Relativistic mechanics

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In physics, relativistic mechanics refers to mechanics compatible with special relativity (SR) and general relativity (GR). It provides a non-quantum mechanical description of a system of particles, or of a fluid, in cases where the velocities of moving objects are comparable to the speed of light c. As a result, classical mechanics is extended correctly to particles traveling at high velocities and energies, and provides a consistent inclusion of electromagnetism 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, 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 accelerations, and "dynamics"; a full description by considering energies, momenta, and angular momenta and their conservation laws, and forces 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" in classical mechanics—depends on the relative motion of observers who measure in frames of reference.

Although some definitions and concepts from classical mechanics do carry over to SR, such as force as the time derivative of momentum (Newton's second law), the work done by a particle as the line integral 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 reference frames. In addition to modifying notions of space and time, SR forces one to reconsider the concepts of mass, momentum, and energy all of which are important constructs in Newtonian mechanics. 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 for details.

The equations become more complicated in the more familiar three-dimensional vector calculus formalism, due to the nonlinearity in the Lorentz factor, which accurately accounts for relativistic velocity dependence and the speed limit of all particles and fields. However, they have a simpler and elegant form in four-dimensional spacetime, which includes flat Minkowski space (SR) and curved spacetime (GR), because three-dimensional vectors derived from space and scalars derived from time can be collected into four vectors, or four-dimensional tensors. 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 of the path through spacetime, 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, 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:

(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

Relativistic energy and momentum

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 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 m0 (also called rest mass), moving with velocity v with respect to a given frame of reference, are respectively given by

The factor of γ(v) comes from the definition of the four-velocity described above. The appearance of the γ factor has an alternative way of being stated, explained in the next section.

The kinetic energy, K, is defined as

And the speed as a function of kinetic energy is given by

Rest mass and relativistic mass

The quantity

is often called the relativistic mass of the object in the given frame of reference. [1]

This makes the relativistic relation between the spatial velocity and the spatial momentum look identical. However, this can be misleading, as it is not appropriate in special relativity in all circumstances. For instance, kinetic energy and force in special relativity can not be written exactly like their classical analogues by only replacing the mass with the relativistic mass. Moreover, under Lorentz transformations, this relativistic mass is not invariant, while the rest mass is. For this reason many people find it easier use the rest mass (thereby introduce γ through the 4-velocity or coordinate time), and discard the concept of relativistic mass.

Lev B. Okun suggested that "this terminology ... has no rational justification today", and should no longer be taught. [2]

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. [3] See mass in special relativity for more information on this debate.

Some authors use m for relativistic mass and m0 for rest mass, [4] others simply use m for rest mass. This article uses the former convention for clarity.

The energy and momentum of an object with invariant mass m0 are related by the formulas

The first is referred to as the relativistic energy–momentum relation . It can be derived by considering that can be written as where the denominator can be written as . Now, gamma can be replaced in the expression of energy. While the energy E and the momentum p depend on the frame of reference in which they are measured, the quantity E2 − (pc)2 is invariant, and arises as −c2 times the squared magnitude of the 4-momentum vector which is −(m0c)2.

The invariant mass of a system

is different from the sum of the rest masses of the particles of which it is composed due to kinetic energy and binding energy. 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 with surroundings, then its rest mass will not change, and can be calculated with the same result in any frame of reference.

A particle whose rest mass is zero is called massless. Photons and gravitons are thought to be massless; and neutrinos are nearly so.

Mass–energy equivalence

The relativistic energy–momentum equation holds for all particles, even for massless particles for which m0 = 0. In this case:

When substituted into Ev = c2p, 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: m0 = E/c2. 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 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 c2

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, causes an increase in energy and momentum without an increase in invariant mass. E = m0c2, 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." [5]

Closed (isolated) systems

In a "totally-closed" system (i.e., isolated 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 = Δmc2 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. [6] 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.

Center of momentum frame

The equation E = m0c2 applies only to isolated systems 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. 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) of the system.

Historically, confusion about mass being "converted" to energy has been aided by confusion between mass and "matter", where matter is defined as fermion 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. [7]

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: [8] [9]

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.


In special relativity, Newton's second law does not hold in the form F = ma, but it does if it is expressed as

where p = γ(v)m0v is the momentum as defined above and m0 is the invariant mass. Thus, the force is given by

Consequently, in some old texts, γ(v)3m0 is referred to as the longitudinal mass, and γ(v)m0 is referred to as the transverse mass, which is numerically the same as the relativistic mass. See mass in special relativity.

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. This 3-D force is the appropriate concept of force since it is the force which obeys Newton's third law of motion. It should not be confused with the so-called four-force 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 of weight +1) when combined with the negative of the density of power transferred.


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: [10] [11]

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 says [12] the change in kinetic energy is equal to the work done on the body. In special relativity:

If in the initial state the body was at rest, so v0 = 0 and γ0(v0) = 1, and in the final state it has speed v1 = v, setting γ1(v1) = γ(v), the kinetic energy is then;

a result that can be directly obtained by subtracting the rest energy m0c2 from the total relativistic energy γ(v)m0c2.

Newtonian limit

The Lorentz factor γ(v) can be expanded into a Taylor series 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 c2 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.

See also

Related Research Articles

Kinetic energy energy of a moving physical body

In physics, the kinetic energy (KE) 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 acceleration, the body maintains this kinetic energy unless its speed changes. The same amount of work is done by the body when decelerating from its current speed to a state of rest.

Lorentz transformation space-time coordinates transformation that conserves the form of electromagnetism laws

In physics, the Lorentz transformations are a one-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former. The respective inverse transformation is then parametrized by the negative of this velocity. The transformations are named after the Dutch physicist Hendrik Lorentz.

Momentum Conserved physical quantity related to the motion of a body

In Newtonian mechanics, linear momentum, translational momentum, or simply momentum is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If m is an object's mass and v is its velocity, then the object’s momentum is:

In SI units, momentum is measured in kilogram meters per second (kg⋅m/s).

Special relativity Theory of interwoven space and time by Albert Einstein

In physics, special relativity is the generally accepted and experimentally confirmed physical theory regarding the relationship between space and time. In Albert Einstein's original pedagogical treatment, it is based on two postulates:

  1. the laws of physics are invariant in all inertial frames of reference ; and
  2. the speed of light in a vacuum is the same for all observers, regardless of the motion of the light source or observer.
Spacetime Mathematical model combining space and time

In physics, spacetime is any mathematical model which fuses the three dimensions of space and the one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why different observers perceive where and when events occur differently.

Dirac equation Relativistic quantum mechanical wave equation

In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin-1/2 massive particles such as electrons and quarks for which parity is a symmetry. It is consistent with both the principles of quantum mechanics and the theory of special relativity, and was the first theory to account fully for special relativity in the context of quantum mechanics. It was validated by accounting for the fine details of the hydrogen spectrum in a completely rigorous way.

Invariant mass bodys intrinsic mass (or equivalent intrinsic energy), motion-independent (Lorentz invariant); equals total mass (or energy) when at rest

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, it is a characteristic of the system's total energy and momentum that is the same in all frames of reference related by Lorentz transformations. If a center-of-momentum frame exists for the system, then the invariant mass of a system is equal to its total mass in that "rest frame". In other reference frames, where the system's momentum is nonzero, the total mass of the system is greater than the invariant mass, but the invariant mass remains unchanged.

Four-momentum relativistic generalization from the classical momentum of three-dimensional space to four-dimensional spacetime

In special relativity, four-momentum is the generalization of the classical three-dimensional momentum to four-dimensional spacetime. Momentum is a vector in three dimensions; similarly four-momentum is a four-vector in spacetime. The contravariant four-momentum of a particle with relativistic energy E and three-momentum p = = γmv, where v is the particle's three-velocity and γ the Lorentz factor, is

Four-vector Vector in special relativity well-behaved with respect to Lorentz transformations

In special relativity, a four-vector is an object with four components, which transform in a specific way under Lorentz transformation. Specifically, a four-vector is an element of a four-dimensional vector space considered as a representation space of the standard representation of the Lorentz group, the (½,½) representation. It differs from a Euclidean vector in how its magnitude is determined. The transformations that preserve this magnitude are the Lorentz transformations, which include spatial rotations and boosts.

In the theory of relativity, four-acceleration is a four-vector that is analogous to classical acceleration. Four-acceleration has applications in areas such as the annihilation of antiprotons, resonance of strange particles and radiation of an accelerated charge.

In the special theory of relativity, four-force is a four-vector that replaces the classical force.

The word mass has two meanings in special relativity: rest mass or invariant mass is an invariant quantity which is the same for all observers in all reference frames, while relativistic mass is dependent on the velocity of the observer. According to the concept of mass–energy equivalence, the rest mass and relativistic mass are equivalent to the rest energy and total energy of the body, respectively. The term relativistic mass tends not to be used in particle and nuclear physics and is often avoided by writers on special relativity, in favor of using the body's total energy. In contrast, rest mass is usually preferred over rest energy. The measurable inertia and gravitational attraction of a body in a given frame of reference is determined by its relativistic mass, not merely its rest mass. For example, light has zero rest mass but contributes to the inertia of any system containing it.

In differential geometry, the four-gradient is the four-vector analogue of the gradient from vector calculus.

Thomas precession

In physics, the Thomas precession, named after Llewellyn Thomas, is a relativistic correction that applies to the spin of an elementary particle or the rotation of a macroscopic gyroscope and relates the angular velocity of the spin of a particle following a curvilinear orbit to the angular velocity of the orbital motion.

In a relativistic theory of physics, a Lorentz scalar is an expression, formed from items of the theory, which evaluates to a scalar, invariant under any Lorentz transformation. A Lorentz scalar may be generated from e.g., the scalar product of vectors, or from contracting tensors of the theory. While the components of vectors and tensors are in general altered under Lorentz transformations, Lorentz scalars remain unchanged.

In physics, the energy–momentum relation, or relativistic dispersion relation, is the relativistic equation relating any object's rest (intrinsic) mass, total energy, and momentum:

Proper velocity alternative to velocity for measuring motion in relativity theory

In relativity, proper velocity, also known as celerity, is an alternative to velocity for measuring motion. Whereas velocity relative to an observer is distance per unit time where both distance and time are measured by the observer, proper velocity relative to an observer divides observer-measured distance by the time elapsed on the clocks of the traveling object. Proper velocity equals velocity at low speeds. Proper velocity at high speeds, moreover, retains many of the properties that velocity loses in relativity compared with Newtonian theory.

In physics, relativistic quantum mechanics (RQM) is any Poincaré covariant formulation of quantum mechanics (QM). This theory is applicable to massive particles propagating at all velocities up to those comparable to the speed of light c, and can accommodate massless particles. The theory has application in high energy physics, particle physics and accelerator physics, as well as atomic physics, chemistry and condensed matter physics. Non-relativistic quantum mechanics refers to the mathematical formulation of quantum mechanics applied in the context of Galilean relativity, more specifically quantizing the equations of classical mechanics by replacing dynamical variables by operators. Relativistic quantum mechanics (RQM) is quantum mechanics applied with special relativity. Although the earlier formulations, like the Schrödinger picture and Heisenberg picture were originally formulated in a non-relativistic background, a few of them also work with special relativity.

Relativistic Lagrangian mechanics Mathematical formulation of special and general relativity

In theoretical physics, relativistic Lagrangian mechanics is Lagrangian mechanics applied in the context of special relativity and general relativity.

Relativistic angular momentum Angular momentum in special and general relativity

In physics, relativistic angular momentum refers to the mathematical formalisms and physical concepts that define angular momentum in special relativity (SR) and general relativity (GR). The relativistic quantity is subtly different from the three-dimensional quantity in classical mechanics.



  1. Philip Gibbs, Jim Carr & Don Koks (2008). "What is relativistic mass?". Usenet Physics FAQ. Retrieved 2008-09-19. 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: Philip Gibbs & Jim Carr (1998). "Does mass change with speed?". Usenet Physics FAQ. Archived from the original on 2007-06-30.
  2. Lev B. Okun (July 1989). "The Concept of Mass" (subscription required). Physics Today. 42 (6): 31–36. Bibcode:1989PhT....42f..31O. doi:10.1063/1.881171.
  3. T. R. Sandin (November 1991). "In defense of relativistic mass". American Journal of Physics. 59 (11): 1032–1036. Bibcode:1991AmJPh..59.1032S. doi:10.1119/1.16642.
  4. See, for example: Feynman, Richard (1998). "The special theory of relativity". Six Not-So-Easy Pieces. Cambridge, Massachusetts: Perseus Books. ISBN   0-201-32842-9.
  5. Einstein on Newton
  6. Randy Harris (2008). Modern Physics: Second Edition. Pearson Addison-Wesley. p. 38. ISBN   978-0-8053-0308-7.
  7. E. F. Taylor and J. A. Wheeler, Spacetime Physics, W.H. Freeman and Co., New York. 1992. ISBN   0-7167-2327-1, see pp. 248–9 for discussion of mass remaining constant after detonation of nuclear bombs, until heat is allowed to escape.
  8. R. Penrose (2005). The Road to Reality. Vintage books. pp. 437–438, 566–569. ISBN   978-0-09-944068-0.Note: Some authors, including Penrose, use Latin letters in this definition, even though it is conventional to use Greek indices for vectors and tensors in spacetime.
  9. M. Fayngold (2008). Special Relativity and How it Works. John Wiley & Sons. pp. 137–139. ISBN   978-3-527-40607-4.
  10. S. Aranoff (1969). "Torque and angular momentum on a system at equilibrium in special relativity". American Journal of Physics. 37 (4): 453–454. Bibcode:1969AmJPh..37..453A. doi:10.1119/1.1975612. This author uses T for torque, here we use capital Gamma Γ since T is most often reserved for the stress–energy tensor.
  11. S. Aranoff (1972). "Equilibrium in special relativity" (PDF). Nuovo Cimento. 10 (1): 159. Bibcode:1972NCimB..10..155A. doi:10.1007/BF02911417.
  12. R.C.Tolman "Relativity Thermodynamics and Cosmology" pp 47–48

Further reading

General scope and special/general relativity
Electromagnetism and special relativity
Classical mechanics and special relativity
General relativity