Four-gradient

In differential geometry, the four-gradient (4-gradient) $\mathbf {\partial }$ is the four-vector (4-vector) analogue of the gradient ${\vec {\mathbf {\nabla } }}$ from Gibbs-Heaviside vector calculus.

In special relativity and in quantum mechanics, the 4-gradient is used to define the properties and relations between the various physical 4-vectors and tensors.

This article uses the (+---) metric signature and tensor index notation in the language of 4-vectors.

Definition

The covariant components compactly written in tensor index notation are:

${\dfrac {\partial }{\partial x^{\alpha }}}=\left({\frac {1}{c}}{\frac {\partial }{\partial t}},{\vec {\nabla }}\right)=\left({\frac {\partial _{t}}{c}},{\vec {\nabla }}\right)=\partial _{\alpha }={}_{,\alpha }$

The comma in the last part above ${}_{,\alpha }$ implies the partial differentiation with respect to $x^{\alpha }$. This is not the same as a semi-colon, used for the covariant derivative.

The contravariant components are:

$\mathbf {\partial } =\partial ^{\alpha }=g^{\alpha \beta }\partial _{\beta }=\left({\frac {1}{c}}{\frac {\partial }{\partial t}},-{\vec {\nabla }}\right)=\left({\frac {\partial _{t}}{c}},-{\vec {\nabla }}\right)=\left({\frac {\partial _{t}}{c}},-\partial _{x},-\partial _{y},-\partial _{z}\right)$

where g^aß is the metric tensor, which here has been chosen for flat spacetime with the metric signature (+,-,-,-).

Alternative symbols to $\partial _{\alpha }$ are $\Box$ and D.

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.

There are alternate ways of writing the expressions:

$\mathbf {\partial } \cdot \mathbf {X}$ is a 4-vector style, which is typically more compact and can use vector notation, (such as the inner product "dot"), always using bold uppercase to represent the 4-vector.

$\partial ^{\mu }\eta _{\mu \nu }X^{\nu }$ is a tensor index style, which is sometimes required in 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 tensor contraction used in the Minkowski metric can go to either side (see Einstein notation):

$\mathbf {A} \cdot \mathbf {B} =A^{\mu }\eta _{\mu \nu }B^{\nu }=A_{\nu }B^{\nu }=A^{\mu }B_{\mu }=\sum _{\nu \mathop {=} 0..3}[a_{\nu }*b^{\nu }]=\sum _{\mu \mathop {=} 0..3}[a^{\mu }*b_{\mu }]=a^{0}*b^{0}-\sum _{i\mathop {=} 1..3}[a^{i}*b^{i}]=a^{0}b^{0}-{\vec {\mathbf {a} }}\cdot {\vec {\mathbf {b} }}$

As a 4-divergence

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 $X^{\mu }=(ct,{\vec {\mathbf {x} }})$ gives the dimension of spacetime:

$\mathbf {\partial } \cdot \mathbf {X} =\partial ^{\mu }\eta _{\mu \nu }X^{\nu }=\left({\frac {\partial _{t}}{c}},-{\vec {\nabla }}\right)\cdot (ct,{\vec {x}})={\frac {\partial _{t}}{c}}(ct)+{\vec {\nabla }}\cdot {\vec {x}}=(\partial _{t}t)+(\partial _{x}x+\partial _{y}y+\partial _{z}z)=(1)+(3)=4$

The 4-divergence of the 4-current $J^{\mu }=(c\rho ,{\vec {\mathbf {j} }})=\rho _{o}U^{\mu }$ gives a conservation law - the conservation of charge:

$\mathbf {\partial } \cdot \mathbf {J} =\partial ^{\mu }\eta _{\mu \nu }J^{\nu }=\left({\frac {\partial _{t}}{c}},-{\vec {\nabla }}\right)\cdot (\rho c,{\vec {j}})={\frac {\partial _{t}}{c}}(\rho c)+{\vec {\nabla }}\cdot {\vec {j}}=\partial _{t}\rho +{\vec {\nabla }}\cdot {\vec {j}}=0$

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 electromagnetic 4-potential $A^{\mu }=({\frac {\phi }{c}},{\vec {\mathbf {a} }})$ is used in the Lorenz gauge condition:

$\mathbf {\partial } \cdot \mathbf {A} =\partial ^{\mu }\eta _{\mu \nu }A^{\nu }=\left({\frac {\partial _{t}}{c}},-{\vec {\nabla }}\right)\cdot \left({\frac {\phi }{c}},{\vec {a}}\right)={\frac {\partial _{t}}{c}}\left({\frac {\phi }{c}}\right)+{\vec {\nabla }}\cdot {\vec {a}}={\frac {\partial _{t}\phi }{c^{2}}}+{\vec {\nabla }}\cdot {\vec {a}}=0$

This is the equivalent of a conservation law for the EM 4-potential.

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 $\partial ^{\mu }$ acting on the 4-position $X^{\nu }$ gives the SR Minkowski space metric $\eta ^{\mu \nu }$:

$\mathbf {\partial } [\mathbf {X} ]=\partial ^{\mu }[X^{\nu }]=X^{\nu _{,}\mu }=\left({\frac {\partial _{t}}{c}},-{\vec {\nabla }}\right)[(ct,{\vec {x}})]=\left({\frac {\partial _{t}}{c}},-\partial _{x},-\partial _{y},-\partial _{z}\right)[(ct,x,y,z)],$

$={\begin{bmatrix}{\frac {\partial _{t}}{c}}ct&{\frac {\partial _{t}}{c}}x&{\frac {\partial _{t}}{c}}y&{\frac {\partial _{t}}{c}}z\\-\partial _{x}ct&-\partial _{x}x&-\partial _{x}y&-\partial _{x}z\\-\partial _{y}ct&-\partial _{y}x&-\partial _{y}y&-\partial _{y}z\\-\partial _{z}ct&-\partial _{z}x&-\partial _{z}y&-\partial _{z}z\end{bmatrix}}={\begin{bmatrix}1&0&0&0\\0&-1&0&0\\0&0&-1&0\\0&0&0&-1\end{bmatrix}}=\mathrm {diag} [1,-1,-1,-1]$

$\mathbf {\partial } [\mathbf {X} ]=\eta ^{\mu \nu }.$

For the Minkowski metric, the components $[\eta ^{\mu \nu }]=[\eta _{\mu \nu }]$

As part of the total proper time derivative

The scalar product of 4-velocity $U^{\mu }$ with the 4-gradient gives the total derivative with respect to proper time ${\frac {d}{d\tau }}$:

$\mathbf {U} \cdot \mathbf {\partial } =U^{\mu }\eta _{\mu \nu }\partial ^{\nu }=\gamma (c,{\vec {u}})\cdot \left({\frac {\partial _{t}}{c}},-{\vec {\nabla }}\right)=\gamma \left(c{\frac {\partial _{t}}{c}}+{\vec {u}}\cdot {\vec {\nabla }}\right)=\gamma \left(\partial _{t}+{\frac {dx}{dt}}\partial _{x}+{\frac {dy}{dt}}\partial _{y}+{\frac {dz}{dt}}\partial _{z}\right)=\gamma {\frac {d}{dt}}={\frac {d}{d\tau }}$

The fact that $\mathbf {U} \cdot \mathbf {\partial }$ 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 proper-time derivative of the 4-position $X^{\mu }$:

${\frac {d}{d\tau }}\mathbf {X} =(\mathbf {U} \cdot \mathbf {\partial } )\mathbf {X} =\mathbf {U} \cdot \mathbf {\partial } [\mathbf {X} ]=U^{\alpha }\cdot \eta ^{\mu \nu }=U^{\alpha }\eta _{\alpha \nu }\eta ^{\mu \nu }=U^{\alpha }\delta _{\alpha }^{\mu }=U^{\mu }=\mathbf {U}$

or

${\frac {d}{d\tau }}\mathbf {X} =\gamma {\frac {d}{dt}}\mathbf {X} =\gamma {\frac {d}{dt}}(ct,{\vec {x}})=\gamma \left({\frac {d}{dt}}ct,{\frac {d}{dt}}{\vec {x}}\right)=\gamma (c,{\vec {u}})=\mathbf {U}$

Another example, the 4-acceleration $A^{\mu }$ is the proper-time derivative of the 4-velocity $U^{\mu }$:

${\frac {d}{d\tau }}\mathbf {U} =(\mathbf {U} \cdot \mathbf {\partial } )\mathbf {U} =\mathbf {U} \cdot \mathbf {\partial } [\mathbf {U} ]=U^{\alpha }\eta _{\alpha \mu }\partial ^{\mu }[U^{\nu }]$

$=U^{\alpha }\eta _{\alpha \mu }{\begin{bmatrix}{\frac {\partial _{t}}{c}}\gamma c&{\frac {\partial _{t}}{c}}\gamma {\vec {u}}\\-{\vec {\nabla }}\gamma c&-{\vec {\nabla }}\gamma {\vec {u}}\end{bmatrix}}=U^{\alpha }{\begin{bmatrix}\ {\frac {\partial _{t}}{c}}\gamma c&0\\0&{\vec {\nabla }}\gamma {\vec {u}}\end{bmatrix}}$

$=\gamma \left(c{\frac {\partial _{t}}{c}}\gamma c,{\vec {u}}\cdot \nabla \gamma {\vec {u}}\right)=\gamma \left(c\partial _{t}\gamma ,{\frac {d}{dt}}[\gamma {\vec {u}}]\right)=\gamma (c{\dot {\gamma }},{\dot {\gamma }}{\vec {u}}+\gamma {\dot {\vec {u}}})=\mathbf {A}$

or

${\frac {d}{d\tau }}\mathbf {U} =\gamma {\frac {d}{dt}}(\gamma c,\gamma {\vec {u}})=\gamma \left({\frac {d}{dt}}[\gamma c],{\frac {d}{dt}}[\gamma {\vec {u}}]\right)=\gamma (c{\dot {\gamma }},{\dot {\gamma }}{\vec {u}}+\gamma {\dot {\vec {u}}})=\mathbf {A}$

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 space-time of a physical system.

Applying the 4-gradient to make an antisymmetric tensor, one gets:

$F^{\mu \nu }=\partial ^{\mu }A^{\nu }-\partial ^{\nu }A^{\mu }={\begin{bmatrix}0&-E_{x}/c&-E_{y}/c&-E_{z}/c\\E_{x}/c&0&-B_{z}&B_{y}\\E_{y}/c&B_{z}&0&-B_{x}\\E_{z}/c&-B_{y}&B_{x}&0\end{bmatrix}}$

where:

Electromagnetic 4-potential $A^{\mu }=\mathbf {A} =\left({\frac {\phi }{c}},{\vec {\mathbf {a} }}\right)$, not to be confused with the 4-acceleration $\mathbf {A} =\gamma (c{\dot {\gamma }},{\dot {\gamma }}{\vec {u}}+\gamma {\dot {\vec {u}}})$

$\phi$ is the electric scalar potential, and ${\vec {\mathbf {a} }}$ is the magnetic 3-vector potential.

By applying the 4-gradient again, and defining the 4-current as $J^{\beta }=\mathbf {J} =(c\rho ,{\vec {\mathbf {j} }})$ one can derive the tensor form of the Maxwell equations:

$\partial _{\alpha }F^{\alpha \beta }=\mu _{0}J^{\beta }$

$\partial _{\gamma }F_{\alpha \beta }+\partial _{\alpha }F_{\beta \gamma }+\partial _{\beta }F_{\gamma \alpha }=0$

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 (inversely proportional to the wavelength), and its direction is ordinarily the direction of wave propagation

The 4-wavevector $K^{\mu }$ is the 4-gradient of the negative phase $\Phi$ (or the negative 4-gradient of the phase) of a wave in Minkowski Space:

$K^{\mu }=\mathbf {K} =\left({\frac {\omega }{c}},{\vec {\mathbf {k} }}\right)=\mathbf {\partial } [-\Phi ]=-\mathbf {\partial } [\Phi ]$

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 $\mathbf {X} =(ct,{\vec {\mathbf {x} }})$, $\omega$ is the temporal angular frequency, ${\vec {\mathbf {k} }}$ is the spatial 3-wavevector, and $\Phi$ is the Lorentz scalar invariant phase.

$\partial [\mathbf {K} \cdot \mathbf {X} ]=\partial [\omega t-{\vec {\mathbf {k} }}\cdot {\vec {\mathbf {x} }}]=({\frac {\partial _{t}}{c}},-\nabla )[\omega t-{\vec {\mathbf {k} }}\cdot {\vec {\mathbf {x} }}]=({\frac {\partial _{t}}{c}}[\omega t-{\vec {\mathbf {k} }}\cdot {\vec {\mathbf {x} }}],-\nabla [\omega t-{\vec {\mathbf {k} }}\cdot {\vec {\mathbf {x} }}])=({\frac {\partial _{t}}{c}}[\omega t],-\nabla [-{\vec {\mathbf {k} }}\cdot {\vec {\mathbf {x} }}])=({\frac {\omega }{c}},{\vec {\mathbf {k} }})=\mathbf {K}$

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 the plane wave can be written as:

$\Psi (\mathbf {X} )=Ae^{-i(\mathbf {K} \cdot \mathbf {X} )}=Ae^{i(\Phi )}$ where $A$ is a (possibly complex) amplitude.

A general wave would be the superposition of multiple plane waves:

$\Psi (\mathbf {X} )=\sum _{n}[A_{n}e^{-i(\mathbf {K_{n}} \cdot \mathbf {X} )}]=\sum _{n}[A_{n}e^{i(\Phi _{n})}]$

As the d'Alembertian operator

The square of $\mathbf {\partial }$ is the 4-Laplacian, which is called the d'Alembert operator:

$\mathbf {\partial } \cdot \mathbf {\partial } =\partial ^{\mu }\cdot \partial ^{\nu }=\partial ^{\mu }\eta _{\mu \nu }\partial ^{\nu }=\partial _{\nu }\partial ^{\nu }={\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}-\nabla ^{2}=\left({\frac {\partial _{t}}{c}}\right)^{2}-\nabla ^{2}$.

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):

$[(\mathbf {\partial } \cdot \mathbf {\partial } )+\left({\frac {m_{o}c}{\hbar }}\right)^{2}]\psi =[\left({\frac {\partial _{t}^{2}}{c^{2}}}-{\vec {\nabla }}^{2}\right)+\left({\frac {m_{o}c}{\hbar }}\right)^{2}]\psi =0$

In the wave equation for the electromagnetic field { using Lorenz gauge $(\mathbf {\partial } \cdot \mathbf {A} )=0$ }:

$(\mathbf {\partial } \cdot \mathbf {\partial } )\mathbf {A} =0$ {in vacuum}

$(\mathbf {\partial } \cdot \mathbf {\partial } )\mathbf {A} =\mu _{0}\mathbf {J}$ {with a 4-current source}

where:

Electromagnetic 4-potential $\mathbf {A} =A^{\alpha }=\left({\frac {\phi }{c}},{\vec {a}}\right)$ is an electromagnetic vector potential

4-current $\mathbf {J} =J^{\alpha }=(\rho c,{\vec {j}})$ is an electromagnetic current density

In the 4-dimensional version of Green's function:

$(\mathbf {\partial } \cdot \mathbf {\partial } )G(x-x')=\delta ^{4}(x-x')$

As a component of the Schrödinger relations in quantum mechanics

The 4-gradient is connected with quantum mechanics.

The relation between the 4-momentum $\mathbf {P}$ and the 4-gradient $\mathbf {\partial }$ gives the Schrödinger QM relations.

$\mathbf {P} =\left({\frac {E}{c}},{\vec {p}}\right)=i\hbar \mathbf {\partial } =i\hbar \left({\frac {\partial _{t}}{c}},-{\vec {\nabla }}\right)$

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:

$\mathbf {K} =\left({\frac {\omega }{c}},{\vec {k}}\right)=i\mathbf {\partial } =i\left({\frac {\partial _{t}}{c}},-{\vec {\nabla }}\right)$ 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: ${\vec {k}}=-i{\vec {\nabla }}$

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).

$[P^{\mu },X^{\nu }]=i\hbar [\partial ^{\mu },X^{\nu }]=i\hbar \eta ^{\mu \nu }$

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):

$[(\partial ^{\mu }\partial _{\mu })+\left({\frac {m_{o}c}{\hbar }}\right)^{2}]\psi =0$

In the Dirac relativistic quantum wave equation for spin-1/2 particles (ex. electrons):

$[i\gamma ^{\mu }\partial _{\mu }-{\frac {m_{o}c}{\hbar }}]\psi =0$

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:

$\{\gamma ^{\mu },\gamma ^{\nu }\}=\gamma ^{\mu }\gamma ^{\nu }+\gamma ^{\nu }\gamma ^{\mu }=2\eta ^{\mu \nu }I_{4}\,$

Conservation of 4-probability current density follows from the continuity equation:

$\mathbf {\partial } \cdot \mathbf {J} =\partial _{t}\rho +{\vec {\mathbf {\nabla } }}\cdot {\vec {\mathbf {j} }}=0$

The 4-probability current density has the relativistically covariant expression:

$J^{\mu }={\frac {i\hbar }{2m_{o}}}(\psi ^{*}\partial ^{\mu }\psi -\psi \partial ^{\mu }\psi ^{*})$

The 4-charge current density is just the charge (q) times the 4-probability current density:

$J^{\mu }={\frac {i\hbar q}{2m_{o}}}(\psi ^{*}\partial ^{\mu }\psi -\psi \partial ^{\mu }\psi ^{*})$

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

Start with the standard SR 4-vectors:

4-position $\mathbf {X} =(ct,{\vec {\mathbf {x} }})$

4-velocity $\mathbf {U} =\gamma (c,{\vec {\mathbf {u} }})$

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 $\mathbf {\partial } =\left({\frac {\partial _{t}}{c}},-{\vec {\mathbf {\nabla } }}\right)$

Note the following simple relations from the previous sections, where each 4-vector is related to another by a Lorentz scalar:

$\mathbf {U} ={\frac {d}{d\tau }}\mathbf {X}$, where $\tau$ is the proper time

$\mathbf {P} =m_{o}\mathbf {U}$, where $m_{o}$ is the rest mass

$\mathbf {K} =(1/\hbar )\mathbf {P}$, which is the 4-vector version of the Planck-Einstein relation & the de Broglie matter wave relation

$\mathbf {\partial } =-i\mathbf {K}$, which is the gradient version of complex-valued plane waves

Now, just apply the standard Lorentz scalar product rule to each one:

$\mathbf {U} \cdot \mathbf {U} =(c)^{2}$

$\mathbf {P} \cdot \mathbf {P} =(m_{o}c)^{2}$

$\mathbf {K} \cdot \mathbf {K} =\left({\frac {m_{o}c}{\hbar }}\right)^{2}$

$\mathbf {\partial } \cdot \mathbf {\partial } =\left({\frac {-im_{o}c}{\hbar }}\right)^{2}=-\left({\frac {m_{o}c}{\hbar }}\right)^{2}$

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.

$[\mathbf {\partial } \cdot \mathbf {\partial } +\left({\frac {m_{o}c}{\hbar }}\right)^{2}]\psi =0$

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 $A^{\mu }$ instead of a Lorentz scalar field $\psi$, then one gets the Proca equation:

$[\mathbf {\partial } \cdot \mathbf {\partial } +\left({\frac {m_{o}c}{\hbar }}\right)^{2}]A^{\mu }=0$

If the rest mass term is set to zero (light-like particles), then this gives the free Maxwell equation:

$[\mathbf {\partial } \cdot \mathbf {\partial } ]A^{\mu }=0$

More complicated forms and interactions can be derived by using the minimal coupling rule:

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:

$\partial ^{\alpha }\ =?=\left({\frac {\partial }{\partial t}},{\vec {\nabla }}\right)$

incorrect

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 $\eta ^{\mu \nu }=Diag[1,-1,-1,-1]$). 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:

$\partial ^{\alpha }\ =\left({\frac {1}{c}}{\frac {\partial }{\partial t}},-{\vec {\nabla }}\right)$

correct

See also

• four-vector
• Ricci calculus
• Index notation
• Tensor
• Antisymmetric tensor
• Einstein notation
• Raising and lowering indices
• Abstract index notation
• Covariance and contravariance of vectors

References

• S. Hildebrandt, "Analysis II" (Calculus II), ISBN 3-540-43970-6, 2003
• L.C. Evans, "Partial differential equations", A.M.Society, Grad.Studies Vol.19, 1988
• J.D. Jackson, "Classical Electrodynamics" Chapter 11, Wiley ISBN 0-471-30932-X