Relativistic quantum chemistry

From Wikipedia, the free encyclopedia

Relativistic quantum chemistry combines relativistic mechanics with quantum chemistry to explain elemental properties and structure, especially for the heavier elements of the periodic table. A prominent example of such an explanation is the color of gold; due to relativistic effects, it is not silvery like most other metals.

The term "relativistic effects" was developed in light of the history of quantum mechanics. Initially quantum mechanics was developed without considering the theory of relativity.^[1] By convention, "relativistic effects" are those discrepancies between values calculated by models considering and not considering relativity.^[2] Relativistic effects are important for the heavier elements with high atomic numbers. In the most common layout of the periodic table, these elements are shown in the lower area. Examples are the lanthanides and actinides.^[3]

Relativistic effects in chemistry can be considered to be perturbations, or small corrections, to the non-relativistic theory of chemistry, which is developed from the solutions of the Schrödinger equation. These corrections affect the electrons differently depending on the electron speed relative to the speed of light. Relativistic effects are more prominent in heavy elements because only in these elements do electrons attain more pronounced relativistic speeds.^{[citation needed]}

History

Beginning in 1935, Bertha Swirles described a relativistic treatment of a many-electron system,^[4] in spite of Paul Dirac's 1929 assertion that the only imperfections remaining in quantum mechanics "...give rise to difficulties only when high-speed particles are involved, and are therefore of no importance in the consideration of atomic and molecular structure and ordinary chemical reactions in which it is, indeed, usually sufficiently accurate if one neglects relativity variation of mass and velocity and assumes only Coulomb forces between the various electrons and atomic nuclei."^[5]
Theoretical chemists by and large agreed with Dirac's sentiment until the 1970s, when relativistic effects began to become realized in heavy elements.^[6] The Schrödinger equation had been developed without considering relativity in Schrödinger's 1926 paper.^[7] Relativistic corrections were made to the Schrödinger equation (see Klein–Gordon equation) in order to explain the fine structure of atomic spectra, but this development and others did not immediately trickle into the chemical community. Since atomic spectral lines were largely in the realm of physics and not in that of chemistry, most chemists were unfamiliar with relativistic quantum mechanics, and their attention was on lighter elements typical for the organic chemistry focus of the time.^{[8][page needed]}

Dirac's opinion on the role relativistic quantum mechanics would play for chemical systems is wrong for two reasons: the first being that electrons in s and p atomic orbitals travel at a significant fraction of the speed of light and the second being that there are indirect consequences of relativistic effects which are especially evident for d and f atomic orbitals.^[6]

Qualitative treatment

Relativistic γ as a function of velocity. For a small velocity, the ${\displaystyle E_{rel}}$ (ordinate) is equal to ${\displaystyle E_{0}=mc^{2}}$ but as ${\displaystyle v_{e}\to c}$ the ${\displaystyle E_{rel}}$ goes to infinity.

One of the most important and familiar results of relativity is that the relativistic mass of the electron increases by

${\displaystyle m_{rel}={\frac {m_{e}}{\sqrt {1-(v_{e}/c)^{2}}}}}$

where ${\displaystyle \displaystyle m_{e},v_{e},c}$ are the electron rest mass, velocity of the electron, and speed of light respectively. The figure at the right illustrates the relativistic effects on the mass of an electron as a function of its velocity.

This has an immediate implication on the Bohr radius (${\displaystyle \displaystyle a_{0}}$) which is given by

${\displaystyle a_{0}={\frac {\hbar }{m_{e}c\alpha }}}$

where ${\displaystyle \hbar }$ is the reduced Planck's constant and α is the fine-structure constant (a relativistic correction for the Bohr model).

Arnold Sommerfeld calculated that, for a 1s electron of a hydrogen atom with an orbiting radius of 0.0529 nm, α ≈ 1/137. That is to say, the fine-structure constant shows the electron traveling at nearly 1/137 the speed of light.^[9] One can extend this to a larger element by using the expression v ≈ Zc/137 for a 1s electron where v is its radial velocity. For gold with (Z = 79) the 1s electron will be going (α = 0.58c) 58% of the speed of light. Plugging this in for v/c for the relativistic mass one finds that m_rel = 1.22m_e and in turn putting this in for the Bohr radius above one finds that the radius shrinks by 22%.

If one substitutes in the relativistic mass into the equation for the Bohr radius it can be written

${\displaystyle a_{rel}={\frac {\hbar {\sqrt {1-(v_{e}/c)^{2}}}}{m_{e}c\alpha }}}$

Ratio of relativistic and nonrelativistic Bohr radii, as a function of electron velocity

It follows that

${\displaystyle {\frac {a_{rel}}{a_{0}}}={\sqrt {1-(v_{e}/c)^{2}}}}$

At right, the above ratio of the relativistic and nonrelativistic Bohr radii has been plotted as a function of the electron velocity. Notice how the relativistic model shows the radius decreasing with increasing velocity.

When the Bohr treatment is extended to hydrogenic-like atoms using the Quantum Rule, the Bohr radius becomes

${\displaystyle r={\frac {n^{2}\hbar ^{2}4\pi \epsilon _{0}}{m_{e}Ze^{2}}}}$

where ${\displaystyle n}$ is the principal quantum number and Z is an integer for the atomic number. From quantum mechanics the angular momentum is given as ${\displaystyle mv_{e}r=n\hbar }$. Substituting into the equation above and solving for ${\displaystyle v}$ gives

${\displaystyle r={\frac {mv_{e}rn\hbar 4\pi \epsilon _{0}}{mZe^{2}}}}$

${\displaystyle 1={\frac {v_{e}n\hbar 4\pi \epsilon _{0}}{Ze^{2}}}}$

${\displaystyle v_{e}={\frac {Ze^{2}}{n\hbar 4\pi \epsilon _{0}}}}$

From this point atomic units can be used to simplify the expression into

${\displaystyle v_{e}={\frac {Z}{n}}}$

Substituting this into the expression for the Bohr ratio mentioned above gives

${\displaystyle {\frac {a_{rel}}{a_{0}}}={\sqrt {1-\left({\frac {Z}{nc}}\right)^{2}}}}$

At this point one can see that for a low value of ${\displaystyle n}$ and a high value of ${\displaystyle Z}$ that ${\displaystyle {\frac {a_{rel}}{a_{0}}}<1 annotation="">$. This fits with intuition: electrons with lower principal quantum numbers will have a higher probability density of being nearer to the nucleus. A nucleus with a large charge will cause an electron to have a high velocity. A higher electron velocity means an increased electron relativistic mass, as a result the electrons will be near the nucleus more of the time and thereby contract the radius for small principal quantum numbers.^[10]

Periodic table deviations

The periodic table was constructed by scientists who noticed periodic trends in known elements of the time. Indeed, the patterns found in it is what gives the periodic table its power. Many of the chemical and physical differences between the 6th period (Cs–Rn) and the 5th period (Rb–Xe) arise from the larger relativistic effects for the former. These relativistic effects are particularly large for gold and its neighbors, platinum and mercury.

Mercury

Mercury (Hg) is a liquid down to −39°C (see Melting Point (m.p.)). Bonding forces are weaker for Hg–Hg bonds than for its immediate neighbors such as cadmium (m.p. 321°C) and gold (m.p. 1064°C). The lanthanide contraction is a partial explanation; however, it does not entirely account for this anomaly.^[9] In the gas phase mercury is alone in metals in that it is quite typically found in a monomeric form as Hg(g). Hg₂²⁺(g) also forms and it is a stable species due to the relativistic shortening of the bond.

Hg₂(g) does not form because the 6s² orbital is contracted by relativistic effects and may therefore only weakly contribute to any bonding; in fact Hg–Hg bonding must be mostly the result of van der Waals forces, which explains why the bonding for Hg–Hg is weak enough to allow for Hg to be a liquid at room temperature.^[9]

Au₂(g) and Hg(g) are analogous, at the least in having the same nature of difference, to H₂(g) and He(g). It is for the relativistic contraction of the 6s² orbital that gaseous mercury can be called a pseudo noble gas.^[9]

Color of gold and caesium

Spectral reflectance curves for aluminum (Al), silver (Ag), and gold (Au) metal mirrors

The reflectivity of Au, Ag, Al is shown on the figure to the right. The human eye sees electromagnetic radiation with a wavelength near 600 nm as yellow. As is clear from its reflectance spectrum, gold appears yellow because it absorbs blue light more than it absorbs other visible wavelengths of light; the reflected light reaching the eye is therefore lacking in blue compared to the incident light. Since yellow is complementary to blue, this makes a piece of gold under white light appear yellow to human eyes.

The electronic transition responsible for this absorption is a transition from the 5d to the 6s level. An analogous transition occurs in Ag but the relativistic effects are lower in Ag so while the 4d experiences some expansion and the 5s some contraction, the 4d-5s distance in Ag is still much greater than the 5d-6s distance in Au because the relativistic effects in Ag are smaller than those in Au. Thus, non-relativistic gold would be white. The relativistic effects are raising the 5d orbital and lowering the 6s orbital.^[11]

A similar effect occurs in caesium metal, the heaviest of the alkali metals which can be collected in quantities sufficient to allow viewing. Whereas the other alkali metals are silver-white, caesium metal has a distinctly golden hue.

Lead-acid battery

Without relativity, lead would be expected to behave much alike tin, so tin-acid batteries should work equally well as lead-acid batteries that are commonly used in cars. However, calculations show that about 10 V of the 12 V produced by a lead-acid battery arise purely from relativistic effects, explaining why tin-acid batteries do not work.^[12]

Inert pair effect

In Tl(I) (thallium), Pb(II) (lead), and Bi(III) (bismuth) complexes there is a 6s² electron pair. The 'inert pair effect' refers to the tendency for this pair of electrons to resist oxidation due to a relativistic contraction of the 6s orbital.^[6]

Others

Some of the phenomena commonly attributed to relativistic effects are:

• Aurophilicity
• The stability of the gold anion, Au⁻, in compounds such as CsAu
• The crystal structure of lead, which is face-centered cubic instead of diamond-like
• The striking similarity between zirconium and hafnium
• The stability of the uranyl cation, as well as other high oxidation states in the early actinides (Pa-Am)
• The small atomic radii of francium and radium
• About 10% of the lanthanide contraction is attributed to the relativistic mass of high velocity electrons and the smaller Bohr radius that results.
• In the case of gold, a lot more than 10% of its contraction is due to relativistically heavy electrons, and gold (element 79) is almost twice as dense as lead (element 82).

Relativistic mechanics

From Wikipedia, the free encyclopedia

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:

${\displaystyle {\boldsymbol {\mathbf {U} }}={\frac {d{\boldsymbol {\mathbf {X} }}}{d\tau }}=\left({\frac {cdt}{d\tau }},{\frac {d\mathbf {x} }{d\tau }}\right)}$

In the above, τ is the proper time of the path through spacetime, called the world-line, followed by the object velocity the above represents, and

${\displaystyle {\boldsymbol {\mathbf {X} }}=(ct,\mathbf {x} )}$

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:

${\displaystyle {\frac {d\tau }{dt}}={\frac {1}{\gamma (\mathbf {v} )}}}$

where γ(v) is the Lorentz factor:

${\displaystyle \gamma (\mathbf {v} )={\frac {1}{\sqrt {1-\mathbf {v} \cdot \mathbf {v} /c^{2}}}}\,\rightleftharpoons \,\gamma (v)={\frac {1}{\sqrt {1-(v/c)^{2}}}}.}$

(either version may be quoted) so it follows:

${\displaystyle {\boldsymbol {\mathbf {U} }}=\gamma (\mathbf {v} )(c,\mathbf {v} )}$

The first three terms, excepting the factor of γ(v), is the velocity as seen by the observer in their own reference frame. The γ(v) is determined by the velocity v 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:

${\displaystyle {\boldsymbol {\mathbf {P} }}=m_{0}{\boldsymbol {\mathbf {U} }}=(E/c,\mathbf {p} )}$

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

${\displaystyle {\begin{aligned}E&=\gamma (\mathbf {v} )m_{0}c^{2}\\\mathbf {p} &=\gamma (\mathbf {v} )m_{0}\mathbf {v} \end{aligned}}}$

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

${\displaystyle K=(\gamma -1)m_{0}c^{2}=E-m_{0}c^{2}\,,}$

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

${\displaystyle v={\frac {c{\sqrt {K(K+2m_{0}c^{2})}}}{K+m_{0}c^{2}}}={\frac {c{\sqrt {(E-m_{0}c^{2})(E+m_{0}c^{2})}}}{E}}={\frac {pc^{2}}{E}}\,.}$

Rest mass and relativistic mass

The quantity

${\displaystyle m=\gamma (\mathbf {v} )m_{0}}$

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

${\displaystyle E^{2}-(pc)^{2}=(m_{0}c^{2})^{2}\,}$

${\displaystyle \mathbf {p} c^{2}=E\mathbf {v} \,.}$

The first is referred to as the relativistic energy–momentum relation. It can be derived by considering that ${\displaystyle {\frac {v^{2}}{c^{2}}}}$ can be written as ${\displaystyle {\frac {p^{2}}{\gamma ^{2}m_{0}^{2}c^{2}}}}$ where the denominator can be written as ${\displaystyle {\frac {E^{2}}{c^{2}}}}$. 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.

It should be noted that the invariant mass of a system

${\displaystyle {m_{0}}_{\text{tot}}={\frac {\sqrt {E_{\text{tot}}^{2}-(p_{\text{tot}}c)^{2}}}{c^{2}}}}$

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:

${\displaystyle E=pc}$

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:

${\displaystyle E^{2}-\mathbf {p} \cdot \mathbf {p} c^{2}=m_{0}^{2}c^{4}}$

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

${\displaystyle m_{0,\,{\rm {system}}}=\sum _{n}E_{n}/c^{2}}$

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, 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

${\displaystyle \mathbf {N} =m\left(\mathbf {x} -t\mathbf {v} \right)}$

and orbital 3-angular momentum

${\displaystyle \mathbf {L} =\mathbf {x} \times \mathbf {p} }$

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]

${\displaystyle \mathbf {M} =\mathbf {X} \wedge \mathbf {P} }$

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 does not hold in the form F = ma, but it does if it is expressed as

${\displaystyle \mathbf {F} ={\frac {d\mathbf {p} }{dt}}}$

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

${\displaystyle \mathbf {F} =\gamma (\mathbf {v} )^{3}m_{0}\,\mathbf {a} _{\parallel }+\gamma (\mathbf {v} )m_{0}\,\mathbf {a} _{\perp }}$

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

${\displaystyle \mathbf {a} ={\frac {1}{m_{0}\gamma (\mathbf {v} )}}\left(\mathbf {F} -{\frac {(\mathbf {v} \cdot \mathbf {F} )\mathbf {v} }{c^{2}}}\right)\,.}$

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.

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:^[10][11]

${\displaystyle {\boldsymbol {\Gamma }}={\frac {d\mathbf {M} }{d\tau }}=\mathbf {X} \wedge \mathbf {F} }$

or in tensor components:

${\displaystyle \Gamma _{\alpha \beta }=X_{\alpha }F_{\beta }-X_{\beta }F_{\alpha }}$

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:

${\displaystyle {\begin{aligned}\Delta K=W=[\gamma _{1}-\gamma _{0}]m_{0}c^{2}.\end{aligned}}}$

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;

${\displaystyle K=[\gamma (v)-1]m_{0}c^{2}\,,}$

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:

${\displaystyle \gamma ={\dfrac {1}{\sqrt {1-(v/c)^{2}}}}=\sum _{n=0}^{\infty }\left({\dfrac {v}{c}}\right)^{2n}\prod _{k=1}^{n}\left({\dfrac {2k-1}{2k}}\right)=1+{\dfrac {1}{2}}\left({\dfrac {v}{c}}\right)^{2}+{\dfrac {3}{8}}\left({\dfrac {v}{c}}\right)^{4}+{\dfrac {5}{16}}\left({\dfrac {v}{c}}\right)^{6}+\cdots }$

and consequently

${\displaystyle E-m_{0}c^{2}={\frac {1}{2}}m_{0}v^{2}+{\frac {3}{8}}{\frac {m_{0}v^{4}}{c^{2}}}+{\frac {5}{16}}{\frac {m_{0}v^{6}}{c^{4}}}+\cdots ;}$

${\displaystyle \mathbf {p} =m_{0}\mathbf {v} +{\frac {1}{2}}{\frac {m_{0}v^{2}\mathbf {v} }{c^{2}}}+{\frac {3}{8}}{\frac {m_{0}v^{4}\mathbf {v} }{c^{4}}}+{\frac {5}{16}}{\frac {m_{0}v^{6}\mathbf {v} }{c^{6}}}+\cdots .}$

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.