Four-gradient

In differential geometry, the four-gradient (or 4-gradient) \boldsymbol{\partial} is the four-vector analogue of the gradient \vec{\boldsymbol{\nabla}} from vector calculus. In special relativity and in quantum mechanics, the four-gradient is used to define the properties and relations between the various physical four-vectors and tensors.

In differential geometry, the four-gradient (or 4-gradient) \boldsymbol{\partial} is the four-vector analogue of the gradient \vec{\boldsymbol{\nabla}} from vector calculus. In special relativity and in quantum mechanics, the four-gradient is used to define the properties and relations between the various physical four-vectors and tensors. ==Notation== This article uses the metric signature. SR and GR are abbreviations for special relativity and general relativity respectively. c indicates the speed of light in vacuum. \eta_{\mu\nu} = \operatorname{diag}[1,-1,-1,-1] is the flat spacetime metric of SR. There are alternate ways of writing four-vector expressions in physics: The four-vector style can be used: \mathbf{A} \cdot \mathbf{B} , which is typically more compact and can use vector notation, (such as the inner product "dot"), always using bold uppercase to represent the four-vector, and bold lowercase to represent 3-space vectors, e.g. \vec{\mathbf{a}} \cdot \vec{\mathbf{b}} . Most of the 3-space vector rules have analogues in four-vector mathematics. The Ricci calculus style can be used: A^\mu \eta_{\mu\nu} B^\nu , which uses tensor index notation and is useful for more complicated expressions, especially those involving tensors with more than one index, such as F^{\mu\nu} = \partial^\mu A^\nu - \partial^\nu A^\mu . The Latin tensor index ranges in and represents a 3-space vector, e.g. A^i = \left(a^1, a^2, a^3\right) = \vec{\mathbf{a}} . The Greek tensor index ranges in and represents a 4-vector, e.g. A^\mu = \left(a^0, a^1, a^2, a^3\right) = \mathbf{A} . In SR physics, one typically uses a concise blend, e.g. \mathbf{A} = \left(a^0, \vec{\mathbf{a}}\right) , where a^0 represents the temporal component and \vec{\mathbf{a}} represents the spatial 3-component. Tensors in SR are typically 4D (m,n) -tensors, with m upper indices and n lower indices, with the 4D indicating 4 dimensions = the number of values each index can take. The tensor contraction used in the Minkowski metric can go to either side (see Einstein notation): \mathbf{A} \cdot \mathbf{B} = A^\mu \eta_{\mu\nu} B^\nu = A_\nu B^\nu = A^\mu B_\mu = \sum_{\mu=0}^{3} a^\mu b_\mu = a^0 b^0 - \sum_{i=1}^{3} a^i b^i = a^0 b^0 - \vec{\mathbf{a}} \cdot \vec{\mathbf{b}} ==Definition== The 4-gradient covariant components compactly written in four-vector and Ricci calculus notation are: \dfrac{\partial}{\partial X^\mu} = \left(\partial_0,\partial_1,\partial_2,\partial_3\right) = \left(\partial_0,\partial_i\right) = \left(\frac{1}{c}\frac{\partial}{\partial t}, \vec{\nabla}\right) = \left(\frac{\partial_t}{c}, \vec{\nabla}\right) = \left(\frac{\partial_t}{c}, \partial_x,\partial_y,\partial_z\right) = \partial_\mu = {}_{,\mu} The comma in the last part above {}_{,\mu} implies the partial differentiation with respect to 4-position X^\mu . The contravariant components are: \boldsymbol{\partial} = \partial^\alpha = \eta^{\alpha \beta} \partial_\beta = \left(\partial^0,\partial^1,\partial^2,\partial^3\right) = \left(\partial^0,\partial^i\right) = \left(\frac{1}{c} \frac{\partial}{\partial t}, -\vec{\nabla} \right) = \left(\frac{\partial_t}{c}, -\vec{\nabla}\right) = \left(\frac{\partial_t}{c}, -\partial_x,-\partial_y,-\partial_z\right) Alternative symbols to \partial_\alpha are \Box and D (although \Box can also signify \partial^\mu \partial_\mu as the d'Alembert operator). In GR, one must use the more general metric tensor g^{\alpha \beta} and the tensor covariant derivative \nabla_{\mu} = {}_{;\mu} (not to be confused with the vector 3-gradient \vec{\nabla} ). The covariant derivative \nabla_{\nu} incorporates the 4-gradient \partial_\nu plus spacetime curvature effects via the Christoffel symbols \Gamma^{\mu}{}_{\sigma \nu} The strong equivalence principle can be stated as: "Any physical law which can be expressed in tensor notation in SR has exactly the same form in a locally inertial frame of a curved spacetime." The 4-gradient commas (,) in SR are simply changed to covariant derivative semi-colons (;) in GR, with the connection between the two using Christoffel symbols. This is known in relativity physics as the "comma to semi-colon rule". So, for example, if T^{\mu\nu}{}_{,\mu} = 0 in SR, then T^{\mu\nu}{}_{;\mu} = 0 in GR. On a (1,0)-tensor or 4-vector this would be: \begin{align} \nabla_\beta V^\alpha &= \partial_\beta V^\alpha + V^\mu \Gamma^{\alpha}{}_{\mu\beta} \\[0.1ex] V^{\alpha}{}_{ ;\beta} &= V^{\alpha}{}_{ ,\beta} + V^\mu \Gamma^{\alpha}{}_{\mu\beta} \end{align} On a (2,0)-tensor this would be: \begin{align} \nabla_{\nu} T^{\mu \nu} &= \partial_\nu T^{\mu \nu} + \Gamma^{\mu}{}_{\sigma \nu}T^{\sigma \nu} + \Gamma^{\nu}{}_{\sigma \nu} T^{\mu \sigma} \\ T^{\mu \nu}{}_{;\nu} &= T^{\mu \nu}{}_{,\nu} + \Gamma^{\mu}{}_{\sigma \nu}T^{\sigma \nu} + \Gamma^{\nu}{}_{\sigma \nu} T^{\mu \sigma} \end{align} ==Usage== The 4-gradient is used in a number of different ways in special relativity (SR): Throughout this article the formulas are all correct for the flat spacetime Minkowski coordinates of SR, but have to be modified for the more general curved space coordinates of general relativity (GR). The 4-divergence of the 4-position X^\mu = \left(ct, \vec{\mathbf{x}}\right) gives the dimension of spacetime: \boldsymbol{\partial} \cdot \mathbf{X} = \partial^\mu \eta_{\mu\nu} X^\nu = \partial_\nu X^\nu = \left(\frac{\partial_t}{c}, -\vec{\nabla}\right) \cdot (ct,\vec{x}) = \frac{\partial_t}{c}(ct) + \vec{\nabla}\cdot \vec{x} = (\partial_t t) + (\partial_x x + \partial_y y + \partial_z z) = (1) + (3) = 4 The 4-divergence of the 4-current density J^\mu = \left(\rho c, \vec{\mathbf{j}}\right) = \rho_o U^\mu = \rho_o \gamma\left(c, \vec{\mathbf{u}}\right) = \left(\rho c, \rho \vec{\mathbf{u}}\right) gives a conservation law – the conservation of charge: \boldsymbol{\partial} \cdot \mathbf{J} = \partial^\mu \eta_{\mu\nu} J^\nu = \partial_\nu J^\nu = \left(\frac{\partial_t}{c}, -\vec{\nabla}\right) \cdot (\rho c,\vec{j}) = \frac{\partial_t}{c} (\rho c) + \vec{\nabla} \cdot \vec{j} = \partial_t \rho + \vec{\nabla} \cdot \vec{j} = 0 This means that the time rate of change of the charge density must equal the negative spatial divergence of the current density \partial_t \rho = -\vec{\nabla}\cdot \vec{j} . In other words, the charge inside a box cannot just change arbitrarily, it must enter and leave the box via a current. This is a continuity equation. The 4-divergence of the 4-number flux (4-dust) N^\mu = \left(nc, \vec{\mathbf{n}}\right) = n_o U^\mu = n_o \gamma\left(c, \vec{\mathbf{u}}\right) = \left(nc, n\vec{\mathbf{u}}\right) is used in particle conservation: \boldsymbol{\partial} \cdot \mathbf{N} = \partial^\mu \eta_{\mu\nu} N^\nu = \partial_\nu N^\nu = \left(\frac{\partial_t}{c}, -\vec{\nabla}\right) \cdot \left(nc, n\vec{\mathbf{u}}\right) = \frac{\partial_t}{c} \left(nc\right) + \vec{\nabla} \cdot n \vec{\mathbf{u}} = \partial_t n + \vec{\nabla}\cdot n\vec{\mathbf{u}} = 0 This is a conservation law for the particle number density, typically something like baryon number density. The 4-divergence of the electromagnetic 4-potential A^\mu = \left(\frac{\phi}{c}, \vec{\mathbf{a}}\right) is used in the Lorenz gauge condition: \boldsymbol{\partial} \cdot \mathbf{A} = \partial^\mu \eta_{\mu\nu} A^\nu = \partial_\nu A^\nu = \left(\frac{\partial_t}{c}, -\vec{\nabla}\right) \cdot \left(\frac{\phi}{c}, \vec{a}\right) = \frac{\partial_t}{c} \left(\frac{\phi}{c}\right) + \vec{\nabla} \cdot \vec{a} = \frac{\partial_t \phi}{c^2} + \vec{\nabla} \cdot \vec{a} = 0 This is the equivalent of a conservation law for the EM 4-potential. The 4-divergence of the transverse traceless 4D (2,0)-tensor h^{\mu\nu}_{TT} representing gravitational radiation in the weak-field limit (i.e. freely propagating far from the source). The transverse condition \boldsymbol{\partial} \cdot h^{\mu\nu}_{TT} = \partial_\mu h^{\mu\nu}_{TT} = 0 is the equivalent of a conservation equation for freely propagating gravitational waves. The 4-divergence of the stress–energy tensor T^{\mu \nu} as the conserved Noether current associated with spacetime translations, gives four conservation laws in SR: The conservation of energy (temporal direction) and the conservation of linear momentum (3 separate spatial directions). \boldsymbol{\partial} \cdot T^{\mu \nu} = \partial_{\nu} T^{\mu \nu} = T^{\mu \nu}{}_{,\nu} = 0^\mu = (0,0,0,0) It is often written as: \partial_{\nu} T^{\mu \nu} = T^{\mu \nu}{}_{,\nu} = 0 where it is understood that the single zero is actually a 4-vector zero 0^\mu = (0,0,0,0) . When the conservation of the stress–energy tensor for a perfect fluid is combined with the conservation of particle number density ( \boldsymbol{\partial} \cdot \mathbf{N} = 0 ), both utilizing the 4-gradient, one can derive the relativistic Euler equations, which in fluid mechanics and astrophysics are a generalization of the Euler equations that account for the effects of special relativity. These equations reduce to the classical Euler equations if the fluid 3-space velocity is much less than the speed of light, the pressure is much less than the energy density, and the latter is dominated by the rest mass density. In flat spacetime and using Cartesian coordinates, if one combines this with the symmetry of the stress–energy tensor, one can show that angular momentum (relativistic angular momentum) is also conserved: \partial_\nu \left(x^{\alpha} T^{\mu \nu} - x^{\mu} T^{\alpha \nu}\right) = \left(x^{\alpha} T^{\mu \nu} - x^{\mu} T^{\alpha \nu}\right)_{,\nu} = 0^{\alpha \mu} where this zero is actually a (2,0)-tensor zero. \begin{align} \boldsymbol{\partial} [\mathbf{X}] = \partial^\mu[X^\nu] = X^{\nu_,\mu} &= \left(\frac{\partial_t}{c}, -\vec{\nabla}\right)\left[\left(ct, \vec{x}\right)\right] = \left(\frac{\partial_t}{c}, -\partial_x, -\partial_y, -\partial_z\right)[(ct, x, y, z)], \\[3pt] &= \begin{bmatrix} \frac{\partial_t}{c} ct & \frac{\partial_t}{c} x & \frac{\partial_t}{c} y & \frac{\partial_t}{c} z \\ -\partial_x ct & -\partial_x x & -\partial_x y & -\partial_x z \\ -\partial_y ct & -\partial_y x & -\partial_y y & -\partial_y z \\ -\partial_z ct & -\partial_z x & -\partial_z y & -\partial_z z \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{bmatrix} \\[3pt] &= \operatorname{diag}[1,-1,-1,-1] = \eta^{\mu\nu}. \end{align} For the Minkowski metric, the components \left[\eta^{\mu\mu}\right] = 1/\left[\eta_{\mu\mu}\right] ( \mu not summed), with non-diagonal components all zero. For the Cartesian Minkowski Metric, this gives \eta^{\mu\nu} = \eta_{\mu\nu} = \operatorname{diag}[1, -1, -1, -1] . Generally, \eta_\mu^\nu = \delta_\mu^\nu = \operatorname{diag}[1,1,1,1] , where \delta_\mu^\nu is the 4D Kronecker delta. X^{\mu'} = \Lambda^{~\mu'}_{\nu} X^\nu and since \Lambda^{\mu'}_\nu are just constants, then \dfrac{\partial X^{\mu'} }{ \partial X^\nu} = \Lambda^{\mu'}_\nu Thus, by definition of the 4-gradient \partial_\nu \left[X^{\mu'}\right] = \left(\dfrac{\partial}{\partial X^\nu}\right)\left[X^{\mu'}\right] = \dfrac{\partial X^{\mu'}}{\partial X^\nu} = \Lambda^{\mu'}_\nu This identity is fundamental. Components of the 4-gradient transform according to the inverse of the components of 4-vectors. So the 4-gradient is the "archetypal" one-form. \begin{align} \mathbf{U} \cdot \boldsymbol{\partial} &= U^\mu \eta_{\mu\nu} \partial^\nu = \gamma \left(c, \vec{u}\right) \cdot \left(\frac{\partial_t}{c}, -\vec{\nabla}\right) = \gamma \left(c \frac{\partial_t}{c} + \vec{u} \cdot \vec{\nabla} \right) = \gamma \left(\partial_t + \frac{dx}{dt} \partial_x + \frac{dy}{dt} \partial_y + \frac{dz}{dt} \partial_z \right) = \gamma \frac{d}{dt} = \frac{d}{d\tau} \\ \frac{d}{d\tau} &= \frac{dX^\mu}{dX^\mu} \frac{d}{d\tau} = \frac{dX^\mu}{d\tau} \frac{d}{dX^\mu} = U^\mu \partial_\mu = \mathbf{U} \cdot \boldsymbol{\partial} \end{align} The fact that \mathbf{U} \cdot \boldsymbol{\partial} is a Lorentz scalar invariant shows that the total derivative with respect to proper time \frac{d}{d\tau} is likewise a Lorentz scalar invariant. So, for example, the 4-velocity U^\mu is the derivative of the 4-position X^\mu with respect to proper time: \frac{d}{d\tau} \mathbf{X} = (\mathbf{U} \cdot \boldsymbol{\partial})\mathbf{X} = \mathbf{U} \cdot \boldsymbol{\partial}[\mathbf{X}] = U^\alpha \cdot \eta^{\mu\nu} = U^\alpha \eta_{\alpha \nu} \eta^{\mu\nu} = U^\alpha \delta_\alpha^\mu = U^\mu = \mathbf{U} or \frac{d}{d\tau} \mathbf{X} = \gamma\frac{d}{dt} \mathbf{X} = \gamma\frac{d}{dt} \left(ct, \vec{x}\right) = \gamma \left(\frac{d}{dt}ct,\frac{d}{dt}\vec{x}\right) = \gamma \left(c, \vec{u}\right) = \mathbf{U} Another example, the 4-acceleration A^\mu is the proper-time derivative of the 4-velocity U^\mu : \begin{align} \frac{d}{d\tau} \mathbf{U} &= (\mathbf{U} \cdot \boldsymbol{\partial})\mathbf{U} = \mathbf{U} \cdot \boldsymbol{\partial}[\mathbf{U}] = U^\alpha \eta_{\alpha\mu}\partial^\mu\left[U^\nu\right] \\ &= U^\alpha \eta_{\alpha\mu}\begin{bmatrix} \frac{\partial_t}{c} \gamma c & \frac{\partial_t}{c} \gamma \vec{u} \\ -\vec{\nabla}\gamma c & -\vec{\nabla}\gamma \vec{u} \end{bmatrix} = U^\alpha \begin{bmatrix}\ \frac{\partial_t}{c} \gamma c & 0 \\ 0 & \vec{\nabla}\gamma \vec{u} \end{bmatrix} \\[3pt] &= \gamma \left(c \frac{\partial_t}{c} \gamma c, \vec{u} \cdot \nabla\gamma \vec{u}\right) = \gamma \left(c \partial_t \gamma, \frac{d}{dt}\left[\gamma \vec{u}\right]\right) = \gamma \left(c \dot{\gamma}, \dot{\gamma} \vec{u} + \gamma \dot{\vec{u}}\right) = \mathbf{A} \end{align} or \frac{d}{d\tau} \mathbf{U} = \gamma \frac{d}{dt} (\gamma c,\gamma \vec{u}) =\gamma \left(\frac{d}{dt} [\gamma c],\frac{d}{dt} [\gamma \vec{u}] \right) = \gamma (c \dot{\gamma}, \dot{\gamma} \vec{u} + \gamma \dot{\vec{u}} ) = \mathbf{A} F^{\mu\nu} = \partial^\mu A^\nu - \partial^\nu A^\mu = \begin{bmatrix} 0 & -E_x/c & -E_y/c & -E_z/c \\ E_x/c & 0 & -B_z & B_y \\ E_y/c & B_z & 0 & -B_x \\ E_z/c & -B_y & B_x & 0 \end{bmatrix} where: Electromagnetic 4-potential A^\mu = \mathbf{A} = \left(\frac{\phi}{c}, \vec{\mathbf{a}}\right) , not to be confused with the 4-acceleration \mathbf{A} = \gamma \left(c \dot{\gamma}, \dot{\gamma} \vec{u} + \gamma \dot{\vec{u}}\right) The electric scalar potential is \phi The magnetic 3-space vector potential is \vec{\mathbf{a}} By applying the 4-gradient again, and defining the 4-current density as J^{\beta} = \mathbf{J} = \left(c\rho, \vec{\mathbf{j}}\right) one can derive the tensor form of the Maxwell equations: \partial_{\alpha} F^{\alpha\beta} = \mu_o J^{\beta} \partial_\gamma F_{ \alpha \beta } + \partial_\alpha F_{ \beta \gamma } + \partial_\beta F_{ \gamma \alpha } = 0_{\alpha \beta \gamma} where the second line is a version of the Bianchi identity (Jacobi identity). K^\mu = \mathbf{K} = \left(\frac{\omega}{c}, \vec{\mathbf{k}}\right) = \boldsymbol{\partial} [-\Phi] = -\boldsymbol{\partial} [\Phi] This is mathematically equivalent to the definition of the phase of a wave (or more specifically a plane wave): \mathbf{K} \cdot \mathbf{X} = \omega t - \vec{\mathbf{k}} \cdot \vec{\mathbf{x}} = -\Phi where 4-position \mathbf{X} = \left(ct, \vec{\mathbf{x}}\right) , \omega is the temporal angular frequency, \vec{\mathbf{k}} is the spatial 3-space wavevector, and \Phi is the Lorentz scalar invariant phase. \partial [\mathbf{K} \cdot \mathbf{X}] = \partial \left[\omega t - \vec{\mathbf{k}} \cdot \vec{\mathbf{x}}\right] = \left(\frac{\partial_t}{c}, -\nabla\right)\left[\omega t - \vec{\mathbf{k}} \cdot \vec{\mathbf{x}}\right] = \left(\frac{\partial_t}{c}\left[\omega t - \vec{\mathbf{k}} \cdot \vec{\mathbf{x}}\right], -\nabla\left[\omega t - \vec{\mathbf{k}} \cdot \vec{\mathbf{x}}\right]\right) = \left(\frac{\partial_t}{c}[\omega t], -\nabla\left[- \vec{\mathbf{k}} \cdot \vec{\mathbf{x}}\right]\right) = \left(\frac{\omega}{c}, \vec{\mathbf{k}}\right) = \mathbf{K} with the assumption that the plane wave \omega and \vec{\mathbf{k}} are not explicit functions of t or \vec{\mathbf{x}} . The explicit form of an SR plane wave \Psi_n(\mathbf{X}) can be written as: \Psi_n(\mathbf{X}) = A_ne^{-i(\mathbf{K_n}\cdot\mathbf{X})} = A_ne^{i(\Phi_n)} where A_n is a (possibly complex) amplitude. A general wave \Psi(\mathbf{X}) would be the superposition of multiple plane waves: \Psi(\mathbf{X}) = \sum_{ n }[\Psi_n(\mathbf{X})] = \sum_{ n }\left[A_{n} e^{-i(\mathbf{K_n}\cdot \mathbf{X})}\right] = \sum_{ n }\left[A_{n} e^{i(\Phi_n)}\right] Again using the 4-gradient, \partial [\Psi(\mathbf{X})] = \partial\left[Ae^{-i(\mathbf{K}\cdot\mathbf{X})}\right] = -i\mathbf{K} \left[Ae^{-i(\mathbf{K}\cdot\mathbf{X})}\right] = -i\mathbf{K} [\Psi(\mathbf{X})] or \boldsymbol{\partial} = -i \mathbf{K} which is the 4-gradient version of complex-valued plane waves \boldsymbol{\partial} \cdot \boldsymbol{\partial} = \partial^\mu \cdot \partial^\nu = \partial^\mu \eta_{\mu\nu} \partial^\nu = \partial_\nu \partial^\nu = \frac{1}{c^2}\frac{\partial^2}{\partial t^2} - \vec{\nabla}^2 = \left(\frac{\partial_t}{c}\right)^2 - \vec{\nabla}^2. As it is the dot product of two 4-vectors, the d'Alembertian is a Lorentz invariant scalar. Occasionally, in analogy with the 3-dimensional notation, the symbols \Box and \Box^2 are used for the 4-gradient and d'Alembertian respectively. More commonly however, the symbol \Box is reserved for the d'Alembertian. Some examples of the 4-gradient as used in the d'Alembertian follow: In the Klein–Gordon relativistic quantum wave equation for spin-0 particles (ex. Higgs boson): \left[(\boldsymbol{\partial} \cdot \boldsymbol{\partial}) + \left(\frac{m_0 c}{\hbar}\right)^2\right]\psi = \left[\left(\frac{\partial_t^2}{c^2} - \vec{\nabla}^2\right) + \left(\frac{m_0 c}{\hbar}\right)^2\right]\psi = 0 In the wave equation for the electromagnetic field (using Lorenz gauge (\boldsymbol{\partial} \cdot \mathbf{A}) = \left(\partial_\mu A^\mu\right) = 0 ): In vacuum: (\boldsymbol{\partial} \cdot \boldsymbol{\partial}) \mathbf{A} = (\boldsymbol{\partial} \cdot \boldsymbol{\partial}) A^{\alpha} = \mathbf{0} = 0^{\alpha} With a 4-current source, not including the effects of spin: (\boldsymbol{\partial} \cdot \boldsymbol{\partial}) \mathbf{A} = (\boldsymbol{\partial} \cdot \boldsymbol{\partial}) A^{\alpha} = \mu_0 \mathbf{J} = \mu_0 J^{\alpha} With quantum electrodynamics source, including effects of spin: (\boldsymbol{\partial} \cdot \boldsymbol{\partial}) \mathbf{A} = (\boldsymbol{\partial} \cdot \boldsymbol{\partial}) A^{\alpha} = e\bar{\psi} \gamma^{\alpha} \psi where: Electromagnetic 4-potential \mathbf{A} = A^{\alpha} = \left(\frac{\phi}{c}, \mathbf{\vec{a}}\right) is an electromagnetic vector potential 4-current density \mathbf{J} = J^{\alpha} = \left(\rho c, \mathbf{\vec{j}}\right) is an electromagnetic current density Dirac Gamma matrices \gamma^\alpha = \left(\gamma^0, \gamma^1, \gamma^2, \gamma^3\right) provide the effects of spin In the wave equation of a gravitational wave (using a similar Lorenz gauge \left(\partial_\mu h^{\mu\nu}_{TT}\right) = 0 ) (\boldsymbol{\partial} \cdot \boldsymbol{\partial}) h^{\mu\nu}_{TT} = 0 where h^{\mu\nu}_{TT} is the transverse traceless 2-tensor representing gravitational radiation in the weak-field limit (i.e. freely propagating far from the source). Further conditions on h^{\mu\nu}_{TT} are: Purely spatial: \mathbf{U} \cdot h^{\mu\nu}_{TT} = h^{0\nu}_{TT} = 0 Traceless: \eta_{\mu\nu} h^{\mu\nu}_{TT} = h^{\nu}_{TT\nu} = 0 Transverse: \boldsymbol{\partial} \cdot h^{\mu\nu}_{TT} = \partial_\mu h^{\mu\nu}_{TT} = 0 In the 4-dimensional version of Green's function: (\boldsymbol{\partial} \cdot \boldsymbol{\partial}) G\left[\mathbf{X} - \mathbf{X'}\right] = \delta^{(4)}\left[\mathbf{X} - \mathbf{X'}\right] where the 4D Delta function is: \delta^{(4)}[\mathbf{X}] = \frac{1}{(2 \pi)^4} \int d^4 \mathbf{K} e^{-i(\mathbf{K} \cdot \mathbf{X})} \int_\Omega d^4X \left(\partial_\mu V^\mu\right) = \oint_{\partial \Omega} dS \left(V^\mu N_\mu\right) or \int_\Omega d^4X \left(\boldsymbol{\partial} \cdot \mathbf{V}\right) = \oint_{\partial \Omega} dS \left(\mathbf{V} \cdot \mathbf{N}\right) where \mathbf{V} = V^\mu is a 4-vector field defined in \Omega \boldsymbol{\partial}\cdot\mathbf{V} = \partial_\mu V^\mu is the 4-divergence of V \mathbf{V}\cdot\mathbf{N} = V^\mu N_\mu is the component of V along direction N \Omega is a 4D simply connected region of Minkowski spacetime \partial \Omega = S is its 3D boundary with its own 3D volume element dS \mathbf{N} = N^\mu is the outward-pointing normal d^4X = (c\,dt) \left(d^3x\right) = (c\,dt) (dx\,dy\,dz) is the 4D differential volume element \mathbf{P_T} = \mathbf{P} + q\mathbf{A} where \mathbf{P} = \left(\frac{E}{c}, \vec{\mathbf{p}}\right) and \mathbf{A} = \left(\frac{\phi}{c}, \vec{\mathbf{a}}\right) This is essentially the 4-total momentum \mathbf{P_T} = \left(\frac{E_T}{c}, \vec{\mathbf{p_T}}\right) of the system; a test particle in a field using the minimal coupling rule. There is the inherent momentum of the particle \mathbf{P} , plus momentum due to interaction with the EM 4-vector potential \mathbf{A} via the particle charge q . The relativistic Hamilton–Jacobi equation is obtained by setting the total momentum equal to the negative 4-gradient of the action S . \mathbf{P_T} = -\boldsymbol{\partial} [S] = \left(\frac{E_T}{c}, \vec{\mathbf{p_T}}\right) = \left(\frac{H}{c}, \vec{\mathbf{p_T}}\right) = -\boldsymbol{\partial} [S] = -\left(\frac{\partial_t}{c}, -\vec{\boldsymbol{\nabla}}\right)[S] The temporal component gives: E_T = H = -\partial_t[S] The spatial components give: \vec{\mathbf{p_T}} = \vec{\boldsymbol{\nabla}}[S] where H is the Hamiltonian. This is actually related to the 4-wavevector being equal the negative 4-gradient of the phase from above. K^\mu = \mathbf{K} = \left(\frac{\omega}{c}, \vec{\mathbf{k}}\right) = -\boldsymbol{\partial} [\Phi] To get the HJE, one first uses the Lorentz scalar invariant rule on the 4-momentum: \mathbf{P} \cdot \mathbf{P} = (m_0 c)^2 But from the minimal coupling rule: \mathbf{P} = \mathbf{P_T} - q\mathbf{A} So: \begin{align} \left(\mathbf{P_T} - q\mathbf{A}\right) \cdot \left(\mathbf{P_T} - q\mathbf{A}\right) = \left(\mathbf{P_T} - q\mathbf{A}\right)^2 &= \left(m_0 c\right)^2 \\ \Rightarrow \left(-\boldsymbol{\partial}[S] - q\mathbf{A}\right)^2 &= \left(m_0 c\right)^2 \end{align} Breaking into the temporal and spatial components: \begin{align} && \left(-\frac{\partial_t[S]}{c} - \frac{q \phi}{c}\right)^2 - (\boldsymbol{\nabla}[S] - q \mathbf{a})^2 &= (m_0 c)^2 \\ &\Rightarrow & (\boldsymbol{\nabla}[S] - q \mathbf{a})^2 - \frac{1}{c^2}(-\partial_t[S] - q \phi)^2 + (m_0 c)^2 &= 0 \\ &\Rightarrow & (\boldsymbol{\nabla}[S] - q \mathbf{a})^2 - \frac{1}{c^2}(\partial_t[S] + q \phi)^2 + (m_0 c)^2 &= 0 \end{align} where the final is the relativistic Hamilton–Jacobi equation. \mathbf{P} = \left(\frac{E}{c}, \vec{p}\right) = i\hbar \boldsymbol{\partial} = i\hbar \left(\frac{\partial_t}{c}, -\vec{\nabla}\right) The temporal component gives: E = i\hbar \partial_t The spatial components give: \vec{p} = -i\hbar \vec{\nabla} This can actually be composed of two separate steps. First: \mathbf{P} = \left(\frac{E}{c},\vec{p}\right) = \hbar \mathbf{K} = \hbar \left(\frac{\omega}{c},\vec{k}\right) which is the full 4-vector version of: The (temporal component) Planck–Einstein relation E = \hbar \omega The (spatial components) de Broglie matter wave relation \vec{p} = \hbar \vec{k} Second: \mathbf{K} = \left(\frac{\omega}{c},\vec{k}\right) = i \boldsymbol{\partial} = i \left(\frac{\partial_t}{c}, -\vec{\nabla}\right) which is just the 4-gradient version of the wave equation for complex-valued plane waves The temporal component gives: \omega = i \partial_t The spatial components give: \vec{k} = - i \vec{\nabla} According to: \left[P^\mu, X^\nu\right] = i \hbar \left[\partial^\mu, X^\nu\right] = i \hbar \partial^\mu\left[X^\nu\right] = i \hbar \eta^{\mu \nu} Taking the spatial components, \left[p^j, x^k\right] = i \hbar \eta^{j k} Since \eta^{\mu \nu} = \operatorname{diag}[1,-1,-1,-1] , \left[p^j, x^k\right] = - i \hbar \delta^{j k} Since [a, b] = -[b, a] , \left[x^k, p^j\right] = i \hbar \delta^{k j} And, relabeling indices gives the usual quantum commutation rules: \left[x^j, p^k\right] = i \hbar \delta^{j k} \left[\left(\partial^\mu \partial_\mu\right) + \left(\frac{m_0 c}{\hbar}\right)^2\right]\psi = 0 In the Dirac relativistic quantum wave equation for spin-1/2 particles (ex. electrons): \left[i \gamma^\mu \partial_\mu - \frac{m_0 c}{\hbar}\right] \psi = 0 where \gamma^\mu are the Dirac gamma matrices and \psi is a relativistic wave function. \psi is Lorentz scalar for the Klein–Gordon equation, and a spinor for the Dirac equation. It is nice that the gamma matrices themselves refer back to the fundamental aspect of SR, the Minkowski metric: \left\{\gamma^\mu, \gamma^\nu\right\} = \gamma^\mu \gamma^\nu + \gamma^\nu \gamma^\mu = 2 \eta^{\mu\nu}I_4 Conservation of 4-probability current density follows from the continuity equation: \boldsymbol{\partial} \cdot \mathbf{J} = \partial_t \rho + \vec{\boldsymbol{\nabla}} \cdot \vec{\mathbf{j}} = 0 The 4-probability current density has the relativistically covariant expression: J_\text{prob}^\mu = \frac{i\hbar}{2m_0}\left(\psi^* \partial^\mu\psi - \psi\partial^\mu \psi^*\right) The 4-charge current density is just the charge ( ) times the 4-probability current density: J_\text{charge}^\mu = \frac{i\hbar q}{2m_0}\left(\psi^* \partial^\mu\psi - \psi\partial^\mu\psi^*\right) 4-position \mathbf{X} = \left(ct, \vec{\mathbf{x}}\right) 4-velocity \mathbf{U} = \gamma\left(c, \vec{\mathbf{u}}\right) 4-momentum \mathbf{P} = \left(\frac{E}{c}, \vec{\mathbf{p}}\right) 4-wavevector \mathbf{K} = \left(\frac{\omega}{c}, \vec{\mathbf{k}}\right) 4-gradient \boldsymbol{\partial} = \left(\frac{\partial_t}{c}, -\vec{\boldsymbol{\nabla}}\right) Note the following simple relations from the previous sections, where each 4-vector is related to another by a Lorentz scalar: 4-velocity \mathbf{U} = \frac{d}{d\tau} \mathbf{X} , where \tau is the proper time 4-momentum \mathbf{P} = m_0 \mathbf{U} , where m_0 is the rest mass 4-wavevector \mathbf{K} = \frac{1}{\hbar} \mathbf{P} , which is the 4-vector version of the Planck–Einstein relation & the de Broglie matter wave relation 4-gradient \boldsymbol{\partial} = -i \mathbf{K} , which is the 4-gradient version of complex-valued plane waves Now, just apply the standard Lorentz scalar product rule to each one: \begin{align} \mathbf{U} \cdot \mathbf{U} &= c^2 \\ \mathbf{P} \cdot \mathbf{P} &= (m_0 c)^2 \\ \mathbf{K} \cdot \mathbf{K} &= \left(\frac{m_0 c}{\hbar}\right)^2 \\ \boldsymbol{\partial} \cdot \boldsymbol{\partial} &= \left(\frac{-i m_0 c}{\hbar}\right)^2 = -\left(\frac{m_0 c}{\hbar}\right)^2 \end{align} The last equation (with the 4-gradient scalar product) is a fundamental quantum relation. When applied to a Lorentz scalar field \psi , one gets the Klein–Gordon equation, the most basic of the quantum relativistic wave equations: \left[\boldsymbol{\partial} \cdot \boldsymbol{\partial} + \left(\frac{m_0 c}{\hbar}\right)^2\right]\psi = 0 The Schrödinger equation is the low-velocity limiting case ( ) of the Klein–Gordon equation. If the quantum relation is applied to a 4-vector field A^\mu instead of a Lorentz scalar field \psi , then one gets the Proca equation: \left[\boldsymbol{\partial} \cdot \boldsymbol{\partial} + \left(\frac{m_0 c}{\hbar}\right)^2\right]A^\mu = 0^\mu If the rest mass term is set to zero (light-like particles), then this gives the free Maxwell equation: [\boldsymbol{\partial} \cdot \boldsymbol{\partial}]A^\mu = 0^\mu More complicated forms and interactions can be derived by using the minimal coupling rule: D^\mu = \partial^\mu - i g A^\mu The full covariant derivative for the fundamental interactions of the Standard Model that we are presently aware of (in natural units) is: D^\mu = \partial^\mu - i g_1 \frac{1}{2} Y B^\mu - i g_2 \frac{1}{2}\tau_i \cdot W_i^\mu - i g_3 \frac{1}{2} \lambda_a \cdot G_a^\mu or \mathbf{D} = \boldsymbol{\partial} - i g_1 \frac{1}{2} Y \mathbf{B} - i g_2 \frac{1}{2} \boldsymbol{\tau}_i \cdot \mathbf{W}_i - i g_3 \frac{1}{2} \boldsymbol{\lambda}_a \cdot \mathbf{G}_a where the scalar product summations ( \cdot ) here refer to the internal spaces, not the tensor indices: B^\mu corresponds to U(1) invariance = (1) EM force gauge boson W_i^\mu corresponds to SU(2) invariance = (3) weak force gauge bosons (i = 1, …, 3) G_a^\mu corresponds to SU(3) invariance = (8) color force gauge bosons (a = 1, …, 8) The coupling constants (g_1, g_2, g_3) are arbitrary numbers that must be discovered from experiment. It is worth emphasizing that for the non-abelian transformations once the g_i are fixed for one representation, they are known for all representations. These internal particle spaces have been discovered empirically. ==Derivation== In three dimensions, the gradient operator maps a scalar field to a vector field such that the line integral between any two points in the vector field is equal to the difference between the scalar field at these two points. Based on this, it may appear incorrectly that the natural extension of the gradient to 4 dimensions should be: \partial^\alpha \overset{?}{=} \left( \frac{\partial}{\partial t}, \vec{\nabla} \right), which is incorrect. However, a line integral involves the application of the vector dot product, and when this is extended to 4-dimensional spacetime, a change of sign is introduced to either the spatial co-ordinates or the time co-ordinate depending on the convention used. This is due to the non-Euclidean nature of spacetime. In this article, we place a negative sign on the spatial coordinates (the time-positive metric convention \eta^{\mu\nu} = \operatorname{diag}[1,-1,-1,-1] ). The factor of (1/c) is to keep the correct unit dimensionality, [length] , for all components of the 4-vector and the (−1) is to keep the 4-gradient Lorentz covariant. Adding these two corrections to the above expression gives the correct definition of 4-gradient: \partial^\alpha = \left(\frac{1}{c} \frac{\partial}{\partial t}, -\vec{\nabla} \right) ==Further reading== S. Hildebrandt, "Analysis II" (Calculus II), , 2003 L.C. Evans, "Partial differential equations", A.M.Society, Grad.Studies Vol.19, 1988 J.D. Jackson, "Classical Electrodynamics" Chapter 11, Wiley

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