View source for Physics:Energy–momentum relation

{{short description|Relativistic equation relating total energy to invariant mass and momentum}}
In [[Physics:Physics|physics]], the '''energy–momentum relation''', or '''relativistic dispersion relation''', is the [[Physics:Theory of relativity|relativistic]] [[Equation|equation]] relating total [[Physics:Energy|energy]] (which is also called '''relativistic energy''') to [[Physics:Invariant mass|invariant mass]] (which is also called rest mass) and [[Physics:Momentum|momentum]]. It is the extension of [[Mass–energy equivalence|mass–energy equivalence]] for bodies or systems with non-zero momentum. It can be written as the following equation:
{{NumBlk
 |:| {{Equation box 1
  | indent=
  | equation=<math>E^2 = (p \textrm c)^2 + \left(m_0 \textrm c^2\right)^2\,</math>
  | border = 1
  | border colour = white
  | background colour = white
 }}
 | {{EquationRef|1}}
}} 
This equation holds for a [[Philosophy:Physical object|body]] or [[Physics:Physical system|system]], such as one or more [[Physics:Particle|particle]]s, with total energy {{math|''E''}}, invariant mass {{math|''m''<sub>0</sub>}}, and momentum of [[Magnitude (mathematics)#Euclidean vector space|magnitude]] {{math|''p''}}; the constant {{math|''c''}} is the [[Physics:Speed of light|speed of light]]. It assumes the [[Physics:Special relativity|special relativity]] case of flat spacetime<ref name="KleppnerKolenkow1973" /><ref>{{cite book|title=Dynamics and Relativity|url=https://archive.org/details/dynamicsrelativi00fors|url-access=limited|author1=J.R. Forshaw |author2=A.G. Smith |publisher=Wiley|pages=[https://archive.org/details/dynamicsrelativi00fors/page/n163 149], 249|year=2009|isbn=978-0-470-01460-8}}</ref><ref>{{cite book|title = Relativity|url = https://archive.org/details/relativitydemyst00mcma_490|url-access = limited|author = D. McMahon|publisher = Mc Graw Hill (USA)|series=DeMystified|year = 2006|page=[https://archive.org/details/relativitydemyst00mcma_490/page/n363 20]|isbn = 0-07-145545-0}}</ref> and that the particles are free. Total energy is the sum of rest energy and [[Physics:Kinetic energy|kinetic energy]], while invariant mass is mass measured in a [[Physics:Center-of-momentum frame|center-of-momentum frame]]. 

For bodies or systems with zero momentum, it simplifies to the mass–energy equation <math> E = m_0 \textrm c^2</math>, where total energy in this case is equal to rest energy (also written as {{math|''E''<sub>0</sub>}}).

The [[Physics:Dirac sea|Dirac sea]] model, which was used to predict the existence of [[Physics:Antimatter|antimatter]], is closely related to the energy–momentum relation.

== Connection to <small>{{math|''E'' {{=}} ''mc''<sup>2</sup>}}</small> ==
thumb|Einstein Triangle

The energy–momentum relation is consistent with the familiar [[Mass–energy equivalence|mass–energy relation]] in both its interpretations: {{math|''E'' {{=}} ''mc''<sup>2</sup>}} relates total energy {{math|''E''}} to the (total) [[Physics:Relativistic mass|relativistic mass]] {{math|''m''}} (alternatively denoted {{math|''m''<sub>rel</sub>}} or {{math|''m''<sub>tot</sub>}} ), while {{math|''E''<sub>0</sub> {{=}} ''m''<sub>0</sub>''c''<sup>2</sup>}} relates rest energy {{math|''E''<sub>0</sub>}} to (invariant) rest mass {{math|''m''<sub>0</sub>}}. 

Unlike either of those equations, the energy–momentum equation ({{EquationNote|1}}) relates the ''total'' energy to the ''rest'' mass {{math|''m''<sub>0</sub>}}. All three equations hold true simultaneously.

== Special cases ==

# If the body is a [[Physics:Massless particle|massless particle]] ({{math|''m''<sub>0</sub> {{=}} 0}}), then ({{EquationNote|1}}) reduces to {{math|''E'' {{=}} ''pc''}}.  For [[Physics:Photon|photon]]s, this is the relation, discovered in 19th century [[Physics:Classical electromagnetism|classical electromagnetism]], between radiant momentum (causing [[Astronomy:Radiation pressure|radiation pressure]]) and [[Physics:Radiant energy|radiant energy]].
# If the body's speed {{math|''v''}} is much less than {{math|''c''}}, then ({{EquationNote|1}}) reduces to {{math|''E'' {{=}} {{sfrac|1|2}}''m<sub>0</sub>v''<sup>2</sup> + ''m''<sub>0</sub>''c''<sup>2</sup>}}; that is, the body's total energy is simply its classical [[Physics:Kinetic energy|kinetic energy]] ({{math|{{sfrac|1|2}}''m''<sub>0</sub>''v''<sup>2</sup>}}) plus its rest energy. 
# If the body is at rest ({{math|''v'' {{=}} 0}}), i.e. in its [[Physics:Center-of-momentum frame|center-of-momentum frame]] ({{math|''p'' {{=}} 0}}), we have {{math|''E'' {{=}} ''E''<sub>0</sub>}} and {{math|''m'' {{=}} ''m''<sub>0</sub>}}; 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 ({{EquationNote|1}}) holds for [[Physics:General relativity|general relativity]].

The [[Physics:Invariant mass|invariant mass (or rest mass)]] is an invariant for all [[Astronomy:Frame of reference|frames of reference]] (hence the name), not just in inertial frames in flat spacetime, but also [[Physics:Non-inertial reference frame|accelerated frames]] traveling through curved spacetime (see below). However the total energy of the particle {{math|''E''}} and its relativistic momentum {{math|'''p'''}} 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 {{math|''E''}} and {{math|'''p'''}}, while the other frame measures {{math|''{{prime|E}}''}} and {{math|'''{{prime|p}}'''}}, where {{math|''{{prime|E}}'' ≠ ''E''}} and {{math|'''{{prime|p}}''' ≠ '''p'''}}, 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:
:<math>{E'}^2 - \left(p'c\right)^2 = \left(m_0 c^2\right)^2\,.</math>

The quantities {{math|''E''}}, {{math|'''p'''}}, {{math|''{{prime|E}}''}}, {{math|'''{{prime|p}}'''}} are all related by a [[Physics:Lorentz transformation|Lorentz transformation]]. The relation allows one to sidestep Lorentz transformations when determining only the [[Norm (mathematics)|magnitudes]] of the energy and momenta by equating the relations in the different frames. Again in flat spacetime, this translates to;

:<math>{E}^2 - \left(pc\right)^2 = {E'}^2 - \left(p'c\right)^2 = \left(m_0 c^2\right)^2\,.</math>

Since {{math|''m''<sub>0</sub>}} does not change from frame to frame, the energy–momentum relation is used in [[Physics:Relativistic mechanics|relativistic mechanics]] and [[Physics:Particle physics|particle physics]] calculations, as energy and momentum are given in a particle's rest frame (that is, {{math|''{{prime|E}}''}} and {{math|'''{{prime|p}}'''}} as an observer moving with the particle would conclude to be) and measured in the lab frame (i.e. {{math|''E''}} and {{math|'''p'''}} as determined by particle physicists in a lab, and not moving with the particles).

In [[Physics:Relativistic quantum mechanics|relativistic quantum mechanics]], it is the basis for constructing [[Physics:Relativistic wave equations|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.<ref>{{cite book|title = Quantum Field Theory|url = https://archive.org/details/quantumfieldtheo00mcma_095|url-access = limited|author = D. McMahon|publisher = Mc Graw Hill (USA)|series=DeMystified|year = 2008|pages=[https://archive.org/details/quantumfieldtheo00mcma_095/page/n29 11], 88|isbn = 978-0-07-154382-8}}
</ref>

==Origins and derivation of the equation ==

The energy–momentum relation goes back to [[Biography:Max Planck|Max Planck]]'s article<ref>{{cite journal|author=Planck, Max|year=1906
|title=Das Prinzip der Relativität und die Grundgleichungen der Mechanik|journal=Verhandlungen der Deutschen Physikalischen Gesellschaft|volume=8|number=7|pages=136–141|url=https://archive.org/details/verhandlungende00goog/page/n179}}
</ref>
published in 1906.
It was used by [[Biography:Walter Gordon (physicist)|Walter Gordon]] in 1926 and then by [[Biography:Paul Dirac|Paul Dirac]] in 1928 under the form <math display="inline">E=\sqrt{c^2p^2+(m_0c^2)^2} + V</math>, where ''V'' is the amount of potential energy.<ref>{{cite journal|author=Gordon, Walter|year=1926|title=The Compton effect according to Schrödinger's theory|journal=Z. Phys.|volume=40|pages=117–133|doi=10.1007/BF01390840|s2cid=122254400 }}</ref><ref>{{cite journal|author=Dirac, Paul|year=1928|title=The Quantum Theory of the Electron|journal=Proc. Roy. Soc. London Ser. A|volume=117|number=778|pages=610–624|doi=10.1098/rspa.1928.0023|doi-access=free|bibcode=1928RSPSA.117..610D }}</ref>

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

# From the relativistic dynamics of a massive particle,
# By evaluating the norm of the [[Physics:Four-momentum|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 {{section link||Many-particle systems}} below).

===Heuristic approach for massive particles===

For a massive object moving at three-velocity {{math|'''u''' {{=}} (''u<sub>x</sub>'', ''u<sub>y</sub>'', ''u<sub>z</sub>'')}} with magnitude {{math|{{abs|'''u'''}} {{=}} ''u''}} in the lab frame:<ref name="KleppnerKolenkow1973">{{cite book |last1= Kleppner |first1=Daniel |author2=Robert J. Kolenkow |title=An Introduction to Mechanics |url= https://archive.org/details/introductiontome00klep_881 |url-access= limited |date=2010 |publisher=Cambridge University Press |isbn=978-0-521-19821-9 |pages=[https://archive.org/details/introductiontome00klep_881/page/n522 499]–500 |orig-year=1973 }}</ref>
:<math>E=\gamma_{(\mathbf{u})}m_0c^2</math>
is the total energy of the moving object in the lab frame,
:<math>\mathbf{p} = \gamma_{(\mathbf{u})} m_0\mathbf{u}</math>
is the three dimensional relativistic momentum of the object in the lab frame with magnitude {{math|{{abs|'''p'''}} {{=}} ''p''}}. The relativistic energy {{math|''E''}} and momentum {{math|'''p'''}} include the [[Lorentz factor]] defined by:
:<math>\gamma_{(\mathbf{u})} = \frac{1}{\sqrt{1 - \frac{\mathbf{u} \cdot \mathbf{u}}{c^2}}} = \frac{1}{\sqrt{1 - \left(\frac{u}{c}\right)^2}} </math>

Some authors use [[Physics:Relativistic mass|relativistic mass]] defined by:
:<math>m = \gamma_{(\mathbf{u})} m_0</math>
although rest mass {{math|''m''<sub>0</sub>}} has a more fundamental significance, and will be used primarily over relativistic mass {{math|''m''}} in this article.

Squaring the 3-momentum gives:
: <math>p^2 = \mathbf{p}\cdot\mathbf{p} = \frac{m_0^2 \mathbf{u}\cdot \mathbf{u}}{1 - \frac{\mathbf{u} \cdot \mathbf{u}}{c^2}} = \frac{m_0^2 u^2}{1 - \left(\frac{u}{c}\right)^2}</math>

then solving for {{math|''u''{{sup|2}}}} and substituting into the Lorentz factor one obtains its alternative form in terms of 3-momentum and mass, rather than 3-velocity:
:<math>\gamma = \sqrt{1 + \left(\frac{p}{m_0 c}\right)^2}</math>

Inserting this form of the Lorentz factor into the energy equation gives:
:<math>E = m_0c^2\sqrt{1 + \left(\frac{p}{m_0 c}\right)^2}</math>

followed by more rearrangement it yields ({{EquationRef|1}}). The elimination of the Lorentz factor also eliminates implicit velocity dependence of the particle in ({{EquationRef|1}}), 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 {{math|''m''<sub>0</sub> {{=}} 0}} would mean that {{math|''E'' {{=}} 0}} and {{math|'''p''' {{=}} '''0'''}} and no energy–momentum relation could be derived, which is not correct.

===Norm of the four-momentum===

[[File:Energy momentum space cropped.svg|250px|right|thumb|The energy and momentum of an object measured in two inertial frames in energy–momentum space – the yellow frame measures {{math|''E''}} and {{math|'''p'''}} while the blue frame measures {{math|''{{prime|E}}''}} and {{math|'''{{prime|p}}'''}}. The green arrow is the four-momentum {{math|'''P'''}} of an object with length proportional to its rest mass {{math|''m''<sub>0</sub>}}. 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 {{math|''Φ''}} and {{nowrap|''Φ'' + ''η''}} are the rapidities of the blue and green frames, respectively.]]

====Special relativity====

{{main article|Physics:Special relativity}}

In [[Physics:Minkowski space|Minkowski space]], energy (divided by {{math|''c''}}) and momentum are two components of a Minkowski [[Physics:Four-vector|four-vector]], namely the [[Physics:Four-momentum|four-momentum]];<ref>{{cite book|title=Dynamics and Relativity|url=https://archive.org/details/dynamicsrelativi00fors|url-access=limited|author1=J.R. Forshaw |author2=A.G. Smith |publisher=Wiley|pages=[https://archive.org/details/dynamicsrelativi00fors/page/n272 258]–259|year=2009|isbn=978-0-470-01460-8}}</ref>
:<math>\mathbf{P} = \left(\frac{E}{c}, \mathbf{p}\right)\,,</math>

(these are the [[Covariance and contravariance of vectors|contravariant]] components).

The [[Physics:Minkowski space#The Minkowski inner product|Minkowski inner product]] {{math|{{langle}} , {{rangle}}}} of this vector with itself gives the square of the [[Norm (mathematics)|norm]] of this vector, it is [[Proportionality (mathematics)|proportional]] to the square of the rest mass {{math|''m''}} of the body:

:<math>\left\langle\mathbf{P}, \mathbf{P}\right\rangle = |\mathbf{P}|^2 = \left(m_0 c\right)^2\,,</math>

a Lorentz [[Physics:Invariant|invariant]] quantity, and therefore independent of the [[Astronomy:Frame of reference|frame of reference]]. Using the [[Physics:Minkowski space|Minkowski metric]] {{math|''η''}} with [[Metric signature|metric signature]] {{math|(− + + +)}}, the inner product is

:<math>\left\langle\mathbf{P},\mathbf{P}\right\rangle = |\mathbf{P}|^2 = - \left(m_0 c\right)^2\,,</math>

and

:<math>
  \left\langle\mathbf{P},\mathbf{P}\right\rangle =
  P^\alpha\eta_{\alpha\beta}P^\beta =
  \begin{pmatrix}
    \frac{E}{c} & p_x & p_y & p_z
  \end{pmatrix}
  \begin{pmatrix}
    -1 & 0 & 0 & 0 \\
     0 & 1 & 0 & 0 \\
     0 & 0 & 1 & 0 \\
     0 & 0 & 0 & 1 \\
  \end{pmatrix}
  \begin{pmatrix}
    \frac{E}{c} \\ p_x \\ p_y \\ p_z 
  \end{pmatrix} =
  -\left(\frac{E}{c}\right)^2 + p^2\,,
</math>

so
:<math>-\left(m_0 c\right)^2 = -\left(\frac{E}{c}\right)^2 + p^2</math>
:
or, in natural units where {{math|''c''}} = 1,
:<math>|\mathbf{P}|^2 + (m_0)^2=0</math>.
:

====General relativity====

In [[Physics:General relativity|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,

:<math>\left\langle\mathbf{P},\mathbf{P}\right\rangle = |\mathbf{P}|^2 = \left(m_0 c\right)^2\,,</math>

in which the Minkowski metric {{math|''η''}} is replaced by the [[Physics:Metric tensor|metric]] [[Tensor field|tensor field]] {{math|''g''}}:

:<math>\left\langle\mathbf{P}, \mathbf{P}\right\rangle = |\mathbf{P}|^2 = P^\alpha g_{\alpha\beta}P^\beta \,,</math>

solved from the [[Physics:Einstein field equations|Einstein field equations]]. Then:<ref>{{cite book|title=Gravitation|url=https://archive.org/details/gravitation00misn_003|url-access=limited|author1=J.A. Wheeler |author2=C. Misner |author3=K.S. Thorne |publisher=W.H. Freeman & Co|pages=[https://archive.org/details/gravitation00misn_003/page/n225 201], 649, 1188|year=1973|isbn=0-7167-0344-0}}</ref>

:<math>P^\alpha g_{\alpha\beta}P^\beta = \left(m_0 c\right)^2\,.</math>

Performing the summations over indices followed by collecting "time-like", "spacetime-like", and "space-like" terms gives:

:<math>\underbrace{g_{00}{\left(P^0\right)}^2}_{\text{time-like}} + 2 \underbrace{g_{0i}P^0 P^i}_{\text{spacetime-like}} + \underbrace{g_{ij}P^i P^j}_{\text{space-like}} = \left(m_0 c\right)^2\,.</math>

where the factor of 2 arises because the metric is a [[Symmetric tensor|symmetric tensor]], and the convention of Latin indices {{math|''i'', ''j''}} taking space-like values 1, 2, 3 is used. As each component of the metric has space and time dependence in general; this is significantly more complicated than the formula quoted at the beginning, see [[Physics:Metric tensor (general relativity)|metric tensor (general relativity)]] for more information.

==Units of energy, mass and momentum==

In [[Natural units|natural units]] where {{math|''c'' {{=}} 1}}, the energy–momentum equation reduces to
:<math>E^2 = p^2 + m_0^2 \,.</math>

In [[Physics:Particle physics|particle physics]], energy is typically given in units of electron volts (eV), momentum in units of eV·{{math|''c''}}<sup>−1</sup>, and mass in units of eV·{{math|''c''}}<sup>−2</sup>. In [[Physics:Electromagnetism|electromagnetism]], and because of relativistic invariance, it is useful to have the [[Physics:Electric field|electric field]] {{math|'''E'''}} and the [[Physics:Magnetic field|magnetic field]] {{math|'''B'''}} in the same unit ([[Physics:Gauss (unit)|Gauss]]), using the [[Physics:Gaussian units|cgs (Gaussian) system of units]], where energy is given in units of [[Physics:Erg|erg]], mass in [[Gram|gram]]s (g), and momentum in g·cm·s<sup>−1</sup>.

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&nbsp;gram of [[Physics:Heat|heat]], and the largest thermonuclear bombs have generated a [[Kilogram|kilogram]] or more of heat. Energies of thermonuclear bombs are usually given in tens of [[Physics:TNT equivalent|kiloton]]s and megatons referring to the energy liberated by exploding that amount of trinitrotoluene (TNT).

==Special cases==

===Centre-of-momentum frame (one particle)===

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

:<math> E_0 = m_0 c^2 \,,</math>

where {{math|''m''<sub>0</sub>}} is the rest mass of the body.

===Massless particles===

If the object is massless, as is the case for a [[Physics:Photon|photon]], then the equation reduces to

:<math> E = pc \,.</math>

This is a useful simplification. It can be rewritten in other ways using the [[Physics:Matter wave|de Broglie relation]]s:

:<math> E = \frac{hc}{\lambda} = \hbar c k \,.</math>

if the [[Wavelength|wavelength]] {{math|''λ''}} or [[Physics:Wavenumber|wavenumber]] {{math|''k''}} are given.

===Correspondence principle===

Rewriting the relation for massive particles as:

:<math>E = m_0 c^2 \sqrt{1 + \left(\frac{p}{m_0c}\right)^2}\,,</math>

and expanding into [[Power series|power series]] by the [[Binomial theorem|binomial theorem]] (or a [[Taylor series]]):

:<math>E = m_0 c^2 \left[1 + \frac{1}{2}\left(\frac{p}{m_0c}\right)^2 - \frac{1}{8}\left(\frac{p}{m_0c}\right)^4 + \cdots \right]\,,</math>

in the limit that {{math|''u'' ≪ ''c''}}, we have {{math|''γ''(''u'') ≈ 1}} so the momentum has the classical form {{math|''p'' ≈ ''m''<sub>0</sub>''u''}}, then to first order in {{math|{{big|{{big|(}}}}{{sfrac|''p''|''m''<sub>0</sub>''c''}}{{big|{{big|)}}}}{{su|p=2}}}} (i.e. retain the term {{math|{{big|{{big|(}}}}{{sfrac|''p''|''m''<sub>0</sub>''c''}}{{big|{{big|)}}}}{{su|p=2''n''}}}} for {{math|''n'' {{=}} 1}} and neglect all terms for {{math|''n'' ≥ 2}}) we have

:<math>E \approx m_0 c^2 \left[1 + \frac{1}{2}\left(\frac{m_0u}{m_0c}\right)^2 \right]\,,</math>

or

:<math>E \approx m_0 c^2 + \frac{1}{2} m_0 u^2 \,,</math>

where the second term is the classical [[Physics:Kinetic energy|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 {{math|'''p'''<sub>''n''</sub>}} and energy {{math|''E<sub>n</sub>''}}, where {{math|''n'' {{=}} 1, 2, ...}} (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;

:<math>\sum_n \mathbf{P}_n = \sum_n \left(\frac{E_n}{c} , \mathbf{p}_n\right) = \left( \sum_n \frac{E_n}{c}, \sum_n \mathbf{p}_n \right)\,,</math>

and then take the norm; to obtain the relation for a many particle system:
:<math>\left|\left(\sum_n \mathbf{P}_n \right)\right|^2 = \left(\sum_n \frac{E_n}{c}\right)^2 - \left(\sum_n \mathbf{p}_n\right)^2 = \left(M_0 c\right)^2\,,</math>

where {{math|''M''<sub>0</sub>}} 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 [[Physics:Mass in special relativity#The mass of composite systems|mass in special relativity]] for more detail). Substituting and rearranging gives the generalization of ({{EquationNote|1}});

{{NumBlk
 |:| {{Equation box 1
  | title=
  | indent =
  | equation = <math>\left(\sum_n E_n \right)^2 = \left(\sum_n \mathbf{p}_n c\right)^2  + \left(M_0 c^2\right)^2</math>
  | cellpadding
  | border = 1
  | border colour = black
  | background colour = white
 }}
 | {{EquationRef|2}}
}}

The energies and momenta in the equation are all frame-dependent, while {{math|''M''<sub>0</sub>}} is frame-independent.

===Center-of-momentum frame===

In the [[Physics:Center-of-momentum frame|center-of-momentum frame]] (COM frame), by definition we have:

:<math>\sum_n \mathbf{p}_n = \boldsymbol{0}\,,</math>

with the implication from ({{EquationNote|2}}) that the invariant mass is also the centre of momentum (COM) mass–energy, aside from the {{math|''c''<sup>2</sup>}} factor:

:<math>\left(\sum_n E_n \right)^2 = \left(M_0 c^2\right)^2 \Rightarrow \sum_n E_{\mathrm{COM}\,n} = E_\mathrm{COM} = M_0 c^2 \,,</math>

and this is true for ''all'' frames since {{math|''M''<sub>0</sub>}} is frame-independent. The energies {{math|''E''<sub>COM ''n''</sub>}} 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:
:<math>E^2_n - \left(\mathbf{p}_n c\right)^2 = \left(m_n c^2\right)^2 \,,</math>

allowing {{math|''M''<sub>0</sub>}} 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):
:<math>
   \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\,,
</math>

:<math>
  \left(\sum_n \mathbf{p}_n \right)^2 =
  \left(\sum_n \mathbf{p}_n \right)\cdot\left(\sum_k \mathbf{p}_k \right) =
    \sum_{n,k} \mathbf{p}_n \cdot \mathbf{p}_k =
   2\sum_{n<k} \mathbf{p}_n \cdot \mathbf{p}_k + \sum_{n}\mathbf{p}_n^2\,,
</math>

so substituting the sums, we can introduce their rest masses {{math|''m<sub>n</sub>''}} in ({{EquationNote|2}}):

:<math> \sum_n \left(m_n c^2\right)^2 + 2\sum_{n<k}\left(E_n E_k - c^2 \mathbf{p}_n \cdot \mathbf{p}_k\right) = \left(M_0 c^2\right)^2 \,.</math>

The energies can be eliminated by:

:<math>E_n = \sqrt{\left(\mathbf{p}_n c\right)^2 + \left(m_n c^2\right)^2} \,,\quad E_k = \sqrt{\left(\mathbf{p}_k c\right)^2 + \left(m_k c^2\right)^2} \,,</math>

similarly the momenta can be eliminated by:

:<math>
  \mathbf{p}_n \cdot \mathbf{p}_k =
  \left|\mathbf{p}_n\right|\left|\mathbf{p}_k\right| \cos\theta_{nk}\,,\quad |\mathbf{p}_n| =
  \frac{1}{c}\sqrt{E_n^2 - \left(m_n c^2\right)^2}\,,\quad |\mathbf{p}_k| =
  \frac{1}{c}\sqrt{E_k^2 - \left(m_k c^2\right)^2}\,,
</math>

where {{math|''θ<sub>nk</sub>''}} is the angle between the momentum vectors {{math|'''p'''<sub>''n''</sub>}} and {{math|'''p'''<sub>''k''</sub>}}.

Rearranging:
:<math> \left(M_0 c^2\right)^2 -\sum_n \left(m_n c^2\right)^2 = 2\sum_{n<k}\left(E_n E_k - c^2 \mathbf{p}_n \cdot \mathbf{p}_k\right) \,.</math>

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 [[Physics:Matter wave|matter wave]]s,
:<math>E = \hbar \omega \,, \quad \mathbf{p} = \hbar\mathbf{k}\,,</math>

where {{math|''ω''}} is the [[Physics:Angular frequency|angular frequency]] and {{math|'''k'''}} is the [[Physics:Wavevector|wavevector]] with magnitude {{math|{{abs|'''k'''}} {{=}} ''k''}}, equal to the wave number, the energy–momentum relation can be expressed in terms of wave quantities:
:<math>\left(\hbar\omega\right)^2 = \left(c \hbar k\right)^2 + \left(m_0 c^2\right)^2 \,,</math>

and tidying up by dividing by {{math|(''ħc'')<sup>2</sup>}} throughout:
{{NumBlk
 |:| {{Equation box 1
 | indent=
 | equation=<math> \left(\frac{\omega}{c}\right)^2 = k^2 + \left(\frac{m_0c}{\hbar}\right)^2 \,.</math>
 |border = 1
 |border colour = black
 |background colour = white
 }}
 | {{EquationRef|3}}
}}

This can also be derived from the magnitude of the [[Physics:Four-vector#Four-wavevector|four-wavevector]]
:<math>\mathbf{K} = \left(\frac{\omega}{c}, \mathbf{k}\right)\,,</math>

in a similar way to the four-momentum above.

Since the reduced Planck constant {{math|''ħ''}} and the [[Physics:Speed of light|speed of light]] {{math|''c''}} both appear and clutter this equation, this is where [[Natural units|natural units]] are especially helpful. Normalizing them so that {{math|''ħ'' {{=}} ''c'' {{=}} 1}}, we have:
:<math>\omega^2 = k^2  + m_0^2 \,.</math>

==Tachyon and exotic matter==
{{main article|Physics:Tachyon|Physics:Exotic matter}}
The velocity of a bradyon with the relativistic energy–momentum relation
:<math>E^2 = p^2 c^2 + m_0^2 c^4\,.</math>

can never exceed {{math|''c''}}. On the contrary, it is always greater than  {{math|''c''}} for a  [[Physics:Tachyon|tachyon]] whose energy–momentum equation is<ref>{{cite journal|author=G. Feinberg|title=Possibility of faster-than-light particles|journal=Physical Review|volume=159|pages=1089–1105|date=1967|issue=5|doi=10.1103/PhysRev.159.1089|bibcode = 1967PhRv..159.1089F }}</ref>
:<math>E^2 = p^2 c^2 - m_0^2 c^4\,.</math>

By contrast, the hypothetical [[Physics:Exotic matter|exotic matter]] has a [[Physics:Negative mass|negative mass]]<ref>{{cite journal| author=Z.Y.Wang| title=Modern Theory for Electromagnetic Metamaterials| journal=Plasmonics| date=2016| volume=11| pages=503–508| doi=10.1007/s11468-015-0071-7| issue=2| s2cid=122346519}}</ref>  and the energy–momentum equation is
:<math>E^2 = -p^2 c^2 + m_0^2 c^4\,.</math>

==See also==

* [[Mass–energy equivalence]]
* [[Physics:Four-momentum|Four-momentum]]
* [[Physics:Mass in special relativity|Mass in special relativity]]

==References==

{{reflist}}

* {{cite book| author=A. Halpern| title=3000 Solved Problems in Physics, Schaum Series| publisher=McGraw-Hill|pages=704–705| year=1988| isbn=978-0-07-025734-4}}
* {{cite book| author=G. Woan| title=The Cambridge Handbook of Physics Formulas| url=https://archive.org/details/cambridgehandboo0000woan| url-access=registration| publisher=Cambridge University Press|page=[https://archive.org/details/cambridgehandboo0000woan/page/65 65]| year=2010| isbn=978-0-521-57507-2}}
* {{cite book| author=C.B. Parker| title=McGraw-Hill Encyclopaedia of Physics| publisher=McGraw-Hill| edition=2nd| pages=[https://archive.org/details/mcgrawhillencycl1993park/page/1192 1192, 1193]| year=1994| isbn=0-07-051400-3| url=https://archive.org/details/mcgrawhillencycl1993park/page/1192}}
* {{cite book| author1=R.G. Lerner| author2=G.L. Trigg| title=Encyclopaedia of Physics| publisher=VHC Publishers| edition=2nd| page=[https://archive.org/details/encyclopediaofph00lern/page/1052 1052]| year=1991| isbn=0-89573-752-3| url=https://archive.org/details/encyclopediaofph00lern/page/1052}}

{{DEFAULTSORT:Energy-Momentum Relation}}

[[Category:Momentum]]
[[Category:Special relativity]]

{{Sourceattribution|Energy–momentum relation}}