- Four-vector
In relativity, a four-vector is a vector in a four-dimensional real
vector space , calledMinkowski space . It differs from a vector in that it can be transformed byLorentz transformations . The usage of the "four-vector" name tacitly assumes that its components refer to a standard basis. The components transform between these bases as thespace andtime coordinate differences, under spatial translations, rotations, and "boosts" (a change by a constant velocity to anotherinertial reference frame ). The set of all such translations, rotations, and boosts (calledPoincaré transformation s) forms thePoincaré group . The set of rotations and boosts (Lorentz transformation s, described by 4×4 matrices) forms theLorentz group .This article considers four-vectors in the context of
special relativity . Although the concept of four-vectors also extends togeneral relativity , some of the results stated in this article require modification in general relativity.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:
where = 0, 1, 2, 3, labels the
spacetime dimension s and where "c" is thespeed of light . The definition ensures that all the coordinates have the same units (of distance). [Jean-Bernard Zuber & Claude Itzykson, "Quantum Field Theory", pg 5 , ISBN 0070320713] [Charles W. Misner ,Kip S. Thorne &John A. Wheeler ,"Gravitation", pg 51, ISBN 0716703440] [George Sterman, "An Introduction to Quantum Field Theory", pg 4 , ISBN 0521311322] 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::
(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 (pseudo-)
inner product of two four-vectors and is defined (usingEinstein notation ) as:
where "η" is the
Minkowski metric . Sometimes this inner product is called the Minkowski inner product. It is not a true inner product in the mathematical sense because it is not positive definite. Note: some authors define "η" with the opposite sign::in which case :An important property of the inner product is that it is invariant (that is, a scalar): a change of coordinates does not result in a change in value of the inner product.
The inner product is often expressed as the effect of the dual vector of one vector on the other:
:
Here the s are the components of the dual vector of in the
dual basis and called thecovariant coordinates of , while the original components are called thecontravariant coordinates. Lower and upper indices indicate always covariant and contravariant coordinates, respectively.The relation between the covariant and contravariant coordinates is:
:.
The four-vectors are arrows on the
spacetime diagram orMinkowski diagram . In this article, four-vectors will be referred to simply as vectors.Four-vectors may be classified as either spacelike, timelike or null. Spacelike, timelike, and null vectors are ones whose inner product with themselves is greater than, less than, and equal to zero respectively.
In special relativity (but not general relativity), the derivative of a four-vector with respect to a scalar (invariant) is itself a four-vector.
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). As proper time is an invariant, this guarantees that the proper-time-derivative of any four-vector is itself a four-vector. It is then important to find a relation between this proper-time-derivative and another time derivative (using the time of an inertial reference frame). This relation is provided by the time transformation in the Lorentz transformations and is:
:
where γ is the
Lorentz factor . Important four-vectors in relativity theory can now be defined, such as thefour-velocity of anworld line is defined by::
where
:
for "i" = 1, 2, 3. Notice that
:
The
four-acceleration is given by::
Since the magnitude of is a constant, the four acceleration is (pseudo-)orthogonal to the four velocity, i.e. the Minkowski inner product of the four-acceleration and the four-velocity is zero:
:
which is true for all world lines.
The
four-momentum for a massive particle is given by::
where "m" is the
invariant mass of the particle and is the relativistic momentum.The
four-force is defined by::
For a particle of constant mass, this is equivalent to
:
where
:.
Physics of four-vectors
The power and elegance of the four-vector formalism may be demonstrated by seeing that known relations between energy and matter are embedded into it.
="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
:
with f as above. Noticing that and expanding this out we get
:
Hence
:
which yields
:
for some constant "S". When the particle is at rest (u = 0), we take its kinetic energy to be zero ("K" = 0). This gives
:
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"2 = "p"2"c"2 + "m"2"c"4=Using the relation , we can write the four-momentum as
:.
Taking the inner product of the four-momentum with itself in two different ways, we obtain the relation
:
i.e.
:
Hence
:
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:
formed from the current density j and charge density ρ, and the
electromagnetic four-potential defined by:
formed from the vector potential a and the scalar potential .
A plane electromagnetic wave can be described by the
four-frequency defined as:
where is the frequency of the wave and n is a unit vector in the travel direction of the wave. Notice that
:
so that the four-frequency is always a null vector.
A wave packet of nearly
monochromatic light can be characterized by thewave vector , or four-wavevector:ee also
*
four-velocity
*four-acceleration
*four-momentum
*four-force
*four-current
*electromagnetic four-potential
*four-gradient
*four-frequency
*wave vector
*Basic introduction to the mathematics of curved spacetime
*Minkowski space References
*Rindler, W. "Introduction to Special Relativity (2nd edn.)" (1991) Clarendon Press Oxford ISBN 0-19-853952-5
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