Download SR 52(12) 14-20

COVER STORY
BIMAN BASU
Looking back at hundred years of general relativity, one may ask:
“How has general relativity changed the world?” The answer can be
seen everywhere in the cosmos.
G
RAVITY is so ubiquitous and so
familiar to us that we hardly ever
give it a serious thought. When anything
falls from a height or something topples
over we take it for granted. Scientists
consider gravity as one of the four
fundamental forces of nature – the other
three being electromagnetism, weak
force, and strong force. The uniqueness of
gravity that sets itself apart from the other
three is its universality – it represents the
universal tendency of all matter to attract
all other matter and it is effective over
extremely large distances. It is also the
weakest of the fundamental forces.
We learnt in school that the English
physicist Isaac Newton discovered
gravity in 1666 after observing an apple
fall in a garden and that he also concluded
that the same force also keeps the Moon
in orbit around Earth. His detailed work
on gravity was published in Principia
Mathematica, in 1687. Newton presented
gravity as a pull to Earth and later, in
a more generalised way, as a force off
attraction that acts universally between
any two masses and the strength of which
is proportional to the product of the two
masses and inversely proportional to the
distance between them.
Left: The equivalence principle.
The windowless chamber on
the left is standing on ground
and the one at right is moving
up with uniform acceleration.
But the occupants in both will
feel the same force of gravity.
Right: Einstein as a patent
examiner at the Swiss Patent
Office in Bern, where he
completed four revolutionary
pieces of work including the
special theory of relativity in
1905.
SCIENCE REPORTER, DECEMBER 2015
14
COVER STORY
Einstein described what happens when
mass is present in spacetime, causing
it to curve and forcing objects travelling
through it to follow a bent (and longer)
path. If enough mass is packed into a very
small region, spacetime becomes infinitely
curved, creating a ‘black hole’.
An artist’s drawing of a black hole named Cygnus X-1. It formed when a large star caved in.
This black hole pulls matter from a blue star beside it. (Credit: NASA/CXC/M. Weiss)
General relativity redefined the concept of gravity; rather than a
force pulling masses together, the theory exposed it as a simple
consequence of the geometry of space and time.
Since then, for almost 250 years,
gravity was regarded as an attractive
force that holds the universe together,
till the German physicist Albert Einstein
came up with a better explanation with
his general theory of relativity in 1915.
Einstein showed that gravity is not a force
but a consequence of curving spacetime.
In other words, the popular idea about
gravity from everyday experience
does not hold water. General relativity
changed the entire concept of how the
universe works.
Anomalous Gravity
There was a serious anomaly in the
Newtonian concept of gravity. If gravity
is a force then it could not explain how
the value of the acceleration due to
gravity, denoted by ‘g’, could be the same
irrespective of the mass of a falling object,
which makes objects with vastly different
masses such as a feather and a hammer
fall at the same rate in a vacuum. Yet this
has been verified experimentally up to an
accuracy of one part in one trillion, i.e., 1
in 1012.
15
On the other hand, according to
Newton’s second law of motion, if the
acting force is the same, the acceleration
produced in a body should depend
inversely on its mass, a condition that
does not appear to hold in case of a falling
body where the acceleration is the same
irrespective of mass.
The question arises: How does the
Earth’s gravity know with how much
force to pull to make objects of different
masses fall at the same rate? There ought
to be something wrong with the concept
of gravity as a force. Einstein decided to
find out.
Einstein published his special
theory of relativity in 1905 in which he
SCIENCE REPORTER, DECEMBER 2015
COVER STORY
Left: According to general relativity, mass
curves spacetime which brings in the effects
observed due to gravity.
Equivalence Principle
established that nothing can travel faster
than the speed of light, which was finite.
Einstein came to this conclusion after the
famous Michelson-Morley experiment
showed that the speed of light did not
change as the Earth swung around
the Sun. But this was contradictory to
Newton’s theory of gravity, in which
gravity exerts its influence across space
instantly over large distances; that is, it
presumed the force of gravitation could
travel with infinite speed.
This looming contradiction set
Einstein thinking and made him to
rewrite the centuries-old rules of
Newtonian gravity. Einstein also saw a
parallel between the relative nature of
motion in his theory of special relativity
and the relative nature of gravity, and
so he decided to generalise relativity to
include both gravity and motion.
Einstein made several unsuccessful
attempts between 1908 and 1914 to
formulate a theory of gravity that was
consistent with special relativity. The
German physicist Max Planck is said to
have told Einstein, “As an older friend,
I must advise you against it….. You will
not succeed, and even if you succeed, no
one will believe you.” Never one to yield
to authority, Einstein proved him wrong!
In 1915, he came out with the general
theory of relativity that could answer
many questions Newton’s theory could
not. It revolutionised our concept of the
universe.
In 1915, Einstein came out with
the general theory of relativity
that could answer many questions
Newton’s theory could not.
Einstein’s equation linking the distribution of
matter to the curvature of space. The cause
(mass) is on the right; the effect (the curvature
of spacetime) is on the left. Gμν and Tμν
are components of tensors. (Credit: http://
manyworldstheory.com)
David Hilbert, who also tried to work out the
field equations of general relativity.
SCIENCE REPORTER, DECEMBER 2015
16
There is a long story behind the evolution
of the general theory of relativity. It all
began with a sudden thought – some
‘gedanken (thought) experiments’ by
Einstein. He had a habit of crafting new
theories about the natural world by using
his mental ability to push beyond the
limitations of laboratory measurements.
It was late 1907; two years after the
“miracle year” in which Einstein had
produced his special theory of relativity
and three other revolutionary pieces of
work, but he was still a patent examiner
in the Swiss patent office. While sitting
in his office in Bern, a thought suddenly
came to his mind. “If a person falls freely
under gravity,” he thought, “he will not
feel his own weight.” Einstein would later
call it “the happiest thought in my life.”
As any swimmer would know, one really
feels weightless while falling into the
pool after jumping from the diving board
because any object in free fall becomes
weightless.
Einstein soon refined his thought
experiment. He imagined the falling man
to be confined in an enclosed chamber,
such as an elevator in free-fall. While
falling, the man in the elevator would feel
weightless, as do today’s astronauts in an
orbiting spacecraft (because an orbiting
spacecraft is also a body in free-fall). Any
object dropped within the falling elevator
would not ‘fall’ but float alongside the
passenger because both are falling at the
same rate. Under such a situation, there
would be no way for the man to tell if
the elevator was falling at an accelerated
rate under gravity or it was floating in a
gravity-free region of outer space.
To extend the concept further,
Einstein imagined that if the elevator is at
rest in a gravitational field (for example,
on Earth) or is instead being hauled up
with constant acceleration, the man inside
would not be able to tell the difference. He
then conjectured that the laws of physics
must be identical in both situations, which
he called the “principle of equivalence”.
Accordingly, he concluded that within the
windowless elevator chamber, the effects
of gravitation would be indistinguishable
from that of uniform acceleration in the
absence of gravity. This realisation that
gravity and acceleration are equivalent
COVER STORY
Left: General relativity could correct
the observed anomalies in the
precession of Mercury’s perihelion.
Right: General relativity predicted
bending of light by gravity due to
curving of spacetime.
put Einstein on an arduous path to
generalise his special theory of relativity,
but he had to literally ‘race’ to find the
correct mathematical formulas for his
theory before the German mathematician
David Hilbert could do so.
Tragically, Einstein also had to face
struggle on the home front during the
same period, as he went through a divorce
from his first wife and a separation from
his sons. But in the end he came out
triumphant, delivering one of the world’s
most revolutionary scientific works in the
shape of his general theory of relativity.
General Relativity Takes Shape
The special theory of relativity is valid for
systems that are not accelerating, which
means it is valid only for situations where
no force is involved. Since gravity involves
acceleration, which involves a force, the
special theory of relativity cannot be used
when there is gravitational field present.
In the general theory of relativity, Einstein
tried to remove this restriction so that he
could apply his ideas to the gravitational
force also. But the task was not simple
and it took Einstein almost 10 years to
fully understand how to do this.
In his general theory of relativity
Einstein was attempting to formulate
a new theory of gravitation in place of
Newtonian gravitation. His goal was
to find the mathematical equations
describing two interwoven processes,
namely (i) how a gravitational field acts
on matter, telling it how to move, and (ii)
how matter generates gravitational fields
in spacetime, telling spacetime how to
curve. The struggle Einstein had to face
in trying to find a mathematical model to
explain gravity incorporating these two
was really daunting. For more than three
years Einstein tried hard to formulate
Einstein made several
unsuccessful
attempts
between 1908 and 1914 to
formulate a theory of gravity
that was consistent with
special relativity.
drafts and outlines that turned out to have
flaws. Then, beginning in the summer of
1915, “the math and the physics began to
come together”.
By late June 1915, many elements
of general relativity were almost ready
and Einstein decided to give a weeklong series of lectures later that month
on his evolving ideas at the University
of Göttingen in Germany. Foremost
among the learned audience was the
German mathematician David Hilbert
and Einstein was particularly eager
to explain all the intricacies of general
relativity to him. As he told his friends
later, he was quite enchanted with Hilbert
and “was able to convince Hilbert of the
general theory of relativity”. Hilbert was
likewise enchanted with Einstein and
with his theory, so much so that he “soon
set out to see if he could do what Einstein
had so far not accomplished: produce
the mathematical equations that would
complete the formulation of general
relativity”. So a race was on.
By early October 1915, Einstein started
worrying about Hilbert’s attempt to work
out general relativity, just as he realised
that his current version of the theory,
which he had been refining continuously,
still had serious flaws. His equations did
not account properly for rotating motion,
nor did they fully explain an anomaly that
astronomers had observed in the orbit
of the planet Mercury. In fact, he could
sense that Hilbert was closing in on the
correct equations. He decided to give a
series of four formal Thursday lectures
on his theory to the members of the
Prussian Academy in Berlin, beginning
4 November 1915. The result was an
17
exhausting month-long rush of activity
during which Einstein wrestled with a
succession of equations, corrections and
updates that he wanted to complete.
Finally, on 25 November 1915,
Einstein delivered the 4th and last of
the Thursday lectures to the Prussian
Academy of Sciences in which he
presented the final version of his
gravitational equations in a paper titled
“The Field Equations of Gravitation”,
in which he wrote, ‘Finally the general
theory of relativity is closed as a logical
structure’. The general theory of relativity
combines special relativity and Newton’s
law of universal gravitation, providing
a unified description of gravity as a
geometric property of space and time, or
spacetime. It characterises gravity not as
an innate force acting on objects but rather
the consequence of spacetime’s curvature.
It was a hard-fought battle.
Five days before Einstein made his
presentation to the Prussian Academy
of Sciences, Hilbert had submitted a
paper to the Königliche Gesellschaft
der Wissenschaften (Royal Scientific
Society) in Göttingen, which contained
the identical field equations but with
one small difference. But Hilbert never
claimed priority for this theory. Abraham
Pais, biographer of Einstein, wrote in
Subtle is the Lord: The Science and the Life of
Albert Einstein, “I do believe that Einstein
was the sole creator of the physical theory
of general relativity and that both he
and Hilbert should be credited for the
discovery of the fundamental equation (of
general relativity).”
General relativity redefined the
concept of gravity; rather than a force
SCIENCE REPORTER, DECEMBER 2015
COVER STORY
The path of totality during the 29 May solar eclipse.
When Einstein presented his general theory of relativity to the Prussian
Academy of Sciences in 1915, Eddington was in England. At that time, Britain
and Germany were at war; so there was no direct scientific communication
between the two countries.
pulling masses together, the theory
exposed it as a simple consequence of the
geometry of space and time. It established
space and time as a single entity,
spacetime. It launched new strands of
research that scientists are still pursuing
and made its creator a star. In his theory,
Einstein described what happens when
mass is present in spacetime, causing it
to curve and forcing objects travelling
through
g it to follow a bent (and longer)
g
English astronomer Arthur Eddington, who
observed the total solar eclipse of 29 May 1919
and proved Einstein’s theory.
SCIENCE REPORTER, DECEMBER 2015
path. If enough mass is packed into a
very small region, spacetime becomes
infinitely curved, creating a ‘black hole’.
Testing General Relativity
The first test of Einstein’s general
relativity came in 1915 and involved
Mercury – the planet closest to the Sun.
Like the other planets, Mercury also
moves around the Sun in an elliptical orbit
with the Sun at one of the foci. The point
of closest approach to the Sun is called
the perihelion. Normally, according to
Newtonian physics, one would expect
the orientation of the orbit to remain
unchanged; that is, the perihelion should
stay where it is, as it does for all solar
system planets except Mercury.
The perihelion of Mercury’s orbit
was known to move slightly with each
successive orbit of the planet round the
Sun. This was called the ‘precession’
of Mercury’s perihelion. Most of this
movement was easily accounted for in
terms of the gravitational attraction of
the other planets in the solar system.
However, it had been noted since 1845
that the actual rate of the precession
differed from the expected value by about
43 seconds of arc per century, which was
not much. But it was definitely there,
and was worrying because it could not
be explained by Newtonian physics.
18
Surprisingly, Einstein’s general relativity
could account for the difference and
correct it. In a way, it appeared to unify
Newton’s law of universal gravitation
with special relativity. Einstein was later
to declare that, on hearing the news of
the verification of his prediction, he “was
beside himself with ecstasy for days”.
The next proof of general relativity
was more dramatic. One consequence
of general relativity was the bending of
light by gravity. This was evident from
Einstein’s thought experiment involving
an elevator being hauled up with uniform
acceleration. If we imagine that the
elevator is being accelerated upward
and a light beam comes in through a
pinhole on one wall, by the time it reaches
the opposite wall, the light would be
a little closer to the floor because the
elevator would have moved upward.
And if it were possible to plot the beam’s
trajectory across the elevator chamber,
it would be curved because of the
upward acceleration of the elevator. The
equivalence principle says that this effect
should be the same whether the elevator
is accelerating upward or is resting still
in a gravitational field. In other words,
light should bend when passing through
a gravitational field.
Historically, the first calculation of
the deflection of light by mass is reported
to have been published by the German
physicist and mathematician Johann
Georg von Soldner in 1804. Basing his
calculations on Newton’s laws of motion
and gravitation and the assumption
that light behaves like very fast moving
particles, Soldner showed that rays from
a distant star skimming the Sun’s surface
would be deflected through an angle of
about 0.9 arc seconds, or one quarter of
a thousandth of a degree. The main flaw
in Soldner’s calculation was that light
particles or photons do not carry mass
and so cannot be deflected by Newtonian
gravity.
More than a century later, in his
general theory of relativity, Einstein
calculated that the deflection predicted
by his theory would be twice the
value calculated by Soldner. But the
experimental proof was yet to come,
and the proof finally came from a British
astronomer named Arthur Stanley
Eddington, who was then Secretary of the
Royal Astronomical Society.
COVER STORY
A timeline of the big bang, with cosmic
inflation on the far left. (Credit: NASA)
world over trumpeted the just released
astronomical measurements establishing
that the positions of stars in the sky
around the eclipsed Sun did appear
slightly different than what Newtonian
physics had predicted. Rather it was just
as it ought to be according to Einstein’s
theory. Eddington’s results triumphantly
confirmed Einstein’s theory and turned
him overnight into an international
superstar.
After the eclipse expedition, there
was some controversy that Eddington’s
analysis had been biased towards
general relativity. But matters were
put to rest in the late 1970s when the
photographic plates exposed during the
1919 eclipse plates were analysed again
and Eddington’s findings were shown to
be correct.
When Einstein presented his general
theory of relativity to the Prussian
Academy of Sciences in 1915, Eddington
was in England. At that time, Britain
and Germany were at war; so there
was no direct scientific communication
between the two countries. But the
Dutch mathematician and astronomer
Willem de Sitter, who was in Holland
at that time, was Eddington’s friend
and he passed on a copy of Einstein’s
paper to him. After reading the paper,
Eddington was immediately aware of its
ramifications and in a report to the Royal
Astronomical Society in early 1917, he
particularly stressed “the importance of
testing the theory using measurement
of light bending”. The upcoming total
solar eclipse of 29 May 1919 offered an
appropriate occasion to test the theory.
A few weeks later, the then
Astronomer Royal, Sir Frank Watson
Dyson, decided to send two scientific
expeditions to observe the eclipse. One,
led by Eddington, was to travel to the
Principe Island off the coast of Spanish
Guinea in West Africa. The other
expedition, led by Royal Greenwich
Observatory
astronomer
Andrew
Crommelin, was to travel to Sobral in
northern Brazil across the Atlantic.
Eddington’s plan was to observe a
bright cluster of stars called the Hyades
in the constellation of Taurus as the Sun
passed in front of them, as seen from Earth
during totality on 29 May. If Einstein’s
theory was correct, the positions of the
stars in the Hyades would appear to
shift by about 1/2000th of a degree from
their normal positions. To pinpoint the
normal position of the Hyades in the
sky, Eddington first took a picture of the
cluster at night from Oxford, UK. Then,
on the day of the eclipse at Principe, he
photographed the Hyades as they lay
almost directly behind the dark Sun, as
the Moon covered the bright solar disc
during totality, turning the sky dark.
Comparing the two measurements,
Eddington was able to show that the shift
was exactly as Einstein had predicted
(1.74 arc seconds) and too large to be
explained by Newton’s theory (0.09 arc
seconds).
Eddington’s
observations,
published in Philosophical Transactions of
the Royal Society, confirmed Einstein’s
theory, and were hailed at the time as a
conclusive proof of general relativity.
Einstein had become a famous
name in the scientific community for
his works on special theory of relativity,
photoelectric effect, Brownian motion,
and his famous equation E = mc2, which
he completed in 1905, although his work
did not immediately make any impact on
the public mind. But it was different after
it was proved that gravity indeed bends
light, exactly as predicted by general
relativity.
On 6 November 1919, four years
after Einstein announced his general
theory of relativity, newspapers the
19
Hundred Years Later
Looking back at hundred years of general
relativity, one may ask: “How has general
relativity changed the world?” The
answer can be seen everywhere in the
cosmos. Modern cosmology – the study of
the origin and evolution of the universe –
owes its origin to general relativity. It was
Einstein’s equations of general relativity
that first predicted an expanding universe,
although Einstein resisted this conclusion
and even modified his equations by
inserting the infamous “cosmological
constant” to ensure a static universe.
However, some of the earliest models
of the Universe to incorporate Einstein’s
general relativity were proposed by
Russian physicist Aleksandr Friedmann
and Belgian priest and physicist Georges
Lemaître. They independently derived
solutions to Einstein’s field equations,
which predicted expansion of space.
When subsequent observations by
Edwin Hubble showed that distant
galaxies are all rushing away and the
universe was indeed expanding, Einstein
abandoned the constant, calling it the
“biggest blunder” of his life. The idea of
an expanding universe gave rise to the
big bang theory, now almost universally
accepted.
The big bang theory has undergone
many changes in the decades since
Lemaître first proposed it, the most widely
held version today being the inflationary
theory. In the inflationary theory, the
SCIENCE REPORTER, DECEMBER 2015
COVER STORY
Gravitational lensing (diagrammatic)
universe is supposed to have expanded
for a fleeting instant at its beginning at
a much faster rate than that expected
for the original big bang concept. This
period, which is called the ‘inflationary
epoch’, is a consequence of the nuclear
force breaking away from the weak and
electromagnetic forces that it was unified
with at higher temperatures in what is
called a phase transition.
Refinements in the big bang theory
have also come to reconcile with results of
observational tests. One such observation,
which received the 2011 Nobel Prize in
Physics, revealed that for the past seven
billion years not only has space been
expanding, but the rate of expansion
has been speeding up. Interestingly,
according to some physicists, the best
explanation for this could be a version
of
Einstein’s
long-ago-discarded
cosmological constant!
In the formation known as Einstein’s Cross,
four images of the same distant quasar appear
around a foreground galaxy due to strong
gravitational lensing. (Credit: Wikimedia)
SCIENCE REPORTER, DECEMBER 2015
Another interesting consequence of
general relativity is a cosmic phenomenon
called gravitational lensing – an effect
of light bending due to gravity, which
often gives rise to multiple images of a
single distant astronomical object. The
gravitational field of a massive object
such as a galaxy or a galaxy cluster
extends far into space, and causes light
rays passing close to that object to be
bent and refocussed somewhere else. The
more massive the object, the stronger its
gravitational field and hence the greater
the bending of light rays – just as using
denser materials to make optical lenses
results in a greater amount of refraction.
Several instances of gravitational
lensing have already been observed
in images of the cosmos. Gravitational
lensing is useful to cosmologists because
it is directly sensitive to the amount and
distribution of dark matter, which cannot
be seen but exerts gravitational
effect. In fact, gravitational lensing
has been used to help verify the
existence of dark matter.
General relativity has also
given birth to the concept of
the black hole. The German
astronomer Karl Schwarzschild,
so the story goes, while carrying
out an analysis during his stint at
the Russian front in the midst of
World War I, derived the first exact
solution of Einstein’s equations,
which gave a precise description
of the warped spacetime produced
by a spherical body like the Sun.
Schwarzschild’s
results
revealed something extraordinary.
His calculations showed that if
a star like our Sun is compressed
to a sufficiently small size, say, to
less than five kilometres across,
20
the resulting spacetime warp “will be
so severe that anything approaching
too closely, including light itself, will be
trapped”. In other words, Schwarzschild’s
work revealed for the first time the
possibility of black holes, the existence of
which has since been established.
A black hole is formed when the
radius of the shrinking star reaches the
‘Schwarzschild radius’, which is the
radius of a sphere such that, if all the mass
of an object were to be compressed within
that sphere, the escape velocity from the
surface of the sphere would equal the
speed of light. The Einstein equations of
general relativity predict that something
weird and horrifying happens when a
mass is squeezed down to the size of its
Schwarzschild radius.
Subsequent to English theoretical
physicist and cosmologist Stephen
Hawking’s calculations in the 1970s,
physicists have become increasingly
convinced that “the extreme nature of
black holes makes them an ideal proving
ground for attempts to push general
relativity forward and, most notably, to
meld it with quantum mechanics”. In
fact, one of today’s most hotly debated
issues concerns how quantum processes
may affect our understanding of the outer
edge of a black hole – its event horizon
– as well as the nature of a black hole’s
interior.
As Columbia University physicist
and author Brian Greene wrote in Scientific
American (September 2015), “Einstein
had the right mind at the right moment
to crack a collection of deep problems of
physics. His numerous but comparatively
modest contributions in the decades after
the discovery of general relativity suggest
that the timeliness of the particular
intellectual nexus he brought to bear on
physics had passed”.
But it cannot be denied that, as
more consequences of his theory were
discovered, general relativity has become
entrenched in the popular imagination,
with its descriptions of expanding
universes and black holes. It is also
“tightly woven into the tapestry of today’s
leading-edge research”.
Mr. Biman Basu is an author, science
communicator & consultant. Address: C-203
Hindon Apartments, 23 Vasundhara Enclave,
Delhi-110096; Email: [email protected]