Four-momentum

In special relativity, four-momentum (also called momentum–energy or momenergy) is the generalization of the classical three-dimensional momentum to four-dimensional spacetime. 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=\left(p^{0},p^{1},p^{2},p^{3}\right)=\left({\frac {E}{c}},p_{x},p_{y},p_{z}\right).}

The quantity mv of above is the 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 covariant vector. This means that it is easy to keep track of how it transforms under Lorentz transformations.

Minkowski norm

Calculating the Minkowski norm squared of the four-momentum gives a 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 p\cdot 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 the following denote:

p

{\textstyle p}

, the four-momentum vector of a particle,

p

⋅

p

{\textstyle p\cdot p}

, the Minkowski inner product of the four-momentum with itself,

p

μ

{\textstyle p^{\mu }}

and

p

ν

{\textstyle p^{\nu }}

, the contravariant components of the four-momentum vector,

p

ν

{\textstyle p_{\nu }}

, the covariant form,

E

{\textstyle E}

, the energy of the particle,

c

{\textstyle c}

, the speed of light,

|

p

|

{\textstyle |\mathbf {p} |}

, the magnitude of the four-momentum vector,

m

{\textstyle m}

, the invariant mass (rest) of the particle,

and

η

μ

ν

=

(

−

1

0

0

0

0

1

0

0

0

0

1

0

0

0

0

1

)

{\displaystyle \eta _{\mu \nu }={\begin{pmatrix}-1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}}}

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.