Four-gradient


In differential geometry, the four-gradient is the four-vector analogue of the gradient 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.
indicates the speed of light in vacuum.
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..
The Greek tensor index ranges in and represents a 4-vector, e.g..
In SR physics, one typically uses a concise blend, e.g., where represents the temporal component and represents the spatial 3-component.
The tensor contraction used in the Minkowski metric can go to either side :

Definition

The 4-gradient covariant components compactly written in four-vector and Ricci calculus notation are:
The comma in the last part above implies the partial differentiation with respect to 4-position.
The contravariant components are:
Alternative symbols to are and D.
In GR, one must use the more general metric tensor, and the tensor covariant derivative,.
The covariant derivative incorporates the 4-gradient plus spacetime curvature effects via the Christoffel symbols
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 in SR, then in GR.
On a -tensor or 4-vector this would be:
On a -tensor this would be:

Usage

The 4-gradient is used in a number of different ways in special relativity :
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.

As a 4-divergence and source of conservation laws

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 gives the dimension of spacetime:
The 4-divergence of the 4-current density 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.
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 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 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 2-tensor representing gravitational radiation in the weak-field limit.
is the equivalent of a conservation equation for freely propagating gravitational waves.
The 4-divergence of the stress–energy tensor, the conserved Noether current associated with spacetime translations, gives four conservation laws in SR:
The conservation of energy and the conservation of linear momentum.
It is often written as:
where it is understood that the single zero is actually a 4-vector zero ).
When the conservation of the stress–energy tensor for a perfect fluid is combined with the conservation of particle number density, 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 is also conserved:
where this zero is actually a -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 acting on the 4-position gives the SR Minkowski space metric :
For the Minkowski metric, the components , with non-diagonal components all zero.
For the Cartesian Minkowski Metric, this gives.
Generally,, where is the 4D Kronecker delta.

As a way to define the Lorentz transformations

The Lorentz transformation is written in tensor form as
and since 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 with the 4-gradient gives the total derivative with respect to proper time :
The fact that is a Lorentz scalar invariant shows that the total derivative with respect to proper time is likewise a Lorentz scalar invariant.
So, for example, the 4-velocity is the derivative of the 4-position with respect to proper time:
or
Another example, the 4-acceleration is the proper-time derivative of the 4-velocity :
or

As a way to define the Faraday electromagnetic tensor and derive the Maxwell equations

The Faraday electromagnetic tensor 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:
is the electric scalar potential, and is the magnetic 3-space vector potential.
By applying the 4-gradient again, and defining the 4-current density as one can derive the tensor form of the Maxwell equations:
where the second line is a version of the Bianchi 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, and its direction is ordinarily the direction of wave propagation
The 4-wavevector is the 4-gradient of the negative phase of a wave in Minkowski Space:
This is mathematically equivalent to the definition of the phase of a wave :
where 4-position, is the temporal angular frequency, is the spatial 3-space wavevector, and is the Lorentz scalar invariant phase.
with the assumption that the plane wave and are not explicit functions of or
The explicit form of an SR plane wave can be written as:
A general wave 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 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 and are used for the 4-gradient and d'Alembertian respectively. More commonly however, the symbol 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 :
In the wave equation for the electromagnetic field :
where:
In the wave equation of a gravitational wave
where is the transverse traceless 2-tensor representing gravitational radiation in the weak-field limit.
Further conditions on 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 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 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 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 of finding an analogy between the propagation of light and the motion of a particle
The generalized relativistic momentum of a particle can be written as
where and
This is essentially the 4-total momentum of the system; a test particle in a field using the minimal coupling rule. There is the inherent momentum of the particle, plus momentum due to interaction with the EM 4-vector potential via the particle charge.
The relativistic Hamilton–Jacobi equation is obtained by setting the total momentum equal to the negative 4-gradient of the action.
The temporal component gives:
The spatial components give:
where is the Hamiltonian.
This is actually related to the 4-wavevector being equal the negative 4-gradient of the phase from above.
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 and the 4-gradient gives the Schrödinger QM relations.
The temporal component gives:
The spatial components give:
This can actually be composed of two separate steps.
First:
which is the full 4-vector version of:
The Planck–Einstein relation
The de Broglie matter wave relation
Second:
which is just the 4-gradient version of the wave equation for complex-valued plane waves
The temporal component gives:
The spatial components give:

As a component of the covariant form of the quantum commutation relation

In quantum mechanics, the canonical commutation relation is the fundamental relation between canonical conjugate quantities.

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 :
In the Dirac relativistic quantum wave equation for spin-1/2 particles :
where are the Dirac gamma matrices and is a relativistic wave function.
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 times the 4-probability current density:

As a key component in deriving quantum mechanics and relativistic quantum wave equations from special relativity

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 is a fundamental quantum relation.
When applied to a Lorentz scalar field, one gets the Klein–Gordon equation, the most basic of the quantum relativistic wave equations:
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 instead of a Lorentz scalar field, then one gets the Proca equation:
If the rest mass term is set to zero, 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 now known to exist.
The version known from classical EM is:
The full covariant derivative for the fundamental interactions of the Standard Model that we are presently aware of is:
or
where:
The coupling constants are arbitrary numbers that must be discovered from experiment. It is worth emphasizing that for the non-abelian transformations once the 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 factor of is to keep the correct unit dimensionality for all components of the 4-vector and the is to keep the 4-gradient Lorentz covariant. Adding these two corrections to the above expression gives the correct definition of 4-gradient: