continues to defy distinctive explanation and robust
definition, despite hundreds, even thousands of years’ endeavour . An elusive
explanation and vague definition of time is forwarded as it is “enigmatic”,
meaning thereby – difficult to interpret and understand, as well as “ineffable”
or very difficult to express in words. But that does not help us much in our
search for the true nature of time. Nearly two and a half thousand years ago,
Aristotle, the esteemed philosopher and great thinker of ancient Greece,
contended, “time is the most unknown of all unknown things” and arguably not
much has changed since then.
At a first glance, it seems obvious that time means: it is
the tick tock of the clock, the turning of the pages of a calendar. But really
these are just incidental physical manifestations of the underlying concept.
Time is “ubiquitous” means present everywhere, and essential ingredient of both
everyday life and all manner of academic thought, but its fundamental nature
remains tantalizingly difficult to encapsulate.
Another way of looking time is as the totality of three
separate elements: the past, the present and the future. The past may be
defined as those events which occurred before a point of time and are
considered to be fixed and immutable. It can be assessed through memory or
recorded in writing. The study of the past, in particular as it relates to humans,
is called “history”.
The present may be defined as time associated with events perceived
directly and for the first time. It is equivalent to the word “now” and the
period of time located between the past and the future.
The future is the indefinite period after the present
moment. It is the portion of the projected time line that is anticipated to
occur, and may be considered as potentially infinite in its context. Some
people may consider it as fixed and predetermined while many other think it to
be quite unknown and open to host of a number of possibilities.
Time appears to be unidirectional, that is, moves only in
one direction like a projected arrow. The past lies behind us and is fixed and
immutable; the future on the other hand, lies ahead and is not necessarily
fixed, although we can perhaps predict it to some extent. Most of the events we
experience are irreversible. For an example, it is easy for us to break an egg
and hard, if not impossible, to get back an already broken egg. This one way direction
or symmetry of time is often referred as the “arrow of time” and what gives us
impression of flow of time into the future.
Physicists, who study the time explicitly, agree that time
is one of the most difficult properties of our universe to understand. They
consider time as basic or primary concept, and not made up of or dependent on
anything else.
In science, generally time is usually defined by its
measurement and it is simply what a clock reads. Physics, in particular, often
requires extreme levels of precision in time measurement which has led to the requirement that time be
considered an infinitely divisible linear continuum and not quantized, i.e.
composed of discrete and divisible units.
According to Sir Isaac Newton the greatest physicist and
mathematician of all times thought that time is “absolute” or not changeable.
It exists independently of any perceiver; progresses at a consistent pace
throughout the universe; is measurable but imperceptible and can only be
understood mathematically. For Newton, absolute time and space were independent
and separate aspects of objective reality. It is not dependent of physical
events or each other.
The general view of the physicists is that time started at a
specific point about 13.8 billion years ago with the Big Bang, when the entire
universe suddenly expanded out of an infinitely hot, indefinitely dense state,
named as singularity, a point where the laws of physics as we understand them
simply break down, it has already been mentioned. This can be considered the
“birth” of the universe and beginning of time. Before the Big Bang there was no
space or time and you cannot go further back in time before Big Bang.
As the eminent theoretical physicist of present day Stephen
Hawking (1942-2018) noted in his famous book “A Brief History of Time” (1988)
that if there was any time-frame, at all, before the Big Bang, we are in
complete dark about it and hence any event occurred then would have no effect
on our present time-frame. So any hypothetical event from before the Big Bang
can, therefore, be considered meaningless.
Einstein’s theory of relativity created a fundamental link
between space and time. As we know, the universe may be viewed as having 3-
space dimensions – up/down, left/right, forward/backward (or read as: having length,
breadth and height). Einstein added time as another distinct dimension to it,
creating a 4 – dimensional space referred to as “Space-time continuum”.
This epoch - making idea has changed our understanding of
time dramatically. The traditional Newtonian idea of absolute time and space
have been superseded by the notion of time as one dimension of space-time
incorporated in special relativity and
dynamically curved space-time in general relativity.
It was Einstein’s genius to realise that speed of light is
absolute, invariable and cannot be exceeded. In fact he rightly perceived that
the speed of light is actually more fundamental than either time or space. In
reality, time is certainly an integral part of the very fabric of the universe.
But if the speed of light is invariable and absolute, Einstein realized, both –
space and time must be flexible and relative to accommodate this.
If you drop something, it falls down instead up. Well,
everybody knows that but what is the reason behind it? It is because of
gravity. Again the question is: what is gravity?
Perhaps all of us have heard the famous story, having
doubtful authenticity but circulated widely as true, that of the legendary Scientist
Isaac Newton and fall of an apple on his head while he was contemplating the
mysterious universe sitting under an apple tree, leading to the discovery of
the epoch making laws of Gravitation.
In fact, the authentic story behind this discovery was that
he was encouraged and got idea from the discoveries of the Laws of Falling
Bodies by Galileo and Laws of Planetary Motion by Kepler ---- two predecessors
of him.
Galileo Galilei (1564-1642), considered to be the father of
modern science, was one of the first modern scientists to state that the Laws
of nature are mathematical. He made observations and then tried to determine
the math that explains them.
In 1589, to demonstrate that all bodies, irrespective of
their weight fall at the same rate towards the centre of the earth as long as
air resistance was negligible, he dropped balls of different weights from the
Leaning Tower of Pisa, Italy. This discovery encouraged Newton and showed him
the way to generalize it and state that --- each and every body in this
universe attracts every other body.
Johannes Kepler (1571-1630) was a German mathematician,
astronomer, astrologer, is famous for his Laws of Planetary motion, known as Kepler’s
law. It explains how the planets move around the Sun but not why. Newton filled
the gap by supposing there was a force, acting between the bodies which make
them moving around each other. We now know this force as gravity.
Newton published his Theory of universal gravitation* in
1689s. It basically set forth the idea that gravity was a predictable force that
acts on all masses in the universe, and depends on the masses of the two bodies
attracting each other and also on the distance between them.
[* Newton’s law of universal
gravitation states that every mass attracts every other mass in the universe,
and the gravitational force between two bodies is proportional to the product
of their masses and inversely proportional to the square of the distance
between them. Mathematically if (M) and (m) be the masses of two bodies
attracting each other with a force (F) remaining at a distance (d), then
here G, called universal
gravitational constant which is same for everything
thought the universe.]
So, the more massive something is, the more of a
gravitational pull it exerts. As we walk on the surface of the Earth, it pulls
on us and we pull back. But since the Earth is so much more massive than we
are, the pull from us is not strong enough to move the Earth while the pull
from the Earth can make us fall that on our faces.
In addition, gravity also depends on how far you are from
something. This is why we are stick to the surface of the Earth instead of
being pulled off in the Sun, which has many more times the gravity of the
Earth.
There is gravity everywhere in the universe. It has played a
great part in making the universe the way it is. Gravity is what makes pieces
of matter clump together into planets, moons, and stars. Gravity is what makes
the planets orbit the stars --- like the Earth orbit our star, the Sun. Gravity
is what makes stars clump together in huge, swirling galaxies.
Newton’s laws of gravitation assume that gravity is an
innate or natural force of an object that can act over a distance. In Newton’s
theory, time and space are absolute, separate from each other, which, as we
common people believe.
The great Scientist Albert Einstein had a new idea of
gravity. He thought that gravity is what happens when space itself is curved or
wrapped around a mass, such as a star or a planet would cause kind of a dip in
space(for a crude analogy, let us think space, as if, a thin sheet of rubber).
So that any other object that came to near would tend to fall in to the dip.
So, Gravity is not a force, but as a consequence of the
curvature of space-time caused by uneven distribution of mass/energy.
All of these ideas are incorporated in his General Theory of
Relativity.
We have seen before some very useful applications of Quantum
theory, a matter of modern physics, but what about Einstein’s theories of
special and general relativity? When Einstein completed his theory of
relativity in 1915, he had little concern for practical or observable
consequences. But after his discovery of relativity scientists are able to come
up with many theories regarding how the universe works from general relativity
and instances in advancement of physics, astronomy and cosmology.
All these relate to academic and scientific advancement
which is, of course, good. But it will be even better to see instances of
general relativity used in practical purposes. One could hardly imagine a
branch of fundamental physics less likely to have practical consequences. But
strangely enough, relativity plays a key role in a multi –million dollar growth
industry centred around the “Global Positioning System” or “GPS”.
GPS is primarily a navigational system. So a background on
navigation will give insight as to how extraordinary GPS is.
People first navigate only by means of landmarks, such as, mountains,
trees or leading trails of stones. This would work only, within a local area
and the environment was subject to change due to environment factors, such as,
disasters.
For travelling across sea, a magnetic compass is required
and calculations of how fast the ship was going, was applied. The measurement
tools were crude and inaccurate. It was also a very complicated process.
Consider for a moment that you are riding a commercial
airliner, the pilot and crew are navigating to destination with the aid of GPS.
If your Plane approaches an airport in bad weather, and you just happened to be
worried, think about Einstein’s theory of relativity and GPS tracker in the
cockpit helping the pilots guide you to safe landing. Further, many luxury cars
now come up with built-in navigation systems that includes GPS receivers with
digital maps and you can purchase hand-hold GPS navigation units that will give
your position on Earth (latitude, longitude and altitude) to an accuracy of 5
to 10 meters that weigh only few grammes at an affordable cost.
GPS was built mainly for military navigation and became
fully operational in 1958. It has rapidly transformed into a thriving
commercial industry.
The system is based on an array of 24 satellites orbiting
the Earth, each carrying a precise atomic clock as the operation of GPS
requires very precise time synchronisation to a level of accuracy that requires
this atomic clock. These clocks are built on the basis the principle of quantum
mechanics. By ultra-precise atomic clock, time now is measured very very
accurately (to about 10– 15 seconds).
Using hand- held GPS receiver which detects radio emissions
from any of the satellites which happen to be over head, users of even
moderately priced devices, such as internet connected smart phones, can
determine latitude, longitude and altitude to an accuracy which currently reach
15 meters and local time of 50 billionth of a second. Apart from military uses,
GPS is finding applications in airplane navigation, oil exploration, a trust worthy
guide during an adventure in wilderness, such as a wild and uncultivated
region, as of forest or desert, uninhabited or inhabited only by wild animals,
a tract of wasteland, bridge construction, sailing and interstate trucking, to
name just a few.
But in relativistic world, things are not so simple. The problems of time dilations are of two types, one due the speed of the satellite
and the other for its location. But what is meant by “time dilation”?
It is observed that as an object moves with relativistic
speed, a speed comparable to speed of light (an unbelievable speed of 300000 km
per second), a strange thing seems to happen to its time as observed by us, the
stationary observer relative to Earth what we see is that the clock in motion
slows down according to our clock, therefore we read two different times. Which
time is correct? Both of them! Because time is not absolute unit but is
relative, it depends on the reference frame (in this case one is in motion and
the other is stationary). Let’s look at the following classic example:
·
There is a set of twins, one an astronaut and the
other works for the mission control of NASA. The astronaut leaves on deep space
trip travelling, if possible, at a speed of almost that of light. Upon
returning, the astronaut’s clock measured 10 years, so his age has increased by
10 years. However, when the astronaut reunites with his Earth bound twin he
sees that his brother is 32 years older than him. Is it magic or a miracle? No,
nothing of that sort at all. It is the effect of relativity, it is the effect
of time dilation, it is the role of math behind relativity. This is explained
due to the fact that the astronaut was travelling at relativistic speeds and
therefore his clock slows down, which is termed ”time dilation”. The actual calculations may
be performed by make use of the equation provided us by the theory of
relativity connecting the two different ages of the twins, speed of space craft
for the space journey and speed of light in vacuum. For sake of simplicity, let
us keep aside the nitty- gritty of that calculation.
There is another type of time dilation due to effect of
gravity. By Einstein’s theory of relativity, time runs slower whenever gravity
is stronger.
For example, let us look to the moon, our nearest natural
satellite. The moon is one fourth the size of the Earth, so its force of
gravity is much less than Earth’s force of gravity. Actually, it is one sixth
that of the Earth.
These two factors of time dilation affect the functioning of
the GPS receiver. So, to receive flawless guidance regarding correct time and
exact location monitored by a GPS tracker, a proper correction for time
dilations must be taken into account.
The satellite clocks are moving at 14000 km per hour in
orbits that circle the Earth twice per day, much faster than clocks on the
surface of the Earth, and Einstein’s theory of special relativity says that
rapidly moving clocks tick more slowly by about seven micro seconds (millionths
of a second) per day.
Also the orbiting clocks are 20,000 km above the Earth and
experience gravity that is four times weaker than that on the ground.
Einstein’s general relativity theory says that gravity curves space and time,
resulting in a tendency for the orbiting clocks to tick slightly faster, by
about 45 microseconds per day. The net result is that time on a GPS Satellite
clock advances faster than a clock on the ground by about 38 microseconds per
day.
But as 38 microseconds per day, the relativistic effect in
the rates of the satellite clocks is so large that if left uncompensated, it
would cause navigational errors that accumulate faster than 10 km per day. GPS
accounts for relativity by electronically adjusting the rates of the satellite
clocks, and by building mathematical corrections into the computer chips which
solve for user’s location. Without proper application of relativity, GPS would
fail in its navigational functions within about 2 minutes.
The benefits from the theory of fundamental physics like
that of thermodynamics, quantum mechanics
and relativity in our everyday life mentioned so far are only a few of the
gifts of science to us for enriching our living standard, diagnosing critical
diseases, helping in its treatment, in relieving pain, making life pleasurable
and joyful and much more. But one thing is to be borne in mind that whenever
there is a science there is math working in one way or the other, overtly or
covertly. So we may think that these are
the gifts of math in disguise.
[To continue]
