Fourvector
From Academic Kids

In relativity, a fourvector is a vector in a fourdimensional real vector space, called Minkowski space, whose components transform like the space and time coordinates (t, x, y, z) under spatial rotations and boosts (a change by a constant velocity to another inertial reference frame). The set of all such rotations and boosts, called Lorentz transformations and described by 4×4 matrices, forms the Lorentz group.
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Mathematics of fourvectors
A point in Minkowski space is called an "event" and is described by the position fourvector defined as
 <math> x^a := \left(ct, x, y, z \right) <math>
for a = 0, 1, 2, 3, where c is the speed of light.
The inner product of two fourvectors x and y is defined (using Einstein notation) as
 <math>
x \cdot y = x^a \eta_{ab} y^b = \left( \begin{matrix}x^0 & x^1 & x^2 & x^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}y^0 \\ y^1 \\ y^2 \\ y^3 \end{matrix} \right) =  x^0 y^0 + x^1 y^1 + x^2 y^2 + x^3 y^3 <math>
where η is the Minkowski metric. Sometimes this inner product is called the Minkowski inner product. The fourvectors make up the spacetime diagram or Minkowski diagram.
Four vectors may be classified as either spacelike, timelike or null. In this article, fourvectors will be referred to simply as vectors. Spacelike, timelike, and null vectors are ones whose inner product is greater than, less than, and equal to zero respectively.
Examples of fourvectors in dynamics
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:
 <math>\frac{d \tau}{dt}=\frac{1}{\gamma}<math>
where γ is the gamma factor of relativity. Important fourvectors in relativity theory can now be defined, such as the fourvelocity defined by:
 <math>U^a := \frac{dx^a}{d \tau}= \frac{dx^a}{dt}\frac{dt}{d \tau}= \left(\gamma c, \gamma \mathbf{u} \right)<math>
where
 <math>u^i = \frac{dx^i}{dt}<math>
for i = 1, 2, 3. Notice that
 <math> U^a U_a = c^2 \,<math>
The fouracceleration is defined by:
 <math>A^a := \frac{dU^a}{d \tau} = \left(\gamma \dot{\gamma} c, \gamma \dot{\gamma} \mathbf{u} + \gamma^2 \mathbf{\dot{u}} \right)<math>
Note that by direct calculation, it is always true that
 <math>A^a U_a = 0 \,<math>
The fourmomentum is defined by:
 <math>P^a :=m_0 U^a = \left(mc, \mathbf{p} \right)<math>
where m_{0} is the rest mass of the particle (with m = γm_{0}) and p = mu.
The fourforce is defined by:
 <math> F^a := m_0 A^a = \left(\gamma \dot{m} c, \gamma \mathbf{f} \right) <math>
where
 <math> \mathbf{f} = m_0 \dot{\gamma} \mathbf{u} + m_0 \gamma \mathbf{\dot{u}} <math>.
Physics of fourvectors
The power and elegance of the fourvector formalism may be demonstrated by deriving some important relations between the physical quantities energy, mass and momentum.
Deriving E = mc^{2}
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
 <math> \frac{dK}{dt}= \mathbf{f} \cdot \mathbf{u} <math>
with f as above. Noticing that F^{a}U_{a} = 0 and expanding this out we get
 <math> \gamma^2 \left(\mathbf{f} \cdot \mathbf{u}  \dot{m} c^2 \right) = 0 <math>
Hence
 <math> \frac{dK}{dt} = c^2 \frac{dm}{dt}<math>
which yields
 <math> K = m c^2 + S \,<math>
for some constant S. When the particle is at rest (u = 0), we take its kinetic energy to be zero (K = 0). This gives
 <math> S = m_0 c^2 \,<math>
Thus, we interpret the total energy E of the particle as composed of its kinetic energy K and its rest energy m_{0}c^{2. Thus, we have }
 <math> E = m c^2 \,<math>
Deriving E^{2} = p^{2}c^{2} + m_{0}^{2}c^{4}
Using the relation E = mc^{2}, we can write the fourmomentum as
 <math> P^a = \left(\frac{E}{c}, \mathbf{p} \right)<math>.
Taking the inner product of the fourmomentum with itself in two different ways, we obtain the relation
 <math> p^2  \frac{E^2}{c^2} = P^a P_a = m_0^2 U^a U_a = m_0^2 c^2 <math>
i.e.
 <math> p^2  \frac{E^2}{c^2} = m_0^2 c^2<math>
Hence
 <math> E^2 = p^2 c^2 + m_0^2 c^4. <math>
This last relation is useful in many areas of physics.
Examples of fourvectors in electromagnetism
Examples of fourvectors in electromagnetism include the fourcurrent defined by
 <math> J^a := \left( \rho c, \mathbf{j} \right) <math>
formed from the current density j and charge density ρ, and the electromagnetic fourpotential defined by
 <math>\Phi^a :=\left(\phi, \mathbf{A} c \right)<math>
formed from the vector potential A and the scalar potential φ.
A plane electromagnetic wave can be described by the fourfrequency defined as
 <math>N ^a :=\left(\nu, \nu \mathbf{n} \right)<math>
where <math>\nu<math> is the frequency of the wave and n is a unit vector in the travel direction of the wave. Notice that
 <math> N^a N_a = \nu ^2 \left(n^2  1 \right) = 0 <math>
so that the fourfrequency 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 fourmomentum of a particle, the fourmomentum of a photon is defined by
 <math>\tilde{P}^a := \left( \frac{E}{c}, \mathbf{p} \right)<math>
where:
 <math>E \,<math> is the photon's energy
 <math>\mathbf{p}=p \mathbf{n}<math> is the photon's momentum
 <math>\mathbf{n}<math> a unit vector in the direction of motion of the photon.
Note that <math>\tilde{P}^a\tilde{P}_a=0<math> by virtue of the relation <math>E=pc \,<math> which comes from electromagnetic theory. Given that <math>\tilde{P}^a<math> and <math>N^a<math> are both null vectors (with each one clearly nonzero, and noting that <math>\tilde{P}^aN_a=0<math>, this means that <math>\tilde{P}<math> and <math>N^a<math> must be proportional, i.e.
 <math>\tilde{P}^a=sN^a<math>
for some real number s. Multiplying the above relation by <math>\frac{\partial x^{'b}}{\partial x^a}<math> gives
 <math>\tilde{P}^{'a}=sN^{'a}.<math>
Considering the 0th 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.
 <math>E=h \nu \,<math>
which is Planck's law, the constant traditionally being denoted by <math>h<math> and called Planck's constant.
Note that by combining <math>E=pc \,<math> with Planck's law, the momentum of a photon may be written as the famous de Broglie equation:
 <math>p=\frac{h}{\lambda}<math>
See also
 fourvelocity
 fouracceleration
 fourmomentum
 fourforce
 fourcurrent
 electromagnetic fourpotential
References
 Rindler, W. Introduction to Special Relativity (2nd edn.) (1991) Clarendon Press Oxford ISBN 0198539525de:Vierervektor