User:JMvanDijk/Sandbox 16/Box1

Electric fields
General Classical Equations


 * {| class="wikitable"

! scope="col" width="200" | Physical situation ! scope="col" width="10" | Equations !Electric potential gradient and field
 * $$ \mathbf{E} = - \nabla V $$

$$ \Delta V = -\int_{r_1}^{r_2} \mathbf{E} \cdot d\mathbf{r}\,\!$$ !Point charge !At a point in a local array of point charges !At a point due to a continuum of charge !Electrostatic torque and potential energy due to non-uniform fields and dipole moments
 * $$ \mathbf{E} = \frac{q}{4 \pi \epsilon_0 \left | \mathbf{r} \right |^2 }\mathbf{\hat{r}} \,\!$$
 * $$\mathbf{E} = \sum \mathbf{E}_i = \frac{1}{4 \pi \epsilon_0} \sum_i \frac{q_i}{\left | \mathbf{r}_i - \mathbf{r} \right |^2}\mathbf{\hat{r}}_i \,\!$$
 * $$ \mathbf{E} = \frac{1}{4\pi\epsilon_0} \int_V \frac{\mathbf{r} \rho \mathrm{d}V}{\left | \mathbf{r} \right |^3} \,\!$$
 * $$ \boldsymbol{\tau} = \int_V \mathrm{d} \mathbf{p} \times \mathbf{E} $$

$$ U = \int_V \mathrm{d} \mathbf{p} \cdot \mathbf{E} $$
 * }
 * }

Magnetic fields and moments
General classical equations


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! scope="col" width="200" | Physical situation ! scope="col" width="10" | Equations !Magnetic potential, EM vector potential !Due to a magnetic moment
 * $$ \mathbf{B} = \nabla \times \mathbf{A} $$
 * $$ \mathbf{A} = \frac{\mu_0}{4\pi}\frac{\mathbf{m}\times\mathbf{r}}{\left | \mathbf{r} \right |^3} $$

$$\mathbf{B}({\mathbf{r}})=\nabla\times{\mathbf{A}}=\frac{\mu_{0}}{4\pi}\left(\frac{3\mathbf{r}(\mathbf{m}\cdot\mathbf{r})}{\left | \mathbf{r} \right |^{5}}-\frac{\left | \mathbf{r} \right |^{3}}\right)$$ !Magnetic moment due to a current distribution !Magnetostatic torque and potential energy due to non-uniform fields and dipole moments
 * $$ \mathbf{m} = \frac{1}{2}\int_V \mathbf{r}\times\mathbf{J} \mathrm{d} V $$
 * $$ \boldsymbol{\tau} = \int_V \mathrm{d} \mathbf{m} \times \mathbf{B} $$

$$ U = \int_V \mathrm{d} \mathbf{m} \cdot \mathbf{B} $$
 * }
 * }

Formulation in Gaussian units convention
The definitions of charge, electric field, and magnetic field can be altered to simplify theoretical calculation, by absorbing dimensioned factors of $ε_{0}$ and $μ_{0}$ into the units of calculation, by convention. With a corresponding change in convention for the Lorentz force law this yields the same physics, i.e. trajectories of charged particles, or work done by an electric motor. These definitions are often preferred in theoretical and high energy physics where it is natural to take the electric and magnetic field with the same units, to simplify the appearance of the electromagnetic tensor: the Lorentz covariant object unifying electric and magnetic field would then contain components with uniform unit and dimension. Such modified definitions are conventionally used with the Gaussian (CGS) units. Using these definitions and conventions, colloquially "in Gaussian units", the Maxwell equations become:

The equations are particularly readable when length and time are measured in compatible units like seconds and lightseconds i.e. in units such that c = 1 unit of length/unit of time. Ever since 1983 (see International System of Units), metres and seconds are compatible except for historical legacy since by definition c = 299 792 458 m/s (≈ 1.0 feet/nanosecond).

Further cosmetic changes, called rationalisations, are possible by absorbing factors of $4π$ depending on whether we want Coulomb's law or Gauss's law to come out nicely, see Lorentz-Heaviside units (used mainly in particle physics). In theoretical physics it is often useful to choose units such that Planck's constant, the elementary charge, and even Newton's constant are 1. See Planck units.

Alternative formulations
Following is a summary of some of the numerous other mathematical formalisms to write the microscopic Maxwell's equations, with the columns separating the two homogeneous Maxwell equations from the two inhomogeneous ones involving charge and current. Each formulation has versions directly in terms of the electric and magnetic fields, and indirectly in terms of the electrical potential $φ$ and the vector potential $A$. Potentials were introduced as a convenient way to solve the homogeneous equations, but it was thought that all observable physics was contained in the electric and magnetic fields (or relativistically, the Faraday tensor). The potentials play a central role in quantum mechanics, however, and act quantum mechanically with observable consequences even when the electric and magnetic fields vanish (Aharonov–Bohm effect).

Each table describes one formalism. See the main article for details of each formulation. SI units are used throughout.

Special Relativity: The metric and four-vectors
In what follows, bold sans serif is used for 4-vectors while normal bold roman is used for ordinary 3-vectors.


 * Inner product (i.e. notion of length):


 * $$ \boldsymbol{\mathsf{a}} \cdot \boldsymbol{\mathsf{b}} =\eta (\boldsymbol{\mathsf{a}}, \boldsymbol{\mathsf{b}})$$

where $$\eta$$ is known as the metric tensor. In special relativity, the metric tensor is the Minkowski metric:


 * $$\eta = \begin{pmatrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}$$


 * Space-time interval


 * $$ds^2 = dx^2 + dy^2 + dz^2 - c^2 dt^2 = \begin{pmatrix} cdt & dx & dy & dz \end{pmatrix} \begin{pmatrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} cdt \\ dx \\ dy \\ dz \end{pmatrix}$$

In the above, ds2 is known as the spacetime interval. This inner product is invariant under the Lorentz transformation, that is,


 * $$ \eta ( \boldsymbol{\mathsf{a}}', \boldsymbol{\mathsf{b}}' ) = \eta \left ( \Lambda \boldsymbol{\mathsf{a}} , \Lambda \boldsymbol{\mathsf{b}} \right ) = \eta ( \boldsymbol{\mathsf{a}} , \boldsymbol{\mathsf{b}} )$$

The sign of the metric and the placement of the ct, ct', cdt, and cdt&prime; time-based terms can vary depending on the author's choice. For instance, many times the time-based terms are placed first in the four-vectors, with the spatial terms following. Also, sometimes η is replaced with −η, making the spatial terms produce negative contributions to the dot product or spacetime interval, while the time term makes a positive contribution. These differences can be used in any combination, so long as the choice of standards is followed completely throughout the computations performed.

Lorentz transforms
It is possible to express the above coordinate transformation via a matrix. To simplify things, it can be best to replace t, t&prime;, dt, and dt&prime; with ct, ct', cdt, and cdt&prime;, which has the dimensions of distance. So:


 * $$x' = \gamma x - \gamma \beta c t \,$$
 * $$y' = y \,$$
 * $$z' = z \,$$
 * $$c t' = \gamma c t - \gamma \beta x \,$$

then in matrix form:


 * $$\begin{pmatrix} c t' \\ x' \\ y' \\ z' \end{pmatrix} = \begin{pmatrix} \gamma & - \gamma \beta & 0 & 0 \\ - \gamma \beta & \gamma & 0 & 0\\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}\begin{pmatrix} c t \\ x \\ y \\ z \end{pmatrix}$$

The vectors in the above transformation equation are known as four-vectors, in this case they are specifically the position four-vectors. In general, in special relativity, four-vectors can be transformed from one reference frame to another as follows:


 * $$\boldsymbol{\mathsf{a}}' = \Lambda \boldsymbol{\mathsf{a}}$$

In the above, $$\boldsymbol{\mathsf{a}}'$$ and $$\boldsymbol{\mathsf{a}}$$ are the four-vector and the transformed four-vector, respectively, and Λ is the transformation matrix, which, for a given transformation is the same for all four-vectors one might want to transform. So $$\boldsymbol{\mathsf{a}}'$$ can be a four-vector representing position, velocity, or momentum, and the same Λ can be used when transforming between the same two frames. The most general Lorentz transformation includes boosts and rotations; the components are complicated and the transformation requires spinors.

4-vectors and frame-invariant results
Invariance and unification of physical quantities both arise from four-vectors. The inner product of a 4-vector with itself is equal to a scalar (by definition of the inner product), and since the 4-vectors are physical quantities their magnitudes correspond to physical quantities also.

Non-relativistic time-independent Schrödinger equation
Summarized below are the various forms the Hamiltonian takes, with the corresponding Schrödinger equations and forms of wavefunction solutions. Notice in the case of one spatial dimension, for one particle, the partial derivative reduces to an ordinary derivative.

Non-relativistic time-dependent Schrödinger equation
Again, summarized below are the various forms the Hamiltonian takes, with the corresponding Schrödinger equations and forms of solutions.

Angular momentum

 * Magnetic moments

In what follows, B is an applied external magnetic field and the quantum numbers above are used.

Fundamental forces
These equations need to be refined such that the notation is defined as has been done for the previous sets of equations.
 * {| class="wikitable"

! Name ! Equations !Strong force \begin{align} \mathcal{L}_\mathrm{QCD} & = \bar{\psi}_i\left(i \gamma^\mu (D_\mu)_{ij} - m\, \delta_{ij}\right) \psi_j - \frac{1}{4}G^a_{\mu \nu} G^{\mu \nu}_a \\ & = \bar{\psi}_i (i \gamma^\mu \partial_\mu -  m )\psi_i - g G^a_\mu \bar{\psi}_i \gamma^\mu T^a_{ij} \psi_j - \frac{1}{4}G^a_{\mu \nu} G^{\mu \nu}_a \,,\\ \end{align} \,\!$$ !Electroweak interaction
 * $$\mathcal{L}_{EW} = \mathcal{L}_g + \mathcal{L}_f + \mathcal{L}_h + \mathcal{L}_y.\,\!$$
 * $$\mathcal{L}_g = -\frac{1}{4}W_a^{\mu\nu}W_{\mu\nu}^a - \frac{1}{4}B^{\mu\nu}B_{\mu\nu}\,\!$$
 * $$\mathcal{L}_f =  \overline{Q}_i iD\!\!\!\!/\; Q_i+ \overline{u}_i^c iD\!\!\!\!/\; u^c_i+ \overline{d}_i^c iD\!\!\!\!/\; d^c_i+ \overline{L}_i iD\!\!\!\!/\; L_i+ \overline{e}^c_i iD\!\!\!\!/\; e^c_i \,\!$$
 * $$\mathcal{L}_h = |D_\mu h|^2 - \lambda \left(|h|^2 - \frac{v^2}{2}\right)^2\,\!$$
 * $$\mathcal{L}_y = - y_{u\, ij} \epsilon^{ab} \,h_b^\dagger\, \overline{Q}_{ia} u_j^c - y_{d\, ij}\, h\, \overline{Q}_i d^c_j - y_{e\,ij} \,h\, \overline{L}_i e^c_j + h.c.\,\!$$

!Quantum electrodynamics
 * $$\mathcal{L}=\bar\psi(i\gamma^\mu D_\mu-m)\psi -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\;,\,\!$$
 * }

General Relativity
see: General relativity, Einstein field equations, List of equations in gravitation

The Einstein field equations (EFE) may be written in the form:

where $Rμν$ is the Ricci curvature tensor, $R$ is the scalar curvature, $gμν$ is the metric tensor, $Λ$ is the cosmological constant, $G$ is Newton's gravitational constant, $c$ is the speed of light in vacuum, and $Tμν$ is the stress–energy tensor.

The EFE is a tensor equation relating a set of symmetric 4 × 4 tensors. Each tensor has 10 independent components. The four Bianchi identities reduce the number of independent equations from 10 to 6, leaving the metric with four gauge fixing degrees of freedom, which correspond to the freedom to choose a coordinate system.

The EFE is a tensor equation relating a set of symmetric 4 × 4 tensors. Each tensor has 10 independent components. The four Bianchi identities reduce the number of independent equations from 10 to 6, leaving the metric with four gauge fixing degrees of freedom, which correspond to the freedom to choose a coordinate system.

Although the Einstein field equations were initially formulated in the context of a four-dimensional theory, some theorists have explored their consequences in $n$ dimensions. The equations in contexts outside of general relativity are still referred to as the Einstein field equations. The vacuum field equations (obtained when $T$ is identically zero) define Einstein manifolds.

Despite the simple appearance of the equations they are actually quite complicated. Given a specified distribution of matter and energy in the form of a stress–energy tensor, the EFE are understood to be equations for the metric tensor $gμν$, as both the Ricci tensor and scalar curvature depend on the metric in a complicated nonlinear manner. In fact, when fully written out, the EFE are a system of ten coupled, nonlinear, hyperbolic-elliptic partial differential equations.

One can write the EFE in a more compact form by defining the Einstein tensor
 * $$G_{\mu \nu} = R_{\mu \nu} - \tfrac{1}{2} R g_{\mu \nu},$$

which is a symmetric second-rank tensor that is a function of the metric. The EFE can then be written as
 * $$G_{\mu \nu} + \Lambda g_{\mu \nu} = \frac{8 \pi G}{c^4} T_{\mu \nu}.$$

In standard units, each term on the left has units of 1/length2. With this choice of Einstein constant as 8πG/c4, then the stress-energy tensor on the right side of the equation must be written with each component in units of energy-density (i.e., energy per volume = pressure).

Using geometrized units where $G = c = 1$, this can be rewritten as
 * $$G_{\mu \nu} + \Lambda g_{\mu \nu} = 8 \pi T_{\mu \nu}\,.$$

The expression on the left represents the curvature of spacetime as determined by the metric; the expression on the right represents the matter/energy content of spacetime. The EFE can then be interpreted as a set of equations dictating how matter/energy determines the curvature of spacetime.

These equations, together with the geodesic equation, which dictates how freely-falling matter moves through space-time, form the core of the mathematical formulation of general relativity.

Sign convention
The above form of the EFE is the standard established by Misner, Thorne, and Wheeler. The authors analyzed all conventions that exist and classified according to the following three signs (S1, S2, S3):



\begin{align} g_{\mu \nu} & = [S1] \times \operatorname{diag}(-1,+1,+1,+1) \\[6pt] {R^\mu}_{\alpha \beta \gamma} & = [S2] \times \left(\Gamma^\mu_{\alpha \gamma,\beta}-\Gamma^\mu_{\alpha \beta,\gamma}+\Gamma^\mu_{\sigma \beta}\Gamma^\sigma_{\gamma \alpha}-\Gamma^\mu_{\sigma \gamma}\Gamma^\sigma_{\beta \alpha}\right) \\[6pt] G_{\mu \nu} & = [S3] \times \frac{8 \pi G}{c^4} T_{\mu \nu} \end{align} $$

The third sign above is related to the choice of convention for the Ricci tensor:
 * $$R_{\mu \nu}=[S2]\times [S3] \times {R^\alpha}_{\mu\alpha\nu} $$

With these definitions Misner, Thorne, and Wheeler classify themselves as $(+ + +)$, whereas Weinberg (1972) and Peacock (1994) are $(+ − −)$, Peebles (1980) and Efstathiou et al. (1990) are $(− + +)$, Rindler (1977), Atwater (1974), Collins Martin & Squires (1989) are $(− + −)$.

Authors including Einstein have used a different sign in their definition for the Ricci tensor which results in the sign of the constant on the right side being negative
 * $$R_{\mu \nu} - \tfrac{1}{2} R g_{\mu \nu} - \Lambda g_{\mu \nu} = -\frac{8 \pi G}{c^4} T_{\mu \nu}.$$

The sign of the (very small) cosmological term would change in both these versions, if the $(+ − − −)$ metric sign convention is used rather than the MTW $(− + + +)$ metric sign convention adopted here.

Equivalent formulations
Taking the trace with respect to the metric of both sides of the EFE one gets
 * $$R - \frac{D}{2} R + D \Lambda = \frac{8 \pi G}{c^4} T \,$$

where $D$ is the spacetime dimension. This expression can be rewritten as
 * $$-R + \frac{D \Lambda}{\frac{D}{2} -1} = \frac{8 \pi G}{c^4} \frac{T}{\frac{D}{2}-1} \,.$$

If one adds $−1⁄2gμν$ times this to the EFE, one gets the following equivalent "trace-reversed" form
 * $$R_{\mu \nu} - \frac{ \Lambda g_{\mu \nu}}{\frac{D}{2}-1} = \frac{8 \pi G}{c^4} \left(T_{\mu \nu} - \frac{1}{D-2}Tg_{\mu \nu}\right) .\,$$

For example, in $D = 4$ dimensions this reduces to
 * $$R_{\mu \nu} - \Lambda g_{\mu \nu} = \frac{8 \pi G}{c^4} \left(T_{\mu \nu} - \tfrac{1}{2}T\,g_{\mu \nu}\right) .\,$$

Reversing the trace again would restore the original EFE. The trace-reversed form may be more convenient in some cases (for example, when one is interested in weak-field limit and can replace $gμν$ in the expression on the right with the Minkowski metric without significant loss of accuracy).