In differential geometry, the four-gradient (or 4-gradient)
Contents
- Notation
- Definition
- Usage
- As a 4 divergence and source of conservation laws
- As a Jacobian matrix for the SR metric tensor
- As part of the total proper time derivative
- As a way to define the Faraday electromagnetic tensor and derive the Maxwell equations
- As a way to define the 4 wavevector
- As the dAlembertian operator
- As a component of the 4D Gauss Theorem Stokes Theorem Divergence Theorem
- As a component of the SR Hamilton Jacobi equation in relativistic analytic mechanics
- As a component of the Schrdinger relations in quantum mechanics
- As a component of the covariant form of the quantum commutation relation
- As a component of the wave equations and probability currents in relativistic quantum mechanics
- As a key component in deriving quantum mechanics and relativistic quantum wave equations from special relativity
- As a component of the RQM covariant derivative internal particle spaces
- Derivation
- References
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 and tensor index notation in the language of 4-vectors.
SR and GR are abbreviations for Special Relativity and General Relativity respectively.
(
There are alternate ways of writing 4-vector expressions in physics:
The Latin tensor index ranges from (1..3), and represents a 3-vector, eg.
The Greek tensor index ranges from (0..3), and represents a 4-vector, eg.
In SR physics, one typically uses a concise blend, eg.
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 tensor index notation are:
The comma in the last part above
The contravariant components are:
Alternative symbols to
In GR, one must use the more general metric tensor
The covariant derivative
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
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 correct for Minkowski coordinates in SR, but may need to be modified for other coordinates.
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
The 4-divergence of the 4-current density
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 (4-dust)
This is a conservation law for the particle number density, typically something like baryon number density.
The 4-divergence of the electromagnetic 4-potential
This is the equivalent of a conservation law for the EM 4-potential.
The 4-divergence of the stress-energy tensor
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
When the conservation of the stress-energy tensor (
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 metric tensor
The Jacobian matrix is the matrix of all first-order partial derivatives of a vector-valued function.
The 4-gradient
For the Minkowski metric, the components
For the Cartesian Minkowski Metric, this gives
Generally,
As part of the total proper time derivative
The scalar product of 4-velocity
The fact that
So, for example, the 4-velocity
or
Another example, the 4-acceleration
or
As a way to define the Faraday electromagnetic tensor and derive the Maxwell equations
The Faraday electromagnetic tensor
Applying the 4-gradient to make an antisymmetric tensor, one gets:
where:
Electromagnetic 4-potential
By applying the 4-gradient again, and defining the 4-current density as
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
This is mathematically equivalent to the definition of the phase of a wave (or more specifically a plane wave):
where 4-position
with the assumption that the plane wave
The explicit form of an SR plane wave
A general wave
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
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
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
where:
Electromagnetic 4-potentialIn 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
where
This is essentially the 4-total momentum
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
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
The temporal component gives:
The spatial components give:
This can actually be composed of two separate steps.
First:
The (temporal component) Planck–Einstein relation
The (spatial components) de Broglie matter wave relation
Second:
The temporal component gives:
The spatial components give:
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
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:
4-positionNote 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
The Schrödinger equation is the low-velocity limiting case (v<<c) of the Klein-Gordon equation.
If the last part is applied to a 4-vector field
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 scalar product summations (The coupling constants
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 space-time, 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 space-time. In this article, we place a negative sign on the spatial coordinates (the time-positive Metric convention