I What is relativity? How did the concept of space-time arise?

What is relativity? How did the concept of space-time arise?
I like Andrew Zimmerman Jones’s approach to answering the first question (in About.com
Guide). According to him classical relativity (defined initially by Galileo and refined by
Newton) involves a simple transformation between two inertial reference frames- one
frame containing, say, a moving object and another containing an observer. To quote him:
“If you are walking in a moving train, and someone stationary on the ground is watching,
your speed relative to the observer will be the sum of your speed relative to the train and the
train's speed relative to the observer. You're in one inertial frame of reference, the train itself
(and anyone sitting still on it) are in another, and the observer is in still another. The problem
with this is that light was believed, in the majority of the 1800s, to propagate as a wave
through a universal substance known as the ether, which would have counted as a separate
frame of reference (similar to the train in the above example). The famed Michelson-Morley
experiment, however, had failed to detect Earth's motion relative to the ether and no one
could explain why. Something was wrong with the classical interpretation of relativity as it
applied to light ... and so the field was ripe for a new interpretation when Einstein came
along (with his Special Theory of Relativity)”
Special Theory of Relativity deals with phenomena that become noticeable when
travelling near the speed of light and reference frames that are moving at a constant
velocity ( inertial reference frames). The result of Einstein's paper was to introduce new
coordinate transformations, called Lorentz transformations, between inertial frames of
reference. At slow speeds, these transformations were essentially identical to the classical
model, but at high speeds, near the speed of light, they produced radically different results,
among them, relativistic mass, relativistic momentum, time dilation, length contraction and
relativity of simultaneity.
The Special Theory of Relativity is based on two postulates:
1. The laws of physics are the same in all inertial (=non-accelerating) reference frames.
This Relativity Principle postulate implies that there is no experiment whether based on the
laws of mechanics or the laws of electromagnetism from which it is possible to determine
whether or not a frame of reference is in a state of uniform motion.
2. The speed of light in free space is constant and independent of the motion of its
source.
Special relativity is formulated so as to not assume that any particular frame of reference is
special; rather, in relativity, any reference frame moving with uniform motion will observe
the same laws of physics. In particular, the speed of light in a vacuum is always measured to
be c, even when measured by multiple systems that are moving at different (but constant)
velocities.
In framing this postulate on the constancy of the speed of light, Einstein was mindful of
Maxwell’s elegant unification of electricity and magnetism in 1862 which showed for the
first time that varying electric and magnetic fields could feed off each other- these fields
could propagate indefinitely through space in a self-sustaining fashion, far from the
varying charges and currents where they originated. It is not hard to show that Maxwell’s
equations imply a speed for electromagnetic waves given by 1/ єoµo , where єo and µo are
respectively the permittivity and permeability of free space. This when evaluated is in fact
the speed of light (about 300,000 km/s). In spite of its stunning successes, it had one huge
problem. Not all inertial observers saw the same Maxwell's laws. Physics was no longer
relative. Two inertial observers at uniform motion relative to each other would measure
different speeds of light. Maxwell himself was aware of this and attempted to solve the
problem by defining his theory to be applicable only to observers stationary relative to a
frame defined by a substance called ether that filled all space. The failure to detect this
frame of reference termed ether continued to haunt physicists searching for a better
explanation. It was Einstein who boldly proclaimed that there was nothing wrong with
Maxwell's laws but only with Galilean relativity. He restored relativity to all of physics by
defining a new set of inertial observers who all agreed on the laws of physics.
One immediate consequence of Einstein’s two postulates is that the speed of light is the
same in all inertial frames of reference. Inherent also in the theory is the idea of a spacetime continuum, that is, a four-dimensional coordinate system containing three Euclidian
spatial dimensions and one time dimension. It was Hermann Minowski, a former professor
of Einstein, who formally introduced this concept of a single manifold combining space and
time to the physics community. In relativistic contexts, time cannot be separated from the
three dimensions of space, because the observed rate at which time passes for an object
depends on the object's velocity relative to the observer and also on the strength
of gravitational fields, which can slow the passage of time. Just as there is an infinite number
of 3-D reference frames in Galilean relativity, there is an infinite number of 4-D space-time
reference frames in Einstein's theory. This is to say that Einstein put an end to absolute
time. His revolutionary insight was that the flow of time in the universe differs depending on
one's reference frame.
An outstanding feature of Special Relativity is its mass-energy relation, which is expressed
in the well-known formula: E=mc². "Mass" is defined in two different ways in special
relativity – either as an invariant quantity ("rest mass" or "invariant mass") which is the
same for all observers in all reference frames, or as a quantity which is dependent on the
velocity of the observer ("relativistic mass").
Einstein derived this relation in an attempt to reconcile Maxwell's electromagnetic theory
with the conservation of energy and momentum. Maxwell said that light carries a
momentum, which is to say that a wave carries an amount of energy. Due to the principle of
conservation of momentum, if a body emits energy in the form of radiation, the body loses
an equivalent amount of mass that is given by E/c². This describes the relation between
energy and mass.
The relativistic expressions for energy (E) and momentum (p) obey the relativistic energymomentum equation: E2 – (pc)2 = (mc2)2 , where the m is the rest mass, c is the speed of
light and E is the total energy.
The equation is also valid for light photons, which have m = 0:
E2 – (pc) 2 = 0
and, therefore, E = pc.
A photon is a discrete energy packet of light. Each photon contains a packet of energy, hf,
where f is the frequency of light and h is Planck’s constant (6.63 x 10 -34 joules second).
Note that light can be treated as a wave or as a particle. As a wave, we can talk about its
wavelength (λ), and as a particle, we can talk about its momentum (p = mc). That is,
E = hf = h c/λ = pc, whence λ = h/p, which connects the momentum of a particle with the
length of its matter wave.
A photon's momentum is a function of its energy, but it is not proportional to the velocity,
which is always c.
For an object at rest, the momentum p is zero, therefore, E0 = mc2. This is a special case of
Einstein’s equation.
The rest mass is only proportional to the total energy in the rest frame of the object.
When the object is moving, the total energy is given by
Nobody knows why the speed of light remains unchanged from one inertial frame to
another. One way to look at it (as Minkowski did a century ago) is that light speed is not a
velocity at all but simply the number of metres in one second. Light travels along the
shortest path between two points in space-time (a geodesic). If the geodesic is curved,
then the path of light is curved. Light is affected by gravity, but not in its speed. Gravity
has two effects on light, namely, bending of light and changing light’s energy (by shifting the
frequency of light: gravitational red shift).
Light has no mass and based on Newtonian theory it would not be affected at all by gravity.
But it has energy and is subject to the effects of gravity. Hence, in Einstein’s theory, light
will fall towards the earth like any other object. That is to say, a photon of light suffers
acceleration in a gravitational field. However, instead of the photon of light travelling
faster and faster as would be the case of a falling ball, the distance the photon of light
travels (and the time it takes the photon to travel that distance) compresses in a manner
to offset the acceleration by gravity so that the velocity of the photon of light always
remains the same, equal to c, relative to the external observer. This space-time
compression effect though is miniscule in the earth’s gravitational field, but not elsewhere
in the universe such as at a black hole where the warping of space-time is infinitely strong
(resulting in a ‘singularity’ at its centre which is governed by the laws of quantum gravity)
caused by the immense energy of the warpage itself. “Warpage begets warpage in a
nonlinear
self-bootstrapping
manner”,
asserts
physicist
Kip
Thorne
(http://www.its.caltech.edu/~kip/scripts/PubScans/VI-47.pdf). We now believe that some stars
end their lives as black holes. “Black holes are like cosmic vacuum cleaners that suck up
everything they encounter, from giant stars to particles of space dust and the photons that
make up light.” (John Boslough: In “Stephen Hawking’s Universe”, Fontana, 1986). In our
own Milky Way Galaxy, recent evidence based on clues from intense radio emissions
(matter near a black hole can release extraordinary amounts of light, including radio waves,
as it gets super-heated rushing toward the point of no return) suggests that there is a black
hole (named Sagittarius A*) approximately 4.3 million times the mass of the sun. Black
holes, large or small, have an event horizon at their edge, where quantum transactions can
occur, and thus allow for emission of particles as first proposed in 1974 by Stephen
Hawking, a legend in the field of astrophysics. Since that period, mathematical evidence has
accumulated that black holes do emit particles and some of them can even explode.
The space-time concept has significantly impacted physical theories and particularly
workings of the universe at both the super-galactic and subatomic levels. The modern
theory of inflation of the Universe actually indicates that space was once accelerating faster
than the speed of light ... nothing in space can move faster than the speed of light, but
space itself can move faster.
Since the speed of light is the same for every observer in an inertial frame of reference, two
things that are simultaneous to one observer will not be so according to other observers.
That is to say, simultaneity is not an absolute concept. The statement, therefore, that “two
events at different places occurred at the same time" is true only in a certain inertial
reference frame and will be found to be incorrect in other frames.
Before we leave the subject, let us examine the case of case of Time dilation citing the
famous twin paradox example, predicted by Einstein’s Special Theory of Relativity.
Two twin brothers, Tom & Jones, are celebrating their 25th birthday. On the same day,
following the birthday party, Tom goes on a space journey in a spaceship that travels at 99%
of the speed of light. Tom stays on his journey for precisely one year and returns to Earth on
his 26th birthday. On Earth, however, seven years have elapsed, so his twin brother Jones is
32 years old at the time of his arrival. This is due to the fact that time is stretched by factor 7
at approx. 99% of the speed of light, which means that in the space traveller Tom’s
reference frame, one year is equivalent to seven years on earth. Yet, time appears to have
passed normally to both brothers, i.e. both still need five minutes to shave each morning in
their respective reference frame. This feature of time dilation can be expressed
mathematically. Relative to the stationary observer, time in the moving system will be
observed to be running slower by a factor γ, called the Lorentz factor. That is, t’ = t γ,
where γ = 1/ (1-v2/c2)1/2, where v= relative velocity between inertial reference frames. At
common speeds, such as that of a car or jet plane, the time dilation is insignificant, but there
is a dramatic increase as v approaches the velocity of light, c. The twin paradox has been
verified experimentally by precise measurements of atomic clocks flown in aircraft and
satellites.
Interestingly, while time expands from the perspective of the stationary observer, space
contracts from the perspective of the moving observer. This phenomenon is known as the
Lorentz contraction, which is exactly the reciprocal of the above time dilation formula,
namely, l’ = l / γ. The contraction is only in the direction parallel to the direction in which
the observed body is travelling. Once again, this effect is negligible at everyday speeds, but
is important at speeds approaching the velocity of light.
{Ref: http://physics.mq.edu.au/~jcresser/Phys378/LectureNotes/SpecialRelativityNotes.pdf}
A postscript on space-time
Despite the overwhelming success of the theory of General Relativity, an impasse remained
between it and quantum theory, largely as a result of Einstein’s deep-rooted detest of
imprecise description of physical quantities with an associated element of uncertainty. In
general relativity, mass and energy are treated in a purely classical manner in the sense that
physical quantities such as the strengths and directions of various fields and the positions
and velocities of particles have definite values. These quantities are represented by tensor
fields, sets of (real) numbers associated with each space-time point. On the other hand, in
quantum mechanics the greater the accuracy of a measurement of location the more
uncertainty there is in the measurement of energy. The Heisenberg uncertainty principle
applies not just to particles but also to empty space. Over very short intervals phantom or
virtual particles can appear. The shorter the time, the more massive the particles can be. At
very short intervals, virtual particles will be massive enough to form black holes. It seemed
an impossibility to extrapolate simultaneously both quantum mechanics and general
relativity to minute distances. The theories simply diverge. A first big step forward to
bridging the divide came in 1974 with Stephen Hawking’s theoretical discovery of exploding
black holes. Very strong gravity could in effect squeeze particles of matter out of empty
space, in accordance with quantum theory. Einstein’s notions of space could no longer be
absolute because of the breakdown of laws of geometry at microscopic distances; space
becomes ‘foamy’ and unpredictable at the Planck length scale ~ 10-33cm.
There is wide agreement among physicists today that the general relativistic treatment of
gravity must be replaced by one in which the gravitational force is replaced by one dealing
with a quanta. Quantum Gravity is the name given to any theory that describes gravity in
the regimes where quantum effects cannot be disregarded. At present, there is no such a
theory which is universally accepted and confirmed by experience. Therefore the term
"Quantum Gravity" indicates more an open problem than a specific theory. To quote
Professor Carlo Rovelli of the Centre of Theoretical Physics, Marseilles
(http://www.scholarpedia.org/article/Quantum_gravity):
“In general relativity, space and time lose their properties of being a fixed framework in
which the dynamical world is immersed. They are identified with the gravitational field and
acquire dynamical properties: space-time can "fold" and "stretch" like a rubber sheet. When
we take quantum mechanics into account, we realize that this "rubber sheet" is in fact a
quantum field, and therefore, like all quantum fields, it must have a microscopic granular
structure (like the photons forming the electromagnetic field) and a probabilistic dynamics.
Therefore quantum gravity is likely to be the theory of granular and probabilistic "quantum
space" and "quantum time". Building the mathematical language and the conceptual
structure for making sense of such notions of quantum space and quantum time is the
challenge for a quantum theory of gravity.”
A microscopic granular structure for space-time appears to be mind-boggling, but the
subject has engaged serious study among theoretical physicists. Brian Greene in his
bestselling book “The Fabric of the Cosmos” (Penguin Books, 2004) asserts that there are
presently two promising directions in the search for the most elementary particles of spacetime: the string theory and the loop quantum theory. Both theories start at opposing ends
of the dimensional scale: the string theory starts with the small (quantum theory) and
moves to embrace the large (gravity), while loop quantum gravity takes the reverse
approach. To some extent, the main achievements of each prove to be the failings of the
other. To quote Greene:
“String theory merges all forces and all matter, including gravity, by describing everything in
the language of vibrating strings (a complete unification that eludes loop theory). The
particle of gravity, the graviton, is but one particular string vibrational pattern, and hence
the theory naturally describes how these elemental bundles of gravity move and interact
quantum mechanically. The failure of current formulations of string theory is that they
presuppose a background space-time within which strings move and vibrate. Loop quantum
gravity does not assume a background space-time. It is a background-independent
framework….”
There is little doubt that either theory, if they remain front-runners, has to address the
complex mathematics that is involved. Eventually some convergence is anticipated, and
when that happens, we might have the seeds of a “theory of everything” for the universe, a
long-cherished desire of Einstein.
vg kumar das (20 July 2012)
[email protected]