Four-velocity

The path of an object in three-dimensional space (in an inertial frame) may be expressed in terms of three spatial coordinate functions of time , where is an index which takes values 1, 2, 3. The three coordinates form the 3d position vector, written as a column vector \vec{x}(t) = \begin{bmatrix} x^1(t) \\[0.7ex] x^2(t) \\[0.7ex] x^3(t) \end{bmatrix} \, . The components of the velocity \vec{u} (tangent to the curve) at any point on the world line are \vec{u} = \begin{bmatrix} u^1 \\ u^2 \\ u^3 \end{bmatrix} = \frac{d \vec{x}}{dt} = \begin{bmatrix} \tfrac{dx^1}{dt} \\ \tfrac{dx^2}{dt} \\ \tfrac{dx^3}{dt} \end{bmatrix}. Each component is simply written u^i = \frac{dx^i}{dt} == Theory of relativity ==

In Einstein's theory of relativity, the path of an object moving relative to a particular frame of reference is defined by four coordinate functions , where is a spacetime index which takes the value 0 for the timelike component, and 1, 2, 3 for the spacelike coordinates. The zeroth component is defined as the time coordinate multiplied by , x^0 = ct\,, Each function depends on one parameter τ called its proper time. As a column vector, \mathbf{x} = \begin{bmatrix} x^0(\tau) \\ x^1(\tau) \\ x^2(\tau) \\ x^3(\tau) \\ \end{bmatrix}\,. Time dilation From time dilation, the differentials in coordinate time and proper time are related by dt = \gamma(u) d\tau where the Lorentz factor, \gamma(u) = \frac{1}{\sqrt{1-\frac{u^2}{c^2}}}\,, is a function of the Euclidean norm of the 3d velocity vector {{nowrap|\vec{u}:}} u = \left\|\ \vec{u}\ \right\| = \sqrt{ \left(u^1\right)^2 + \left(u^2\right)^2 + \left(u^3\right)^2} \,. Definition of the four-velocity The four-velocity is the tangent four-vector of a timelike world line. The four-velocity \mathbf{U} at any point of world line \mathbf{X}(\tau) is defined as: \mathbf{U} = \frac{d\mathbf{X}}{d \tau} where \mathbf{X} is the four-position and \tau is the proper time. The four-velocity defined here using the proper time of an object does not exist for world lines for massless objects such as photons travelling at the speed of light; nor is it defined for tachyonic world lines, where the tangent vector is spacelike. Components of the four-velocity The relationship between the time and the coordinate time is defined by x^0 = ct . Taking the derivative of this with respect to the proper time , we find the velocity component for : U^0 = \frac{dx^0}{d\tau} = \frac{d(ct)}{d\tau} = c\frac{dt}{d\tau} = c \gamma(u) and for the other 3 components to proper time we get the velocity component for : U^i = \frac{dx^i}{d\tau} = \frac{dx^i}{dt} \frac{dt}{d\tau} = \frac{dx^i}{dt} \gamma(u) = \gamma(u) u^i where we have used the chain rule and the relationships u^i = {dx^i \over dt } \,,\quad \frac{dt}{d\tau} = \gamma (u) Thus, we find for the four-velocity {{nowrap|\mathbf{U}:}} \mathbf{U} = \gamma \begin{bmatrix} c \\ \vec{u} \\ \end{bmatrix}. Written in standard four-vector notation this is: \mathbf{U} = \gamma \left(c, \vec{u}\right) = \left(\gamma c, \gamma \vec{u}\right) where \gamma c is the temporal component and \gamma \vec{u} is the spatial component. In terms of the synchronized clocks and rulers associated with a particular slice of flat spacetime, the three spacelike components of four-velocity define a traveling object's proper velocity \gamma \vec{u} = d\vec{x} / d\tau i.e. the rate at which distance is covered in the reference map frame per unit proper time elapsed on clocks traveling with the object. Unlike most other four-vectors, the four-velocity has only 3 independent components u_x, u_y, u_z instead of 4. The \gamma factor is a function of the three-dimensional velocity \vec{u}. When certain Lorentz scalars are multiplied by the four-velocity, one then gets new physical four-vectors that have 4 independent components. For example: • Four-momentum: \mathbf{P} = m_o\mathbf{U} = \gamma m_o\left(c, \vec{u}\right) = m\left(c, \vec{u}\right) = \left(mc, m\vec{u}\right) = \left(mc, \vec{p}\right) = \left(\frac{E}{c},\vec{p}\right), where m_o is the rest mass • Four-current density: \mathbf{J} = \rho_o\mathbf{U} = \gamma \rho_o\left(c, \vec{u}\right) = \rho\left(c, \vec{u}\right) = \left(\rho c, \rho\vec{u}\right) = \left(\rho c, \vec{j}\right) , where \rho_o is the charge density Effectively, the \gamma factor combines with the Lorentz scalar term to make the 4th independent component m = \gamma m_o and \rho = \gamma \rho_o. Magnitude Using the differential of the four-position in the rest frame, the magnitude of the four-velocity can be obtained by the Minkowski metric with signature : \left\|\mathbf{U}\right\|^2 = \eta_{\mu\nu} U^\mu U^\nu = \eta_{\mu\nu} \frac{dX^\mu}{d\tau} \frac{dX^\nu}{d\tau} = - c^2 \,, in short, the magnitude of the four-velocity for any object is always a fixed constant: \left\|\mathbf{U}\right\|^2 = - c^2 In a moving frame, the same norm is: \left\|\mathbf{U}\right\|^2 = {\gamma(u)}^2 \left( - c^2 + \vec{u} \cdot \vec{u} \right) , so that: - c^2 = {\gamma(u)}^2 \left( - c^2 + \vec{u} \cdot \vec{u} \right) , which reduces to the definition of the Lorentz factor. ==See also==