Time as systematized in
modern scientific society

Atomic time

Relativistic effects

A clock displaying TAI
on Earth will have periodic, relativistic deviations from the dynamical scale
TDB and from a pulsar time scale PS (see below Pulsar time). These variations,
denoted R above, were demonstrated in 1982–84 by measurements of the pulsar PSR
1937+21.

The main contributions
to R result from the continuous changes in the Earth's speed and distance from
the Sun. These cause variations in the transverse Doppler effect
andin the red shift due to the Sun's gravitational
potential. The frequency of TAI is higher at aphelion (about July 3) than at
perihelion (about January 4) by about 6.6 parts in 1010, and TAI is more
advanced in epoch by about 3.3 milliseconds on October 1 than on April 1.

By Einstein's theory of
general relativity a photon produced near the Earth's surface should be higher
in frequency by 1.09 parts in 1016 for each metre above sea level. In 1960 the
U.S. physicists Robert V. Pound and Glen A. Rebka
measured the difference between the frequencies of photons produced at
different elevations and found that it agreed very closely with what was
predicted. The primary standards used to form the frequency of TAI are
corrected for height above sea level.

Two-way,
round-the-world flights of atomic clocks in 1971 produced changes in clock
epochs that agreed well with the predictions of special and general relativity.
The results have been cited as proof that the gravitational red shift in the
frequency of a photon is produced when the photon is formed, as predicted by
Einstein, and not later, as the photon moves in a gravitational field. In
effect, gravitational potential is a perturbation that lowers the energy of a
quantum state.

relativistic mechanics

science concerned with the
motion of bodies whose relative velocities approach the speed of light c, or
whose kinetic energies are comparable with the product of their masses m and
the square of the velocity of light, or mc2. Such bodies are said to be
relativistic, and when their motion is studied, it is necessary to take into
account Einstein's special theory of relativity. As long as gravitational
effects can be ignored, which is true so long as gravitational potential energy
differences are small compared with mc2, the effects of Einstein's general
theory of relativity may be safely ignored.

The bodies concerned
may be sufficiently small that one may ignore their internal structure and size
and regard them as point particles, in which case one speaks of relativistic
point-particle mechanics; or one may need to take into account their internal
structure, in which case one speaks of relativistic continuum mechanics. This
article is concerned only with relativistic point-particle mechanics. It is
also assumed that quantum mechanical effects are unimportant, otherwise
relativistic quantum mechanics or relativistic quantum field theory—the latter
theory being a quantum mechanical extension of relativistic continuum
mechanics—would have to be considered. The condition that allows quantum
effects to be safely ignored is that the sizes and separations of the bodies
concerned are larger than their Compton wavelengths. (The Compton wavelength of
a body of mass m is given by h/mc, where his Planck's constant.) Despite these
restrictions, there are nevertheless a number of situations in nature where
relativistic mechanics is applicable. For example, it is essential to take into
account the effects of relativity when calculating the motion of elementary particles
accelerated to higher energies in particle accelerators, such as those at CERN
(European Organization for Nuclear Research) near Geneva or at Fermilab (Fermi National Accelerator Laboratory) near
Chicago. Moreover, such particles are caused to collide, thus creating further
particles; although this creation process can only be understood through
quantum mechanics, once the particles are well separated, they are subject to
the laws of special relativity.

Similar remarks apply
to cosmic rays that reach the Earth from outer space. In some cases, these have
energies as high as 1020 electron volts (eV). An
electron of that energy has a velocity that differs from that of light by about
1 part in 1028, as can be seen from the relativistic relation between energy
and velocity, which will be given later. For a proton of the same energy, the
velocity would differ from that of light by about 1 part in 1022. At a more
mundane level, relativistic mechanics must be used to calculatethe energies of electrons or positrons emitted by
the decay of radioactive nuclei. Astrophysicists need to use relativistic
mechanics when dealing with the energy sources of stars, the energy released in
supernova explosions, and the motion of electrons
moving in the atmospheres of pulsars or when considering the hot big bang. At
temperatures in the very early universe above 1010 kelvins
(K), at which typical thermal energies kT (where k is
Boltzmann's constant and T is temperature) are
comparable with the rest mass energy of the electron, the primordial plasma
must have been relativistic. Relativistic mechanics also must be considered
when dealing with satellite navigational systems used, for example, by the
military, such as the Global Positioning System (GPS). In this case, however, it
is the purely kinematic effecton
the rate of clocks on board the satellites (i.e., time dilation) that is
important rather than the dynamic effects of relativity on the motion of the
satellites themselves.

Development of the
special theory of relativity

Since the time of
Galileo it has been realized that there exists a class of so-called inertial
frames of reference—i.e., in a state of uniform motion with respect to one
another such that one cannot, by purely mechanical means, distinguish one from
the other. It follows that the laws of mechanics must take the same form in
every inertial frame of reference. To the accuracy of present-day technology,
the class of inertial frames may be regarded as those that are neither
accelerating nor rotating with respectto the distant
galaxies. To specify the motion of a body relative to a frame of reference, one
gives its position x as a function of a time coordinate t(x is called the
position vector and has the components x, y, and z).

Newton's first law of
motion (which remains true in special relativity) states that a body acted upon
by no external forces will continue to move in a state of uniform motion
relative to an inertial frame. It follows from this that the transformation
between the coordinates (t, x) and (t´, x´) of two inertial frames with
relative velocity u must be relatedby a linear
transformation. Before Einstein's special theory of relativity was published in
1905, it was usually assumed that the time coordinates measured in all inertial
frames were identical and equal to an “absolute time.” Thus,

The position
coordinates x and x´ were then assumed to be related by

The two formulas (97)
and (98) are called a Galilean transformation. The laws of nonrelativistic
mechanics take the same form in all frames related by Galilean transformations.
This is the restricted, or Galilean, principle of relativity.

The position of a
light-wave front speeding from the origin at time zero should satisfy

in the frame (t, x) and

in the frame (t´, x´). Formula
(100) does not transform into formula (99) using the Galilean transformations
(97) and (98), however. Put another way, if one uses Galilean transformations
one finds that the velocity of light depends on one's inertial frame, which is
contrary to the Michelson-Morley experiment (see relativity). Einstein realized
that either it is possible to determine a unique absolute
frame of rest relative to which the motion of a light wave is given by equation
(99) and its velocity is c only in that frameor
the assumption that all inertial observers measure the same absolute time
t—i.e., formula (97)—must be wrong. Since he believed in (and experiment
confirmed) the (extended) principle of relativity, which meant that one cannot,
by any means, including the use of light waves, distinguish between two
inertial frames in uniform relative motion, Einstein chose to give up the
Galilean transformations (97) and (98) and replaced them with the Lorentz transformations:

where xï
and x⊥ are the projections of
x parallel and perpendicular to the velocity u, respectively, and similarly for
x´.

The reader may check
that substitution of the Lorentz transformation
formulas (101) and (102) into the left-hand side of equation (100) results in
the left-hand side of equation (99). For simplicity, it has been assumed here
and throughout this discussion, that the spatial axes are not rotated with
respect to one another. Even in this case one sometimes considers Lorentz transformations that are more general than those of
equations (101) and (102). These more general transformations may reverse the
sense of time; i.e., t and t´ may have opposite signs or may reverse spatial
orientation or parity. To distinguish this more general class of
transformations from those of equations (101) and (102), one sometimes refers
to (101) and (102) as properLorentz transformations.

The laws of light
propagation are the same in all frames related by Lorentz
transformations, and the velocity of light is the same in all such frames. The
same is true of Maxwell's laws of electromagnetism. However, the usual laws of
mechanics are not the same in all frames related by Lorentz
transformations and thus must be altered to agree with the principle of
relativity.

The unique absolute
frame of rest with respect to which light waves had velocity c according to the
prerelativistic viewpoint was often regarded, before
Einstein, as being at rest relative to a hypothesized
all-pervading ether. The vibrations of this ether were held to explain the
phenomenon of electromagnetic radiation. The failure of experimenters to detect
motion relative to this ether, together with the widespread acceptance of
Einstein's special theory of relativity, led to the abandonment of the theory
of the ether. It is ironic therefore to note that the discovery in 1964 by the
American astrophysicists Arno Penzias
and Robert Wilson of a universal cosmic microwave 3 K radiation background
shows that the universe does indeed possess a privileged inertial frame.
Nevertheless, this does not contradict special relativity because one cannot
measure the Earth's velocity relative to it by experiments in a closed
laboratory. One must actually detect the microwaves themselves.

If the relative
velocity u between inertial frames is small in magnitude compared with the
velocity of light, then Galilean transformations and Lorentz
transformations agree, as do the usual laws of nonrelativistic
mechanics and the more accurate laws of relativistic mechanics. The requirement
that the laws of physics take the same form in all inertial reference frames
related by Lorentz transformations is called for the
sake of brevity the requirement of relativistic invariance. It has become a
powerful guide in the formation of new physical theories.

Relativistic space-time

The modification of the
usual laws of mechanics may be understood purely in terms of the Lorentz transformation formulas (101) and (102). It was
pointed out, however, by theGerman mathematician
Hermann Minkowski in 1908, that the Lorentz transformations have a simple geometric
interpretation that is both beautiful and useful. The motion of aparticle may be regarded as forming a curve made up of
points, called events, in a four-dimensional space whose four coordinates
comprise the three spatial coordinates x ≡ (x, y, z) and the time t.

The four-dimensional
space is called Minkowski space-time and the curve a
world line. It is frequently useful to represent physical processes by
space-time diagrams in which time runs vertically and the spatial coordinates
run horizontally. Of course, since space-time is four-dimensional, at least one
of the spatial dimensions in the diagram must be suppressed.

Newton's first law can
be interpreted in four-dimensional space as the statement that
the world lines of particles suffering no external forces are straight lines in
space-time. Linear transformations take straight lines to straight lines, and Lorentz transformations have the additional property that
they leave invariant the invariant interval τ through two events (t1, x1)
and (t2, x2) given by

If the right-hand side
of equation (103) is zero, the two events may be joined by a light ray and are
said to be on each other's light cones because the light cone of any event (t,
x) inspace-time is the set of points reachable from
it by light rays (see Figure 1). Thus the set of all events (t2, x2) satisfying
equation (103) with zero on the right-hand side is the light cone of the event
(t1, x1). Because Lorentz transformations leave invariant
the space-time interval (103), all inertial observers agree on what the light
cones are. In space-time diagrams it is customary to adopt a scaling of the
time coordinate such that the light cones have a half angle of 45°.

If the right-hand side
of equation (103) is strictly positive, in which case one says that the two
events are timelike separated, or have a timelike interval, then one can find an inertial frame with
respect to which the two events have the same spatial position. The straight
world line joining the two events corresponds to the time axis of this inertial
frame of reference. The quantity τ is equal to the difference in time
between the two events in this inertial frame and is called the proper time
between the two events. The proper time would be measured by any clock moving
along the straight world line between the two events.

An accelerating body
will have a curved world line that may bespecified by
giving its coordinates t and x as a function of the proper time τ along
the world line. The laws of either may be phrased in terms of the more familiar
velocity v = dx/dt and acceleration a = d2x/dt2 or in
terms of the 4-velocity (dt/dτ, dx/dτ) and 4-acceleration (d2t/dτ2, dx/dτ2).
Just as an ordinaryvector like v has three
components, vx, vy, and vz, a 4-vector has four components. Geometrically the
4-velocity and 4-acceleration correspond, respectively, to the tangent vector
and the curvature vector of the world line (see Figure 2). If the particle
moves slower than light, the tangent, or velocity, vector at each event on the
world line points inside the light cone of that event, and the acceleration, or
curvature, vector points outside the light cone. If the particle moves with the
speed of light, then the tangent vector lies on the light cone at each event on
the world line. The proper time τ along a world line moving with a speed
less than light is not an independent quantity from t and x: it satisfies

For a particle moving
with exactly the speed of light, one cannot define a proper time τ. One
can, however, define a so-called affine parameter that satisfies equation (104)
with zero on the right-hand side. For the time being this discussion will be
restricted to particles moving with speeds less than light.

Equation (104) does not
fix the sign of τ relative to that of t. It is usual to resolve this
ambiguity by demanding that the proper time τ increase as the time t
increases. This requirement is invariant under Lorentz
transformations of the form of equations (101) and (102). The tangent vector
then points inside the future light cone and is said to be future-directed and timelike (see Figure 3). One may if one wishes
attach an arrow to the world line to indicate this fact. One says that the
particle moves forward in time. It was pointed out by the Swiss physicist
Ernest C.G. Stückelberg deBreidenbach
and by the American physicist Richard Feynmanthat a
meaning can be attached to world lines moving backward in time—i.e., for those
for which ordinary time t decreases as proper time τ increases. Since, as
shall be shown later, the energy E of a particle is mc2dt/dτ, such world
lines correspond to the motion of particles with negative energy. It is
possible to interpret these world lines interms of
antiparticles, as will be seen when particles moving in a background
electromagnetic field are considered.

The fundamental laws of
motion for a body of mass m in relativistic mechanics are

and

where m is the constant
so-called rest mass of the body and the quantities (f0, f) are the components
of the force 4-vector. Equations (105) and (106), which relate the curvature of
the world line to the applied forces, are the same in all inertial frames
related by Lorentz transformations. The quantities (mdt/dτ, mdx/dτ) make up
the 4-momentum of the particle. According to Minkowski's
reformulation of special relativity,a
Lorentz transformation may be thought of as a
generalized rotation of points of Minkowski space-time
into themselves. It induces an identical rotation on the 4-acceleration and
force 4-vectors. To say that both of these 4-vectors experience the same
generalized rotation or Lorentz transformation is
simply to say that the fundamental laws of motion (105) and (106) are the same
in all inertial frames related by Lorentz
transformations. Minkowski's geometric ideas provided
a powerful tool for checking the mathematical consistency of special relativity
and for calculating its experimental consequences. They also have a natural
generalization in the general theory of relativity, which incorporates the
effects of gravity.

Relativistic momentum,
mass, and energy

The law of motion (106)
may also be expressed as:

where F = f Ö1 - v2/c2 .
Equation (107) is of the same form as Newton's second law of motion, which
states that the rate of change of momentum equals the applied force. F is the
Newtonian force, but the Newtonian relation between momentum p and velocity vin which p = mv
is modified to become

Consider a relativistic
particle with positive energy and electric charge q moving in an electric field
E and magnetic field B; it will experience an electromagnetic, or Lorentz, force given by F = qE + qv × B. If t(τ) and x(τ)
are the time and space coordinates of the particle, it follows from equations
(105) and (106), with f 0 = (qE · v)dt/dτ and f = q(E + v × B)dt/dτ,
that -t(-τ) and -x(-τ) are the coordinates of a particle with
positive energy and the opposite electric charge -q moving in the same electric
and magnetic field. A particle of the opposite charge but with the same rest
mass as the original particle is called the original particle's antiparticle.
It is in this sense that Feynman and Stückelberg
spoke of antiparticles as particles moving backward in time. This idea is a
consequence of special relativity alone. It really comes into its own, however,
when oneconsiders relativistic quantum mechanics.

Just as in nonrelativistic mechanics, the rate of work done when the
point of application of a force F is moved with velocity v equals F · v when
measured with respect to the time coordinate t. This work goes into increasing
the energy E of the particle. Taking the dot product of equation (107) with v
gives

The reader should note
that the 4-momentum is just (E/c2, p). It was once fairly common to encounter
the use of a “velocity-dependent mass” equal to E/c2. However, experience has
shown that its introduction serves no useful purpose and may lead to confusion,
and it is not used in this article. The invariant quantity is the rest mass m.
For that reason it has not been thought necessary to add a subscript or
superscript to m to emphasize that it is the rest mass rather than a
velocity-dependent quantity. When subscripts are attached to a mass, they
indicate the particular particle of which it is the rest mass.

If the applied force F
is perpendicular to the velocity v, it follows from equation (109) thatthe energy E, or, equivalently, the velocity squared
v2, will be constant, just as in Newtonian mechanics. This will be true, for
example, for a particle moving in a purely magnetic field with no electric
field present. It then follows from equation (107) that the shape of the orbits
of the particle are the same according to the classical and the relativistic
equations. However, the rate at which the orbits are traversed differs
according to the two theories. If w is the speed according to the nonrelativistic theory and v that according to special
relativity, then w = v Ö1 − v2/c2.

For velocities that are
small compared with that of light,

The
first term, mc2, which remains even when the particle is at rest, is called the
rest mass energy. For a single particle, its inclusion in the expression for energy
might seem to be a matter of convention: it appears as an arbitrary constant of
integration. However, for systems of particles that undergo collisions, its
inclusion is essential.

Both theory and
experiment agree that, in a process in which particles of rest masses m1, m2, .
. . mn collide or decay or transmute one into
another, both the total energy E1 + E2 + . . . + En and the total momentum p1 +
p2 + . . . + pn are the same before and after the
process, even though the number of particles may not be the same before andafter. This corresponds to conservation of the total
4-momentum (E1 + E2 + . . . + En)/c2,p1 + p2 + . . . +
pn).

The relativistic law of
energy-momentum conservation thus combines and generalizesin
one relativistically invariant expression the
separate conservation laws of prerelativistic
physics: the conservation of mass, the conservation of momentum, and the conservation
of energy. In fact, the law of conservation of mass becomes incorporated in the
law of conservation of energy and is modified if the amount of energy exchanged
is comparable with the rest mass energy of any of the particles.

For example, if a particle
of mass M at rest decays into two particles the sum of whose rest masses m1 +
m2 is smaller than M (see Figure 4), then the two momenta
p1 and p2 must be equal in magnitude and opposite in direction. The quantityT = E - mc2 is the kinetic energy of the particle.
In such a decay the initial kinetic energy is zero.
Since the conservation of energy implies that in the process Mc2 = T1 + T2 +
m1c2 + m2c2, one speaks of the conversion of an amount (M - m1 - m2)c2 of rest mass energy to kinetic energy. It is precisely
this process that provides the large amount of energy available during nuclear
fission, for example, in the spontaneous fission of the uranium-235 isotope.
The opposite process occurs in nuclear fusion when two particles fuse to form a
particle of smaller total rest mass. The difference (m1 + m2 - M) multiplied by
c2 is called the binding energy. If the two initial particles are both at rest,
a fourth particle is required to satisfy the conservation of energyand momentum. The rest mass of this fourth particle
will not change, but it will acquire kinetic energy equal to the binding energy
minus the kinetic energy of the fused particles. Perhaps the most important
examples are the conversion of hydrogen to helium in the centre of stars, such
as the Sun, and during thermonuclear reactions used in atomic bombs.

This article has so far
dealt only with particles with non-vanishing rest mass whose velocities must
always be less than that of light. One may always find an inertial reference
frame with respect to which they are at rest and their energy in that frame
equals mc2. However, special relativity allows a generalization of classical
ideas to include particles with vanishing rest masses that can move only with
the velocity of light. Particles in nature that correspond to this possibility
and that could not, therefore, be incorporated into the classical scheme are
the photon, which is associated with the transmission of electromagnetic
radiation, three species of neutrinos associated with the weak interaction
responsible for radioactive decay, and—more speculatively—the graviton, which
plays the same role with respect to gravitational waves as does the photon with
respect to electromagnetic waves. Strictly speaking, it is not yet known
whether the rest masses of the three neutrino species vanish, but in many
processes their energies exceed their possible rest masses by so much that they
may be regarded as effectively massless. The velocity
v of any particle in relativistic mechanics is given by v = pc2/E, and the
relation between energy E and momentum is E2 = m2c4 + p2c2. Thus for massless particles E =|p|c and
the 4-momentum is given by (|p|/c, p). Itfollows from
the relativistic laws of energy and momentum conservation that, if a massless particle were to decay, it could do so only if the
particles produced were all strictly massless and
their momenta p1, p2, . . . pn
were all strictly aligned with the momentum p of the original massless particle. Since this is a situation of vanishing
likelihood, it follows that strictly massless
particles are absolutely stable.

It also follows that
one or more massive particles cannot decay into a single massless
particle, conserving both energy and momentum. They can, however, decay into
two or more massless particles, and indeed this is
observed in the decay of the neutral pion into
photons and in the annihilation of an electron and a positron pair into
photons. In the latter case, the world lines of the annihilating particles meet
at the space-time event where they annihilate. Using the interpretation of
Feynman and Stückelberg, one may view these two world
lines as a single continuous world line with two portions, one moving forward
in time and one moving backward in time (see Figure 5). This interpretation
plays an important role in the quantum theory of such processes.

Gary William Gibbons

**Additional reading**

An outstanding work
containing an account of the special theory of relativity is Abraham Pais, “Subtle Is the Lord—”: The Science and Life of Albert
Einstein (1982). Some good introductions at the undergraduate level are W. Rindler, Essential Relativity: Special, General, and
Cosmological, 2nd ed. (1977); James H. Smith, Introduction to Special
Relativity (1965); Edwin F. Taylor and John Archibald Wheeler, Spacetime Physics (1966). More substantial treatises are J.
Aharoni, The Special Theory
of Relativity, 2nd ed. (1965, reprinted 1985); and J.L. Synge,
Relativity: The Special Theory, 2nd ed. (1965).Gary William Gibbons