Four-gradient
In differential geometry, the four-gradient (or 4-gradient) formula_1 is the four-vector analogue of the gradient formula_2 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. formula_3 indicates the speed of light in vacuum. formula_4 is the flat spacetime metric of SR. There are alternate ways of writing four-vector expressions in physics: The Latin tensor index  ranges in and represents a 3-space vector, e.g. formula_9. The Greek tensor index ranges in and represents a 4-vector, e.g. formula_10. In SR physics, one typically uses a concise blend, e.g. formula_11, where formula_12 represents the temporal component and formula_13 represents the spatial 3-component. Tensors in SR are typically 4D formula_14-tensors, with formula_15 upper indices and formula_16 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): formula_17 Definition. The 4-gradient covariant  components compactly written in four-vector and Ricci calculus notation are: formula_18 The "comma" in the last part above formula_19 implies the "partial differentiation" with respect to 4-position formula_20. The contravariant components are: formula_21 Alternative symbols to formula_22 are formula_23 and "D" (although formula_23 can also signify formula_25, the d'Alembert operator). In GR, one must use the more general metric tensor formula_26, and the tensor covariant derivative formula_27, (not to be confused with the vector 3-gradient formula_28). The covariant derivative formula_29 incorporates the 4-gradient formula_30 plus spacetime curvature effects via the Christoffel symbols formula_31 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 formula_32 in SR, then formula_33 in GR. On a (1,0)-tensor or 4-vector this would be: formula_34 On a (2,0)-tensor this would be: formula_35 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). As a 4-divergence and source of conservation laws. Divergence is a vector operator that produces a signed scalar field giving the quantity of a vector field's source at each point. Note that in this metric signature [+,−,−,−] the 4-Gradient has a negative spatial component. It gets canceled when taking the 4D  dot product since the Minkowski Metric is Diagonal[+1,−1,−1,−1]. The 4-divergence of the 4-position formula_36 gives the dimension of spacetime: formula_37 The 4-divergence of the 4-current density formula_38 gives a conservation law – the conservation of charge: formula_39 This means that the time rate of change of the charge density must equal the negative spatial divergence of the current density formula_40. 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) formula_41 is used in particle conservation: formula_42 This is a conservation law for the particle number density, typically something like baryon number density. The 4-divergence of the electromagnetic 4-potential formula_43 is used in the Lorenz gauge condition: formula_44 This is the equivalent of a conservation law for the EM 4-potential. The 4-divergence of the transverse traceless 4D (2,0)-tensor formula_45 representing gravitational radiation in the weak-field limit (i.e. freely propagating far from the source). is the equivalent of a conservation equation for freely propagating gravitational waves. The 4-divergence of the stress–energy tensor formula_47 , 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). formula_48 It is often written as: formula_49 where it is understood that the single zero is actually a 4-vector zero formula_50. When the conservation of the stress–energy tensor for a perfect fluid is combined with the conservation of particle number density (formula_51), 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: formula_52 where this zero is actually a (2,0)-tensor zero. As a Jacobian matrix for the SR Minkowski metric tensor. The Jacobian  matrix is the matrix of all first-order partial derivatives of a vector-valued function. The 4-gradient formula_53 acting on the 4-position formula_54 gives the SR Minkowski space metric formula_55: formula_56 For the Minkowski metric, the components formula_57 (formula_58 not summed), with non-diagonal components all zero. For the Cartesian Minkowski Metric, this gives formula_59. Generally, formula_60, where formula_61 is the 4D Kronecker delta. As a way to define the Lorentz transformations. The Lorentz transformation is written in tensor form as formula_62 and since formula_63 are just constants, then formula_64 Thus, by definition of the 4-gradient formula_65  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. As part of the total proper time derivative. The scalar product of 4-velocity formula_66 with the 4-gradient gives the total derivative with respect to proper time formula_67: formula_68 The fact that formula_69 is a Lorentz scalar invariant shows that the total derivative with respect to proper time formula_67 is likewise a Lorentz scalar invariant. So, for example, the 4-velocity formula_66 is the derivative of the 4-position formula_20  with respect to proper time: formula_73 or formula_74 Another example, the 4-acceleration formula_75 is the proper-time derivative of the 4-velocity formula_66: formula_77 or formula_78 As a way to define the Faraday electromagnetic tensor and derive the Maxwell equations. The Faraday electromagnetic tensor formula_79 is a mathematical object that describes the electromagnetic field in spacetime of a physical system. Applying the 4-gradient to make an antisymmetric tensor, one gets: formula_80 where: By applying the 4-gradient again, and defining the 4-current density as formula_85 one can derive the tensor form of the Maxwell equations: formula_86 formula_87  where the second line is a version of the Bianchi identity (Jacobi identity). As a way to define the 4-wavevector. A wavevector is a vector which helps describe a wave. Like any vector, it has a magnitude and direction, both of which are important: Its magnitude is either the wavenumber or angular wavenumber of the wave (inversely proportional to the wavelength), and its direction is ordinarily the direction of wave propagation The 4-wavevector formula_88 is the 4-gradient of the negative phase formula_89 (or the negative 4-gradient of the phase) of a wave in Minkowski Space: formula_90  This is mathematically equivalent to the definition of the phase of a wave (or more specifically a plane wave): formula_91 where 4-position formula_92, formula_93 is the temporal angular frequency, formula_94 is the spatial 3-space wavevector, and formula_89 is the Lorentz scalar invariant phase. formula_96 with the assumption that the plane wave formula_93 and formula_94 are not explicit functions of formula_99 or formula_100. The explicit form of an SR plane wave formula_101 can be written as: A general wave formula_104 would be the superposition of multiple plane waves: formula_105 Again using the 4-gradient, formula_106 or As the d'Alembertian  operator. In special relativity, electromagnetism and wave theory, the d'Alembert operator, also called the d'Alembertian or the wave operator, is the Laplace operator of Minkowski space. The operator is named after French mathematician and physicist Jean le Rond d'Alembert. The square of formula_108 is the 4-Laplacian, which is called the d'Alembert operator: formula_109 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 formula_23 and formula_111 are used for the 4-gradient and d'Alembertian respectively. More commonly however, the symbol formula_23 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): formula_113 In the wave equation for the electromagnetic field (using Lorenz gauge formula_114): where: In the wave equation of a gravitational wave (using a similar Lorenz gauge formula_121) formula_122 where formula_45 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 formula_45 are: In the 4-dimensional version of Green's function: formula_128 where the 4D  Delta function is: formula_129 As a component of the 4D Gauss' Theorem / Stokes' Theorem / Divergence Theorem. In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, is a result that relates the flow (that is, flux) of a vector field through a surface to the behavior of the vector field inside the surface. More precisely, the divergence theorem states that the outward flux of a vector field through a closed surface is equal to the volume integral of the divergence over the region inside the surface. Intuitively, it states that "the sum of all  sources minus the sum of all sinks gives the net flow out of a region". In vector calculus, and more generally differential geometry, Stokes' theorem (also called the generalized Stokes' theorem) is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus. formula_130 or formula_131 where As a component of the SR Hamilton–Jacobi equation in relativistic analytic mechanics. The Hamilton–Jacobi equation (HJE) is a formulation of classical mechanics, equivalent to other formulations such as Newton's laws of motion, Lagrangian mechanics and Hamiltonian mechanics. The Hamilton–Jacobi equation  is particularly useful in identifying conserved quantities for mechanical systems, which may be possible even when the mechanical problem itself cannot be solved completely. The HJE is also the only formulation of mechanics in which the motion of a particle can be represented as a wave. In this sense, the HJE fulfilled a long-held goal of theoretical physics (dating at least to Johann Bernoulli in the 18th century) of finding an analogy between the propagation of light and the motion of a particle The generalized relativistic momentum formula_144 of a particle can be written as formula_145 where formula_146  and formula_147 This is essentially the 4-total momentum formula_148 of the system; a test particle in a field using the minimal coupling rule. There is the inherent momentum of the particle formula_149, plus momentum due to interaction with the EM 4-vector potential formula_150 via the particle charge formula_151. The relativistic Hamilton–Jacobi equation is obtained by setting the total momentum equal to the negative 4-gradient of the action formula_152. formula_153 The temporal component gives: formula_154 The spatial components give: formula_155 where formula_156 is the Hamiltonian. This is actually related to the 4-wavevector being equal the negative  4-gradient of the phase from above. formula_157 To get the HJE, one first uses the Lorentz scalar invariant rule on the 4-momentum: formula_158 But from the minimal coupling rule: formula_159 So: formula_160 Breaking into the temporal and spatial components: formula_161 where the final is the relativistic Hamilton–Jacobi equation. As a component of the Schrödinger relations in quantum mechanics. The 4-gradient is connected with quantum mechanics. The relation between the 4-momentum formula_149 and the 4-gradient formula_108 gives the Schrödinger QM relations. formula_164 The temporal component gives: formula_165 The spatial components give: formula_166 This can actually  be composed of two separate steps. First: formula_167 which is the full 4-vector version of: The (temporal component) Planck–Einstein relation formula_168 The (spatial components) de Broglie matter wave relation formula_169 Second: formula_170 which is just the 4-gradient version of the wave equation for complex-valued plane waves The temporal component gives: formula_171 The spatial components give: formula_172 As a component of the covariant form of the quantum commutation relation. In quantum mechanics (physics), the canonical commutation relation is the fundamental relation between canonical conjugate quantities (quantities which are related by definition such that one is the Fourier  transform of another). As a component of the wave equations and probability currents in relativistic quantum mechanics. The 4-gradient is a component in several of the relativistic wave equations: In the Klein–Gordon relativistic quantum wave equation for spin-0 particles (ex. Higgs boson): formula_180 In the Dirac relativistic quantum wave equation for spin-1/2 particles (ex. electrons): formula_181 where formula_182 are the Dirac gamma matrices and formula_183 is a relativistic wave function. formula_183 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: formula_185 Conservation of 4-probability current density follows from the continuity equation: formula_186 The 4-probability current density has the relativistically covariant expression: formula_187 The 4-charge current density is just the charge (q) times the 4-probability current density: formula_188 As a key component in deriving quantum mechanics and relativistic quantum wave equations from special relativity. Relativistic wave equations use 4-vectors in order to be covariant. Start with the standard SR 4-vectors: Note the following simple relations from the previous sections, where each 4-vector is related to another by  a Lorentz scalar: Now, just apply the standard Lorentz scalar product rule to each one: The last equation (with the 4-gradient scalar product) is a fundamental quantum relation. When applied to a Lorentz scalar field formula_183, one gets the Klein–Gordon equation, the most basic of the quantum relativistic wave equations: formula_205 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 formula_75 instead of a Lorentz scalar field formula_183, then one gets the Proca equation: formula_208 If the rest mass term is set  to zero (light-like particles), then this gives the free Maxwell equation: formula_209 More complicated forms and interactions can be derived by using the minimal coupling rule: As a component of the RQM covariant derivative (internal particle spaces). In modern elementary particle physics, one can define a gauge covariant derivative which utilizes the extra RQM fields (internal particle spaces) now known to exist. The version known from classical EM (in natural units) is: formula_210 The full covariant derivative for the fundamental interactions of the Standard Model that we are presently aware of (in natural units) is: formula_211 or formula_212 where  the scalar product summations (formula_213) here refer to the internal spaces, not the tensor indices: The coupling constants formula_217 are arbitrary numbers that must be discovered from experiment. It is worth emphasizing that for the non-abelian transformations once the formula_218 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: 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 formula_220). 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: Note about References. Regarding the use of scalars, 4-vectors and tensors in physics, various authors use slightly different notations for the same equations. For instance, some use formula_15 for invariant rest mass, others use formula_197 for invariant rest mass and use formula_15 for relativistic mass. Many authors set factors of formula_3 and formula_226 and formula_227 to dimensionless unity. Others show some or all  the constants. Some authors use formula_228 for velocity, others use formula_229. Some use formula_230 as a 4-wavevector (to pick an arbitrary example). Others use formula_231 or formula_232 or formula_233 or formula_234 or formula_235 or formula_138, etc. Some write the 4-wavevector as formula_237, some as formula_238 or formula_239 or formula_240 or formula_241 or formula_242 or formula_243. Some will make sure that the dimensional units match across the 4-vector, others do not. Some refer to the temporal component in the 4-vector name, others refer to the spatial component in the 4-vector name. Some mix it throughout the  book, sometimes using one then later on the other. Some use the metric , others use the metric . Some don't use 4-vectors, but do everything as the old style "E" and 3-space vector p. The thing is, all of these are just notational styles, with some more clear and concise than the others. The physics is the same as long as one uses a consistent style throughout the whole derivation.