USSR postage stamp dedicated to Albert
Einstein
Special relativity (SR) (also known as the
special theory of relativity or
STR) is the
physical
theory of measurement in
inertial frames of reference
proposed in 1905 by
Albert Einstein
(after the considerable and independent contributions of
Hendrik Lorentz,
Henri Poincaré and others) in the paper
"
On the
Electrodynamics of Moving Bodies". It generalizes
Galileo's principle of relativity–that
all
uniform motion is
relative, and that there is no absolute and welldefined state of
rest (no
privileged reference
frames)–from
mechanics to all the
laws of physics, including both the
laws of mechanics and of
electrodynamics, whatever they may be.
Special relativity incorporates the principle that the
speed of light is the same for all inertial
observers regardless
of the state of motion of the source.
This theory has a wide range of consequences which have been
experimentally verified, including counterintuitive ones such as
length contraction,
time dilation and
relativity of simultaneity,
contradicting the classical notion that the duration of the time
interval between two events is equal for all observers. (On the
other hand, it introduces the
spacetime interval, which
is
invariant.) Combined with other laws of physics, the two postulates
of special relativity predict the equivalence of
matter and
energy, as expressed
in the
massenergy
equivalence formula
E =
mc^{2}, where
c is
the
speed of light in a vacuum. The
predictions of special relativity agree well with Newtonian
mechanics in their common realm of applicability, specifically in
experiments in which all velocities are small compared to the speed
of light. Special relativity reveals that
c is not just
the velocity of a certain phenomenon—namely the propagation of
electromagnetic radiation
(light)—but rather a fundamental feature of the way space and time
are unified as
spacetime. One of the
consequences of the theory is that it is impossible for any
particle that has
rest mass to be
accelerated to the speed of light.
The theory is termed "special" because it applies the
principle of relativity only to
frames in uniform relative
motion. Einstein developed
general relativity to apply the principle
more generally, that is, to any frame so as to handle
general coordinate transformations, and
that theory includes the effects of
gravity.
From the theory of general relativity it follows that special
relativity will still apply locally (i.e., to
first order) to observers moving on
arbitrary
trajectories, and hence to any
relativistic situation where gravity is not a significant
factor.
Postulates
Einstein discerned two fundamental propositions that seemed to be
the most assured, regardless of the exact validity of the (then)
known laws of either mechanics or electrodynamics. These
propositions were the constancy of the speed of light and the
independence of physical laws (especially the constancy of the
speed of light) from the choice of inertial system. In his initial
presentation of special relativity in 1905 he expressed these
postulates as:
 The Principle of Relativity – The laws by which the states of
physical systems undergo change are not affected, whether these
changes of state be referred to the one or the other of two systems
in uniform translatory motion relative to each other.
 The Principle of Invariant Light Speed – Light in vacuum
propagates with the speed c (a fixed constant) in terms of
any system of inertial coordinates, regardless of the state of
motion of the light source.
It should be noted that the derivation of special relativity
depends not only on these two explicit postulates, but also on
several tacit assumptions (which are made in almost all theories of
physics), including the
isotropy and
homogeneity of space and the
independence of measuring rods and clocks from their past
history.
Following Einstein's original presentation of special relativity in
1905, many different sets of postulates have been proposed in
various alternative derivations. However, the most common set of
postulates remains those employed by Einstein in his original
paper. A more mathematical statement of the Principle of Relativity
made later by Einstein, which introduces the concept of simplicity
not mentioned above is:
Henri Poincaré provided the
mathematical framework for relativity theory by proving that
Lorentz transformations are a subset of his
Poincaré group of symmetry
transformations. Einstein later derived these transformations from
his axioms.
Many of Einstein's papers present derivations of the Lorentz
transformation based upon these two principles.
Einstein consistently based the derivation of Lorentz invariance
(the essential core of special relativity) on just the two basic
principles of relativity and lightspeed invariance. He
wrote:
Thus many modern treatments of special relativity base it on the
single postulate of universal Lorentz covariance, or, equivalently,
on the single postulate of
Minkowski
spacetime.
From the principle of relativity alone without assuming the
constancy of the speed of light, i.e. using the isotropy of space
and the symmetry implied by the principle of special relativity,
one can show that the spacetime transformations between inertial
frames are either Euclidean, Galilean, or Lorentzian. In the
Lorentzian case, one can then obtain relativistic interval
conservation and a certain finite limiting speed. Experiments
suggest that this speed is the speed of light in vacuum.
Massenergy equivalence
In addition to the papers referenced above—which give derivations
of the Lorentz transformation and describe the foundations of
special relativity—Einstein also wrote at least four papers giving
heuristic arguments for the equivalence (and transmutability) of
mass and energy, for
E =
mc^{2}.
Massenergy equivalence is a consequence of special relativity. The
energy and momentum, which are separate in Newtonian mechanics,
form a
fourvector in relativity, and
this relates the time component (the energy) to the space
components (the momentum) in a nontrivial way. For an object at
rest, the energymomentum fourvector is (
E, 0, 0, 0): it
has a time component which is the energy, and three space
components which are zero. By changing frames with a Lorentz
transformation in the x direction with a small value of the
velocity v, the energy momentum fourvector becomes (
E,
Ev/
c^{2}, 0, 0). The momentum is equal to
the energy divided by
c^{2} times the velocity. So
the newtonian mass of an object, which is the ratio of the momentum
to the velocity for slow velocities, is equal to
E/
c^{2}.
The energy and momentum are properties of matter, and it is
impossible to deduce that they form a fourvector just from the two
basic postulates of special relativity by themselves, because these
don't talk about matter, they only talk about space and time. The
derivation therefore requires some additional physical reasoning.
In his 1905 paper, Einstein used the additional principles that
Newtonian mechanics should hold for slow velocities, so that there
is one energy scalar and one threevector momentum at slow
velocities, and that the conservation law for energy and momentum
is exactly true in relativity. Furthermore, he assumed that the
energy/momentum of light transforms like the energy/momentum of
massless particles, which was known to be true from Maxwell's
equations. The first of Einstein's papers on this subject was "Does
the Inertia of a Body Depend upon its Energy Content?" in 1905.
Although Einstein's argument in this paper is nearly universally
accepted by physicists as correct, even selfevident, many authors
over the years have suggested that it is wrong. Other authors
suggest that the argument was merely inconclusive because it relied
on some implicit assumptions.
Einstein acknowledged the controversy over his derivation in his
1907 survey paper on special relativity. There he notes that it is
problematic to rely on Maxwell's equations for the heuristic
massenergy argument. The argument in his 1905 paper can be carried
out with the emission of any massless particles, but the Maxwell
equations are implicitly used to make it obvious that the emission
of light in particular can be achieved only by doing work. To emit
electromagnetic waves, all you have to do is shake a charged
particle, and this is clearly doing work, so that the emission is
of energy.
Lack of an absolute reference frame
The
principle of relativity,
which states that there is no preferred
inertial reference frame, dates
back to
Galileo, and was
incorporated into Newtonian Physics. However, in the late
19
^{th} century, the existence of
electromagnetic waves led
physicists to suggest that the universe was filled with a substance
known as "
aether", which would
act as the medium through which these waves, or vibrations
traveled. The aether was thought to constitute an absolute
reference frame against which speeds could be measured. In other
words, the aether was the only fixed or motionless thing in the
universe. Aether supposedly had some wonderful properties: it was
sufficiently elastic that it could support electromagnetic waves,
and those waves could interact with matter, yet it offered no
resistance to bodies passing through it. The results of various
experiments, including the
MichelsonMorley experiment,
indicated that the Earth was always 'stationary' relative to the
aether–something that was difficult to explain, since the Earth is
in orbit around the Sun. Einstein's solution was to discard the
notion of an aether and an absolute state of rest. 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.
Consequences
Einstein has said that all of the consequences of special
relativity can be derived from examination of the
Lorentz transformations.
These transformations, and hence special relativity, lead to
different physical predictions than Newtonian mechanics when
relative velocities become comparable to the speed of light. The
speed of light is so much larger than anything humans encounter
that some of the effectspredicted by relativity are initially
counterintuitive:
 Time dilation –
the time lapse between two events is not invariant from one
observer to another, but is dependent on the relative speeds of the
observers' reference frames (e.g., the twin
paradox which concerns a twin who flies off in a spaceship
traveling near the speed of light and returns to discover that his
or her twin sibling has aged much more).
 Relativity of
simultaneity – two events happening in two different
locations that occur simultaneously in the reference frame of one
inertial observer, may occur nonsimultaneously in the reference
frame of another inertial observer (lack of absolute simultaneity).
 Lorentz
contraction – the dimensions (e.g., length) of an
object as measured by one observer may be smaller than the results
of measurements of the same object made by another observer (e.g.,
the ladder paradox involves a long
ladder traveling near the speed of light and being contained within
a smaller garage).
 Composition of
velocities – velocities (and speeds) do not simply
'add', for example if a rocket is moving at the speed of light
relative to an observer, and the rocket fires a missile at of the
speed of light relative to the rocket, the missile does not exceed
the speed of light relative to the observer. (In this example, the
observer would see the missile travel with a speed of the speed of
light.)
 Inertia and momentum
– as an object's speed approaches the speed of light from an
observer's point of view, its mass appears to increase thereby
making it more and more difficult to accelerate it from within the
observer's frame of reference.
 Equivalence of mass and energy, E = mc^{2}
– The energy content of an object at rest with mass m
equals mc^{2}. Conservation of energy implies that
in any reaction a decrease of the sum of the masses of particles
must be accompanied by an increase in kinetic energies of the
particles after the reaction. Similarly, the mass of an object can
be increased by taking in kinetic energies.
Reference frames, coordinates and the Lorentz
transformation
[[Image:Lorentz transform of world line.gifrightframedDiagram 1.
Changing views of spacetime along the
world
line of a rapidly accelerating observer.In this animation, the
vertical direction indicates time and the horizontal direction
indicates distance, the dashed line is the spacetime trajectory
("world line") of the observer. The lower quarter of the diagram
shows the events that are visible to the observer, and the upper
quarter shows the
light cone those that
will be able to see the observer. The small dots are arbitrary
events in spacetime.The slope of the world line (deviation from
being vertical) gives the relative velocity to the observer. Note
how the view of spacetime changes when the observer
accelerates.]]Relativity theory depends on "
reference frame". The term reference
frame as used here is an observational perspective in space at
rest, or in uniform motion, from which a position can be measured
along 3 spatial axes. In addition, a reference frame has the
ability to determine measurements of the time of events using a
'clock' (any reference device with uniform periodicity).
An event is an occurrence that can be assigned a single unique time
and location in space relative to a reference frame: it is a
"point" in
spacetime. Since the speed of
light is constant in relativity in each and every reference frame,
pulses of light can be used to unambiguously measure distances and
refer back the times that events occurred to the clock, even though
light takes time to reach the clock after the event has
transpired.
For example, the explosion of a firecracker may be considered to be
an "event". We can completely specify an event by its four
spacetime coordinates: The time of occurrence and its
3dimensional spatial location define a reference point. Let's call
this reference frame
S.
In relativity theory we often want to calculate the position of a
point from a different reference point.
Suppose we have a second reference frame
S′, whose spatial
axes and clock exactly coincide with that of
S at time
zero, but it is moving at a constant velocity
v with
respect to
S along the
xaxis.
Since there is no absolute reference frame in relativity theory, a
concept of 'moving' doesn't strictly exist, as everything is always
moving with respect to some other reference frame. Instead, any two
frames that move at the same speed in the same direction are said
to be
comoving. Therefore
S and
S′ are
not
comoving.
Let's define the
event to
have spacetime coordinates
(
t,
x,
y,
z) in system
S
and (
t′,
x′,
y′,
z′) in
S′. Then the
Lorentz
transformation specifies that these coordinates are related in
the following way:
 \begin{align}
t' &= \gamma (t  vx/c^2) \\x' &= \gamma (x  v t) \\y'
&= y \\z' &= z ,\end{align}where
 \gamma = \frac{1}{\sqrt{1  \frac{v^2}{c^2}}}
is the
Lorentz factor and
c
is the
speed of light in a
vacuum.
The
y and
z coordinates are unaffected; only the
x and
t axes transformed. These Lorentz
transformations form a
oneparameter
group of
linear mappings, that
parameter being called
rapidity.
A quantity invariant under
Lorentz transformations is known as
a
Lorentz scalar.
The Lorentz transformation given above is for the particular case
in which the velocity
v of
S′ with respect to S
is parallel to the
xaxis. We now give the Lorentz
transformation in the general case. Suppose the velocity of
S′ with respect to
S is
v.
Denote the spacetime coordinates of an event in
S by
(
t,
r) (instead of
(
t,
x,
y,
z)). Then the
coordinates (
t′,
r′) of this event in
S′ are given by:
 \left ( \begin{array}{l} t' \\ \mathbf{r}' \end{array} \right )
=\gamma(\mathbf{v}) \left ( \begin{array}{ll}1 &
\mathbf{v}^T/c^2 \\ \mathbf{v} &
P_{\mathbf{v}}+\alpha_{\mathbf{v}}(IP_{\mathbf{v}}) \end{array}
\right ) \left ( \begin{array}{l} t \\ \mathbf{r} \end{array}
\right ),
where
v^{T} denotes the transpose
of
v, , and
P(
v) denotes
the projection onto the direction of
v.
Simultaneity
From the first equation of the Lorentz transformation in terms of
coordinate differences
 \Delta t' = \gamma \left(\Delta t  \frac{v \,\Delta x}{c^{2}}
\right)
it is clear that two events that are simultaneous in frame
S (satisfying ), are not necessarily simultaneous in
another inertial frame
S′ (satisfying ). Only if these
events are colocal in frame
S (satisfying ), will they be
simultaneous in another frame
S′.
Time dilation and length contraction
Writing the Lorentz transformation and its inverse in terms of
coordinate differences we get
 \begin{cases}
\Delta t' = \gamma \left(\Delta t  \frac{v \,\Delta x}{c^{2}}
\right) \\\Delta x' = \gamma (\Delta x  v \,\Delta
t)\,\end{cases}and
 \begin{cases}
\Delta t = \gamma \left(\Delta t' + \frac{v \,\Delta x'}{c^{2}}
\right) \\\Delta x = \gamma (\Delta x' + v \,\Delta
t')\,\end{cases}
Suppose we have a
clock at rest in the
unprimed system S. Two consecutive ticks of this clock are then
characterized by
\Delta x = 0. If we want to know
the relation between the times between these ticks as measured in
both systems, we can use the first equation and find:
 \Delta t' = \gamma\, \Delta t \qquad ( \, for events satisfying
\Delta x = 0 )\,
This shows that the time \Delta t' between the two ticks as seen in
the 'moving' frame S' is larger than the time \Delta t between
these ticks as measured in the rest frame of the clock. This
phenomenon is called
time
dilation.
Similarly, suppose we have a
measuring
rod at rest in the unprimed system. In this system, the length
of this rod is written as \Delta x. If we want to find the length
of this rod as measured in the 'moving' system S', we must make
sure to measure the distances x' to the end points of the rod
simultaneously in the primed frame S'. In other words, the
measurement is characterized by
\Delta t' = 0,
which we can combine with the fourth equation to find the relation
between the lengths \Delta x and \Delta x':
 \Delta x' = \frac{\Delta x}{\gamma} \qquad ( \, for events
satisfying \Delta t' = 0 )\,
This shows that the length \Delta x' of the rod as measured in the
'moving' frame S' is shorter than the length \Delta x in its own
rest frame. This phenomenon is called
length contraction or
Lorentz
contraction.
These effects are not merely appearances; they are explicitly
related to our way of measuring
time intervals between
events which occur at the same place in a given coordinate system
(called "colocal" events). These time intervals will be
different in another coordinate system moving with respect
to the first, unless the events are also simultaneous. Similarly,
these effects also relate to our measured distances between
separated but simultaneous events in a given coordinate system of
choice. If these events are not colocal, but are separated by
distance (space), they will
not occur at the same
spatial distance from each other when seen from another
moving coordinate system. However, the
spacetime interval will be the same for
all observers. The underlying reality remains the same. Only our
perspective changes.
Causality and prohibition of motion faster than light
Diagram 2.
diagram 2 the interval AB is 'timelike';
i.e., there is a
frame of reference in which events A and B occur at the same
location in space, separated only by occurring at different times.
If A precedes B in that frame, then A precedes B in all frames. It
is hypothetically possible for matter (or information) to travel
from A to B, so there can be a causal relationship (with A the
cause and B the effect).
The interval AC in the diagram is 'spacelike';
i.e.,
there is a frame of reference in which events A and C occur
simultaneously, separated only in space. However there are also
frames in which A precedes C (as shown) and frames in which C
precedes A. If it were possible for a causeandeffect relationship
to exist between events A and C, then paradoxes of causality would
result. For example, if A was the cause, and C the effect, then
there would be frames of reference in which the effect preceded the
cause. Although this in itself won't give rise to a paradox, one
can show that faster than light signals can be sent back into one's
own past. A causal paradox can then be constructed by sending the
signal if and only if no signal was received previously.
Therefore, one of the consequences of special relativity is that
(assuming
causality is to be preserved),
no information or material object can travel
faster than light. On the other hand, the
logical situation is not as clear in the case of general
relativity, so it is an open question whether there is some
fundamental
principle that preserves causality (and therefore prevents
motion faster than light) in general relativity.
Even without considerations of causality, there are other strong
reasons why fasterthanlight travel is forbidden by special
relativity. For example, if a constant force is applied to an
object for a limitless amount of time, then integrating
F =
dp/
dt gives a momentum
that grows without bound, but this is simply because p = m \gamma v
\, approaches
infinity as
v
approaches
c. To an observer who is not accelerating, it
appears as though the object's inertia is increasing, so as to
produce a smaller acceleration in response to the same force. This
behavior is in fact observed in
particle accelerators.
Composition of velocities
If the observer in
S sees an object moving along the
x axis at velocity
w, then the observer in
the
S' system, a frame of reference moving at velocity
v in the
x direction with respect to
S, will see the object moving with velocity
w' where
 w'=\frac{wv}{1wv/c^2}.
This equation can be derived from the space and time
transformations above.
 w'=\frac{dx'}{dt'}=\frac{\gamma(dxv dt)}{\gamma(dtv
dx/c^2)}=\frac{(dx/dt)v}{1(v/c^2)(dx/dt)}
Notice that if the object were moving at the speed of light in the
S system (i.e. w=c), then it would also be moving at the
speed of light in the
S' system. Also, if both
w and
v are small with respect to the speed
of light, we will recover the intuitive Galilean transformation of
velocities: w' \approx wv.
The usual example given is that of a train (call it system K)
travelling due east with a velocity v with respect to the tracks
(system K'). A child inside the train throws a baseball due east
with a velocity u with respect to the train. In classical physics,
an observer at rest on the tracks will measure the velocity of the
baseball as v+u.
In special relativity, this is no longer true. Instead, an observer
on the tracks will measure the velocity of the baseball as
\frac{v+u}{1+\frac{vu}{c^2}}. If u and v are small compared to c,
then the above expression approaches the classical sum v+u.
In the more general case, the baseball is not necessarily
travelling in the same direction as the train. To obtain the
general formula for Einstein velocity addition, suppose an observer
at rest in system K measures the velocity of an object as
\mathbf{u}. Let K' be an inertial system such that the relative
velocity of K to K' is \mathbf{v}, where \mathbf{u} and \mathbf{v}
are now vectors in R^3. An observer at rest in K' will then measure
the velocity of the object as
 \mathbf{v} \oplus_E
\mathbf{u}=\frac{\mathbf{v}+\mathbf{u}_{\parallel} +
\alpha_{\mathbf{v}}\mathbf{u}_{\perp}}{1+\frac{\mathbf{v}\cdot\mathbf{u}}{c^2}},
where \mathbf{u}_{\parallel} and \mathbf{u}_{\perp} are the
components of \mathbf{u} parallel and perpendicular, respectively,
to \mathbf{v}, and
\alpha_{\mathbf{v}}=\frac{1}{\gamma(\mathbf{v})}=\sqrt{1\frac{\mathbf{v}^2}{c^2}}.
Einstein velocity addition is commutative
only when
\mathbf{v} and \mathbf{u} are
parallel. In fact,
 \mathbf{v} \oplus
\mathbf{u}=gyr[\mathbf{v},\mathbf{u}](\mathbf{u} \oplus
\mathbf{v}),
,where
gyr is the mathematical abstraction of
Thomas precession into an operator called
Thomas gyration and given by
 gyr[\mathbf{u},\mathbf{v}]\mathbf{w}=\ominus(\mathbf{u} \oplus
\mathbf{v}) \oplus (\mathbf{u} \oplus (\mathbf{v} \oplus
\mathbf{w}))
for all
w.
The
gyr operator forms the foundation of
gyrovector spaces.
Einstein's addition of colinear velocites is consistent with the
Fizeau experiment which determined
the speed of light in a fluid moving parallel to the light, but no
experiment has ever tested the formula for the general case of
nonparallel velocities.
Relativistic mechanics
In addition to modifying notions of space and time, special
relativity forces one to reconsider the concepts of
mass,
momentum, and
energy, all of which are important constructs in
Newtonian mechanics. Special
relativity shows, in fact, that these concepts are all different
aspects of the same physical quantity in much the same way that it
shows space and time to be interrelated.
There are a couple of (equivalent) ways to define momentum and
energy in SR. One method uses
conservation laws. If these laws are to
remain valid in SR they must be true in every possible reference
frame. However, if one does some simple
thought experiments using the Newtonian
definitions of momentum and energy, one sees that these quantities
are not conserved in SR. One can rescue the idea of conservation by
making some small modifications to the definitions to account for
relativistic velocities. It is these new definitions which are
taken as the correct ones for momentum and energy in SR.
The energy and momentum of an object with
invariant mass m (also called
rest mass in the case of a single particle), moving with
velocity v with respect to
a given frame of reference, are given by
 \begin{array}{r l}
E &= \gamma m c^2 \\\mathbf{p} &= \gamma m
\mathbf{v}\end{array}respectively, where
γ (the
Lorentz factor) is given by
 \gamma = \frac{1}{\sqrt{1  (v/c)^2}}.
The quantity
γm is often called the
relativistic
mass of the object in the given frame of reference,although
recently this concept is falling in disuse, and
Lev B. Okun suggested
that "this terminology [...] has no rational justification today",
and should no longer be taught.Other physicists, including
Wolfgang Rindler and
T. R. Sandin, have argued that relativistic mass is a
useful concept and there is little reason to stop using it.See
Mass in special
relativity for more information on this debate. Some authors
use the symbol
m to refer to relativistic mass, and the
symbol
m_{0} to refer to rest mass.
The energy and momentum of an object with invariant mass
m
are related by the formulas
 E^2  (p c)^2 = (m c^2)^2 \,
 \mathbf{p} c^2 = E \mathbf{v} \,.
The first is referred to as the
relativistic energymomentum
equation. While the energy
E and the momentum
p depend on the frame of reference in which they
are measured, the quantity
E^{2} −
(
pc)
^{2} is invariant, being equal to the squared
invariant mass of the object (
up to the
multiplicative constant
c^{4}).
It should be noted that the invariant mass of a system
 m_\text{tot} = \frac {\sqrt{E_\text{tot}^2 
(p_\text{tot}c)^2}} {c^2}
is
greater than the sum of the rest masses of the
particles it is composed of (unless they are all stationary with
respect to the
center of mass of the
system, and hence to each other). The sum of rest masses is not
even always conserved in
closed
systems, since rest mass may be converted to particles which
individually have no mass, such as photons. Invariant mass,
however, is conserved and invariant for all observers, so long as
the system remains closed. This is due to the fact that even
massless particles contribute invariant mass to systems, as also
does the kinetic energy of particles. Thus, even under
transformations of rest mass to photons or kinetic energy, the
invariant mass of a system which contains these energies still
reflects the invariant mass associated with them.
Mass–energy equivalence
For
massless particles,
m
is zero. The relativistic energymomentum equation still holds,
however, and by substituting
m with 0, the relation
E =
pc is obtained; when substituted into
Ev =
c^{2}p, it gives
v =
c: massless particles (such as
photons) always travel at the speed of
light.
A particle which has no rest mass (for example, a photon) can
nevertheless contribute to the total invariant mass of a system,
since some or all of its momentum is canceled by another particle,
causing a contribution to the system's invariant mass due to the
photon's energy. For single photons this does not happen, since the
energy and momentum terms exactly cancel.
Looking at the above formula for invariant mass of a system, one
sees that, when a single massive object is at rest
(
v = 0,
p = 0), there is a
nonzero mass remaining:
m_{rest} =
E/
c^{2}.The corresponding energy, which
is also the total energy when a single particle is at rest, is
referred to as "rest energy". In systems of particles which are
seen from a moving inertial frame, total energy increases and so
does momentum. However, for single particles the rest mass remains
constant, and for systems of particles the invariant mass remain
constant, because in both cases, the energy and momentum increases
subtract from each other, and cancel. Thus, the invariant mass of
systems of particles is a calculated constant for all observers, as
is the rest mass of single particles.
The mass of systems and conservation of invariant mass
For systems, the inertial frame in which the momenta of all
particles sums to zero is called the
center of momentum frame. In this
special frame, the relativistic energymomentum equation has
p = 0, and thus gives the invariant mass of the
system as merely the total energy of all parts of the system,
divided by
c^{2}
 m = \sum E/c^2
This is the invariant mass of any system which is measured in a
frame where it has zero total momentum, such as a bottle of hot gas
on a scale. In such a system, the mass which the scale weighs is
the invariant mass, and it depends on the total energy of the
system. It is thus more than the sum of the rest masses of the
molecules, but also includes all the totaled energies in the system
as well. Like energy and momentum, the invariant mass of closed
systems cannot be changed so long as the system is closed, because
the total relativistic energy of the system remains constant so
long as nothing can enter or leave it.
An increase in the energy of such a system which is caused by
translating the system to an inertial frame which is not the
center of momentum frame,
causes an increase in energy and momentum without an increase in
invariant mass.
E =
mc^{2}, however,
applies only to closed systems in their centerofmomentum frame
where momentum sums to zero.
Taking this formula at face value, we see that in relativity,
mass is simply another form of energy. In 1927 Einstein
remarked about special relativity, "Under this theory mass is not
an unalterable magnitude, but a magnitude dependent on (and,
indeed, identical with) the amount of energy."
Einstein was not referring to closed systems in this remark,
however. For, even in his 1905 paper, which first derived the
relationship between mass and energy, Einstein showed that the
energy of an object had to be increased for its invariant mass
(rest mass) to increase. In such cases, the system is not closed
(in Einstein's thought experiment, for example, a mass gives off
two photons, which are lost).
Closed systems
In a closed system the total energy, the total momentum, and hence
the total invariant mass are conserved. Einstein's formula for
change in mass translates to its simplest ΔE = mc
^{2} form,
however, in nonclosed systems in which energy, and thus invariant
mass, is allowed to escape (for example, as heat and light).
Einstein's equation shows that such systems must lose mass, in
accordance with the above formula, in proportion to the energy they
lose to the surroundings. Conversely, if one can measure the
differences in mass between a system before it undergoes a reaction
which releases heat and light, and the system after the reaction
when heat and light have escaped, one can estimate the amount of
energy which escapes the system. In both nuclear and chemical
reactions, such energy represents the difference in binding
energies of electrons in atoms (for chemistry) or between nucleons
in nuclei (in atomic reactions). In both cases, the mass difference
between reactants and (cooled) products measures the mass of heat
and light which will escape the reaction, and thus (using the
equation) give the equivalent energy of heat and light which may be
emitted if the reaction proceeds.
In chemistry, the mass differences associated with the emitted
energy are around onebillionth of the molecular mass. However, in
nuclear reactions the energies are so large that they are
associated with mass differences, which can be estimated in
advance, if the products and reactants have been weighed (atoms can
be weighed indirectly by using atomic masses, which are always the
same for each
nuclide). Thus, Einstein's
formula becomes important when one has measured the masses of
different atomic nuclei. By looking at the difference in masses,
one can predict which nuclei have stored energy that can be
released by certain
nuclear
reactions, providing important information which was useful in
the development of nuclear energy and, consequently, the
nuclear bomb. Historically, for example,
Lise Meitner was able to use the mass
differences in nuclei to estimate that there was enough energy
available to make nuclear fission a favorable process. The
implications of this special form of Einstein's formula have thus
made it one of the most famous equations in all of science.
Because the
E =
mc^{2} equation
applies to systems only in their
center of momentum frame, it has
been popularly misunderstood to mean that mass may be
converted to energy, after which the
mass
disappears. This is incorrect, as for closed systems, mass never
disappears in the center of momentum frame, because energy cannot
disappear. Instead, this equation, in context, means only that when
any energy is added to, or escapes from, a system in the
centerofmomentum frame, the system will be measured as having
gained or lost mass, in proportion to energy added or removed.
Thus, in theory, if even an atomic bomb were placed in a box strong
enough to hold its blast, and detonated upon a scale, the mass of
this closed system would not change, and the scale would not move.
Only when a transparent "window" was opened in the superstrong
plasmafilled box, and light and heat were allowed to escape in a
beam, and the bomb components to cool, would the system lose the
mass associated with the energy of the blast. In a 21 kiloton bomb,
for example, about a gram of light and heat is created. If this
heat and light were allowed to escape, the remains of the bomb
would lose a gram of mass, as it cooled. However, invariant mass
cannot be destroyed in special relativity, but only moved from
place to place. In this thoughtexperiment, the light and heat
carry away the gram of mass, and would therefore deposit this gram
of mass in the objects that absorb them.
Force
In special relativity, Newton's second law does not hold in its
form
F =
ma', but it
does if it is expressed as
 \mathbf{F} = \frac{d\mathbf{p}}{dt}
where
p is the momentum as defined above (
\mathbf{p}= \gamma m \mathbf{v} ) and "m" is the
invariant mass. Thus, the force is given by
 \mathbf{F} = m \frac{d(\gamma \, \mathbf{v})}{dt} = m \left(
\frac{d \gamma}{dt} \, \mathbf{v} + \gamma \frac{d\mathbf{v}}{dt}
\right).
Carrying out the derivatives gives
 \mathbf{F} = \frac{\gamma^3 m v}{c^2} \frac{dv}{dt} \,
\mathbf{v} + \gamma m\, \mathbf{a}
which, taking into account the identity v \tfrac{dv}{dt}=
\mathbf{v} \cdot \mathbf{a} , can also be expressed as
 \mathbf{F} = \frac{\gamma^3 m \left( \mathbf{v} \cdot
\mathbf{a} \right)}{c^2} \, \mathbf{v} + \gamma m\,
\mathbf{a}.
If the acceleration is separated into
the part
parallel to the velocity and the part perpendicular to it, one
gets
 \mathbf{F} = \frac{\gamma^3 m v^{2}}{c^2} \,
\mathbf{a}_{\parallel} + \gamma m \, (\mathbf{a}_{\parallel} +
\mathbf{a}_{\perp}) \,
 : = \gamma^3 m \left( \frac{v^2}{c^2} + \frac{1}{\gamma^2}
\right) \mathbf{a}_{\parallel} + \gamma m \, \mathbf{a}_{\perp}
\,
 : = \gamma^3 m \left( \frac{v^{2}}{c^2} + 1  \frac{v^{2}}{c^2}
\right) \mathbf{a}_{\parallel} + \gamma m \, \mathbf{a}_{\perp}
\,
 : = \gamma^3 m \, \mathbf{a}_{\parallel} + \gamma m \,
\mathbf{a}_{\perp} \,.
Consequently in some old texts,
γ^{3}m is
referred to as the
longitudinal mass, and
γm is
referred to as the
transverse mass, which is the same as
the
relativistic mass. See
mass in special
relativity.
For the
fourforce, see
below.
Kinetic energy
The
Workenergy Theorem says the change in kinetic energy
is equal to the work done on the body, that is
 \Delta K = W = \int_{\mathbf{r}_0}^{\mathbf{r}_1} \mathbf{F}
\cdot d\mathbf{r}
 := \int_{t_0}^{t_1} \frac{d}{dt}(\gamma
m\mathbf{v})\cdot\mathbf{v}dt = \left. \gamma m \mathbf{v} \cdot
\mathbf{v} \right^{t_1}_{t_0}  \int_{t_0}^{t_1} \gamma
m\mathbf{v} \cdot \frac{d\mathbf{v}}{dt} dt = \left. \gamma m v^2
\right^{t_1}_{t_0}  m\int_{v_0}^{v_1} \gamma v\,dv = m \left(
\left. \gamma v^2 \right^{t_1}_{t_0}  c^2\int_{v_0}^{v_1}
\frac{2v/c^2}{2\sqrt{1v^2/c^2}}\,dv \right) = \left. m\left(\frac
{v^2}{\sqrt{1v^2/c^2}} + c^2 \sqrt{1v^2/c^2} \right)
\right^{t_1}_{t_0} = \left. \frac {mc^2}{\sqrt{1v^2/c^2}}
\right^{t_1}_{t_0} = \left. {\gamma mc^2}\right^{t_1}_{t_0}
 ::\displaystyle= \gamma_1 mc^2  \gamma_0 mc^2.
If in the initial state the body was at rest
(
γ_{0} = 1) and in the final state it
has speed
v
(
γ_{1} =
γ), the kinetic energy
is
K = (
γ − 1)
mc^{2} , a result
that can be directly obtained by subtracting the rest energy
mc^{2} from the total relativistic energy
γmc^{2}.
Application in cyclotrons
The application of the above in
cyclotrons
is immediate:
 \frac {dW}{dt}=mc^2 \frac{d \gamma}{dt}
In the presence of a magnetic field only, the Lorentz force is:
 \mathbf{F}=q \mathbf{v \times B}
Since:
 \frac {dW}{dt}=\mathbf{F \cdot v}=0
it follows that:
 \frac{d \gamma}{dt}=0
meaning that
γ is constant, and so is
v. This is
instrumental in solving the equation of motion for a charge
particle of charge
q in a magnetic field of induction
B as follows:
 \mathbf{F}=\frac{d \gamma m_0 \mathbf{v}}{dt}=\gamma m_0
\frac{d \mathbf{v}}{dt}
On the other hand:
 \mathbf{F}=q \mathbf{v \times B}
Thus:
 \gamma m_0 \frac{d \mathbf{v}}{dt}=q \mathbf{v \times B}
Separating by components, we obtain:
 qBv_y=\gamma m_0 \frac{d v_x}{dt}
 qBv_x=\gamma m_0 \frac{d v_y}{dt}
 0=\gamma m_0 \frac{d v_z}{dt}
The solutions are:
 v_x = r \omega \cos(\omega t)\
 v_y =  r \omega \sin(\omega t)\
 \omega=\frac{qB}{\gamma(v_0) m_0}
By integrating one more time with respect to t the differential
equations above we obtain the equations of motion: a circle of
radius r=\frac{\gamma(v_0)m_0v_0}{qB} in the plane z=constant,
where v_0 is the initial speed of the particle entering the
cyclotron. Notice that this calculation
ignores the
AbrahamLorentz
force which is the reaction to the emission of electromagnetic
radiation by the particle. If the speed is held constant by
applying an electric field, then the magnitude of the acceleration
is constant, a = \frac{{v_0}^2}{r}\,, but its direction keeps
changing in a cyclotron. The jerk is proportional with the second
time derivative of speed:
 \frac{d^2 v_x}{dt^2} = r \omega^3 \cos(\omega t)
 \frac{d^2 v_y}{dt^2} = r \omega^3 \sin(\omega t)
Because the
jerk is directed opposite
to the velocity, the AbrahamLorentz force tends to slow the
particle down. Note that the AbrahamLorentz force is much smaller
than the
Lorentz force:
 :\mathbf{F}_\mathrm{rad} = \frac{\mu_0 q^2}{6 \pi c}
\mathbf{\dot{a}} =  \frac{\mu_0 q^4 B^2}{6 \pi c \gamma^2 {m_0}^2}
\mathbf{v} \,
 :\frac{F_{rad}}{F_{Lorentz}}=\frac{\mu_0 q^3 B}{6 \pi c
\gamma^2 {m_0}^2}
so, it can be ignored in most computations.
Classical limit
Notice that
γ can be expanded into a
Taylor series for \frac{v^2}{c^2} 1,
obtaining:
 \gamma = \sum_{n=0}^{\infty} \prod_{k=1}^n \frac{(2k  1)
v^2}{2k c^2} = 1 + \frac{1}{2} \frac{v^2}{c^2} + \frac{3}{8}
\frac{v^4}{c^4} + \frac{5}{16} \frac{v^6}{c^6} + \ldots
and consequently
 E  m c^2 = \frac{1}{2} m v^2 + \frac{3}{8} \frac{m v^4}{c^2} +
\frac{5}{16} \frac{m v^6}{c^4} + \ldots ;
 \mathbf{p} = m \mathbf{v} + \frac{1}{2} \frac{m v^2
\mathbf{v}}{c^2} + \frac{3}{8} \frac{m v^4 \mathbf{v}}{c^4} +
\frac{5}{16} \frac{m v^6 \mathbf{v}}{c^6} + \cdots .
For velocities much smaller than that of light, one can neglect the
terms with
c^{2} and higher in the denominator.
These formulas then reduce to the standard definitions of Newtonian
kinetic energy and momentum. This is
as it should be, for special relativity must agree with Newtonian
mechanics at low velocities.
The geometry of spacetime
SR uses a 'flat' 4dimensional Minkowski space, which is an example
of a
spacetime. This space, however, is
very similar to the standard 3 dimensional
Euclidean space.
The
differential of
distance (
ds) in
cartesian 3D
space is defined as:
 ds^2 = dx_1^2 + dx_2^2 + dx_3^2
where (dx_1,dx_2,dx_3) are the differentials of the three spatial
dimensions. In the geometry of special relativity, a fourth
dimension is added, derived from time, so that the equation for the
differential of distance becomes:
 ds^2 = dx_1^2 + dx_2^2 + dx_3^2  c^2 dt^2 .
If we wished to make the time coordinate look like the space
coordinates, we could treat time as
imaginary:
x_{4} = ict .
In this case the above equation becomes symmetric:
 ds^2 = dx_1^2 + dx_2^2 + dx_3^2 + dx_4^2 .
This suggests what is in fact a profound theoretical insight as it
shows that special relativity is simply a
rotational symmetry of our
spacetime, very similar to rotational symmetry
of
Euclidean space. Just as
Euclidean space uses a
Euclidean
metric, so spacetime uses a
Minkowski metric. Basically, SR can be
stated in terms of the invariance of
spacetime
interval (between any two events) as seen from any
inertial reference frame. All equations and effects of special
relativity can be derived from this rotational symmetry (the
Poincaré group) of Minkowski
spacetime. According to Misner (1971 §2.3), ultimately the deeper
understanding of both special and general relativity will come from
the study of the Minkowski metric (described below) rather than a
"disguised" Euclidean metric using
ict as the time
coordinate.
If we reduce the spatial dimensions to 2, so that we can represent
the physics in a 3D space
 ds^2 = dx_1^2 + dx_2^2  c^2 dt^2 ,
we see that the
null geodesics lie along a dualcone:
defined by the equation
 ds^2 = 0 = dx_1^2 + dx_2^2  c^2 dt^2
or simply
 dx_1^2 + dx_2^2 = c^2 dt^2
— which is the equation of a circle with
r=c×dt.If we
extend this to three spatial dimensions, the null geodesics are
the4dimensional cone:
 ds^2 = 0 = dx_1^2 + dx_2^2 + dx_3^2  c^2 dt^2
 dx_1^2 + dx_2^2 + dx_3^2 = c^2 dt^2 .
This null dualcone represents the "line of sight" of a point in
space. That is, when we look at the
stars and
say "The light from that star which I am receiving is X years old",
we are looking down this line of sight: a null geodesic. We are
looking at an event a distance d = \sqrt{x_1^2+x_2^2+x_3^2} away
and a time
d/c in the past. For this reason the null dual
cone is also known as the 'light cone'. (The point in the lower
left of the picture below represents the star, the origin
represents the observer, and the line represents the null geodesic
"line of sight".)
The cone in the
t region is the information that the
point is 'receiving', while the cone in the
+t section is
the information that the point is 'sending'.
The geometry of Minkowski space can be depicted using
Minkowski diagrams, which are useful also
in understanding many of the thoughtexperiments in special
relativity.
Physics in spacetime
Here, we see how to write the equations of special relativity in a
manifestly
Lorentz covariant
form. The position of an event in spacetime is given by a
contravariant four
vector whose components are:
 x^\nu=\left(ct, x, y, z\right)
where x^1 = x and x^2 = y and x^3 = z as usual. We define x^0 = ct
so that the time coordinate has the same dimension of distance as
the other spatial dimensions; in accordance with the general
principle that space and time are treated equally, so far as
possible. Superscripts are contravariant indices in this section
rather than exponents except when they indicate a square.
Subscripts are
covariant indices which
also range from zero to three as with the spacetime gradient of a
field φ:
 \partial_0 \phi = \frac{1}{c}\frac{\partial \phi}{\partial t},
\quad \partial_1 \phi = \frac{\partial \phi}{\partial x}, \quad
\partial_2 \phi = \frac{\partial \phi}{\partial y}, \quad
\partial_3 \phi = \frac{\partial \phi}{\partial z}.
Metric and transformations of coordinates
Having recognised the fourdimensional nature of spacetime, we are
driven to employ the Minkowski metric,
η, given in
components (valid in any
inertial reference frame) as:
 \eta_{\alpha\beta} = \begin{pmatrix}
1 & 0 & 0 & 0\\0 & 1 & 0 & 0\\0 & 0
& 1 & 0\\0 & 0 & 0 & 1\end{pmatrix}
which is equal to its reciprocal, \eta^{\alpha\beta}, in those
frames.
Then we recognize that coordinate transformations between inertial
reference frames are given by the
Lorentz transformation tensor Λ. For the special case of motion along the
xaxis, we have:
 \Lambda^{\mu'}{}_\nu = \begin{pmatrix}
\gamma & \beta\gamma & 0 & 0\\\beta\gamma &
\gamma & 0 & 0\\0 & 0 & 1 & 0\\0 & 0 &
0 & 1\end{pmatrix}
which is simply the matrix of a boost (like a rotation) between the
x and
ct coordinates. Where μ' indicates the row
and ν indicates the column. Also,
β and
γ are
defined as:
 \beta = \frac{v}{c},\ \gamma = \frac{1}{\sqrt{1\beta^2}}.
More generally, a transformation from one inertial frame (ignoring
translations for simplicity) to another must satisfy:
 \eta_{\alpha\beta} = \eta_{\mu'\nu'} \Lambda^{\mu'}{}_\alpha
\Lambda^{\nu'}{}_\beta \!
where there is an implied summation of \mu' \! and \nu' \! from 0
to 3 on the righthand side in accordance with the
Einstein summation convention. The
Poincaré group is the most
general group of transformations which preserves the
Minkowski metric and this is the physical
symmetry underlying special relativity.
All proper physical quantities are given by tensors. So to
transform from one frame to another, we use the wellknown
tensor transformation law

T^{\left[i_1',i_2',\dots,i_p'\right]}_{\left[j_1',j_2',\dots,j_q'\right]}
=
\Lambda^{i_1'}{}_{i_1}\Lambda^{i_2'}{}_{i_2}\cdots\Lambda^{i_p'}{}_{i_p}\Lambda_{j_1'}{}^{j_1}\Lambda_{j_2'}{}^{j_2}\cdots\Lambda_{j_q'}{}^{j_q}T^{\left[i_1,i_2,\dots,i_p\right]}_{\left[j_1,j_2,\dots,j_q\right]}
Where \Lambda_{j_k'}{}^{j_k} \! is the reciprocal matrix of
\Lambda^{j_k'}{}_{j_k} \!.
To see how this is useful, we transform the position of an event
from an unprimed coordinate system
S to a primed system
S', we calculate
\begin{pmatrix}ct'\\ x'\\ y'\\ z'\end{pmatrix} =
x^{\mu'}=\Lambda^{\mu'}{}_\nu x^\nu=\begin{pmatrix}\gamma &
\beta\gamma & 0 & 0\\\beta\gamma & \gamma & 0
& 0\\0 & 0 & 1 & 0\\0 & 0 & 0 &
1\end{pmatrix}\begin{pmatrix}ct\\ x\\ y\\ z\end{pmatrix}
=\begin{pmatrix}\gamma ct \gamma\beta x\\\gamma x  \beta \gamma
ct \\ y\\ z\end{pmatrix}
which is the Lorentz transformation given above. All tensors
transform by the same rule.
The squared length of the differential of the position fourvector
dx^\mu \! constructed using
 \mathbf{dx}^2 = \eta_{\mu\nu}\,dx^\mu \,dx^\nu = (c \cdot
dt)^2+(dx)^2+(dy)^2+(dz)^2\,
is an invariant. Being invariant means that it takes the same value
in all inertial frames, because it is a scalar (0 rank tensor), and
so no Λ appears in its trivial transformation. Notice that when the
line element \mathbf{dx}^2 is negative
that d\tau=\sqrt{\mathbf{dx}^2} / c is the differential of
proper time, while when \mathbf{dx}^2 is
positive, \sqrt{\mathbf{dx}^2} is differential of the
proper distance.
The primary value of expressing the equations of physics in a
tensor form is that they are then manifestly invariant under the
Poincaré group, so that we do not have to do a special and tedious
calculation to check that fact. Also in constructing such equations
we often find that equations previously thought to be unrelated
are, in fact, closely connected being part of the same tensor
equation.
Velocity and acceleration in 4D
Recognising other physical quantities as tensors also simplifies
their transformation laws. First note that the
velocity fourvector U^{μ} is
given by
 U^\mu = \frac{dx^\mu}{d\tau} = \begin{pmatrix} \gamma c \\
\gamma v_x \\ \gamma v_y \\ \gamma v_z \end{pmatrix}
Recognising this, we can turn the awkward looking law about
composition of velocities into a simple statement about
transforming the velocity fourvector of one particle from one
frame to another.
U^{μ} also has an invariant
form:
 {\mathbf U}^2 = \eta_{\nu\mu} U^\nu U^\mu = c^2 .
So all velocity fourvectors have a magnitude of
c. This
is an expression of the fact that there is no such thing as being
at coordinate rest in relativity: at the least, you are always
moving forward through time. The
acceleration 4vector is given by A^\mu =
d{\mathbf U^\mu}/d\tau. Given this, differentiating the above
equation by
τ produces
 2\eta_{\mu\nu}A^\mu U^\nu = 0. \!
So in relativity, the acceleration fourvector and the velocity
fourvector are orthogonal.
Momentum in 4D
The momentum and energy combine into a covariant 4vector:
 p_\nu = m \cdot \eta_{\nu\mu} U^\mu = \begin{pmatrix}
E/c \\ p_x\\ p_y\\ p_z\end{pmatrix}.
where
m is the
invariant
mass.
The invariant magnitude of the
momentum
4vector is:
 \mathbf{p}^2 = \eta^{\mu\nu}p_\mu p_\nu = (E/c)^2 + p^2 .
We can work out what this invariant is by first arguing that, since
it is a scalar, it doesn't matter which reference frame we
calculate it, and then by transforming to a frame where the total
momentum is zero.
 \mathbf{p}^2 =  (E_{rest}/c)^2 =  (m \cdot c)^2 .
We see that the rest energy is an independent invariant. A rest
energy can be calculated even for particles and systems in motion,
by translating to a frame in which momentum is zero.
The rest energy is related to the mass according to the celebrated
equation discussed above:
 E_{rest} = m c^2\,
Note that the mass of systems measured in their center of momentum
frame (where total momentum is zero) is given by the total energy
of the system in this frame. It may not be equal to the sum of
individual system masses measured in other frames.
Force in 4D
To use
Newton's third law
of motion, both forces must be defined as the rate of change of
momentum with respect to the same time coordinate. That is, it
requires the 3D force defined above. Unfortunately, there is no
tensor in 4D which contains the components of the 3D force vector
among its components.
If a particle is not traveling at
c, one can transform the
3D force from the particle's comoving reference frame into the
observer's reference frame. This yields a 4vector called the
fourforce. It is the rate of change of
the above energy momentum
fourvector
with respect to proper time. The covariant version of the
fourforce is:
 F_\nu = \frac{d p_{\nu}}{d \tau} = \begin{pmatrix} {d
(E/c)}/{d \tau} \\ {d p_x}/{d \tau} \\ {d p_y}/{d \tau} \\ {d
p_z}/{d \tau} \end{pmatrix}
where \tau \, is the proper time.
In the rest frame of the object, the time component of the four
force is zero unless the "
invariant
mass" of the object is changing (this requires a nonclosed
system in which energy/mass is being directly added or removed from
the object) in which case it is the negative of that rate of change
of mass, times
c. In general, though, the components of
the four force are not equal to the components of the threeforce,
because the three force is defined by the rate of change of
momentum with respect to coordinate time, i.e. \frac{d p}{d t}
while the four force is defined by the rate of change of momentum
with respect to proper time, i.e. \frac{d p} {d \tau} .
In a continuous medium, the 3D
density of force combines
with the
density of power to form a covariant 4vector.
The spatial part is the result of dividing the force on a small
cell (in 3space) by the volume of that cell. The time component is
−1/
c times the power transferred to that cell divided by
the volume of the cell. This will be used below in the section on
electromagnetism.
Relativity and unifying electromagnetism
Theoretical investigation in
classical electromagnetism led to
the discovery of wave propagation. Equations generalizing the
electromagnetic effects found that finite propagationspeed of the
E and B fields required certain behaviors on charged particles. The
general study of moving charges forms the
Liénard–Wiechert
potential, which is a step towards special relativity.
The Lorentz transformation of the
electric field of a moving charge into a
nonmoving observer's reference frame results in the appearance of
a mathematical term commonly called the
magnetic field. Conversely, the
magnetic field generated by a moving charge disappears and
becomes a purely
electrostatic field in a comoving frame
of reference.
Maxwell's
equations are thus simply an empirical fit to special
relativistic effects in a classical model of the Universe. As
electric and magnetic fields are reference frame dependent and thus
intertwined, one speaks of
electromagnetic fields. Special
relativity provides the transformation rules for how an
electromagnetic field in one inertial frame appears in another
inertial frame.
Electromagnetism in 4D
Maxwell's equations in the 3D
form are already consistent with the physical content of special
relativity. But we must rewrite them to make them manifestly
invariant.
The
charge density \rho \! and
current density [J_x,J_y,J_z] \! are
unified into the
currentcharge
4vector:
 J^\mu = \begin{pmatrix}
\rho c \\ J_x\\ J_y\\ J_z\end{pmatrix}.
The law of
charge conservation,
\frac{\partial \rho} {\partial t} + \nabla \cdot \mathbf{J} = 0,
becomes:
 \partial_\mu J^\mu = 0. \!
The
electric field [E_x,E_y,E_z] \!
and the
magnetic induction
[B_x,B_y,B_z] \! are now unified into the (rank 2 antisymmetric
covariant)
electromagnetic
field tensor:
F_{\mu\nu} =
\begin{pmatrix}
0 & E_x/c & E_y/c & E_z/c \\
E_x/c & 0 & B_z & B_y \\
E_y/c & B_z & 0 & B_x \\
E_z/c & B_y & B_x & 0
\end{pmatrix}.
The density, f_\mu \!, of the
Lorentz
force, \mathbf{f} = \rho \mathbf{E} + \mathbf{J} \times
\mathbf{B}, exerted on matter by the electromagnetic field
becomes:
 f_\mu = F_{\mu\nu}J^\nu .\!
Faraday's law of
induction, \nabla \times \mathbf{E} = \frac{\partial
\mathbf{B}} {\partial t}, and
Gauss's law for magnetism, \nabla
\cdot \mathbf{B} = 0, combine to form:
 \partial_\lambda F_{\mu\nu}+ \partial _\mu F_{\nu
\lambda}+
\partial_\nu F_{\lambda \mu} = 0. \!
Although there appear to be 64 equations here, it actually reduces
to just four independent equations. Using the antisymmetry of the
electromagnetic field one can either reduce to an identity (0=0) or
render redundant all the equations except for those with λ,μ,ν =
either 1,2,3 or 2,3,0 or 3,0,1 or 0,1,2.
The
electric displacement
[D_x,D_y,D_z] \! and the
magnetic
field [H_x,H_y,H_z] \! are now unified into the (rank 2
antisymmetric contravariant) electromagnetic displacement
tensor:
\mathcal{D}^{\mu\nu} =
\begin{pmatrix}
0 & D_xc & D_yc & D_zc \\
D_xc & 0 & H_z & H_y \\
D_yc & H_z & 0 & H_x \\
D_zc & H_y & H_x & 0
\end{pmatrix}.
Ampère's law, \nabla
\times \mathbf{H} = \mathbf{J} + \frac{\partial \mathbf{D}}
{\partial t}, and
Gauss's law, \nabla
\cdot \mathbf{D} = \rho, combine to form:
 \partial_\nu \mathcal{D}^{\mu \nu} = J^{\mu}. \!
In a vacuum, the
constitutive
equations are:
 \mu_0 \mathcal{D}^{\mu \nu} = \eta^{\mu \alpha} F_{\alpha
\beta} \eta^{\beta \nu} \,.
Antisymmetry reduces these 16 equations to just six independent
equations. Because it is usual to define F^{\mu \nu}\, by
 F^{\mu \nu} = \eta^{\mu \alpha} F_{\alpha \beta} \eta^{\beta
\nu} \,
the constitutive equations may, in a
vacuum, be combined
with Ampère's law etc. to get:
 \partial_\beta F^{\alpha \beta} = \mu_0 J^{\alpha}. \!
The
energy
density of the electromagnetic field combines with
Poynting vector and the
Maxwell stress tensor to form the 4D
electromagnetic
stressenergy tensor. It is the flux (density) of the momentum
4vector and as a rank 2 mixed tensor it is:
 T_\alpha^\pi = F_{\alpha\beta} \mathcal{D}^{\pi\beta} 
\frac{1}{4} \delta_\alpha^\pi F_{\mu\nu} \mathcal{D}^{\mu\nu}
where \delta_\alpha^\pi is the
Kronecker
delta. When upper index is lowered with η, it becomes symmetric
and is part of the source of the gravitational field.
The conservation of linear momentum and energy by the
electromagnetic field is expressed by:
 f_\mu + \partial_\nu T_\mu^\nu = 0\!
where f_\mu \! is again the density of the
Lorentz force. This equation can be deduced
from the equations above (with considerable effort).
Status
Special relativity is accurate only when
gravitational potential is much less
than
c^{2}; in a strong gravitational field one
must use
general relativity
(which becomes special relativity at the limit of weak field). At
very small scales, such as at the
Planck
length and below, quantum effects must be taken into
consideration resulting in
quantum
gravity. However, at macroscopic scales and in the absence of
strong gravitational fields, special relativity is experimentally
tested to extremely high degree of accuracy (10
^{−20})The
number of works is vast, see as example:
Sidney Coleman, Sheldon L. Glashow,
Cosmic Ray and Neutrino
Tests of Special Relativity, Phys. Lett. B405 (1997) 249252,
online
An overview can be found on
this pageand thus accepted by the physics
community. Experimental results which appear to contradict it are
not reproducible and are thus widely believed to be due to
experimental errors.
Special relativity is mathematically selfconsistent, and it is an
organic part of all modern physical theories, most notably
quantum field theory,
string theory, and general relativity (in the
limiting case of negligible gravitational fields).
Newtonian mechanics mathematically follows from special relativity
at small velocities (compared to the speed of light) — thus
Newtonian mechanics can be considered as a special relativity of
slow moving bodies. See
Status of special relativity
for a more detailed discussion.
Several experiments predating Einstein's 1905 paper are now
interpreted as evidence for relativity. (Of these, Einstein was
only aware of the Fizeau experiment before 1905.)
 The Trouton–Noble
experiment showed that the torque on a capacitor is independent
of position and inertial reference frame.
 The famous MichelsonMorley experiment gave
further support to the postulate that detecting an absolute
reference velocity was not achievable. It should be stated here
that, contrary to many alternative claims, it said little about the
invariance of the speed of light with respect to the source and
observer's velocity, as both source and observer were travelling
together at the same velocity at all times.
 The Fizeau experiment measured
the speed of light in moving media, with results that are
consistent with relativistic addition of colinear velocities.
A number of experiments have been conducted to test special
relativity against rival theories. These include:
In addition, particle accelerators routinely accelerate and measure
the properties of particles moving at near the speed of light,
where their behavior is completely consistent with relativity
theory and inconsistent with the earlier
Newtonian mechanics. These machines
would simply not work if they were not engineered according to
relativistic principles.
See also
 People: Arthur
Eddington  Albert Einstein 
Hendrik Lorentz  Hermann Minkowski  Bernhard Riemann  Henri Poincaré  Alexander MacFarlane 
Harry Bateman  Robert S. Shankland  Walter Ritz
 Relativity: Theory of relativity  History of special relativity
 principle of relativity 
general relativity  Fundamental Speed  frame of reference  inertial frame of reference 
Lorentz transformations 
Bondi kcalculus  Einstein synchronisation  RietdijkPutnam Argument
 Physics: Newtonian Mechanics  spacetime  speed of
light  simultaneity  physical cosmology  Doppler effect  relativistic Euler equations 
Aether drag hypothesis 
Lorentz ether theory  Moving magnet and conductor
problem  Shape waves Relativistic heat
conduction
 Maths: Minkowski
space  fourvector  world line  light cone
 Lorentz group  Poincaré group  geometry  tensors 
splitcomplex number  Relativity in the APS
formalism
 Philosophy: actualism
 conventionalism  formalism
 Paradoxes: Twin
paradox  Ehrenfest paradox 
Ladder paradox  Bell's spaceship paradox
 Experiments: Kennedy–Thorndike
experiment  Trouton–Rankine
experiment  Michelson–Morley
experiment  Hafele–Keating experiment
 Ives–Stilwell
experiment  Rossi–Hall experiment
References
 Albert
Einstein (1905) " Zur Elektrodynamik bewegter Körper", Annalen der
Physik 17: 891; English translation On the Electrodynamics of Moving Bodies by
George Barker Jeffery and Wilfrid
Perrett (1923); Another English translation On the
Electrodynamics of Moving Bodies by Megh Nad Saha
(1920).
 Albert Einstein, Relativity  The Special and
General Theory, chapter 18
 Charles W. Misner, Kip S. Thorne &
John A.
Wheeler,Gravitation, pg 172, 6.6 The local
coordinate system of an accelerated observer, ISBN
0716703440
 Einstein, "Fundamental Ideas and Methods of the Theory of
Relativity", 1920)
 For a survey of such derivations, see Lucas and Hodgson,
Spacetime and Electromagnetism, 1990
 Einstein, On the Relativity Principle and the Conclusions Drawn
from It, 1907; "The Principle of Relativity and Its Consequences in
Modern Physics, 1910; "The Theory of Relativity", 1911; Manuscript
on the Special Theory of Relativity, 1912; Theory of Relativity,
1913; Einstein, Relativity, the Special and General Theory, 1916;
The Principle Ideas of the Theory of Relativity, 1916; What Is The
Theory of Relativity?, 1919; The Principle of Relativity (Princeton
Lectures), 1921; Physics and Reality, 1936; The Theory of
Relativity, 1949.
 Das, A., The Special Theory of Relativity, A Mathematical
Exposition, Springer, 1993.
 Schutz, J., Independent Axioms for Minkowski Spacetime,
1997.
 Yaakov Friedman, Physical Applications of Homogeneous
Balls, Progress in Mathematical Physics 40
Birkhäuser, Boston, 2004, pages 121.
 David Morin, Introduction to Classical Mechanics,
Cambridge University Press, Cambridge, 2007, chapter 11, Appendix
I
 Does the inertia of a body depend upon its energy
content? A. Einstein, Annalen der Physik.
18:639, 1905 (English translation by W. Perrett
and G.B. Jeffery)
 On the Inertia of Energy Required by the
Relativity Principle, A. Einstein, Annalen der Physik 23
(1907): 371384
 In a letter to Carl Seelig in 1955, Einstein wrote "I had
already previously found that Maxwell's theory did not account for
the microstructure of radiation and could therefore have no
general validity.", Einstein letter to Carl Seelig, 1955.
 R. C. Tolman, The theory of the Relativity of Motion,
(Berkeley 1917), p. 54
 G. A. Benford, D. L. Book, and W. A. Newcomb, The Tachyonic
Antitelephone, Phys. Rev. D 2, 263–265 (1970)
article
 A.A. Ungar, Beyond the Einstein Addition Law and its
Gyroscopic Thomas Precession: The Theory of Gyrogroups and
Gyrovector Spaces, Kluwer, 2002.
 Note that in 2008 the last editor, Don Koks, rewrote a
significant portion of the page, changing it from a view extremely
dismissive of the usefulness of relativistic mass to one which
hardly questions it. The previous version was:
 See, for example:
 Einstein on Newton
 R.C.Tolman "Relativity Thermodynamics and Cosmology"
pp4748
 C. S. Roberts and S. J. Buchsbaum, “Motion of a chaged particle
in a constant magnetic field and a trasnverse electromagnetic wave
propagating along the field”, Phys. Rev. 135, A381 (1964)
 V.G. Bagrov, D.M. Gitman and A.V. Jushin, Solutions for the
motion of an electron in electromagnetic field, Phys. Rev.D , 12,
3200 (1975)
 H.R.Jory, A.W.Trivelpiece, J.Appl.Phys. "Charged particle
motion in large. amplitude electromagnetic fields",39,3053
(1968)
 R. OndarzaRovira, “Relativistic motion of a charged particle
driven by an elliptically polarized electromagnetic wave
propagating along a static magnetic field” , IEEE Transactions on
Plasma Science, Vol. 29, 6, 903 (2001)
 J Kruger and M Bovyn, “Relativistic motion of a charged
particle in a plane electromagnetic wave with arbitrary amplitude”,
J. Phys. A: Math. Gen., Vol.11, 9 1841 (1976)
 H. Takabe, “ Relativistic motion of charged particles in
ultraintense laser fields”, Journal of Plasma and Fusion Research,
Vol.78, 4, 341(2005)
 H P Zehrfeld, G. Fussmann, B.J. Green, “Electric field effects
on relativistic charged particle motion in Tokamaks”, Plasma Phys.
23 473 (1981)
 R. Giovanelli, “Analytic treatment of the relativistic motion
of charged particles in electric and magnetic field”, Il Nuovo
Cimento D, Vol. 9, 11,1443 (1987)
 A. Bourdier, M.Valentini, J.Valat, “Dynamics of a relativistic
charged particle in a constant homogeneous magnetic field and a
transverse homogeneous rotating electric field”, Phys. Rev. E 54,
5681 (1996)
 S.W.Kim, D.H.Kwon, H.W. Lee, “Relativistic cyclotron motion in
a polarized electric field”, Jour. of Kor. Phys. Soc., Vol. 32, 1,
30 (1998)
 L.B.Kong, P.K. Liu, “Analytical solution for relativistic
charged particle motion in a circularly polarized electromagnetic
wave”, Phys. Plasmas 14, 063101 (2007)
 JeanBernard Zuber & Claude Itzykson, Quantum Field
Theory, pg 5 , ISBN 0070320713
 Charles W. Misner, Kip S. Thorne &
John A.
Wheeler,Gravitation, pg 51, ISBN 0716703440
 George Sterman, An Introduction to Quantum Field
Theory, pg 4 , ISBN 0521311322
Textbooks
 Einstein, Albert (1920). Relativity: The
Special and General Theory.
 Einstein, Albert (1996). The Meaning of Relativity.
Fine Communications. ISBN 1567311369
 Freund, Jűrgen (2008) Special Relativity for Beginners  A Textbook for
Undergraduates World Scientific. ISBN 9812771603
 Robert Geroch (1981). General
Relativity From A to B. University of Chicago Press. ISBN
0226288641
 Logunov, Anatoly A. (2005) Henri
Poincaré and the Relativity Theory (transl. from Russian by G.
Pontocorvo and V. O. Soleviev, edited by V. A. Petrov) Nauka,
Moscow.
 Charles Misner, Kip Thorne, and John Archibald Wheeler (1971)
Gravitation. W. H. Freeman & Co. ISBN
0716703343
 Post, E.J., 1997 (1962) Formal Structure of
Electromagnetics: General Covariance and Electromagnetics.
Dover Publications.
 Wolfgang Rindler (1991).
Introduction to Special Relativity (2nd ed.), Oxford University
Press. ISBN13: 9780198539520; ISBN 0198539525
 Wolfgang Rindler (2006).
Relativity: Special, General, and Cosmological (2nd ed.), Oxford
University Press. ISBN13: 9780198567325; ISBN 0198567324
 Schutz, Bernard F. A First Course in General
Relativity, Cambridge University Press. ISBN
0521277035
 Silberstein, Ludwik (1914) The
Theory of Relativity.
 Taylor, Edwin, and John
Archibald Wheeler (1992) Spacetime Physics (2nd ed.).
W.H. Freeman & Co. ISBN 0716723271
 Tipler, Paul, and Llewellyn, Ralph (2002). Modern
Physics (4th ed.). W. H. Freeman & Co. ISBN
0716743450
Journal articles
 Alvager et al. (1964) "Test of the Second Postulate of Special
Relativity in the GeV region," Physics Letters 12:
260.
 Darrigol, Olivier (2004) "[The Mystery of the PoincaréEinstein
Connection]," Isis 95(4): 61426.
 Mitchell Feigenbaum (2008) "
The Theory
of Relativity  Galileo's Child."
 Gulevich, D. R., et al. (2008) "Shape waves in 2D Josephson
junctions: Exact solutions and time dilation," Phys.
Rev. Lett. 101: 127002.
 Rizzi, G., et al., (2005) " Synchronization Gauges and the Principles of Special
Relativity," Found. Phys. 34: 183587.
 Will, Clifford M. (1992) "Clock synchronization and isotropy of
the oneway speed of light," Physics Review D 45:
40311.
 Wolf, Peter, and Petit, Gerard (1997) "Satellite test of
Special Relativity using the Global Positioning System,"
Physics Review A 56(6): 440509.
External links
Original works
Special relativity for a general audience (no math knowledge
required)
 Wikibooks: Special Relativity
 Einstein Light An awardwinning, nontechnical introduction (film
clips and demonstrations) supported by dozens of pages of further
explanations and animations, at levels with or without
mathematics.
 Einstein Online Introduction to relativity theory,
from the Max Planck Institute for Gravitational Physics.
Special relativity explained (using simple or more advanced
math)
Visualization
Other