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 the extension of mass–energy equivalence for bodies or systems with non-zero momentum.

It can be formulated as: Template:NumBlk This equation holds for a body or system, such as one or more particles, with total energy Template:Math, invariant mass Template:Math, and momentum of magnitude Template:Math; the constant Template:Math is the speed of light. It assumes the special relativity case of flat spacetime and that the particles are free. Total energy is the sum of rest energy $E_0=m_0{c}^2$ and relativistic kinetic energy: $$E_K=E-E_0=\sqrt{(pc{)}^2+{\left(m_0{c}^2\right)}^2}-m_0{c}^2$$ Invariant mass is mass measured in a center-of-momentum frame. For bodies or systems with zero momentum, it simplifies to the mass–energy equation $E_0=m_0{c}^2$, where total energy in this case is equal to rest energy.

The Dirac sea model, which was used to predict the existence of antimatter, is closely related to the energy–momentum relation.

Connection to E = mc²

Einstein Triangle

The energy–momentum relation is consistent with the familiar mass–energy relation in both its interpretations: Template:Math relates total energy Template:Math to the (total) relativistic mass Template:Math (alternatively denoted Template:Math or Template:Math), while Template:Math relates rest energy Template:Math to (invariant) rest mass Template:Math.

Unlike either of those equations, the energy–momentum equation (Template:EquationNote) relates the total energy to the rest mass Template:Math. All three equations hold true simultaneously.

Special cases

1. If the body is a massless particle (Template:Math), then (Template:EquationNote) reduces to Template:Math. For photons, this is the relation, discovered in 19th century classical electromagnetism, between radiant momentum (causing radiation pressure) and radiant energy.
2. If the body's speed Template:Math is much less than Template:Math, then (Template:EquationNote) reduces to Template:Math; that is, the body's total energy is simply its classical kinetic energy (Template:Math) plus its rest energy.
3. If the body is at rest (Template:Math), i.e. in its center-of-momentum frame (Template:Math), we have Template:Math and Template:Math; thus the energy–momentum relation and both forms of the mass–energy relation (mentioned above) all become the same.

A more general form of relation (Template:EquationNote) holds for general relativity.

The invariant mass (or rest mass) is an invariant for all frames of reference (hence the name), not just in inertial frames in flat spacetime, but also accelerated frames traveling through curved spacetime (see below). However the total energy of the particle Template:Math and its relativistic momentum Template:Math are frame-dependent; relative motion between two frames causes the observers in those frames to measure different values of the particle's energy and momentum; one frame measures Template:Math and Template:Math, while the other frame measures Template:Math and Template:Math, where Template:Math and Template:Math, unless there is no relative motion between observers, in which case each observer measures the same energy and momenta. Although we still have, in flat spacetime:

${E^{\prime }}^2-{\left(p^{\prime }c\right)}^2={\left(m_0{c}^2\right)}^2.$

The quantities Template:Math, Template:Math, Template:Math, Template:Math are all related by a Lorentz transformation. The relation allows one to sidestep Lorentz transformations when determining only the magnitudes of the energy and momenta by equating the relations in the different frames. Again in flat spacetime, this translates to;

$E^2-{\left(pc\right)}^2={E^{\prime }}^2-{\left(p^{\prime }c\right)}^2={\left(m_0{c}^2\right)}^2.$

Since Template:Math does not change from frame to frame, the energy–momentum relation is used in relativistic mechanics and particle physics calculations, as energy and momentum are given in a particle's rest frame (that is, Template:Math and Template:Math as an observer moving with the particle would conclude to be) and measured in the lab frame (i.e. Template:Math and Template:Math as determined by particle physicists in a lab, and not moving with the particles).

In relativistic quantum mechanics, it is the basis for constructing relativistic wave equations, since if the relativistic wave equation describing the particle is consistent with this equation – it is consistent with relativistic mechanics, and is Lorentz invariant. In relativistic quantum field theory, it is applicable to all particles and fields.

Origins and derivation of the equation

The energy–momentum relation goes back to Max Planck's article published in 1906. It was used by Walter Gordon in 1926 and then by Paul Dirac in 1928 under the form $E=\sqrt{c^2{p}^2+(m_0{c}^2{)}^2}+V$, where V is the amount of potential energy.

The equation can be derived in a number of ways, two of the simplest include:

1. From the relativistic dynamics of a massive particle,
2. By evaluating the norm of the four-momentum of the system. This method applies to both massive and massless particles, and can be extended to multi-particle systems with relatively little effort (see Template:Section link below).

Heuristic approach for massive particles

For a massive object moving at three-velocity Template:Math with magnitude Template:Math in the lab frame:

$E={\gamma }_{(𝐮)}{m}_0{c}^2$

is the total energy of the moving object in the lab frame,

$𝐩={\gamma }_{(𝐮)}{m}_0𝐮$

is the three dimensional relativistic momentum of the object in the lab frame with magnitude Template:Math. The relativistic energy Template:Math and momentum Template:Math include the Lorentz factor defined by:

${\gamma }_{(𝐮)}=\frac{1}{\sqrt{1-\frac{𝐮\cdot 𝐮}{c^2}}}=\frac{1}{\sqrt{1-{\left(\frac{u}{c}\right)}^2}}$

Some authors use relativistic mass defined by:

$m={\gamma }_{(𝐮)}{m}_0$

although rest mass Template:Math has a more fundamental significance, and will be used primarily over relativistic mass Template:Math in this article.

Squaring the 3-momentum gives:

$p^2=𝐩\cdot 𝐩=\frac{m_0^2𝐮\cdot 𝐮}{1-\frac{𝐮\cdot 𝐮}{c^2}}=\frac{m_0^2{u}^2}{1-{\left(\frac{u}{c}\right)}^2}$

then solving for Template:Math and substituting into the Lorentz factor one obtains its alternative form in terms of 3-momentum and mass, rather than 3-velocity:

$\gamma =\sqrt{1+{\left(\frac{p}{m_{0}c}\right)}^2}$

Inserting this form of the Lorentz factor into the energy equation gives:

$E=m_0{c}^2\sqrt{1+{\left(\frac{p}{m_{0}c}\right)}^2}$

followed by more rearrangement it yields (Template:EquationRef). The elimination of the Lorentz factor also eliminates implicit velocity dependence of the particle in (Template:EquationRef), as well as any inferences to the "relativistic mass" of a massive particle. This approach is not general as massless particles are not considered. Naively setting Template:Math would mean that Template:Math and Template:Math and no energy–momentum relation could be derived, which is not correct.

Norm of the four-momentum

The energy and momentum of an object measured in two inertial frames in energy–momentum space – the yellow frame measures Template:Math and Template:Math while the blue frame measures Template:Math and Template:Math. The green arrow is the four-momentum Template:Math of an object with length proportional to its rest mass Template:Math. The green frame is the centre-of-momentum frame for the object with energy equal to the rest energy. The hyperbolae show the Lorentz transformation from one frame to another is a hyperbolic rotation, and Template:Math and Template:Nowrap are the rapidities of the blue and green frames, respectively.

Special relativity

In Minkowski space, energy (divided by Template:Math) and momentum are two components of a Minkowski four-vector, namely the four-momentum;

$𝐏=(\frac{E}{c},𝐩),$

(these are the contravariant components).

The Minkowski inner product Template:Math of this vector with itself gives the square of the norm of this vector, it is proportional to the square of the rest mass Template:Math of the body:

$⟨𝐏,𝐏⟩=|𝐏{|}^2={\left(m_{0}c\right)}^2,$

a Lorentz invariant quantity, and therefore independent of the frame of reference. Using the Minkowski metric Template:Math with metric signature Template:Math, the inner product is

$⟨𝐏,𝐏⟩=|𝐏{|}^2=-{\left(m_{0}c\right)}^2,$

and

$⟨𝐏,𝐏⟩$ $=P^{\alpha }{\eta }_{\alpha \beta }{P}^{\beta }$ $=\left(\begin{array}{cccc}\frac{E}{c}& p_x& p_y& p_z\end{array}\right)\left(\begin{array}{cccc}-1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right)\left(\begin{array}{c}\frac{E}{c}\\ p_x\\ p_y\\ p_z\end{array}\right)$ $=-{\left(\frac{E}{c}\right)}^2+p^2,$

so

$-{\left(m_{0}c\right)}^2=-{\left(\frac{E}{c}\right)}^2+p^2$

or, in natural units where Template:Math = 1,

$|𝐏{|}^2+(m_0{)}^2=0.$

General relativity

In general relativity, the 4-momentum is a four-vector defined in a local coordinate frame, although by definition the inner product is similar to that of special relativity,

$⟨𝐏,𝐏⟩=|𝐏{|}^2={\left(m_{0}c\right)}^2,$

in which the Minkowski metric Template:Math is replaced by the metric tensor field Template:Math:

$⟨𝐏,𝐏⟩=|𝐏{|}^2=P^{\alpha }{g}_{\alpha \beta }{P}^{\beta },$

solved from the Einstein field equations. Then:

$P^{\alpha }{g}_{\alpha \beta }{P}^{\beta }={\left(m_{0}c\right)}^2.$

Units of energy, mass and momentum

In natural units where Template:Math, the energy–momentum equation reduces to

$E^2=p^2+m_0^2.$

In particle physics, energy is typically given in units of electron volts (eV), momentum in units of eV·Template:Math⁻¹, and mass in units of eV·Template:Math⁻². In electromagnetism, and because of relativistic invariance, it is useful to have the electric field Template:Math and the magnetic field Template:Math in the same unit (Gauss), using the cgs (Gaussian) system of units, where energy is given in units of erg, mass in grams (g), and momentum in g·cm·s⁻¹.

Energy may also in theory be expressed in units of grams, though in practice it requires a large amount of energy to be equivalent to masses in this range. For example, the first atomic bomb liberated about 1 gram of heat, and the largest thermonuclear bombs have generated a kilogram or more of heat. Energies of thermonuclear bombs are usually given in tens of kilotons and megatons referring to the energy liberated by exploding that amount of trinitrotoluene (TNT).

Special cases

Centre-of-momentum frame (one particle)

Template:See also For a body in its rest frame, the momentum is zero, so the equation simplifies to

$E_0=m_0{c}^2,$

where Template:Math is the rest mass of the body.

Massless particles

If the object is massless, as is the case for a photon, then the equation reduces to

$E=pc.$

This is a useful simplification. It can be rewritten in other ways using the de Broglie relations:

$E=\frac{hc}{\lambda }=\hslash ck.$

if the wavelength Template:Math or wavenumber Template:Math are given.

Correspondence principle

Rewriting the relation for massive particles as:

$E=m_0{c}^2\sqrt{1+{\left(\frac{p}{m_{0}c}\right)}^2},$

and expanding into power series by the binomial theorem (or a Taylor series):

$E=m_0{c}^2[1+\frac{1}{2}{\left(\frac{p}{m_{0}c}\right)}^2-\frac{1}{8}{\left(\frac{p}{m_{0}c}\right)}^4+\cdots ],$

in the limit that Template:Math, we have Template:Math so the momentum has the classical form Template:Math, then to first order in Template:Math (i.e. retain the term Template:Math for Template:Math and neglect all terms for Template:Math) we have

$E\approx m_0{c}^2[1+\frac{1}{2}{\left(\frac{m_{0}u}{m_{0}c}\right)}^2],$

or

$E\approx m_0{c}^2+\frac{1}{2}{m}_0{u}^2,$

where the second term is the classical kinetic energy, and the first is the rest energy of the particle. This approximation is not valid for massless particles, since the expansion required the division of momentum by mass. Incidentally, there are no massless particles in classical mechanics.

Many-particle systems

Addition of four momenta

In the case of many particles with relativistic momenta Template:Math and energy Template:Math, where Template:Math (up to the total number of particles) simply labels the particles, as measured in a particular frame, the four-momenta in this frame can be added;

$\sum _n{𝐏}_n=\sum _n(\frac{E_n}{c},{𝐩}_n)=(\sum _n\frac{E_n}{c},\sum _n{𝐩}_n),$

and then take the norm; to obtain the relation for a many particle system:

${\left|\left(\sum _n{𝐏}_n\right)\right|}^2={\left(\sum _n\frac{E_n}{c}\right)}^2-{\left(\sum _n{𝐩}_n\right)}^2={\left(M_{0}c\right)}^2,$

where Template:Math is the invariant mass of the whole system, and is not equal to the sum of the rest masses of the particles unless all particles are at rest (see Template:Slink for more detail). Substituting and rearranging gives the generalization of (Template:EquationNote);

The energies and momenta in the equation are all frame-dependent, while Template:Math is frame-independent.

Center-of-momentum frame

In the center-of-momentum frame (COM frame), by definition we have:

$\sum _n{𝐩}_n=0,$

with the implication from (Template:EquationNote) that the invariant mass is also the centre of momentum (COM) mass–energy, aside from the Template:Math factor:

${\left(\sum _n{E}_n\right)}^2={\left(M_0{c}^2\right)}^2\Rightarrow \sum _n{E}_{\mathrm{C}\mathrm{O}\mathrm{M}n}=E_{\mathrm{C}\mathrm{O}\mathrm{M}}=M_0{c}^2,$

and this is true for all frames since Template:Math is frame-independent. The energies Template:Math are those in the COM frame, not the lab frame. However, many familiar bound systems have the lab frame as COM frame, since the system itself is not in motion and so the momenta all cancel to zero. An example would be a simple object (where vibrational momenta of atoms cancel) or a container of gas where the container is at rest. In such systems, all the energies of the system are measured as mass. For example, the heat in an object on a scale, or the total of kinetic energies in a container of gas on the scale, all are measured by the scale as the mass of the system.

Rest masses and the invariant mass

Either the energies or momenta of the particles, as measured in some frame, can be eliminated using the energy momentum relation for each particle:

$E_n^2-{\left({𝐩}_{n}c\right)}^2={\left(m_n{c}^2\right)}^2,$

allowing Template:Math to be expressed in terms of the energies and rest masses, or momenta and rest masses. In a particular frame, the squares of sums can be rewritten as sums of squares (and products):

${\left(\sum _n{E}_n\right)}^2=\left(\sum _n{E}_n\right)\left(\sum _k{E}_k\right)=\sum _{n,k}{E}_n{E}_k=2\sum _{n<k}{E}_n{E}_k+\sum _n{E}_n^2,$

${\left(\sum _n{𝐩}_n\right)}^2=\left(\sum _n{𝐩}_n\right)\cdot \left(\sum _k{𝐩}_k\right)=\sum _{n,k}{𝐩}_n\cdot {𝐩}_k=2\sum _{n<k}{𝐩}_n\cdot {𝐩}_k+\sum _n{𝐩}_n^2,$

so substituting the sums, we can introduce their rest masses Template:Math in (Template:EquationNote):

$\sum _n{\left(m_n{c}^2\right)}^2+2\sum _{n<k}(E_n{E}_k-c^2{𝐩}_n\cdot {𝐩}_k)={\left(M_0{c}^2\right)}^2.$

The energies can be eliminated by:

$E_n=\sqrt{{\left({𝐩}_{n}c\right)}^2+{\left(m_n{c}^2\right)}^2},E_k=\sqrt{{\left({𝐩}_{k}c\right)}^2+{\left(m_k{c}^2\right)}^2},$

similarly the momenta can be eliminated by:

${𝐩}_n\cdot {𝐩}_k=\left|{𝐩}_n\right|\left|{𝐩}_k\right|\cos {\theta }_{nk},|{𝐩}_n|=\frac{1}{c}\sqrt{E_n^2-{\left(m_n{c}^2\right)}^2},|{𝐩}_k|=\frac{1}{c}\sqrt{E_k^2-{\left(m_k{c}^2\right)}^2},$

where Template:Math is the angle between the momentum vectors Template:Math and Template:Math.

Rearranging:

${\left(M_0{c}^2\right)}^2-\sum _n{\left(m_n{c}^2\right)}^2=2\sum _{n<k}(E_n{E}_k-c^2{𝐩}_n\cdot {𝐩}_k).$

Since the invariant mass of the system and the rest masses of each particle are frame-independent, the right hand side is also an invariant (even though the energies and momenta are all measured in a particular frame).

Matter waves

Using the de Broglie relations for energy and momentum for matter waves,

$E=\hslash \omega ,𝐩=\hslash 𝐤,$

where Template:Math is the angular frequency and Template:Math is the wavevector with magnitude Template:Math, equal to the wave number, the energy–momentum relation can be expressed in terms of wave quantities:

${\left(\hslash \omega \right)}^2={\left(c\hslash k\right)}^2+{\left(m_0{c}^2\right)}^2,$

and tidying up by dividing by Template:Math throughout: Template:NumBlk

This can also be derived from the magnitude of the four-wavevector

$𝐊=(\frac{\omega }{c},𝐤),$

in a similar way to the four-momentum above.

Since the reduced Planck constant Template:Math and the speed of light Template:Math both appear and clutter this equation, this is where natural units are especially helpful. Normalizing them so that Template:Math, we have:

${\omega }^2=k^2+m_0^2.$

Tachyon and exotic matter

Template:Main article The velocity of a bradyon with the relativistic energy–momentum relation

$E^2=p^2{c}^2+m_0^2{c}^4.$

can never exceed Template:Math. On the contrary, it is always greater than Template:Math for a tachyon whose energy–momentum equation is

$E^2=p^2{c}^2-m_0^2{c}^4.$

By contrast, the hypothetical exotic matter has a negative mass and the energy–momentum equation is

$E^2=-p^2{c}^2+m_0^2{c}^4.$

See also

• Mass–energy equivalence
• Four-momentum
• Mass in special relativity

References

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