In relativity, a four-vector is a vector in a four-dimensional real vector space, called Minkowski space, whose components transform like increases in the space and time coordinates, $(\backslash Delta\; t,\; \backslash Delta\; x,\; \backslash Delta\; y,\; \backslash Delta\; z)$, under spatial translations, rotations, and boosts (a change by a constant velocity to another inertial reference frame). The set of all such translations, rotations, and boosts (called Poincaré transformations and described by 4×4 matrices) forms the Poincaré group. The set of rotations and boosts (Lorentz transformations) forms the Lorentz group.

Mathematics of four-vectors

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A point in Minkowski space is called an "event" and is described by a set of four coordinates such as

$x^a\; :=\; \backslash left(ct,\; x,\; y,\; z\; \backslash right)$

for a = 0, 1, 2, 3, where c is the speed of light.

The position four-vector is defined to be an "arrow" linking two events:

$\backslash Delta\; x^a\; :=\; \backslash left(\backslash Delta\; ct,\; \backslash Delta\; x,\; \backslash Delta\; y,\; \backslash Delta\; z\; \backslash right)$

(Note that $(ct,\backslash ,\; x,\backslash ,\; y,\backslash ,\; z)$ is not a four-vector. What is required are coordinate differences, not the coordinates themselves.) The inner product of two four-vectors $u$ and $v$ is defined (using Einstein notation) as

$$

u \cdot v = u^a \eta_{ab} v^b = \left( \begin{matrix}u^0 & u^1 & u^2 & u^3 \end{matrix} \right) \left( \begin{matrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} \right) \left( \begin{matrix}v^0 \\ v^1 \\ v^2 \\ v^3 \end{matrix} \right) = - u^0 v^0 + u^1 v^1 + u^2 v^2 + u^3 v^3

where η is the Minkowski metric. Sometimes this inner product is called the Minkowski inner product. The four-vectors are arrows on the spacetime diagram or Minkowski diagram.

Four-vectors may be classified as either spacelike, timelike or null. In this article, four-vectors will be referred to simply as vectors. Spacelike, timelike, and null vectors are ones whose inner product with themselves is greater than, less than, and equal to zero respectively.

Examples of four-vectors in dynamics

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When considering physical phenomena, differential equations arise naturally; however, when considering space and time derivatives of functions, it is unclear which reference frame these derivatives are taken with respect to. It is agreed that time derivatives are taken with respect to the proper time (τ) in the given reference frame. It is then important to find a relation between this time derivative and another time derivative (taken in another inertial reference frame). This relation is provided by the time transformation in the Lorentz transformations and is:

$\backslash frac\{d\; \backslash tau\}\{dt\}=\backslash frac\{1\}\{\backslash gamma\}$

where γ is the Lorentz factor. Important four-vectors in relativity theory can now be defined, such as the four-velocity defined by:

$U^a\; :=\; \backslash frac\{dx^a\}\{d\; \backslash tau\}=\; \backslash frac\{dx^a\}\{dt\}\backslash frac\{dt\}\{d\; \backslash tau\}=\; \backslash left(\backslash gamma\; c,\; \backslash gamma\; \backslash mathbf\{u\}\; \backslash right)$

where

$u^i\; =\; \backslash frac\{dx^i\}\{dt\}$

for i = 1, 2, 3. Notice that

$U^a\; U\_a\; =\; -c^2\; \backslash ,$

The four-acceleration is defined by:

$A^a\; :=\; U\_\{a,b\}\; U^b\; =\; \backslash frac\{\backslash partial\; U\_a\}\{\backslash partial\; x^b\}\; U^b\; =\backslash frac\{dU^a\}\{d\; \backslash tau\}\; =\; \backslash left(\backslash gamma\; \backslash dot\{\backslash gamma\}\; c,\; \backslash gamma\; \backslash dot\{\backslash gamma\}\; \backslash mathbf\{u\}\; +\; \backslash gamma^2\; \backslash mathbf\{\backslash dot\{u\}\}\; \backslash right)$

Since the magnitude of $U^a$ is a constant

$0\; =\; \backslash frac\{\backslash partial\; U^a\; U\_a\}\{\backslash partial\; x^b\}\; =\; 2\; U\_\{a,b\}\; U^a\; \backslash ,$

multiplying both sides by $U^b$ yields the identity

$A^a\; U\_a\; =\; 0\; \backslash ,$

which is true for all velocity distributions.

The four-momentum is defined by:

$P^a\; :=m\_0\; U^a\; =\; \backslash left(mc,\; \backslash mathbf\{p\}\; \backslash right)$

where m₀ is the rest mass of the particle (with m = γm₀) and p = mu.

The four-force is defined by:

$F^a\; :=\; m\_0\; A^a\; =\; \backslash left(\backslash gamma\; \backslash dot\{m\}\; c,\; \backslash gamma\; \backslash mathbf\{f\}\; \backslash right)$

where

$\backslash mathbf\{f\}\; =\; m\_0\; \backslash dot\{\backslash gamma\}\; \backslash mathbf\{u\}\; +\; m\_0\; \backslash gamma\; \backslash mathbf\{\backslash dot\{u\}\}$.

Physics of four-vectors

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The power and elegance of the four-vector formalism may be demonstrated by deriving some important relations between the physical quantities energy, mass and momentum.

Deriving E = mc²

Here, an expression for the total energy of a particle will be derived. The kinetic energy (K) of a particle is defined analogously to the classical definition, namely as

$\backslash frac\{dK\}\{dt\}=\; \backslash mathbf\{f\}\; \backslash cdot\; \backslash mathbf\{u\}$

with f as above. Noticing that F^aU_a = 0 and expanding this out we get

$\backslash gamma^2\; \backslash left(\backslash mathbf\{f\}\; \backslash cdot\; \backslash mathbf\{u\}\; -\; \backslash dot\{m\}\; c^2\; \backslash right)\; =\; 0$

Hence

$\backslash frac\{dK\}\{dt\}\; =\; c^2\; \backslash frac\{dm\}\{dt\}$

which yields

$K\; =\; m\; c^2\; +\; S\; \backslash ,$

for some constant S. When the particle is at rest (u = 0), we take its kinetic energy to be zero (K = 0). This gives

$S\; =\; -m\_0\; c^2\; \backslash ,$

Thus, we interpret the total energy E of the particle as composed of its kinetic energy K and its rest energy m₀c^{2. Thus, we have}

$E\; =\; m\; c^2\; \backslash ,$

Deriving E² = p²c² + m₀²c⁴

Using the relation E = mc², we can write the four-momentum as

$P^a\; =\; \backslash left(\backslash frac\{E\}\{c\},\; \backslash mathbf\{p\}\; \backslash right)$.

Taking the inner product of the four-momentum with itself in two different ways, we obtain the relation

$p^2\; -\; \backslash frac\{E^2\}\{c^2\}\; =\; P^a\; P\_a\; =\; m\_0^2\; U^a\; U\_a\; =\; -m\_0^2\; c^2$

i.e.

$p^2\; -\; \backslash frac\{E^2\}\{c^2\}\; =\; -m\_0^2\; c^2$

Hence

$E^2\; =\; p^2\; c^2\; +\; m\_0^2\; c^4.$

This last relation is useful in many areas of physics.

Examples of four-vectors in electromagnetism

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Examples of four-vectors in electromagnetism include the four-current defined by

$J^a\; :=\; \backslash left(\; \backslash rho\; c,\; \backslash mathbf\{j\}\; \backslash right)$

formed from the current density j and charge density ρ, and the electromagnetic four-potential defined by

$\backslash Phi^a\; :=\backslash left(\backslash phi,\; \backslash mathbf\{A\}\; c\; \backslash right)$

formed from the vector potential A and the scalar potential φ.

A plane electromagnetic wave can be described by the four-frequency defined as

$N\; ^a\; :=\backslash left(\backslash nu,\; \backslash nu\; \backslash mathbf\{n\}\; \backslash right)$

where $\backslash nu$ is the frequency of the wave and n is a unit vector in the travel direction of the wave. Notice that

$N^a\; N\_a\; =\; \backslash nu\; ^2\; \backslash left(n^2\; -\; 1\; \backslash right)\; =\; 0$

so that the four-frequency is always a null vector.

Deriving Planck's law

It is often assumed that Planck's law relating the energy and frequency of a photon must necessarily come from quantum mechanics. However, Planck's law can be obtained purely within the formalism of special relativity. In analogy with the definition for the four-momentum of a particle, the photon four-momentum is defined by

$\backslash tilde\{P\}^a\; :=\; \backslash left(\; \backslash frac\{E\}\{c\},\; \backslash mathbf\{p\}\; \backslash right)$

where:

$E\; \backslash ,$ is the photon's energy

$\backslash mathbf\{p\}=p\; \backslash mathbf\{n\}$ is the photon's momentum

$\backslash mathbf\{n\}$ a unit vector in the direction of motion of the photon.

Note that $\backslash tilde\{P\}^a\backslash tilde\{P\}\_a=0$ by virtue of the relation $E=pc\; \backslash ,$ which comes from electromagnetic theory. Given that $\backslash tilde\{P\}^a$ and $N^a$ are both null vectors (with each one clearly non-zero, and noting that $\backslash tilde\{P\}^aN\_a=0$, this means that $\backslash tilde\{P\}$ and $N^a$ must be proportional, i.e.

$\backslash tilde\{P\}^a=sN^a$

for some real number s. Multiplying the above relation by $\backslash frac\{\backslash partial\; x^\{\text{'}b\}\}\{\backslash partial\; x^a\}$ gives

$\backslash tilde\{P\}^\{\text{'}a\}=sN^\{\text{'}a\}.$

Considering the 0-th component of the last two relations shows that the ratio of a photon's energy to its frequency is the same in any two inertial reference frames, i.e.

$E=h\; \backslash nu\; \backslash ,$

which is Planck's law, the constant traditionally being denoted by $h$ and called Planck's constant. In fact, by employing the same definitions for the photon four- momentum and the four-frequency in general relativity, the above proof appears to be valid in GR.

Note that by combining $E=pc\; \backslash ,$ with Planck's law, the momentum of a photon may be written as the famous de Broglie equation:

$p=\backslash frac\{h\}\{\backslash lambda\}$

See also

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• four-velocity
• four-acceleration
• four-momentum
• four-force
• four-current
• electromagnetic four-potential

References

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• Rindler, W. Introduction to Special Relativity (2nd edn.) (1991) Clarendon Press Oxford ISBN 0-19-853952-5

Relativity

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