Four-vector

From Example Problems
Jump to navigation Jump to search

In relativity, a four-vector is a vector in a four-dimensional real vector space, called Minkowski space. The usage of the four-vector name tacitly assumes that its components refer to a standard basis. The components transform between these bases as the space and time coordinate differences, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\Delta t, \Delta x, \Delta y, \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) forms the Poincaré group. The set of rotations and boosts (Lorentz transformations, described by 4×4 matrices) forms the Lorentz group.

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{x} := \left(ct, x, y, z \right) }

for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta \mathbf{x}:= \left(\Delta ct, \Delta x, \Delta y, \Delta z \right) }

(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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v} is defined (using Einstein notation) as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u \cdot v = u^{\mu} \eta_{\mu \nu} v^{\nu} = \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 inner product is often expressed as the effect of the dual vector of one vector on the other:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u \cdot v = u^*(v) = u{_\nu}v^{\nu} }

Here the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_{\nu}} -s are the components of the dual vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u^*} of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u} in the dual basis and called the covariant coordinates of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u} , while the original Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u^\nu} components are called the contravariant coordinates. Lower and upper indices indicate always covariant and contravariant coordinates, respectively.

The relation between the covariant and contravariant coordinates is:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_{\mu} = u^{\nu} \eta_{\mu \nu} \,} .


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

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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d \tau}{dt}=\frac{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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{x}(\tau) } world line is defined by:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{U} := \frac{d\mathbf{x}}{d \tau}= \frac{d\mathbf{x}}{dt}\frac{dt}{d \tau}= \left(\gamma c, \gamma \mathbf{u} \right)}

where

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u^i = \frac{dx^i}{dt}}

for i = 1, 2, 3. Notice that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle || \mathbf{U} ||^2 = U^{\mu} U_{\mu} = c^2 \,}

The four-acceleration is given by:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A =\frac{dU }{d \tau} = \left(\gamma \dot{\gamma} c, \gamma \dot{\gamma} \mathbf{u} + \gamma^2 \mathbf{\dot{u}} \right)}

Since the magnitude Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt{ | U_\mu U^\mu | }} of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{U}} 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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_\mu U^\mu =\frac{1}{2} \frac{\partial (U^\mu U_\mu)}{\partial \tau } = 0 \,}

which is true for all world lines.

The four-momentum is for a massive particle is given by:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p :=m U = \left(\gamma mc, \mathbf{p} \right)}

where m is the invariant mass of the particle.

The four-force is defined by:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F := m A = \left(\gamma \dot{\gamma} mc, \gamma \mathbf{f} \right) }

where

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{f} = \frac{d\mathbf{p}}{dt} = m \dot{\gamma} \mathbf{u} + m \gamma \mathbf{\dot{u}} } .

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 = 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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dK}{dt}= \mathbf{f} \cdot \mathbf{u} }

with f as above. Noticing that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F^\mu U_\mu = 0} and expanding this out we get

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma^2 \left(\mathbf{f} \cdot \mathbf{u} - \dot{\gamma} mc^2 \right) = 0 }

Hence

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dK}{dt} = c^2 \frac{d\gamma m}{dt}}

which yields

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 c2. Thus, we have

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E = \gamma m c^2 \,}

Deriving E2 = p2c2 + m2c4

Using the relation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E = \gamma mc^2} , we can write the four-momentum as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p = \left(\frac{E}{c}, \mathbf{p} \right)} .

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{E^2}{c^2} - p^2 = P^\mu P_\mu = m^2 U^\mu U_\mu = m^2 c^2 }

i.e.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{E^2}{c^2} - p^2 = m^2 c^2 }

Hence

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J := \left( \rho c, \mathbf{j} \right) }

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi :=\left(\psi, \mathbf{A} c \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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N :=\left(\nu, \nu \mathbf{n} \right)}

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nu} is the frequency of the wave and n is a unit vector in the travel direction of the wave. Notice that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N^\mu N_\mu = \nu ^2 \left(n^2 - 1 \right) = 0 }

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

See also

References

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