In
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which rel ...
, relativistic mechanics refers to
mechanics
Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to objects ...
compatible with
special relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates:
# The law ...
(SR) and
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics. ...
(GR). It provides a non-
quantum mechanical
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, qu ...
description of a system of particles, or of a
fluid
In physics, a fluid is a liquid, gas, or other material that continuously deforms (''flows'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are substances which cannot resist any shea ...
, in cases where the
velocities of moving objects are comparable to the
speed of light
The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit fo ...
''c''. As a result,
classical mechanics
Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical ...
is extended correctly to particles traveling at high velocities and energies, and provides a consistent inclusion of
electromagnetism
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions o ...
with the mechanics of particles. This was not possible in Galilean relativity, where it would be permitted for particles and light to travel at ''any'' speed, including faster than light. The foundations of relativistic mechanics are the
postulates of special relativity and general relativity. The unification of SR with quantum mechanics is
relativistic quantum mechanics, while attempts for that of GR is
quantum gravity
Quantum gravity (QG) is a field of theoretical physics that seeks to describe gravity according to the principles of quantum mechanics; it deals with environments in which neither gravitational nor quantum effects can be ignored, such as in the v ...
, an
unsolved problem in physics.
As with classical mechanics, the subject can be divided into "
kinematics"; the description of motion by specifying
positions, velocities and
acceleration
In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by ...
s, and "
dynamics"; a full description by considering
energies,
momenta
Momenta is an autonomous driving company headquartered in Beijing, China that aims to build the 'Brains' for autonomous vehicles.
In December 2021, Momenta and BYD established a 100 million yuan ($15.7 million) joint venture to deploy autonomous ...
, and
angular momenta and their
conservation law
In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves over time. Exact conservation laws include conservation of energy, conservation of linear momentum, ...
s, and
force
In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a ...
s acting on particles or exerted by particles. There is however a subtlety; what appears to be "moving" and what is "at rest"—which is termed by "
statics
Statics is the branch of classical mechanics that is concerned with the analysis of force and torque (also called moment) acting on physical systems that do not experience an acceleration (''a''=0), but rather, are in static equilibrium with t ...
" in classical mechanics—depends on the relative motion of
observers who measure in
frames of reference
In physics and astronomy, a frame of reference (or reference frame) is an abstract coordinate system whose origin, orientation, and scale are specified by a set of reference points― geometric points whose position is identified both mathem ...
.
Although some definitions and concepts from classical mechanics do carry over to SR, such as force as the
time derivative
A time derivative is a derivative of a function with respect to time, usually interpreted as the rate of change of the value of the function. The variable denoting time is usually written as t.
Notation
A variety of notations are used to denote th ...
of momentum (
Newton's second law
Newton's laws of motion are three basic Scientific law, laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows:
# A body remains at re ...
), the
work done by a particle as the
line integral
In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms ''path integral'', ''curve integral'', and ''curvilinear integral'' are also used; '' contour integral'' is used as well, ...
of force exerted on the particle along a path, and
power as the time derivative of work done, there are a number of significant modifications to the remaining definitions and formulae. SR states that motion is relative and the laws of physics are the same for all experimenters irrespective of their
inertial
In classical physics and special relativity, an inertial frame of reference (also called inertial reference frame, inertial frame, inertial space, or Galilean reference frame) is a frame of reference that is not undergoing any acceleration. ...
reference frames. In addition to modifying notions of
space and time, SR forces one to reconsider the concepts of
mass
Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different element ...
,
momentum
In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass ...
, and
energy
In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of hea ...
all of which are important constructs in
Newtonian mechanics
Newton's laws of motion are three basic Scientific law, laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows:
# A body remains at re ...
. SR shows 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. Consequently, another modification is the concept of the
center of mass of a system, which is straightforward to define in classical mechanics but much less obvious in relativity – see
relativistic center of mass
Relativity may refer to:
Physics
* Galilean relativity, Galileo's conception of relativity
* Numerical relativity, a subfield of computational physics that aims to establish numerical solutions to Einstein's field equations in general relativity ...
for details.
The equations become more complicated in the more familiar
three-dimensional
Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position of an element (i.e., point). This is the informal ...
vector calculus formalism, due to the
nonlinearity
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...
in the
Lorentz factor
The Lorentz factor or Lorentz term is a quantity expressing how much the measurements of time, length, and other physical properties change for an object while that object is moving. The expression appears in several equations in special relativi ...
, which accurately accounts for relativistic velocity dependence and the
speed limit
Speed limits on road traffic, as used in most countries, set the legal maximum speed at which vehicles may travel on a given stretch of road. Speed limits are generally indicated on a traffic sign reflecting the maximum permitted speed - expre ...
of all particles and fields. However, they have a simpler and elegant form in ''four''-dimensional
spacetime
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why diffe ...
, which includes flat
Minkowski space
In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the ...
(SR) and
curved space
Curved space often refers to a spatial geometry which is not "flat", where a flat space is described by Euclidean geometry. Curved spaces can generally be described by Riemannian geometry though some simple cases can be described in other ways. ...
time (GR), because three-dimensional vectors derived from space and scalars derived from time can be collected into
four vectors, or four-dimensional
tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tens ...
s. However, the six component angular momentum tensor is sometimes called a
bivector because in the 3D viewpoint it is two vectors (one of these, the conventional angular momentum, being an
axial vector).
Relativistic kinematics
The relativistic four-velocity, that is the four-vector representing velocity in relativity, is defined as follows:
:
In the above,
is the
proper time
In relativity, proper time (from Latin, meaning ''own time'') along a timelike world line is defined as the time as measured by a clock following that line. It is thus independent of coordinates, and is a Lorentz scalar. The proper time interval ...
of the path through
spacetime
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why diffe ...
, called the world-line, followed by the object velocity the above represents, and
:
is the
four-position; the coordinates of an
event. Due to
time dilation
In physics and relativity, time dilation is the difference in the elapsed time as measured by two clocks. It is either due to a relative velocity between them ( special relativistic "kinetic" time dilation) or to a difference in gravitational ...
, the proper time is the time between two events in a frame of reference where they take place at the same location. The proper time is related to
coordinate time ''t'' by:
:
where
is the
Lorentz factor
The Lorentz factor or Lorentz term is a quantity expressing how much the measurements of time, length, and other physical properties change for an object while that object is moving. The expression appears in several equations in special relativi ...
:
:
(either version may be quoted) so it follows:
:
The first three terms, excepting the factor of
, is the velocity as seen by the observer in their own reference frame. The
is determined by the velocity
between the observer's reference frame and the object's frame, which is the frame in which its proper time is measured. This quantity is invariant under Lorentz transformation, so to check to see what an observer in a different reference frame sees, one simply multiplies the velocity four-vector by the Lorentz transformation matrix between the two reference frames.
Relativistic dynamics
Rest mass and relativistic mass
The mass of an object as measured in its own frame of reference is called its ''rest mass'' or ''
invariant mass
The invariant mass, rest mass, intrinsic mass, proper mass, or in the case of bound systems simply mass, is the portion of the total mass of an object or system of objects that is independent of the overall motion of the system. More precisely, ...
'' and is sometimes written
. If an object moves with velocity
in some other reference frame, the quantity
is often called the object's "relativistic mass" in that frame.
Some authors use
to denote rest mass, but for the sake of clarity this article will follow the convention of using
for relativistic mass and
for rest mass.
Lev Okun has suggested that the concept of relativistic mass "has no rational justification today" and should no longer be taught.
Other physicists, including
Wolfgang Rindler and T. R. Sandin, contend that the concept is useful.
See
mass in special relativity
The word "mass" has two meanings in special relativity: ''invariant mass'' (also called rest mass) is an invariant quantity which is the same for all observers in all reference frames, while the relativistic mass is dependent on the velocity of ...
for more information on this debate.
A particle whose rest mass is zero is called ''massless''.
Photon
A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are Massless particle, massless ...
s and
graviton
In theories of quantum gravity, the graviton is the hypothetical quantum of gravity, an elementary particle that mediates the force of gravitational interaction. There is no complete quantum field theory of gravitons due to an outstanding mathe ...
s are thought to be massless, and
neutrino
A neutrino ( ; denoted by the Greek letter ) is a fermion (an elementary particle with spin of ) that interacts only via the weak interaction and gravity. The neutrino is so named because it is electrically neutral and because its rest mass ...
s are nearly so.
Relativistic energy and momentum
There are a couple of (equivalent) ways to define momentum and energy in SR. One method uses
conservation law
In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves over time. Exact conservation laws include conservation of energy, conservation of linear momentum, ...
s. 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 experiment
A thought experiment is a hypothetical situation in which a hypothesis, theory, or principle is laid out for the purpose of thinking through its consequences.
History
The ancient Greek ''deiknymi'' (), or thought experiment, "was the most anci ...
s 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
In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates:
# The laws ...
. It is these new definitions which are taken as the correct ones for momentum and energy in SR.
The
four-momentum of an object is straightforward, identical in form to the classical momentum, but replacing 3-vectors with 4-vectors:
:
The energy and momentum of an object with invariant mass
, moving with
velocity
Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity i ...
with respect to a given frame of reference, are respectively given by
:
The factor
comes from the definition of the four-velocity described above. The appearance of
may be stated in an alternative way, which will be explained in the next section.
The kinetic energy,
, is defined as
:
and the speed as a function of kinetic energy is given by
:
The spatial momentum may be written as
, preserving the form from Newtonian mechanics with relativistic mass substituted for Newtonian mass. However, this substitution fails for some quantities, including force and kinetic energy. Moreover, the relativistic mass is not invariant under Lorentz transformations, while the rest mass is. For this reason, many people prefer to use the rest mass and account for
explicitly through the 4-velocity or coordinate time.
A simple relation between energy, momentum, and velocity may be obtained from the definitions of energy and momentum by multiplying the energy by
, multiplying the momentum by
, and noting that the two expressions are equal. This yields
:
may then be eliminated by dividing this equation by
and squaring,
:
dividing the definition of energy by
and squaring,
:
and substituting:
:
This is the ''relativistic
energy–momentum relation
In physics, the energy–momentum relation, or relativistic dispersion relation, is the relativistic equation relating total energy (which is also called relativistic energy) to invariant mass (which is also called rest mass) and momentum. It is ...
''.
While the energy
and the momentum
depend on the frame of reference in which they are measured, the quantity
is invariant. Its value is
times the squared magnitude of the
4-momentum vector.
The invariant mass of a system may be written as
:
Due to kinetic energy and binding energy, this quantity is different from the sum of the rest masses of the particles of which the system is composed. Rest mass is not a conserved quantity in special relativity, unlike the situation in Newtonian physics. However, even if an object is changing internally, so long as it does not exchange energy or momentum with its surroundings, its rest mass will not change and can be calculated with the same result in any reference frame.
Mass–energy equivalence
The relativistic energy–momentum equation holds for all particles, even for
massless particles for which ''m''
0 = 0. In this case:
:
When substituted into ''Ev'' = ''c''
2''p'', this gives ''v'' = ''c'': massless particles (such as
photons) always travel at the speed of light.
Notice that the rest mass of a composite system will generally be slightly different from the sum of the rest masses of its parts since, in its rest frame, their kinetic energy will increase its mass and their (negative) binding energy will decrease its mass. In particular, a hypothetical "box of light" would have rest mass even though made of particles which do not since their momenta would 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 non-zero mass remaining: ''m''
0 = ''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 of particles, the energy–momentum equation requires summing the momentum vectors of the particles:
:
The inertial frame in which the momenta of all particles sums to zero is called the
center of momentum frame
In physics, the center-of-momentum frame (also zero-momentum frame or COM frame) of a system is the unique (up to velocity but not origin) inertial frame in which the total momentum of the system vanishes. The ''center of momentum'' of a system i ...
. In this special frame, the relativistic energy–momentum 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
:
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 isolated systems cannot be changed so long as the system remains totally closed (no mass or energy allowed in or out), 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
In physics, the center-of-momentum frame (also zero-momentum frame or COM frame) of a system is the unique (up to velocity but not origin) inertial frame in which the total momentum of the system vanishes. The ''center of momentum'' of a system i ...
, causes an increase in energy and momentum without an increase in invariant mass. ''E'' = ''m''
0''c''
2, however, applies only to isolated systems in their center-of-momentum frame where momentum sums to zero.
Taking this formula at face value, we see that in relativity, mass is simply energy by another name (and measured in different units). 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."
Closed (isolated) systems
In a "totally-closed" system (i.e.,
isolated system
In physical science, an isolated system is either of the following:
# a physical system so far removed from other systems that it does not interact with them.
# a thermodynamic system enclosed by rigid immovable walls through which neither m ...
) 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 Δ''E'' = Δ''mc''
2 form, however, only in non-closed systems in which energy is allowed to escape (for example, as heat and light), and thus invariant mass is reduced. 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.
Chemical and nuclear reactions
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 10
−9 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 reaction
In nuclear physics and nuclear chemistry, a nuclear reaction is a process in which two nuclei, or a nucleus and an external subatomic particle, collide to produce one or more new nuclides. Thus, a nuclear reaction must cause a transformatio ...
s, providing important information which was useful in the development of nuclear energy and, consequently, the
nuclear bomb
A nuclear weapon is an explosive device that derives its destructive force from nuclear reactions, either fission (fission bomb) or a combination of fission and fusion reactions ( thermonuclear bomb), producing a nuclear explosion. Both bo ...
. Historically, for example,
Lise Meitner
Elise Meitner ( , ; 7 November 1878 – 27 October 1968) was an Austrian-Swedish physicist who was one of those responsible for the discovery of the element protactinium and nuclear fission. While working at the Kaiser Wilhelm Institute on ra ...
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.
Center of momentum frame
The equation ''E'' = ''m''
0''c''
2 applies only to isolated systems in their
center of momentum frame
In physics, the center-of-momentum frame (also zero-momentum frame or COM frame) of a system is the unique (up to velocity but not origin) inertial frame in which the total momentum of the system vanishes. The ''center of momentum'' of a system i ...
. It has been popularly misunderstood to mean that mass may be ''converted'' to energy, after which the ''mass'' disappears. However, popular explanations of the equation as applied to systems include open (non-isolated) systems for which heat and light are allowed to escape, when they otherwise would have contributed to the mass (
invariant mass
The invariant mass, rest mass, intrinsic mass, proper mass, or in the case of bound systems simply mass, is the portion of the total mass of an object or system of objects that is independent of the overall motion of the system. More precisely, ...
) of the system.
Historically, confusion about mass being "converted" to energy has been aided by confusion between mass and "
matter
In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of atoms, which are made up of interacting subatomic par ...
", where matter is defined as
fermion
In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks and ...
particles. In such a definition, electromagnetic radiation and kinetic energy (or heat) are not considered "matter". In some situations, matter may indeed be converted to non-matter forms of energy (see above), but in all these situations, the matter and non-matter forms of energy still retain their original mass.
For isolated systems (closed to all mass and energy exchange), 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 center-of-momentum frame, the system will be measured as having gained or lost mass, in proportion to energy added or removed. Thus, in theory, if 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 super-strong plasma-filled 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. In this thought-experiment, the light and heat carry away the gram of mass, and would therefore deposit this gram of mass in the objects that absorb them.
Angular momentum
In relativistic mechanics, the time-varying mass moment
:
and orbital 3-angular momentum
:
of a point-like particle are combined into a four-dimensional
bivector in terms of the 4-position X and the 4-momentum P of the particle:
:
where ∧ denotes the
exterior product. This tensor is additive: the total angular momentum of a system is the sum of the angular momentum tensors for each constituent of the system. So, for an assembly of discrete particles one sums the angular momentum tensors over the particles, or integrates the density of angular momentum over the extent of a continuous mass distribution.
Each of the six components forms a conserved quantity when aggregated with the corresponding components for other objects and fields.
Force
In special relativity,
Newton's second law
Newton's laws of motion are three basic Scientific law, laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows:
# A body remains at re ...
does not hold in the form F = ''m''a, but it does if it is expressed as
:
where p = γ(v)''m''
0v is the momentum as defined above and ''m''
0 is the
invariant mass
The invariant mass, rest mass, intrinsic mass, proper mass, or in the case of bound systems simply mass, is the portion of the total mass of an object or system of objects that is independent of the overall motion of the system. More precisely, ...
. Thus, the force is given by
:
:
Consequently, in some old texts, γ(v)
3''m''
0 is referred to as the ''longitudinal mass'', and γ(v)''m''
0 is referred to as the ''transverse mass'', which is numerically the same as the
relativistic mass. See
mass in special relativity
The word "mass" has two meanings in special relativity: ''invariant mass'' (also called rest mass) is an invariant quantity which is the same for all observers in all reference frames, while the relativistic mass is dependent on the velocity of ...
.
If one inverts this to calculate acceleration from force, one gets
:
The force described in this section is the classical 3-D force which is not a
four-vector
In special relativity, a four-vector (or 4-vector) is an object with four components, which transform in a specific way under Lorentz transformations. Specifically, a four-vector is an element of a four-dimensional vector space considered as ...
. This 3-D force is the appropriate concept of force since it is the force which obeys
Newton's third law of motion
Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows:
# A body remains at rest, or in moti ...
. It should not be confused with the so-called
four-force In the special theory of relativity, four-force is a four-vector that replaces the classical force.
In special relativity
The four-force is defined as the rate of change in the four-momentum of a particle with respect to the particle's proper ti ...
which is merely the 3-D force in the comoving frame of the object transformed as if it were a four-vector. However, the density of 3-D force (linear momentum transferred per unit
four-volume) ''is'' a four-vector (
density
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...
of weight +1) when combined with the negative of the density of power transferred.
Torque
The torque acting on a point-like particle is defined as the derivative of the angular momentum tensor given above with respect to proper time:
:
or in tensor components:
:
where F is the 4d force acting on the particle at the event X. As with angular momentum, torque is additive, so for an extended object one sums or integrates over the distribution of mass.
Kinetic energy
The ''
work-energy theorem
In physics, work is the energy transferred to or from an object via the application of force along a displacement. In its simplest form, for a constant force aligned with the direction of motion, the work equals the product of the force stre ...
'' says
[R.C.Tolman "Relativity Thermodynamics and Cosmology" pp 47–48] the change in
kinetic energy
In physics, the kinetic energy of an object is the energy that it possesses due to its motion.
It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its a ...
is equal to the work done on the body. In special relativity:
:
:
If in the initial state the body was at rest, so ''v''
0 = 0 and γ
0(''v''
0) = 1, and in the final state it has speed ''v''
1 = ''v'', setting γ
1(''v''
1) = γ(''v''), the kinetic energy is then;
:
a result that can be directly obtained by subtracting the rest energy ''m''
0''c''
2 from the total relativistic energy γ(''v'')''m''
0''c''
2.
Newtonian limit
The Lorentz factor γ(''v'') can be expanded into a
Taylor series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor se ...
or
binomial series for (''v''/''c'')
2 < 1, obtaining:
:
and consequently
:
:
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
In physics, the kinetic energy of an object is the energy that it possesses due to its motion.
It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its a ...
and momentum. This is as it should be, for special relativity must agree with Newtonian mechanics at low velocities.
See also
*
Introduction to special relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates:
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Twin paradox
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Relativistic equations
Following is a list of the frequently occurring equations in the theory of special relativity.
Postulates of Special Relativity
To derive the equations of special relativity, one must start with two other
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Relativistic heat conduction
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Classical electromagnetism and special relativity
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Relativistic system (mathematics)
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Relativistic Lagrangian mechanics
References
Notes
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Further reading
;General scope and special/general relativity
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*Concepts of Modern Physics (4th Edition), A. Beiser, Physics, McGraw-Hill (International), 1987,
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;Electromagnetism and special relativity
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;Classical mechanics and special relativity
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;General relativity
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{{Branches of physics
Theory of relativity