Four-vector

MATHEMATICS OF FOUR-VECTORS

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

:

\mathbf{x} := \left(ct, x, y, z 
ight)

for

\mu

= 0, 1, 2, 3, where ''c'' is the Speed Of Light . These coordinates are the components of the ''position four-vector'' for the event.

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

:

\Delta \mathbf{x}:= \left(\Delta ct, \Delta x, \Delta y, \Delta z 
ight)

(Note that the position vector is the displacement vector when one of the two events is the origin of the coordinate system. Position vectors are relatively trivial; the general theory of four-vectors is concerned with displacement vectors.) The Inner Product of two four-vectors

u

and

v

is defined (using Einstein Notation ) as

:

u \cdot v
= u^{\mu} \eta_{\mu 
u} v^{
u}
= \left( \begin{matrix}u^0 & u^1 & u^2 & u^3 \end{matrix} 
ight)
\left( \begin{matrix} 1 & 0 & 0 & 0 \ 0 & -1 & 0 & 0 \ 0 & 0 & -1 & 0 \ 0 & 0 & 0 & -1 \end{matrix} 
ight)
\left( \begin{matrix}v^0 \ v^1 \ v^2 \ v^3 \end{matrix} 
ight)
=  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 inner product is often expressed as the effect of the Dual Vector of one vector on the other:

(v) = u{_

u}

Here the

u_{

of $u$ in the Dual Basis and called the Covariant coordinates of $u$ , while the original $u^$

Contravariant

The relation between the covariant and contravariant coordinates is:

u_{\mu} =  u^{
u} \eta_{\mu 
u} \,

.

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 Vector s are ones whose inner product with themselves is greater than, less than, and equal to zero respectively.

EXAMPLES OF FOUR-VECTORS 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:

:

rac{d 	au}{dt}=rac{1}{\gamma}

where γ is the Lorentz Factor . Important four-vectors in relativity theory can now be defined, such as the Four-velocity of an

\mathbf{x}(	au)

World Line is defined by:

:

\mathbf{U} := rac{d\mathbf{x}}{d 	au}= rac{d\mathbf{x}}{dt}rac{dt}{d 	au}= \left(\gamma c, \gamma \mathbf{u} 
ight)

where

:

u^i = rac{dx^i}{dt}

for ''i'' = 1, 2, 3. Notice that

PHYSICS OF FOUR-VECTORS

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'' = ''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

:

rac{dK}{dt}= \mathbf{f} \cdot \mathbf{u}

with f as above. Noticing that

F^\mu U_\mu = 0

and expanding this out we get

:

\gamma^2 \left(\mathbf{f} \cdot \mathbf{u} - \dot{\gamma} mc^2 
ight) = 0

Hence

:

rac{dK}{dt} = c^2 rac{d\gamma m}{dt}

which yields

:

K = \gamma m c^2 + S \,

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 c^2 \,

Thus, we interpret the total energy ''E'' of the particle as composed of its kinetic energy ''K'' and its Rest Energy ''m'' ''c''². Thus, we have

:

E = \gamma m c^2 \,

Deriving ''E''² = ''p''²''c''² + ''m''²''c''⁴

Using the relation

E = \gamma mc^2

, we can write the four-momentum as

:

p = \left(rac{E}{c}, \mathbf{p} 
ight)

.

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

:

rac{E^2}{c^2} - p^2 = P^\mu P_\mu = m^2 U^\mu U_\mu = m^2 c^2

i.e.

:

rac{E^2}{c^2} - p^2 = m^2 c^2

Hence

:

E^2 = p^2 c^2 + m^2 c^4 \,

This last relation is useful in many areas of physics.

EXAMPLES OF FOUR-VECTORS IN ELECTROMAGNETISM

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

:

J := \left( 
ho c, \mathbf{j} 
ight)

formed from the current density j and charge density ρ, and the Electromagnetic Four-potential defined by

:

\psi := \left( \phi ,  \mathbf{A} c 
ight)

formed from the vector potential A and the scalar potential

\phi \,

.

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

:

N :=\left(
u, 
u \mathbf{n}  
ight)

where

u

is the frequency of the wave and n is a unit vector in the travel direction of the wave. Notice that

:

N^\mu N_\mu = 
u ^2 \left(n^2 - 1 
ight) = 0

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

A wave packet of nearly Monochromatic light can be characterized by the Wave Vector , or four-wavevector
:

K = \left(rac{\omega}{c}, \mathbf{k} 
ight) \,

	CATEGORIES ABOUT FOUR-VECTOR

	minkowski spacetime

	relativity

	fundamental physics concepts

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Four-vector