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| In [[physics]], the '''energy–momentum relation''' is the [[theory of relativity|relativistic]] [[equation]] relating any object's [[Invariant mass|rest (intrinsic) mass]], total [[energy]], and [[momentum]]:
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| {{Equation box 1|indent=|equation=<math>E^2 = (pc)^2 + (m_0c^2)^2\,</math>|border = 1|border colour = black|background colour = white}}
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| |{{EquationRef|1}}}}
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| holds for a [[Physical system|system]], such as a [[particle]] or [[macroscopic]] [[Physical body|body]], having intrinsic rest mass {{math|''m''<sub>0</sub>}}, total energy {{math|''E''}}, and a momentum of [[Magnitude (mathematics)#Euclidean vectors|magnitude]] {{math|''p''}}, where the constant ''c'' is the [[speed of light]], assuming the [[special relativity|special]] relativity case of [[flat spacetime]].<ref>{{cite book|title = An Introduction to Mechanics|author = D. Kleppner, R.J. Kolenkov|publisher = Cambridge University Press|year = 2010|page=500|isbn = 9-780521-198219}}</ref><ref>{{cite book|title=Dynamics and Relativity|author=J.R. Forshaw, A.G. Smith|publisher=Wiley|pages=149, 249|year=2009|isbn=978-0-470-01460-8}}</ref><ref>{{cite book|title = Relativity|author = D. McMahon|publisher = Mc Graw Hill (USA)|series=DeMystified|year = 2006|page=20|isbn = 0-07-145545-0}}</ref>
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| The energy-momentum relation ({{EquationNote|1}}) 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) [[relativistic mass]] {{math|''m''}} (alternatively denoted {{math|''m''<sub>rel</sub>}} or {{math|''m''<sub>tot</sub>}} ), while {{math|''E''<sub>0</sup> {{=}} ''m''<sub>0</sub>''c''<sup>2</sup>}} relates [[rest energy]] {{math|''E''<sub>0</sup>}} to rest (invariant) mass which we denote {{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.
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| Special cases of the relation ({{EquationNote|1}}) include:
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| #If the body is a [[massless particle]] ({{math|''m''<sub>0</sub> {{=}} 0}}), then ({{EquationNote|1}}) reduces to {{math|''E'' {{=}} ''pc''}}. For [[photon]]s, this is the relation, discovered in 19th century [[classical electromagnetism]], between radiant momentum (causing [[radiation pressure]]) and [[radiant energy]].
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| #If the body's speed {{math|''v''}} is much less than {{math|''c''}}, then ({{EquationNote|1}}) reduces to {{math|''E'' {{=}} ''m<sub>0</sub>v''<sup>2</sup>/2 + ''m''<sub>0</sub>''c''<sup>2</sup>}}; that is, the body's total energy is simply its classical [[kinetic energy]] ({{math|''m''<sub>0</sub>''v''<sup>2</sup>/2}}) plus its rest energy.
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| #If the body is at rest ({{math|''v'' {{=}} 0}}), i.e. in its [[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.
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| A more [[#General relativity|general form]] of relation ({{EquationNote|1}}) holds for [[general relativity]].
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| The [[invariant mass|invariant mass (or rest mass)]] is an invariant for all [[frame of reference|frames of reference]] (hence the name), not just in [[inertial frame]]s in flat spacetime, but also [[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|''E′''}} and {{math|'''p′'''}}, where {{math|''E′'' ≠ ''E''}} and {{math|'''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;
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| :<math>{E'}^2 - (p'c)^2 = (m_0c^2)^2\,.</math>
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| The quantities {{math|''E''}}, {{math|'''p'''}}, {{math|''E′''}}, {{math|'''p′'''}} are all related by a [[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;
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| :<math>{E}^2 - (pc)^2 = {E'}^2 - (p'c)^2 = (m_0c^2)^2\,.</math>
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| Since {{math|''m''<sub>0</sub>}} 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, {{math|''E′''}} and {{math|'''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).
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| 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.<ref>{{cite book|title = Quantum Field Theory|author = D. McMahon|publisher = Mc Graw Hill (USA)|series=DeMystified|year = 2008|page=11, 88|isbn = 978-0-07-154382-8}}
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| </ref>
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| This article will use the conventional notation for the "square of a vector" as the [[dot product]] of a vector with itself: {{math|'''p'''<sup>2</sup> {{=}} '''p''' · '''p''' {{=}} {{!}}'''p'''{{!}}<sup>2</sup>}}.
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| ==Origins of the equation==
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| The equation can be derived in a number of ways, two of the simplest include:
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| #considering the relativistic dynamics of a massive particle,
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| #evaluating the norm of the four-momentum of the system. This is completely general for all particles, and is easy to extend to multi-particle systems ([[#Many-particle systems|see below]]).
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| ===Heuristic approach for massive particles===
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| 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|{{!}}'''u'''{{!}} {{=}} ''u''}} in the [[lab frame]]:<ref>{{cite book|title = An Introduction to Mechanics|author = D. Kleppner, R.J. Kolenkov|publisher = Cambridge University Press|year = 2010|page=499-500|isbn = 9-780521-198219}}</ref>
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| :<math>E=\gamma(\mathbf{u})m_0c^2</math>
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| is the total energy of the moving object in the lab frame,
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| :<math>\mathbf{p}=\gamma(\mathbf{u})m_0\mathbf{u}</math>
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| is the three dimensional [[relativistic momentum]] of the object in the lab frame with magnitude {{math|{{!}}'''p'''{{!}} {{=}} ''p''}}. The relativistic energy {{math|''E''}} and momentum {{math|'''p'''}} include the [[Lorentz factor]] defined by:
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| :<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>
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| Some authors use [[relativistic mass]] defined by:
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| :<math>m=\gamma(\mathbf{u})m_0</math>
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| 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.
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| Squaring the 3-momentum gives:
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| :<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>
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| then solving for {{math|(''u''/''c'')<sup>2</sup>}} and substituting into the Lorentz factor obtains its alternative form in terms of 3-momentum and mass, rather than 3-velocity:
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| :<math>\gamma = \sqrt{1 + \left(\frac{p}{m_0 c}\right)^2}</math>
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| Inserting this form of the Lorentz factor into the energy equation:
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| :<math>E = m_0c^2\sqrt{1 + \left(\frac{p}{m_0 c}\right)^2}</math>
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| followed by more rearrangement 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.
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| ===Norm of the four-momentum===
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| [[File:Energy momentum space cropped.svg|250px|right|thumb|The energy and momentum of an object measured in two [[inertial frame]]s in energy-momentum space - the yellow frame measures {{math|''E''}} and {{math|'''p'''}} while the blue frame measures ''E′'' and {{math|'''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 ''ϕ'' and {{nowrap|''ϕ'' + ''η''}} are the [[rapidity|rapidities]] of the blue and green frames, respectively.]]
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| ====Special relativity====
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| {{main|Special relativity}}
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| In [[Minkowski space]], energy (divided by ''c'') and momentum are two components of a Minkowski [[four-vector]], namely the [[four-momentum]];<ref>{{cite book|title=Dynamics and Relativity|author=J.R. Forshaw, A.G. Smith|publisher=Wiley|pages=258–259|year=2009|isbn=978-0-470-01460-8}}</ref>
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| :<math>\mathbf{P}=(E/c,\mathbf{p})\,,</math> | |
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| (these are the [[Covariance and contravariance of vectors|contravariant]] components).
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| The [[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:
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| :<math>\left\langle\mathbf{P},\mathbf{P}\right\rangle = |\mathbf{P}|^2 = (m_0 c)^2\,,</math>
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| a [[Lorentz]] [[Invariant (physics)|invariant]] quantity, and therefore independent of the [[frame of reference]]. Using the [[Minkowski space|Minkowski metric]] {{math|''η''}} with [[metric signature]] {{math|(+−−−)}}, the inner product in [[tensor index notation]] and as [[matrix multiplication]] can be calculated as:
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| :<math>\left\langle\mathbf{P},\mathbf{P}\right\rangle = P^\alpha\eta_{\alpha\beta}P^\beta
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| = \begin{pmatrix}
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| E/c & p_x & p_y & p_z
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| \end{pmatrix}
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| \begin{pmatrix}
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| 1 & 0 & 0 & 0 \\
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| 0 & -1 & 0 & 0 \\
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| 0 & 0 & -1 & 0 \\
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| 0 & 0 & 0 & -1 \\
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| \end{pmatrix}
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| \begin{pmatrix}
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| E/c \\ p_x \\ p_y \\ p_z
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| \end{pmatrix}
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| = \left(\frac{E}{c}\right)^2 - p^2\,,</math>
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| and so:
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| :<math>(m_0 c)^2 = \left(\frac{E}{c}\right)^2 - p^2\,,</math>
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| which is the energy–momentum relation. If we had the other metric signature {{math|(−+++)}} for {{math|''η''}}, the inner product would be
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| :<math>\left\langle\mathbf{P},\mathbf{P}\right\rangle = |\mathbf{P}|^2 = - (m_0 c)^2\,,</math>
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| and
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| :<math>\left\langle\mathbf{P},\mathbf{P}\right\rangle = P^\alpha\eta_{\alpha\beta}P^\beta
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| = \begin{pmatrix}
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| E/c & p_x & p_y & p_z
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| \end{pmatrix}
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| \begin{pmatrix}
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| -1 & 0 & 0 & 0 \\ | |
| 0 & 1 & 0 & 0 \\
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| 0 & 0 & 1 & 0 \\
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| 0 & 0 & 0 & 1 \\
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| \end{pmatrix}
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| \begin{pmatrix}
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| E/c \\ p_x \\ p_y \\ p_z
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| \end{pmatrix}
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| = -\left(\frac{E}{c}\right)^2 + p^2\,,</math>
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| so:
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| :<math>-(m_0 c)^2 = -\left(\frac{E}{c}\right)^2 + p^2\,,</math>
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| which is still the same equation, as it should be because the inner product is an invariant.
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| ====General relativity====
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| 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,
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| :<math>\left\langle\mathbf{P},\mathbf{P}\right\rangle = |\mathbf{P}|^2 = (m_0 c)^2\,,</math>
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| in which the Minkowski metric {{math|''η''}} is replaced by the [[metric tensor|metric]] [[tensor field]] ''g'':
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| :<math>\left\langle\mathbf{P},\mathbf{P}\right\rangle = |\mathbf{P}|^2 = P^\alpha g_{\alpha\beta}P^\beta \,,</math>
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| solved from the [[Einstein field equations]]. Then:<ref>{{cite book|title=Gravitation|author=J.A. Wheeler, C. Misner, K.S. Thorne|publisher=W.H. Freeman & Co|page=201, 649, 1188|year=1973|isbn=0-7167-0344-0}}</ref>
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| :<math>P^\alpha g_{\alpha\beta}P^\beta = (m_0 c)^2\,.</math>
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| Performing the summations over indices followed by collecting "timelike", "spacetime-like", and "spacelike" terms gives:
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| :<math> \underbrace{g_{00}{(P^0)}^2}_{\text{timelike}} + 2 \underbrace{g_{0i}P^0 P^i}_{\text{spacetime-like}} + \underbrace{g_{ij}P^i P^j}_{\text{spacelike}} = (m_0 c)^2\,.</math>
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| where the factor of 2 arises because the metric is a [[symmetric tensor]], and the convention of Latin indices ''i'', ''j'' taking spacelike 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 [[metric tensor (general relativity)]] for more information.
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| ==Units of energy, mass and momentum==
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| In [[natural units]] where {{math|''c'' {{=}} 1}}, the energy–momentum equation reduces to
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| :<math>E^2 = p^2 + m_0^2 \,.</math>
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| In [[particle physics]], energy is typically given in units of [[electron volt]]s (eV), momentum in units of eV·''c''<sup>−1</sup>, and mass in units of eV·''c''<sup>−2</sup>. In [[electromagnetism]], and because of relativistic invariance, it is useful to have the [[electric field]] {{math|'''E'''}} and the [[magnetic field]] {{math|'''B'''}} in the same unit ([[Gauss (unit)|Gauss]]), using the [[Gaussian units|cgs (Gaussian) system of units]], where energy is given in units of [[erg]], mass in [[gram]]s (g), and momentum in g·cm·s<sup>−1</sup>.
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| 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 bomb]]s have generated a [[kilogram]] or more of heat. Energies of thermonuclear bombs are usually given in tens of [[TNT equivalent|kiloton]]s and megatons referring to the energy liberated by exploding that amount of [[trinitrotoluene]] (TNT).
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| ==Special cases==
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| ===Centre-of-momentum frame (one particle)===
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| {{see also|Center-of-momentum frame}}
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| For a body in its rest frame, the momentum is zero, so the equation simplifies to
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| :<math> E_0 = m_0 c^2 \,,</math>
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| where {{math|''m''<sub>0</sub>}} is the rest mass of the body. | |
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| ===Massless particles===
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| If the object is massless, as is the case for a [[photon]], then the equation reduces to
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| :<math> E = pc \,.</math>
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| This is a useful simplification. It can be rewritten in other ways using the [[matter wave|de Broglie relation]]s:
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| :<math> E = \frac{hc}{\lambda} = \hbar c k \,.</math>
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| if the [[wavelength]] {{math|''λ''}} or [[wavenumber]] {{math|''k''}} are given.
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| ===Correspondence principle===
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| Rewriting the relation for massive particles as:
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| :<math>E = m_0c^2\left[1+\left(\frac{p}{m_0c}\right)^2\right]^{1/2}\,,</math>
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| and expanding into [[power series]] by the [[binomial theorem]] (or a [[Taylor series]]):
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| :<math>E = m_0c^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>
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| 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|(''p''/''m''<sub>0</sub>''c'')<sup>2</sup>}} (i.e. retain the term {{math|(''p''/''m''<sub>0</sub>''c'')<sup>2''n''</sup>}} for {{math|''n'' {{=}} 1}} and neglect all terms for {{math|''n'' ≥ 2}}) we have
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| :<math>E \approx m_0c^2\left[1 + \frac{1}{2}\left(\frac{m_0u}{m_0c}\right)^2 \right]\,,</math>
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| or
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| :<math>E \approx m_0c^2 + \frac{1}{2}mu_0^2 \,,</math>
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| where the second term is the classical [[kinetic energy]], and the first is the [[rest mass]] 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.
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| ==Many-particle systems==
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| {{see also|many-body problem}}
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| ===Addition of four momenta===
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| 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;
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| :<math>\sum_n \mathbf{P}_n = \sum_n (E_n /c , \mathbf{p}_n )= \left(\sum_n E_n /c , \sum_n \mathbf{p}_n \right)\,,</math>
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| and then take the norm; to obtain the relation for a many particle system:
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| :<math>\left|\left(\sum_n \mathbf{P}_n \right)\right|^2 = \left(\sum_n E_n/c \right)^2 - \left(\sum_n \mathbf{p}_n \right)^2 = (M_0 c)^2\,,</math>
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| 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 [[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}});
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| {{NumBlk|:|
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| {{Equation box 1
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| |title=
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| |indent =
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| |equation =
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| <math> \left(\sum_n E_n \right)^2 = \left(\sum_n \mathbf{p}_n c\right)^2 + (M_0 c^2)^2 </math>
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| |cellpadding
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| |border = 1
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| |border colour = black
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| |background colour = white}}
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| |{{EquationRef|2}}}}
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| The energies and momenta in the equation are all frame-dependent, while {{math|''M''<sub>0</sub>}} is frame-independent.
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| ===Center-of-momentum frame===
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| In the [[center-of-momentum frame]] (COM frame), by definition we have:
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| :<math>\sum_n \mathbf{p}_n = \boldsymbol{0}\,,</math>
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| 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:
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| :<math>\left(\sum_n E_n \right)^2 = (M_0 c^2)^2 \Rightarrow \sum_n E_{\mathrm{COM}\,n} = E_\mathrm{COM} = M_0 c^2 \,,</math>
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| 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.
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| ===Rest masses and the invariant mass===
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| Either the energies or momenta of the particles, as measured in some frame, can be eliminated using the energy momentum relation for each particle:
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| :<math>E^2_n - (\mathbf{p}_n c)^2 = (m_n c^2)^2 \,,</math>
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| 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):
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| :<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>
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| :<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>
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| so substituting the sums, we can introduce their rest masses {{math|''m<sub>n</sub>''}} in ({{EquationNote|2}}):
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| :<math> \sum_n (m_n c^2)^2 - 2\sum_{n<k}(E_n E_k - c^2 \mathbf{p}_n \cdot \mathbf{p}_k) = (M_0 c^2)^2 \,. </math>
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| The energies can be eliminated by:
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| :<math>E_n =\sqrt{(\mathbf{p}_n c)^2 + (m_n c^2)^2} \,,\quad E_k =\sqrt{(\mathbf{p}_k c)^2 + (m_k c^2)^2} \,,</math>
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| similarly the momenta can be eliminated by:
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| :<math>\mathbf{p}_n \cdot \mathbf{p}_k = |\mathbf{p}_n||\mathbf{p}_k|\cos\theta_{nk}\,,\quad |\mathbf{p}_n| = \frac{1}{c}\sqrt{E_n^2 - (m_n c^2)^2}\,,\quad |\mathbf{p}_k| = \frac{1}{c}\sqrt{E_k^2 - (m_k c^2)^2} \,,</math>
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| where {{math|''θ<sub>nk</sub>''}} is the angle between the momentum vectors {{math|'''p'''<sub>''n''</sub>}} and {{math|'''p'''<sub>''k''</sub>}}.
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| Rearranging:
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| :<math> \sum_n (m_n c^2)^2 - (M_0 c^2)^2 = 2\sum_{n<k}(E_n E_k - c^2 \mathbf{p}_n \cdot \mathbf{p}_k) \,.</math>
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| 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).
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| ==Matter waves==
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| Using the [[De Broglie relations]] for energy and momentum
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| :<math> E=\hbar \omega \,, \quad \mathbf{p}=\hbar\mathbf{k}\,,</math> | |
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| for [[matter wave]]s of {{math|''ω''}} is [[angular frequency]], {{math|'''k'''}} is the [[wavevector]] with magnitude {{math|{{!}}'''k'''{{!}} {{=}} ''k''}} equal to the [[wave number]], the energy–momentum relation can be expressed in terms of wave quantities:
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| :<math> (\hbar\omega)^2 = (c \hbar k)^2 + (m_0c^2)^2 \,,</math>
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| and tidying up by dividing by {{math|(''ħc'')<sup>2</sup>}} throughout:
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| {{NumBlk|:|
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| {{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}}
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| |{{EquationRef|3}}}}
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| This can also be derived from the magnitude of the [[Four-vector#Four-wavevector|four-wavevector]]
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| :<math> \mathbf{K} = (\omega/c, \mathbf{k})\,,</math>
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| in a similar way to the four-momentum above.
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| Since the [[Planck's constant|reduced Planck constant]] {{math|''ħ''}} and the [[speed of light]] {{math|''c''}} both appear and clutter this equation, this is where [[natural units]] are especially helpful. Normalizing them so that {{math|''ħ'' {{=}} ''c'' {{=}} 1}}, we have:
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| :<math> \omega^2 = k^2 + m_0^2 \,.</math>
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| ==See also==
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| * [[Mass–energy equivalence]]
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| * [[Four-momentum]]
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| * [[Mass in special relativity]]
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| ==References==
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| {{reflist}}
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| * {{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}}
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| * {{cite book| author=G. Woan| title=The Cambridge Handbook of Physics Formulas| publisher=Cambridge University Press|page=65| year=2010| isbn=978-0-521-57507-2}}
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| * {{cite book| author=C.B. Parker| title=McGraw-Hill Encyclopaedia of Physics| publisher=McGraw-Hill|edition=2nd|pages=1192, 1193|year=1994| isbn=0-07-051400-3}}
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| * {{cite book|pages=| author=R.G. Lerner, G.L. Trigg| title=Encyclopaedia of Physics| publisher=VHC Publishers|edition=2nd|page=1052| year=1991| isbn=0-89573-752-3}}
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| {{DEFAULTSORT:Energy-Momentum Relation}}
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| [[Category:Special relativity]]
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