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 (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):
Definition.
The 4-gradient covariant components compactly written in four-vector and Ricci calculus notation are:
The "comma" in the last part above formula_16 implies the "partial differentiation" with respect to 4-position formula_17.
The contravariant components are:
Alternative symbols to formula_19 are formula_20 and "D" (although formula_20 can also signify formula_22, the d'Alembert operator).
In GR, one must use the more general metric tensor formula_23, and the tensor covariant derivative formula_24, (not to be confused with the vector 3-gradient formula_25).
The covariant derivative formula_26 incorporates the 4-gradient formula_27 plus spacetime curvature effects via the Christoffel symbols formula_28
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_29 in SR, then formula_30 in GR.
On a (1,0)-tensor or 4-vector this would be:
On a (2,0)-tensor this would be:
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.
The 4-divergence of the 4-position formula_35 gives the dimension of spacetime:
The 4-divergence of the 4-current density formula_37 gives a conservation law – the conservation of charge:
This means that the time rate of change of the charge density must equal the negative spatial divergence of the current density formula_39.
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_40 is used in particle conservation:
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_42 is used in the Lorenz gauge condition:
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_44 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_46, 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).
It is often written as:
where it is understood that the single zero is actually a 4-vector zero formula_49.
When the conservation of the stress–energy tensor (formula_50) 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:
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:
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
and since formula_63 are just constants, then
Thus, by definition of the 4-gradient
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:
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_17 with respect to proper time:
or
Another example, the 4-acceleration formula_75 is the proper-time derivative of the 4-velocity formula_66:
or
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:
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:
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:
This is mathematically equivalent to the definition of the phase of a wave (or more specifically a plane wave):
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.
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:
Again using the 4-gradient,
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_1 is the 4-Laplacian, which is called the d'Alembert operator:
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_20 and formula_111 are used for the 4-gradient and d'Alembertian respectively. More commonly however, the symbol formula_20 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):
In the wave equation for the electromagnetic field { using Lorenz gauge formula_114 }:
where:
where formula_44 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_44 are:
In the 4-dimensional version of Green's function:
where the 4D Delta function is:
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.
or
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
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.
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:
But from the minimal coupling rule:
So:
Breaking into the temporal and spatial components:
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_1 gives the Schrödinger QM relations.
The temporal component gives: formula_165
The spatial components give: formula_166
This can actually be composed of two separate steps.
First:
The (temporal component) Planck–Einstein relation formula_168
The (spatial components) de Broglie matter wave relation formula_169
Second:
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):
In the Dirac relativistic quantum wave equation for spin-1/2 particles (ex. electrons):
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:
Conservation of 4-probability current density follows from the continuity equation:
The 4-probability current density has the relativistically covariant expression:
The 4-charge current density is just the charge (q) times the 4-probability current density:
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:
The Schrödinger equation is the low-velocity limiting case {|v| ≪ c} 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:
If the rest mass term is set to zero (light-like particles), then this gives the free Maxwell equation:
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:
The full covariant derivative for the fundamental interactions of the Standard Model that we are presently aware of (in natural units) is:
or
where:
The coupling constants formula_214 are arbitrary numbers that must be discovered from experiment. It is worth emphasizing that for the non-abelian transformations once the formula_215 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_217). The factor of (1/"c") is to keep the correct unit dimensionality {1/[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_219 for invariant rest mass, others use formula_197 for invariant rest mass and use formula_219 for relativistic mass. Many authors set factors of formula_3 and formula_223 and formula_224 to dimensionless unity. Others show some or all the constants. Some authors use formula_225 for velocity, others use formula_226. Some use formula_227 as a 4-wavevector (to pick an arbitrary example). Others use formula_228 or formula_229 or formula_230 or formula_231 or formula_232 or formula_138, etc. Some write the 4-wavevector as formula_234, some as formula_235 or formula_236 or formula_237 or formula_238 or formula_239 or formula_240. 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.