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

p = (

p

0

,

p

1

,

p

2

,

p

3

) =

(

E c

,

p

x

,

p

y

,

p

z

)

.

displaystyle p=(p^ 0 ,p^ 1 ,p^ 2 ,p^ 3 )=left( E over c ,p_ x ,p_ y ,p_ z right).

The quantity mv of above is ordinary non-relativistic momentum of the particle and m its rest mass. The four-momentum is useful in relativistic calculations because it is a Lorentz vector. This means that it is easy to keep track of how it transforms under Lorentz transformations. The above definition applies under the coordinate convention that x0 = ct. Some authors use the convention x0 = t, which yields a modified definition with p0 = E/c2. It is also possible to define covariant four-momentum pμ where the sign of the energy is reversed.

Contents

1 Minkowski norm 2 Relation to four-velocity 3 Derivation 4 Conservation of four-momentum 5 Canonical momentum in the presence of an electromagnetic potential 6 See also 7 References

Minkowski norm[edit] Calculating the Minkowski norm squared of the four-momentum gives a Lorentz invariant
Lorentz invariant
quantity equal (up to factors of the speed of light c) to the square of the particle's proper mass:

p ⋅ p =

η

μ ν

p

μ

p

ν

=

p

ν

p

ν

= −

E

2

c

2

+

p

2

= −

m

2

c

2

displaystyle pcdot p=eta _ mu nu p^ mu p^ nu =p_ nu p^ nu =- E^ 2 over c^ 2 +mathbf p ^ 2 =-m^ 2 c^ 2

where

η

μ ν

=

(

− 1

0

0

0

0

1

0

0

0

0

1

0

0

0

0

1

)

displaystyle eta _ mu nu =left( begin matrix -1&0&0&0\0&1&0&0\0&0&1&0\0&0&0&1end matrix right)

is the metric tensor of special relativity with metric signature for definiteness chosen to be (–1, 1, 1, 1). The negativity of the norm reflects that the momentum is a timelike four-vector for massive particles. The other choice of signature would flip signs in certain formulas (like for the norm here). This choice is not important, but once made it must for consistency be kept throughout. The Minkowski norm is Lorentz invariant, meaning its value is not changed by Lorentz transformations/boosting into different frames of reference. More generally, for any two four-momenta p and q, the quantity p ⋅ q is invariant. Relation to four-velocity[edit] For a massive particle, the four-momentum is given by the particle's invariant mass m multiplied by the particle's four-velocity,

p

μ

= m

u

μ

,

displaystyle p^ mu =mu^ mu ,

where the four-velocity u is

u =

(

u

0

,

u

1

,

u

2

,

u

3

)

=

γ

v

(

c ,

v

x

,

v

y

,

v

z

)

,

displaystyle u=left(u^ 0 ,u^ 1 ,u^ 2 ,u^ 3 right)=gamma _ v left(c,v_ x ,v_ y ,v_ z right),

and

γ

v

=

1

1 −

v

2

c

2

displaystyle gamma _ v = frac 1 sqrt 1- frac v^ 2 c^ 2

is the Lorentz factor
Lorentz factor
(associated with the speed v), c is the speed of light. Derivation[edit] There are several ways to arrive at the correct expression for four-momentum. One way is to first define the four-velocity u = dx/dτ and simply define p = mu, being content that it is a four-vector with the correct units and correct behavior. Another, more satisfactory, approach is to begin with the principle of least action and use the Lagrangian framework to derive the four-momentum, including the expression for the energy.[1] One may at once, using the observations detailed below, define four-momentum from the action S. Given that in general for a closed system with generalized coordinates qi and canonical momenta pi,[2]

p

i

=

∂ S

q

i

,

E = −

∂ S

∂ t

,

displaystyle p_ i = frac partial S partial q_ i ,quad E=- frac partial S partial t ,

it is immediate (recalling x0 = ct, x1 = x, x2 = y, x3 = z and x0 = −x0, x1 = x1, x2 = x2, x3 = x3 in the present metric convention) that

p

μ

= −

∂ S

x

μ

=

(

E c

, −

p

)

displaystyle p_ mu =- frac partial S partial x^ mu =left( E over c ,-mathbf p right)

is a covariant four-vector with the three-vector part being the (negative of) canonical momentum.

Observations

Consider initially a system of one degree of freedom q. In the derivation of the equations of motion from the action using Hamilton's principle, one finds (generally) in an intermediate stage for the variation of the action,

δ S =

[

∂ L

q ˙

δ q

]

t

1

t

2

+

t

1

t

2

(

∂ L

∂ q

d

d t

∂ L

q ˙

)

δ q d t .

displaystyle delta S=left.left[ frac partial L partial dot q delta qright]right_ t_ 1 ^ t_ 2 +int _ t_ 1 ^ t_ 2 left( frac partial L partial q - frac d dt frac partial L partial dot q right)delta qdt.

The assumption is then that the varied paths satisfy δq(t1) = δq(t2) = 0, from which Lagrange's equations
Lagrange's equations
follow at once. When the equations of motion are known (or simply assumed to be satisfied), one may let go of the requirement δq(t2) = 0. In this case the path is assumed to satisfy the equations of motion, and the action is a function of the upper integration limit δq(t2), but t2 is still fixed. The above equation becomes with S = S(q), and defining δq(t2) = δq, and letting in more degrees of freedom,

δ S =

i

∂ L

q ˙

i

δ

q

i

=

i

p

i

δ

q

i

.

displaystyle delta S=sum _ i frac partial L partial dot q _ i delta q_ i =sum _ i p_ i delta q_ i .

Observing that

δ S =

i

∂ S

q

i

δ

q

i

,

displaystyle delta S=sum _ i frac partial S partial q _ i delta q_ i ,

one concludes

p

i

=

∂ S

q

i

.

displaystyle p_ i = frac partial S partial q_ i .

In a similar fashion, keep endpoints fixed, but let t2 = t vary. This time, the system is allowed to move through configuration space at "arbitrary speed" or with "more or less energy", the field equations still assumed to hold and variation can be carried out on the integral, but instead observe

d S

d t

= L

displaystyle frac dS dt =L

by the fundamental theorem of calculus. Compute using the above expression for canonical momenta,

d S

d t

=

∂ S

∂ t

+

i

∂ S

q

i

q ˙

i

=

∂ S

∂ t

+

i

p

i

q ˙

i

= L .

displaystyle frac dS dt = frac partial S partial t +sum _ i frac partial S partial q_ i dot q _ i = frac partial S partial t +sum _ i p_ i dot q _ i =L.

Now using

H =

i

p

i

q ˙

i

− L ,

displaystyle H=sum _ i p_ i dot q _ i -L,

where H is the Hamiltonian, leads to, since E = H in the present case,

E = H = −

∂ S

∂ t

.

displaystyle E=H=- frac partial S partial t .

Incidentally, using H = H(q, p, t) with p = ∂S/∂q in the above equation yields the Hamilton–Jacobi equations. In this context, S is called Hamilton's principal function.

The action S is given by

S = − m c ∫ d s = ∫ L d t ,

L = − m

c

2

1 −

v

2

c

2

,

displaystyle S=-mcint ds=int Ldt,quad L=-mc^ 2 sqrt 1- frac v^ 2 c^ 2 ,

where L is the relativistic Lagrangian for a free particle. From this,

glossing over these details,

The variation of the action is

δ S = − m c ∫ δ d s .

displaystyle delta S=-mcint delta ds.

To calculate δds, observe first that δds2 = 2dsδds and that

δ d

s

2

= δ

η

μ ν

d

x

μ

d

x

ν

=

η

μ ν

(

δ

(

d

x

μ

)

d

x

ν

+ d

x

μ

δ

(

d

x

ν

)

)

= 2

η

μ ν

δ

(

d

x

μ

)

d

x

ν

.

displaystyle delta ds^ 2 =delta eta _ mu nu dx^ mu dx^ nu =eta _ mu nu left(delta left(dx^ mu right)dx^ nu +dx^ mu delta left(dx^ nu right)right)=2eta _ mu nu delta left(dx^ mu right)dx^ nu .

So

δ d s =

η

μ ν

δ d

x

μ

d

x

ν

d s

=

η

μ ν

d δ

x

μ

d

x

ν

d s

,

displaystyle delta ds=eta _ mu nu delta dx^ mu frac dx^ nu ds =eta _ mu nu ddelta x^ mu frac dx^ nu ds ,

or

δ d s =

η

μ ν

d δ

x

μ

d τ

d

x

ν

c d τ

d τ ,

displaystyle delta ds=eta _ mu nu frac ddelta x^ mu dtau frac dx^ nu cdtau dtau ,

and thus

δ S = − m ∫

η

μ ν

d δ

x

μ

d τ

d

x

ν

d τ

d τ = − m ∫

η

μ ν

d δ

x

μ

d τ

u

ν

d τ = − m ∫

η

μ ν

[

d

d τ

(

δ

x

μ

u

ν

)

− δ

x

μ

d

d τ

u

ν

]

d τ

displaystyle delta S=-mint eta _ mu nu frac ddelta x^ mu dtau frac dx^ nu dtau dtau =-mint eta _ mu nu frac ddelta x^ mu dtau u^ nu dtau =-mint eta _ mu nu leftlbrack frac d dtau left(delta x^ mu u^ nu right)-delta x^ mu frac d dtau u^ nu rightrbrack dtau

which is just

δ S =

[

− m

u

μ

δ

x

μ

]

t

1

t

2

+ m

t

1

t

2

δ

x

μ

d

u

μ

d s

d s

displaystyle delta S=left[-mu_ mu delta x^ mu right]_ t_ 1 ^ t_ 2 +mint _ t_ 1 ^ t_ 2 delta x^ mu frac du_ mu ds ds

δ S =

[

− m

u

μ

δ

x

μ

]

t

1

t

2

+ m

t

1

t

2

δ

x

μ

d

u

μ

d s

d s = − m

u

μ

δ

x

μ

=

∂ S

x

μ

δ

x

μ

= −

p

μ

δ

x

μ

,

displaystyle delta S=left[-mu_ mu delta x^ mu right]_ t_ 1 ^ t_ 2 +mint _ t_ 1 ^ t_ 2 delta x^ mu frac du_ mu ds ds=-mu_ mu delta x^ mu = frac partial S partial x^ mu delta x^ mu =-p_ mu delta x^ mu ,

where the second step employs the field equations duμ/ds = 0, (δxμ)t1 = 0, and (δxμ)t2 ≡ δxμ as in the observations above. Now compare the last three expressions to find

p

μ

= −

μ

[ S ] = −

∂ S

x

μ

= m

u

μ

= m

(

c

1 −

v

2

c

2

,

v

x

1 −

v

2

c

2

,

v

y

1 −

v

2

c

2

,

v

z

1 −

v

2

c

2

)

,

displaystyle p^ mu =-partial ^ mu [S]=- frac partial S partial x_ mu =mu^ mu =mleft( frac c sqrt 1- frac v^ 2 c^ 2 , frac v_ x sqrt 1- frac v^ 2 c^ 2 , frac v_ y sqrt 1- frac v^ 2 c^ 2 , frac v_ z sqrt 1- frac v^ 2 c^ 2 right),

with norm −m2c2, and the famed result for the relativistic energy,

E =

m

c

2

1 −

v

2

c

2

=

m

r

c

2

,

displaystyle E= frac mc^ 2 sqrt 1- frac v^ 2 c^ 2 =m_ r c^ 2 ,

where mr is the now unfashionable relativistic mass, follows. By comparing the expressions for momentum and energy directly, one has

p

= E

v

c

2

,

displaystyle mathbf p =E frac mathbf v c^ 2 ,

that holds for massless particles as well. Squaring the expressions for energy and three-momentum and relating them gives the energy–momentum relation,

E

2

c

2

=

p

p

+

m

2

c

2

.

displaystyle frac E^ 2 c^ 2 =mathbf p cdot mathbf p +m^ 2 c^ 2 .

Substituting

p

μ

↔ −

∂ S

x

μ

displaystyle p_ mu leftrightarrow - frac partial S partial x^ mu

in the equation for the norm gives the relativistic Hamilton–Jacobi equation,[3]

η

μ ν

∂ S

x

μ

∂ S

x

ν

= −

m

2

c

2

.

displaystyle eta ^ mu nu frac partial S partial x^ mu frac partial S partial x^ nu =-m^ 2 c^ 2 .

It is also possible to derive the results from the Lagrangian directly. By definition,[4]

p

=

∂ L

v

=

(

∂ L

x ˙

,

∂ L

y ˙

,

∂ L

z ˙

)

= m ( γ

v

x

, γ

v

y

, γ

v

z

) = m γ

v

= m

u

,

E

=

p

v

− L =

m

c

2

1 −

v

2

c

2

,

displaystyle begin aligned mathbf p &= frac partial L partial mathbf v =left( partial L over partial dot x , partial L over partial dot y , partial L over partial dot z right)=m(gamma v_ x ,gamma v_ y ,gamma v_ z )=mgamma mathbf v =mmathbf u ,\E&=mathbf p cdot mathbf v -L= frac mc^ 2 sqrt 1- frac v^ 2 c^ 2 ,end aligned

which constitute the standard formulae for canonical momentum and energy of a closed (time-independent Lagrangian) system. With this approach it is less clear that the energy and momentum are parts of a four-vector. The energy and the three-momentum are separately conserved quantities for isolated systems in the Lagrangian framework. Hence four-momentum is conserved as well. More on this below. More pedestrian approaches include expected behavior in electrodynamics.[5] In this approach, the starting point is application of Lorentz force law
Lorentz force law
and Newton's second law
Newton's second law
in the rest frame of the particle. The transformation properties of the electromagnetic field tensor, including invariance of electric charge, are then used to transform to the lab frame, and the resulting expression (again Lorentz force
Lorentz force
law) is interpreted in the spirit of Newton's second law, leading to the correct expression for the relativistic three- momentum. The disadvantage, of course, is that it isn't immediately clear that the result applies to all particles, whether charged or not, and that it doesn't yield the complete four-vector. It is also possible to avoid electromagnetism and use well tuned experiments of thought involving well-trained physicists throwing billiard balls, utilizing knowledge of the velocity addition formula and assuming conservation of momentum.[6][7] This too gives only the three-vector part. Conservation of four-momentum[edit] As shown above, there are three (not independent, the last two implies the first) conservation laws:

The four-momentum p (either covariant or contravariant) is conserved. The total energy E = p0c is conserved. The 3-space
3-space
momentum p is conserved.

Note that the invariant mass of a system of particles may be more than the sum of the particles' rest masses, since kinetic energy in the system center-of-mass frame and potential energy from forces between the particles contribute to the invariant mass. As an example, two particles with four-momenta (5 GeV/c, 4 GeV/c, 0, 0) and (5 GeV/c, −4 GeV/c, 0, 0) each have (rest) mass 3 GeV/c2 separately, but their total mass (the system mass) is 10 GeV/c2. If these particles were to collide and stick, the mass of the composite object would be 10 GeV/c2. One practical application from particle physics of the conservation of the invariant mass involves combining the four-momenta pA and pB of two daughter particles produced in the decay of a heavier particle with four-momentum pC to find the mass of the heavier particle. Conservation of four-momentum gives pCμ = pAμ + pBμ, while the mass M of the heavier particle is given by −PC ⋅ PC = M2c2. By measuring the energies and three-momenta of the daughter particles, one can reconstruct the invariant mass of the two-particle system, which must be equal to M. This technique is used, e.g., in experimental searches for Z′ bosons at high-energy particle colliders, where the Z′ boson would show up as a bump in the invariant mass spectrum of electron–positron or muon–antimuon pairs. If the mass of an object does not change, the Minkowski inner product of its four-momentum and corresponding four-acceleration Aμ is simply zero. The four-acceleration is proportional to the proper time derivative of the four-momentum divided by the particle's mass, so

p

μ

A

μ

=

η

μ ν

p

μ

A

ν

=

η

μ ν

p

μ

d

d τ

p

ν

m

=

1

2 m

d

d τ

p ⋅ p =

1

2 m

d

d τ

(

m

2

c

2

)

= 0.

displaystyle p^ mu A_ mu =eta _ mu nu p^ mu A^ nu =eta _ mu nu p^ mu frac d dtau frac p^ nu m = frac 1 2m frac d dtau pcdot p= frac 1 2m frac d dtau left(-m^ 2 c^ 2 right)=0.

Canonical momentum in the presence of an electromagnetic potential[edit] For a charged particle of charge q, moving in an electromagnetic field given by the electromagnetic four-potential:

A =

(

A

0

,

A

1

,

A

2

,

A

3

)

=

(

ϕ c

,

A

x

,

A

y

,

A

z

)

displaystyle A=left(A^ 0 ,A^ 1 ,A^ 2 ,A^ 3 right)=left( phi over c ,A_ x ,A_ y ,A_ z right)

where Φ is the scalar potential and A = (Ax, Ay, Az) the vector potential, the components of the canonical momentum four-vector P is

P

μ

=

p

μ

+ q

A

μ

.

displaystyle P^ mu =p^ mu +qA^ mu .!

This, in turn, allows the potential energy from the charged particle in an electrostatic potential and the Lorentz force
Lorentz force
on the charged particle moving in a magnetic field to be incorporated in a compact way, in relativistic quantum mechanics. See also[edit]

Physics portal

Four-force Four-gradient Pauli–Lubanski pseudovector

References[edit]

^ Landau & Lifshitz 2002, pp. 25–29 ^ Landau & Lifshitz 1975, pp. 139 ^ Landau & Lifshitz 1975, p. 30 ^ Landau & Lifshitz 1975, pp. 15–16 ^ Sard 1970, Section 3.1 ^ Sard 1970, Section 3.2 ^ Lewis & Tolman 1909 Wikisource version

Goldstein, Herbert (1980). Classical mechanics (2nd ed.). Reading, Mass.: Addison–Wesley Pub. Co. ISBN 0201029189.  Landau, L. D.; Lifshitz, E. M. (1975) [1939]. Mechanics. Translated from Russian by J. B. Sykes and J. S. Bell. (3rd ed.). Amsterdam: Elsevier. ISBN 978-0-7506-28969.  Landau, L.D.; Lifshitz, E.M. (2000). The classical theory of fields. 4th rev. English edition, reprinted with corrections; translated from the Russian by Morton Hamermesh. Oxford: Butterworth Heinemann. ISBN 9780750627689.  Rindler, Wolfgang (1991). Introduction to Special
Special
Relativity (2nd ed.). Oxford: Oxford University Press. ISBN 0-19-853952-5.  Sard, R. D. (1970). Relativistic Mechanics - Special
Special
Relativity and Classical Particle Dynamics. New York: W. A. Benjamin. ISBN 978-0805384918.  Lewis, G. N.; Tolman, R. C. (1909). "The Principle of Relativity, and Non-Newtonian Mechanics". Phil. Mag. 6. Taylor & Francis. 18 (106): 510–523. doi:10.1080/14786441008636725. (Subscription required (help)).  Wikiso

.