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Relativity

Albert Einstein's theory of relativity forms

one of the two pillars of modern physics, the

other being quantum mechanics. It consists of

two parts: the special theory of relativity from

1905 and the general theory of relativity from

1915.

The special theory of relativity describes

how space and time are perceived by

observers in different inertial systems.

Einstein derived this theory from a single

physical principle of relativity. It was

discovered in 1632 by Galileo Galilei that the

laws of mechanics are the same in all inertial

systems—a discovery, known as Galileo's

principle of relativity, that constituted a

radical break with the prevailing Aristotelian

physics. Einstein's principle of relativity

generalized this to all laws of nature,

including Maxwell's laws of

electromagnetism which govern the

propagation of light. It thus follows from

Einstein's principle of relativity that the speed

of light is the same in all inertial systems, a

central result in the theory of relativity. Prior

to Einstein, it was believed that light

propagates through a “luminiferous aether” in

the same way as sound propagates through

air, but all attempts to measure the speed of

the Earth relative to this aether, such as the

Michelson-Morley experiment in 1887, failed.

Special relativity explained the negative

results of these experiments and made the

aether hypothesis superfluous.

The general theory of relativity unifies

special relativity with Isaac Newton's law of

universal gravity. Its basis is Einstein's

equivalence principle according to which an

accelerated system of reference (such as a so-

called Einstein elevator) is indistinguishable

from a system at rest in a gravitational field.

Mathematically, Einstein's field equations

describe how the presence of mass, energy,

and momentum gives rise to a curvature of

space and time. Although this has little

significance in weak gravitational fields such

as that of the Earth, general relativity is

essential in the study of the universe as a

whole. For example, Karl Schwarzschild in

1915 found an exact solution to Einstein's

equations that makes possible the existence of

black holes.

The many surprising consequences of

the theory of relativity have been described in

numerous popularizations, most notably by

George Gamow. Einstein's theory must not be

confused with the various relativist positions

in philosophy such as aesthetic, moral,

cultural, or cognitive relativism.

Special relativity

The Lorentz transformation forms the basis of

the special theory of relativity. It is a set of

equations describing how to translate suitably

chosen coordinates of space and time between

two inertial systems S and S' moving with the

speed v relative to one another:

),( vtxx

−

=

′

γ

)./(

2

cvxtt −=

′γ

Here, c denotes the speed of light of

299,792,458 meters per second, and the

dimensionless number

22

/1

1

cv−

=γ

is the so-called Lorentz factor. In 1908,

Hermann Minkowski gave a mathematical

description of the Lorentz transformation as a

rotation of the coordinate axes in four-

dimensional space-time. When v is much

smaller than c, the Lorentz factor is close to 1,

and the Lorentz transformation reduces to the

classical Galilean transformation. When v

approaches c, however, the Lorentz

transformation has a number of consequences

that radically contradict classical physics as

well as common sense. For example, clocks in

motion are slowed down (relativistic time

dilation), objects in motion are contracted in

the direction of movement (relativistic length

contraction), and clocks in motion which are

seen as synchronized by an observer moving

with the clocks are seen as non-synchronized

by an observer at rest (relativity of

simultaneity).

It is another consequence of special

relativity that no material objects, or signals

of any kind, can travel faster than light. This

is so because anything traveling faster than

light relative to one observer would appear to

be traveling backwards in time relative to

another observer, thus leading to paradoxes

regarding cause and effect. There is a

quantum-mechanical phenomenon, the so-

called Einstein-Podolsky-Rosen paradox,

which seems to contradict this principle:

according to quantum mechanics, the wave

function of two entangled particles is affected

by a measurement of the state of one of the

particles, causing an instantaneous change to

the state of the other even if the two particles

are located in different galaxies. But this

phenomenon, which has since been verified

experimentally, does not really contradict

relativity since it cannot be used to transmit

information from one galaxy to the other.

Special relativity dictates that mass and

energy are connected by the equation

,

2

mcE =

undoubtedly the most famous formula in all

of physics. Any particle with mass m has a

rest energy given by this equation. If the same

particle is accelerated to the speed v, its

energy is multiplied by the Lorentz factor γ,

and its kinetic energy is found as the

difference between total energy and rest

energy:

.

2

1

222

mvmcmcE

kin

≈−= γ

The approximation, valid for v much smaller

than c, equals the expression for kinetic

energy in classical mechanics. This formula

shows that it would require an infinite amount

of energy to accelerate a particle with positive

mass to the speed of light.

General relativity

Einstein noted that special relativity implies

that space appears to be curved, or non-

Euclidean, to observers in accelerated

systems, for example on a rotating disc, and

inferred from the equivalence principle that

the same must be true in gravitational fields.

However, after realizing this fundamental

principle in 1907, it took him eight years to

find the field equations which describe the

exact curvature of space-time.

The idea that physical space might be

curved was not new. Already in 1823, Carl

Friedrich Gauss investigated this question

empirically by measuring the sum of angles of

a triangle formed by three mountain tops but

found no curvature. Bernhard Riemann

further developed the mathematics of curved

space in 1854, and this work would become

an essential part of Einstein's theory.

General relativity predicts that a body

falling freely in a gravitational field, such as

the Earth in its orbit around the Sun, follows a

geodesic in curved space-time. This geodesic

is called the body's world-line. In a curved

space, geodesics are the least curved lines, in

the same way as the equator is a least curved

line on the surface of the Earth. Although the

predictions of general relativity are nearly the

same as those of classical mechanics for

bodies in weak gravitational fields, the

interpretation of gravity is radically different:

whereas classical mechanics explains the

elliptical orbit of the Earth as a consequence

of a gravitational force emanating from the

Sun, general relativity postulates that the mass

of the Sun gives rise to a curvature of space-

time, and that the world-line of the Earth is in

fact a geodesic.

It is a consequence of general relativity

that clocks in gravitational fields are slowed

down. This effect is called gravitational time

dilation. For a clock at rest in the gravitational

field of the Earth, the dilation factor is

,1

2

1

22

rc

GM

rc

GM

−≈−

where G is Newton's gravitational constant, M

is the mass of the Earth, and r is the distance

between the clock and the center of the Earth.

Proofs and applications of relativity

Einstein showed in 1915 that general

relativity explains the perihelion precession of

the planet Mercury. This is the phenomenon,

which had mystified astronomers since its

discovery in 1859, that the elliptical orbit of

Mercury rotates around the Sun with 43 arc

seconds per century.

Also in 1915, Einstein predicted that

light emitted from distant stars is deflected

when passing through the gravitational field

of the Sun. Although this effect had

previously been derived from Newtonian

gravity alone, Einstein showed that the angle

of deflection following from general relativity

is twice the angle following from classical

physics. Einstein's prediction was confirmed

dramatically by Arthur Eddington during the

total solar eclipse of 29 May 1919.

Contrary to quantum mechanics the

technological implementations of which are

ubiquitous, relativity has few practical

applications. One notable exception is the

Global Positioning System (GPS). The GPS

satellites revolve around the Earth twice per

sidereal day in a height of about 20,000

kilometers and with a speed of about 4

kilometers per second. This means that the

atomic clocks aboard the satellites are subject

both to relativistic time dilation due to the

speed, and to a reduced gravitational time

dilation due to the height. The first effect

amounts to a loss of 7 microseconds per day,

the second to a gain of 45 microseconds per

day. In total, therefore, the atomic satellite

clocks gain 38 microseconds per day relative

to clocks on the ground. Failure to take these

relativistic effects into account would render

GPS useless since the resulting positional

error would accumulate to 11 kilometers per

day.

SEE ALSO: Black Holes; Einstein, Albert;

GPS Systems; Geometry of the Universe;

Gravity.

FURTHER READINGS:

Einstein, Albert. Relativity: The Special and

General Theory. New York: Henry Holt,

1920.

Feynman, Richard, Robert Leighton and

Matthew Sands. The Feynman Lectures on

Physics. Reading: Addison-Wesley, 1964.

Gamow, George. Mr. Tompkins in

Wonderland. New York: Macmillan, 1946.

Møller, Christian. The Theory of Relativity.

Oxford: Oxford University Press, 1952.

Russell, Bertrand. The ABC of Relativity.

London: Kegan Paul, Trench, Trubner, 1925.

David Brink, Ph.D.

University College Dublin, Ireland