User:F=q(E+v^B)/Mathematical summary of physics

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Physical laws are often summarized by a single equation, or at least a small set of equations, usually coupled. This article only summarizes the mathematical formalism of the current physical laws from a fundamental perspective.

Classical mechanics
Summary:


 * 1) Start from an appropriate physical law or principle, analyze the classical system and impose boundary and initial conditions,
 * 2) derive the equations of motion for the system,
 * 3) solve the equations to obtain functions describing the motion for all times,
 * 4) Naturally - symmetries and conservation laws are looked for. These occur in parallel.

Generalized dynamics

 * 1) Generalization of Galilean-Newtonian mechanics to a more rigourous, versatile and complete theory of classical mechanics,
 * 2) by introducing the new analytical procedure: generalized coordinates, virtual work, and constraints,
 * 3) leading to understand symmetry and conservation laws.
 * 4) It was the first development of modern theoretical physics,
 * 5) and has many roots into further refinements of physics even when classical mechanics itself was inconsistent with experiment.
 * 6) The reformulation is borderline between classical and quantum mechanics, and even optics, where analogous principles and simalar equations occur
 * 7) General relativity makes use of variational principles of action,
 * 8) Chaotic dynamical systems are analysed using concepts developed from generlized coordinate and momentum spaces.

Generalized quantities

Generalized coordinates:


 * 1) are the minimum number of coordinates, to completley define the configeration of the constituients of the system,
 * 2) choice is arbitary (governed only by conveience),
 * 3) based on lengths, angles, curves in any direction,

For each coordinate is a canonically conjugate generalized momentum, using the Lagranginan function (see below).


 * {| class="wikitable"

!Quantity !Definition
 * Coordinates||$$\mathbf{\dot{q}}\,\!$$
 * Velocities||$$\mathbf{\dot{q}}=\dfrac{\mathrm{d}\mathbf{q}}{\mathrm{d}t}\,\!$$
 * Momenta||$$\mathbf{p}=\dfrac{\partial L}{\partial\mathbf{\dot{q}}}\,\!$$
 * Forces||$$\mathbf{Q}=\mathbf{\dot{p}}\,\!$$
 * }
 * Momenta||$$\mathbf{p}=\dfrac{\partial L}{\partial\mathbf{\dot{q}}}\,\!$$
 * Forces||$$\mathbf{Q}=\mathbf{\dot{p}}\,\!$$
 * }
 * }
 * }

NB: while its an abuse of notation to write:


 * $$ \dfrac{\partial L}{\partial\mathbf{q}}$$

i.e. "divide by a vector", this actually a very conveient notation: it only means collecting a number the expressions


 * $$ \dfrac{\partial L}{\partial q_i }\quad i=1,2\cdots$$

into one equation, the natural power and efficency of vector algebra.

Configuration space:


 * 1) The set/tuple of generalized coordinates: $$\mathbf{q} = \left(q_1, q_2, \cdots q_N \right)\,\!$$.
 * 2) Physiaclly its the set of each object's position in space.
 * 3) At a more abstract mathematical level, the single vector q can be treated as an element of a vector space, for this purpose known as a configeration space of the system.

Lagrangian mechanics
The lagrangian formalism is perhaps the most efficient method in practice to solve for the motion of the system, i.e. where each constiutient of the system will be at time t.

Lagrangian function:


 * 1) The general form is $$L=L(\mathbf{q},\mathbf{\dot{q}},t)$$, i.e. "a function of the system's configeration", though also includes the rate at which the configeration changes, and time.
 * 2) Summarizes the dynamics of a system
 * 3) in terms of positions (generalized coordinates) corresponding to potential energies, and velocities (generalized) corresponding to potential and kinetic energies.

Lagrange's equations:



Usually quoted as


 * $$\dfrac{\mathrm{d}}{\mathrm{d} t}\left(\dfrac{\partial L}{\partial \mathbf{\dot{q}}}\right)=\dfrac{\partial L}{\partial\mathbf{q}}$$

 although the more symmetrical statement is by using the definition of generlized momentum, and observing the symmetry in the position of the dot:



Symmetry in the equations: from definition of p, the transformation $$\mathbf{p}\rightarrow \mathbf{\dot{p}},\quad \mathbf{\dot{p}} \rightarrow \mathbf{q},$$ obtains


 * $$ \mathbf{p}=\frac{\partial L}{\partial\mathbf{\dot{q}}}\quad\xrightarrow{\mathbf{p}\rightarrow\mathbf{\dot{p}},\,\mathbf{\dot{q}}\rightarrow\mathbf{\dot{q}}}\quad\mathbf{\dot{p}}=\frac{\partial L}{\partial\mathbf{q}} $$

which are Lagrange's equations. Obviously reversing the change yeilds only the generalized momentum.



NB: People refer to one of Einstein's mass-energy equivalence $$E = mc^2$$, field equations (see below) or Mandelbrot's $$z=z^2+c$$ as "the simplest equation one could write down". There are no fundamental constants cluttering the Lagrange's equation only to be set equal to unity by suggestive natural units. The equation can be obtained from the definition of generalized momentum simply by changing the position of the dot from q to p, more symmetrical than Hamiltonians equations, which usually gain the attension of their antisymmetric appearance.

Lagrangian inveriance:


 * 1) In Lagrange's equations, since only the partial derivatives with respect to generalized coordinates and velocities are requried,
 * 2) adding the total time derivative of an arbitary function of the generalized coordinates and time describes the same motion:


 * $$ L'(\mathbf{q},\mathbf{\dot{q}},t) = L'(\mathbf{q},\mathbf{\dot{q}},t) + \dfrac{d F(\mathbf{q},t)}{dt} $$

which is analogous to gauge invariance of electromagnetic potentials (see below).

Constants of motion: If


 * $$\dfrac{\mathrm{d}}{\mathrm{d} t}\left(\dfrac{\partial L}{\partial \mathbf{\dot{q}}}\right)=\dfrac{\partial L}{\partial\mathbf{q}}=\boldsymbol{0}$$

then the Hamtilonian


 * $$ H=\mathbf{p}\cdot\mathbf{\dot{q}} - L $$

and p is are constants of the motion (i..e conserved quantities).

Hamiltonian mechanics
Hamiltonian's formalism is usually less practical, but has deeper insights into classical mechanics than Lagrange's formalism.

  Hamiltonian function: $$H=H(\mathbf{q},\mathbf{p},t)$$, analagous to the Lagrangian. The definition is (given above)
 * $$ H=\mathbf{p}\cdot\mathbf{\dot{q}} - L $$

  Hamtilton's equations: can be derived from Lagrange's equations by a Legendre transformation of the Hamtilonian definition to p and q,



 Symmetry in the equations: by the transformation $$p\rightleftharpoons q,\quad H\rightarrow-H$$, the equations are not changed,


 * $$\mathbf{\dot{p}} = -\frac{\partial H}{\partial \mathbf{q}} \quad \xrightarrow[]{\mathbf{p} \rightleftharpoons \mathbf{q},\, H\rightarrow -H} \quad \mathbf{\dot{q}} = -\frac{\partial (-H)}{\partial \mathbf{p}} \quad\rightarrow\quad \mathbf{\dot{q}} = +\frac{\partial H}{\partial \mathbf{p}} $$


 * $$\mathbf{\dot{q}} = +\frac{\partial H}{\partial \mathbf{p}} \quad\xrightarrow[]{\mathbf{p} \rightleftharpoons \mathbf{q},\, H\rightarrow -H} \quad\mathbf{\dot{p}}_i = +\frac{\partial (-H)}{\partial \mathbf{q}} \quad\rightarrow \quad \mathbf{\dot{p}} = -\frac{\partial H}{\partial \mathbf{q}}$$

Phase space: The tuple $$(\mathbf{q},\mathbf{p},t) = (q_1,q_2\cdots p_1,p_2\cdots,t) $$. A specific curve in the phase space is a phase path, the collection of all phase paths (solutions to the equations) is the phase portrait.</li>

<li>Canonical transformations:

The change of variables in Hamtilton's equations can be transformed to new, more conveient variables rendering Hamiltonian's equations easier to solve, at the same time in the same form. The expression for the Hamiltonian changes. This is done using generating functions, intuitivley the generating function "generates the transformation".

Analagous to the Lagrangian, adding the time derivative of another function F to a Hamtiltonian describes the same motion:


 * $$K = H + \dfrac{\partial F}{\partial t}$$

</li> </ol>

Principle of stationary action
This is one of the most fundemental principles of all physics, it applies in Classical and Quantum mechanics, and General relativity. Like other laws which have profoundly simple equations, this can be written very simply indeed.


 * 1) Definition of Action functional: $$S=S[\mathbf{q}]={\displaystyle \int_{t_{1}}^{t_{2}}L(\mathbf{q},\mathbf{\dot{q}},t)dt}$$

The principle of stationary action states the action is stationary to first order:

From this simple expression, all of classical mechanics can be derived.

Hamiltonian-Jacobi mechanics
The equation of which generating function is the simplist leads to the HJE, and it turns out a type-2 generating function is the action:


 * 1) Hamilton–Jacobi equation

where $$H=H\left(\mathbf{q},\dfrac{\partial S}{\partial\mathbf{q}},t\right)$$.

It is a non-linear PDE, making it generally impossible to solve for S exactly, but in fact its not always required to explicitly find S as the problem can be solved for using information generated by the equation. This equation takes classical mechanics to its border, into QM. Many close analogies arise between the SE and HJE.

Statistical mechanics (SM)
Entropy is minimized for a fixed energy.


 * 1) Entropy defnition: $$S=-k{\displaystyle \sum_{i}p_{i}\ln p_{i}}$$.
 * 2) The laws of thermodynamics:

Thermodynamics is a consequence of Statistical mechanics, which can be extended to quantum mechanics.

Summary

 * 1) Analyse the system, impose boundary conditions
 * 2) Use Maxwell's equations to solve for the electromagnetic field, two route to consider:
 * 3) Derive a wave equation for the EM field and solve to predict propagation of EM wave in spacetime, or
 * 4) Use the Lorentz-Heaviside force to obtain an equation of motion for the charge distribution, solve to predict the motion of the electric charges in the external electromagnetic field.

Formalism
Formulations of the EM field:in terms of

<ol> <li>the electric and magnetic fields:

• E and B (two vector fields), or

• the EM field F (one tensor field), or

• the EM field F (multi-vector)</li> <li>the potential fields:

• A and ϕ (seperatley vector and scalar fields), or

• the 4-potential A (4-vector used in conjuction with F), </li> <li> each related to the sources

• J and ρ (seperatley vector and scalar fields), or

• the 4-current J (4-vector).

• the electric current J (multi-vector).</li> </ol>

General relations:


 * $$\begin{align}

-&\mathbf{E}=\nabla\phi + \dfrac{\partial \mathbf{A}}{\partial t}\\ & \mathbf{B}=\nabla\times\mathbf{A} \\ \end{align}$$

The many formalisms are summarized below. These do not include magnetic monopoles (experimentally this is the case so far).


 * {| class="wikitable"

! Formalism !! Maxwell's equations !! Lorentz force
 * rowspan="2"| Vector fields:
 * rowspan="2"| Vector fields:

E and B fields, or

A and ϕ fields

& \nabla\cdot\mathbf{E}=\rho/\varepsilon,\\ & \nabla\cdot\mathbf{B}=0\\ & \nabla\times\mathbf{E}=-\dfrac{\partial \mathbf{B}}{\partial t} \\ & \nabla\times\mathbf{B} = \mu \mathbf{J} + \dfrac{1}{c^2}\dfrac{\partial \mathbf{E}}{\partial t} \\ \end{align}$$ & \nabla^2 \varphi + \frac{\partial}{\partial t} \left ( \mathbf \nabla \cdot \mathbf A \right ) = - \frac{\rho}{\varepsilon_0}\\ & \left ( \nabla^2 \mathbf A - \frac{1}{c^2} \frac{\partial^2 \mathbf A}{\partial t^2} \right ) - \mathbf \nabla \left ( \mathbf \nabla \cdot \mathbf A + \frac{1}{c^2} \frac{\partial \varphi}{\partial t} \right ) = - \mu_0 \mathbf J \end{align}$$ &\dfrac{\partial F^{\alpha\beta}}{\partial x^{\alpha}}=\mu J^{\beta}\\ &\epsilon^{\delta\alpha\beta\gamma}\dfrac{\partial F_{\alpha\beta}}{\partial x^{\gamma}}=0 \end{align}$$ & \mathrm{d}\mathbf{F}=0 \\ & \mathrm{d}\, {*\mathbf{F}}=\mathbf{J} \\ \end{align}$$
 * $$\begin{align}
 * $$\mathbf{F}=q(\mathbf{E}+\mathbf{v}\times\mathbf{B} )$$
 * $$\begin{align}
 * $$\begin{align}
 * $$\mathbf{F}=q\left(-\nabla\phi - \dfrac{\partial \mathbf{A}}{\partial t} + \mathbf{v}\times(\nabla\times\mathbf{A}) \right)$$
 * Tensor fields
 * $$\begin{align}
 * $$\begin{align}
 * $$\dfrac{\mathrm{d}P^{\alpha}}{\mathrm{d}\tau}=F^{\alpha}{}_{\beta}V^{\beta}$$
 * Differential forms
 * $$\begin{align}
 * $$\begin{align}
 * Geometric algebra
 * $$\left(\dfrac{1}{c}\dfrac{\partial}{\partial t} + \nabla\right)F=\mu cJ $$
 * }
 * $$\left(\dfrac{1}{c}\dfrac{\partial}{\partial t} + \nabla\right)F=\mu cJ $$
 * }
 * }
 * }

</li>

</ol>

Postulates
Postulates in QM are analagous to the Newton's laws, treated as axioms. In some sense they are fundamental since they cannot be derived.


 * 1) State of a system completley described by the wavefunction \Psi.
 * 2) Amplitude corresponds to probability of particle in the state given state.
 * 3) Observables are mathematically operators.
 * 4) Wave equations describe the behaviour of the system, i.e. the solutions are \Psi.
 * 5) Quantize observables by replacing the classical poission bracket with a commutator.

Wave mechanics
Summary:


 * 1) Anylize the quantum system, impose boundary conditions
 * 2) Solve the wave equation for the wavefunction, using an approprite Hamiltonian operator
 * 3) The wavefunction describes the behaviour of the quantum state, use with other operators to find dynamical variables and to calculate probabiities and expectation values

Wave equation

The most general wave equation is

which originally arose from Shrodinger's equation. As relativistic equations were developed, such as the Dirac equation and Breit equation, they became more complicated but ultimatley it was the Hamiltonian operator which changed. Otehr wave equations in physics are 2nd order in space and time.

Duality and symmetry. <ol> <li>De Broglie relations:

</li>

<li>Fourier transform symmetry, between momentum p and position r,

$$\begin{array}{c} \psi(\mathbf{r},t)=\dfrac{1}{(\sqrt{2\pi\hbar})^{3}}{\displaystyle \int_{\mathbf{R^{3}}}}\mathrm{d}^{3}\mathbf{r}\phi\left(\mathbf{p},t\right)e^{i\mathbf{p}\cdot\mathbf{r}/\hbar} \end{array}$$

though more natural to use k instead since factors of ħ are removed:


 * $$\begin{array}{c}

\psi(\mathbf{r},t)=\dfrac{1}{(\sqrt{2\pi})^{3}}{\displaystyle \int_{\mathbf{R^{3}}}}\mathrm{d}^{3}\mathbf{r}\phi\left(\mathbf{k},t\right)e^{i\mathbf{k}\cdot\mathbf{r}} \end{array}$$</li> </ol>

Matrix mechanics

 * 1) Heisenberg equation
 * 2) Matrix operator elements: $$A_{ij}=\left\langle \psi_{j}\right|\hat{A}\left|\psi_{i}\right\rangle$$.

Quantum states
<ol> <li> Dirac representation:

The wavefunction can be represented as an abstract vector in a vector space (a Hilbert space),

$$|\psi\rangle =\sum_{s_{z}}\int_{\mathbf{R^{3}}}\mathrm{d}^{3}\mathbf{r}\psi\left(\mathbf{r},t,s_{z}\right)\left|\mathbf{r},t,s_{z}\right\rangle $$

where the descrete variable (spin) are summed and continuous variable (position) are integrated. Time is a parameter, no integration over the time coordinate is to be done. The addition and integration of all basis vectors (basis ket) $$|\mathbf{r},t,s_{z}\rangle $$ superimposes to form the state vector. Alternativley momentum p can replace r. </li>

<li> Probability of the system in the state $$ |\mathbf{r},t,s_{z}\rangle $$ is


 * $$ P=\langle \mathbf{r},t,s_{z}|\psi\rangle $$,

integrated over the region of interest and summed over the necerssary spin values. For normalization $$ \langle \psi |\psi\rangle = 1 $$.</li>

<li> Overlap matrix/integral:


 * $$\left\langle \psi_{j}\right|\hat{A}\left|\psi_{i}\right\rangle ={\displaystyle \int}\psi_{j}^{*}\hat{A}\psi_{i}d^{3}\mathbf{r}$$</li>

<li> Average/Expectation value of observable A:


 * $$\langle \hat{A}\rangle =\left\langle \psi\right|\hat{A}\left|\psi\right\rangle ={\displaystyle \int}\psi^{*}\hat{A}\psi d^{3}\mathbf{r}$$

Note the case for the identity operator $$\hat{A} = \hat{I} = 1 $$, which reduces to the probability. </li>

</ol>

Definitions
<ol> <li> Commutator of operators: $$[\hat{A},\hat{B}]=\hat{A}\hat{B}-\hat{B}\hat{A}$$</li>

<li> Poisson bracket: $$\{A,B\}=\dfrac{\partial A}{\partial q_{i}}\dfrac{\partial B}{\partial p_{i}}-\dfrac{\partial A}{\partial p_{i}}\dfrac{\partial B}{\partial q_{i}}$$</li> </ol>

Formalism
<ol> <li> The general Uncertainty principle

<li>canonical commutation

$$[\hat{A},\hat{B}]=i\hbar\left\{ A,B\right\} $$ </li>

<li>Quantum operators:

The most fundamental operators are position, momentum, and spin. All the others can be derived from position and momentum, but spin can't be derived.


 * {| class="wikitable"

!Quantity !Operator !Eigenvalue
 * Position||$$\hat{\mathbf{r}}=\mathbf{r}$$|| r
 * Momentum||$$\hat{\mathbf{p}}=-i\hbar\nabla $$|| p
 * Spin||$$\hat{\mathbf{S}}=\dfrac{\hbar}{2}\boldsymbol{\sigma}$$||$$\hbar\sqrt{s(s+1)}$$
 * Wavevector||$$\hat{\mathbf{k}}=-i\nabla$$ || k
 * Angular momentum||$$\hat{\mathbf{L}}=\hat{\mathbf{r}}\times\hat{\mathbf{p}}=-i\hbar\mathbf{r}\times\nabla$$||$$\hbar\sqrt{\ell(\ell+1)}$$
 * Potential energy||$$\hat{V}=V$$||V
 * Kinetic energy	||$$\hat{T}=\dfrac{\hat{\mathbf{p}}\cdot\hat{\mathbf{p}}}{2m}=-\dfrac{\hbar^{2}}{2m}\nabla^{2}$$||T
 * Hamiltonian||$$\hat{H}={\displaystyle \sum_{i}}\hat{T_{i}}+\hat{V}$$||
 * Total energy||$$\hat{E}=i\hbar\dfrac{\partial}{\partial t}$$||E
 * Angular frequency||$$\hat{\omega}=i\dfrac{\partial}{\partial t}$$||ω
 * Total angular momentum||$$\hat{\mathbf{J}}=\hat{\mathbf{L}}+\hat{\mathbf{S}}$$||$$\hbar\sqrt{j(j+1)}$$
 * }</li>
 * Kinetic energy	||$$\hat{T}=\dfrac{\hat{\mathbf{p}}\cdot\hat{\mathbf{p}}}{2m}=-\dfrac{\hbar^{2}}{2m}\nabla^{2}$$||T
 * Hamiltonian||$$\hat{H}={\displaystyle \sum_{i}}\hat{T_{i}}+\hat{V}$$||
 * Total energy||$$\hat{E}=i\hbar\dfrac{\partial}{\partial t}$$||E
 * Angular frequency||$$\hat{\omega}=i\dfrac{\partial}{\partial t}$$||ω
 * Total angular momentum||$$\hat{\mathbf{J}}=\hat{\mathbf{L}}+\hat{\mathbf{S}}$$||$$\hbar\sqrt{j(j+1)}$$
 * }</li>
 * Angular frequency||$$\hat{\omega}=i\dfrac{\partial}{\partial t}$$||ω
 * Total angular momentum||$$\hat{\mathbf{J}}=\hat{\mathbf{L}}+\hat{\mathbf{S}}$$||$$\hbar\sqrt{j(j+1)}$$
 * }</li>
 * Total angular momentum||$$\hat{\mathbf{J}}=\hat{\mathbf{L}}+\hat{\mathbf{S}}$$||$$\hbar\sqrt{j(j+1)}$$
 * }</li>
 * }</li>

<li>Canonical commutation and uncertainty relations: Intrinsic quantum phenomenon. It is impossible to know/measure the exact values of two quantities simaltaneously whose mathematical operators do not commuate.


 * {|class="wikitable"

!Quantities !Commutation !Uncertainty </ol>
 * r, p||$$\left[\hat{x}_{i},\hat{p}_{j}\right]=i\hbar\delta_{ij}$$||$$\sigma\left(x_{i}\right)\sigma\left(p_{j}\right)\geq\dfrac{\hbar}{2}\delta_{ij}$$
 * E, t||N/A||$$\sigma\left(E\right)\sigma\left(t\right)\geq\dfrac{\hbar}{2}$$
 * L||$$\left[\hat{L}_{i},\hat{L}_{j}\right]=i\hbar\epsilon_{ijk}\hat{L}_{k}$$||$$\sigma\left(L_{i}\right)\sigma\left(L_{j}\right)\geq\dfrac{\hbar}{2}\left|\langle \epsilon_{ijk}\hat{L}_{k}\rangle \right|$$
 * S||$$\left[\hat{S}_{i},\hat{S}_{j}\right]=i\hbar\epsilon_{ijk}\hat{S}_{k}$$||$$\sigma\left(S_{i}\right)\sigma\left(S_{j}\right)\geq\dfrac{\hbar}{2}\left|\langle \epsilon_{ijk}\hat{S}_{k}\rangle \right|$$
 * σ||$$\left[\hat{\sigma}_{i},\hat{\sigma}_{j}\right]=2i\epsilon_{ijk}\hat{\sigma}_{k}$$||N/A
 * J||$$\left[\hat{J}_{i},\hat{J}_{j}\right]=i\hbar\epsilon_{ijk}\hat{J}_{k}$$||$$\sigma\left(J_{i}\right)\sigma\left(J_{j}\right)\geq\dfrac{\hbar}{2}\left|\langle \epsilon_{ijk}\hat{J}_{k}\rangle \right|$$
 * }</li>
 * S||$$\left[\hat{S}_{i},\hat{S}_{j}\right]=i\hbar\epsilon_{ijk}\hat{S}_{k}$$||$$\sigma\left(S_{i}\right)\sigma\left(S_{j}\right)\geq\dfrac{\hbar}{2}\left|\langle \epsilon_{ijk}\hat{S}_{k}\rangle \right|$$
 * σ||$$\left[\hat{\sigma}_{i},\hat{\sigma}_{j}\right]=2i\epsilon_{ijk}\hat{\sigma}_{k}$$||N/A
 * J||$$\left[\hat{J}_{i},\hat{J}_{j}\right]=i\hbar\epsilon_{ijk}\hat{J}_{k}$$||$$\sigma\left(J_{i}\right)\sigma\left(J_{j}\right)\geq\dfrac{\hbar}{2}\left|\langle \epsilon_{ijk}\hat{J}_{k}\rangle \right|$$
 * }</li>
 * J||$$\left[\hat{J}_{i},\hat{J}_{j}\right]=i\hbar\epsilon_{ijk}\hat{J}_{k}$$||$$\sigma\left(J_{i}\right)\sigma\left(J_{j}\right)\geq\dfrac{\hbar}{2}\left|\langle \epsilon_{ijk}\hat{J}_{k}\rangle \right|$$
 * }</li>

Postulate
There is one simple, intuitive, but important, postulate of relativity. Again it cannot be derived.

The physical laws are the same for all observers.

Metrics in relativity
Various 4-vectors are below.


 * {| class="wikitable"

! Metric !! Signature !! Representation !! -1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \end{pmatrix}$$ || 1 & 0 & 0 & 0\\ 0 & -1 & 0 & 0\\ 0 & 0 & -1 & 0\\ 0 & 0 & 0 & -1 \end{pmatrix}$$ ||
 * rowspan="2"| Minkowski || (−+++) || $$\eta=\begin{pmatrix}
 * rowspan="2"| Minkowski || (−+++) || $$\eta=\begin{pmatrix}
 * (+−−−) || $$\eta=\begin{pmatrix}
 * (+−−−) || $$\eta=\begin{pmatrix}
 * Example || Example || Example || Example
 * }
 * }

Intrinsic definitons
<ol> <li>Rapidity ϕ</li> <li>Velocity ratio: $$\beta = v/c = \tanh \phi$$</li> <li>Lorentz factor: $$\gamma(\mathbf{v}) = \dfrac{1}{\sqrt{1-\dfrac{\mathbf{v}\cdot\mathbf{v}}{c^2}}} = \cosh \phi$$</li> <li> The Velocity-addition formula is:


 * $$\mathbf{v} \oplus\mathbf{u}=\frac{\mathbf{v}+\mathbf{u}_{\parallel} + \mathbf{u}_{\perp}/\gamma(\mathbf{v})}{1+\frac{\mathbf{v}\cdot\mathbf{u}}{c^2}},$$

where $$ \mathbf{u} = \mathbf{u}_{\parallel} + \mathbf{u}_{\perp}$$

(the componets of u are parallel and perpendicular to $$\mathbf{v}$$). </li> </ol>

4-vectors
Various 4-vectors are below, using the Minkowski metric with signature (+−−−),


 * 1) indicies raised by $$A^\mu=\eta^{\mu\lambda}A_\lambda,\,B^{\alpha\beta}=\eta^{\alpha\mu}\eta^{\beta\lambda}B_{\mu\lambda}$$
 * 2) and lowered by $$A_\mu=\eta_{\mu\lambda}A^\lambda,\,B_{\alpha\beta}=\eta_{\alpha\mu}\eta_{\beta\lambda}B^{\mu\lambda}$$

The covariant vector components are given below; use the metric above to obtain contravariant vectors.


 * {|class="wikitable"

!scope="col" width="100"|4-Quantity !General definition !scope="col" width="200"|Components using (+−−−) metric ! (leave for now) 0 & xp_y-yp_x & xp_z-zp_x \\ yp_x-xp_y & 0 & yp_z-zp_y \\ zp_x-xp_z & zp_y-yp_z & 0 \end{pmatrix}$$ $$J_\mu=\left(c\rho,-j_x,-j_y,-j_z\right)$$ 0 & E_x/c & E_y/c & E_z/c \\ -E_x/c & 0 & -B_z & B_y \\ -E_y/c & B_z & 0 & -B_x \\ -E_z/c & -B_y & B_x & 0 \end{pmatrix}$$ \mathcal{M}^{\mu \nu} = \begin{pmatrix} 0    & P_xc & P_yc & P_zc \\ - P_xc & 0     &  - M_z   & M_y    \\ - P_yc & M_z    & 0      & - M_x   \\ - P_zc & - M_y   & M_x    & 0 \end{pmatrix} $$ \mathcal{D}^{\mu \nu} = \begin{pmatrix} 0    & - D_xc & - D_yc & - D_zc \\ D_xc & 0     & - H_z   & H_y    \\ D_yc & H_z   & 0      & - H_x   \\ D_zc & - H_y  & H_x    & 0 \end{pmatrix} $$ {\mu_0} \right)\delta _{ij} $$ \begin{pmatrix} \frac{1}{2}\left(\epsilon_0 E^2 + \dfrac{B^2}{\mu_0} \right) & \epsilon_0E_xE_y + \frac{B_xB_y}{\mu_0} & \epsilon_0E_xE_z + \frac{B_xB_z}{\mu_0} \\ \epsilon_0E_yE_x + \frac{B_yB_x}{\mu_0} & \frac{1}{2}\left(\epsilon_0 E^2  + \dfrac{B^2}{\mu_0} \right) & \epsilon_0E_yE_z + \frac{B_yB_z}{\mu_0} \\ \epsilon_0E_zE_x + \frac{B_zB_x}{\mu_0} & \epsilon_0E_zE_y + \frac{B_zB_y}{\mu_0} & \frac{1}{2}\left(\epsilon_0 E^2  + \dfrac{B^2}{\mu_0} \right) \end{pmatrix}$$ S_y/c & -\sigma_{yx} & -\sigma_{yy} & -\sigma_{yz} \\ S_z/c & -\sigma_{zx} & -\sigma_{zy} & -\sigma_{zz} \end{pmatrix}\,$$ R^{\alpha}{}_{\beta\gamma\delta}= & \Gamma^{\alpha}{}_{\mu\gamma}\Gamma^{\mu}{}_{\beta\delta}-\Gamma^{\alpha}{}_{\mu\delta}\Gamma^{\mu}{}_{\beta\gamma}\\ & +\partial_{\gamma}\Gamma^{\alpha}{}_{\beta\delta}-\partial_{\delta}\Gamma^{\alpha}{}_{\beta\gamma} \end{array}$$
 * 4-position|| ||$$X_\mu=\left(ct,-x,-y,-z\right)$$||
 * 4-velocity||$$V_{\mu}=\dfrac{\mathrm{d}X_{\mu}}{\mathrm{d}\tau}$$||$$V_\mu=\gamma\left( c,-v_x,-v_y,-v_z\right)$$||
 * 4-acceleration	|| $$A_{\mu}=\dfrac{\mathrm{d}V_{\mu}}{\mathrm{d}\tau}$$ || $$A_{\mu}=\gamma \left( c\frac{\mathrm{d}\gamma}{\mathrm{d}t}, \frac{\mathrm{d}\gamma}{\mathrm{d}t} \mathbf{v} + \gamma \mathbf{a} \right)$$||
 * 4-momentum||$$P_\mu=mV_\mu$$||$$P_\mu=\left(E/c,-p_x,-p_y,-p_z\right)$$||
 * 4-force||$$F_{\mu}=\dfrac{\mathrm{d}P_{\mu}}{\mathrm{d}\tau}$$||$$F_\mu=\gamma m \left( c\frac{\mathrm{d}\gamma}{\mathrm{d}t}, \frac{\mathrm{d}\gamma}{\mathrm{d}t} \mathbf{v} + \gamma \mathbf{a} \right)$$||
 * Angular momentum
 * $$ L^{ij} = X^i P^j - X^j P^i = 2 X^{[i}P^{j]} $$
 * $$L^{ij} = \begin{pmatrix}
 * 4-force||$$F_{\mu}=\dfrac{\mathrm{d}P_{\mu}}{\mathrm{d}\tau}$$||$$F_\mu=\gamma m \left( c\frac{\mathrm{d}\gamma}{\mathrm{d}t}, \frac{\mathrm{d}\gamma}{\mathrm{d}t} \mathbf{v} + \gamma \mathbf{a} \right)$$||
 * Angular momentum
 * $$ L^{ij} = X^i P^j - X^j P^i = 2 X^{[i}P^{j]} $$
 * $$L^{ij} = \begin{pmatrix}
 * $$ L^{ij} = X^i P^j - X^j P^i = 2 X^{[i}P^{j]} $$
 * $$L^{ij} = \begin{pmatrix}
 * 4-current
 * 4-current
 * 4-current
 * EM 4-potential
 * $$A_\mu=\left(\phi/c,-A_x,-A_y,-A_z\right)$$
 * EM tensor
 * $$ F^{\alpha \beta} = \frac {\partial A^{\beta}}{\partial x_{\alpha}} - \frac {\partial A^{\alpha}}{\partial x_{\beta}} = 2\partial_{[\alpha} A_{\beta]} $$
 * $$F_{\alpha\beta} = \begin{pmatrix}
 * EM tensor
 * $$ F^{\alpha \beta} = \frac {\partial A^{\beta}}{\partial x_{\alpha}} - \frac {\partial A^{\alpha}}{\partial x_{\beta}} = 2\partial_{[\alpha} A_{\beta]} $$
 * $$F_{\alpha\beta} = \begin{pmatrix}
 * $$ F^{\alpha \beta} = \frac {\partial A^{\beta}}{\partial x_{\alpha}} - \frac {\partial A^{\alpha}}{\partial x_{\beta}} = 2\partial_{[\alpha} A_{\beta]} $$
 * $$F_{\alpha\beta} = \begin{pmatrix}
 * Magnetization-polarization tensor
 * Magnetization-polarization tensor
 * Magnetization-polarization tensor
 * electromagnetic displacement tensor
 * $$\mathcal{D}^{\mu \nu} = \frac{1}{\mu_{0}} F^{\mu \nu} - \mathcal{M}^{\mu \nu} \,$$
 * electromagnetic displacement tensor
 * $$\mathcal{D}^{\mu \nu} = \frac{1}{\mu_{0}} F^{\mu \nu} - \mathcal{M}^{\mu \nu} \,$$
 * Maxwell stress tensor
 * $$\sigma_{ij} = \epsilon_0E_iE_j + \frac{B_iB_j}{\mu_0} - \frac{1}{2}\left(\epsilon_0 E^2 + \dfrac{B^2}
 * Maxwell stress tensor
 * $$\sigma_{ij} = \epsilon_0E_iE_j + \frac{B_iB_j}{\mu_0} - \frac{1}{2}\left(\epsilon_0 E^2 + \dfrac{B^2}
 * $$\sigma_{ij} =
 * Stress–energy tensor
 * Electromagnetic stress-energy tensor
 * $$T^{\alpha\beta} = \frac{1}{\mu_0} \left( \eta_{\gamma \nu}F^{\alpha \gamma}F^{\nu \beta} - \frac{1}{4}\eta^{\alpha\beta}F_{\gamma \nu}F^{\gamma \nu}\right)$$
 * $$T^{\alpha\beta} = \begin{pmatrix} \epsilon_{0}E^2/2 + B^2/2\mu_{0} & S_x/c & S_y/c & S_z/c \\ S_x/c & -\sigma_{xx} & -\sigma_{xy} & -\sigma_{xz} \\
 * Electromagnetic stress-energy tensor
 * $$T^{\alpha\beta} = \frac{1}{\mu_0} \left( \eta_{\gamma \nu}F^{\alpha \gamma}F^{\nu \beta} - \frac{1}{4}\eta^{\alpha\beta}F_{\gamma \nu}F^{\gamma \nu}\right)$$
 * $$T^{\alpha\beta} = \begin{pmatrix} \epsilon_{0}E^2/2 + B^2/2\mu_{0} & S_x/c & S_y/c & S_z/c \\ S_x/c & -\sigma_{xx} & -\sigma_{xy} & -\sigma_{xz} \\
 * Electromagnetic stress-energy tensor
 * $$T^{\alpha\beta} = \frac{1}{\mu_0} \left( \eta_{\gamma \nu}F^{\alpha \gamma}F^{\nu \beta} - \frac{1}{4}\eta^{\alpha\beta}F_{\gamma \nu}F^{\gamma \nu}\right)$$
 * $$T^{\alpha\beta} = \begin{pmatrix} \epsilon_{0}E^2/2 + B^2/2\mu_{0} & S_x/c & S_y/c & S_z/c \\ S_x/c & -\sigma_{xx} & -\sigma_{xy} & -\sigma_{xz} \\
 * Riemann curvature tensor
 * $$\begin{array}{cl}
 * Riemann curvature tensor
 * $$\begin{array}{cl}
 * Ricci curvature tensor
 * $$ R^{\sigma}{}_{\alpha\sigma\beta} = g^{\sigma\delta}R_{\delta\alpha\sigma\beta} = R_{\beta\alpha}$$
 * Einstein tensor
 * $$G^{\mu\lambda} = R^{\mu\lambda} - \frac{1}{2}g^{\mu\lambda}R$$
 * }
 * $$ R^{\sigma}{}_{\alpha\sigma\beta} = g^{\sigma\delta}R_{\delta\alpha\sigma\beta} = R_{\beta\alpha}$$
 * Einstein tensor
 * $$G^{\mu\lambda} = R^{\mu\lambda} - \frac{1}{2}g^{\mu\lambda}R$$
 * }
 * $$G^{\mu\lambda} = R^{\mu\lambda} - \frac{1}{2}g^{\mu\lambda}R$$
 * }
 * }
 * }
 * }

The general definitions are frame-independent.

NB: The Maxwell stress tensor is not to be confused with the electromagnetic stress-energy tensor, the latter is a special case of the general Stress-energy tensor (as in Einstein's field equations, see below).

Useful invariant relations, also frame-independent equations and quantities:


 * {|class="wikitable"

!Quantities !Components !Resultant !Equation !Invariant quantity
 * $$X_\mu$$||$$X_\mu X^\mu=\left(ct\right)^2-r^2$$||$$X_\mu X^\mu=\left(c\tau\right)^2$$||$$\left(ct\right)^2-r^2=\left(c\tau\right)^2$$||Proper time $$\tau$$
 * $$V_\mu$$||$$V_\mu V^\mu=\gamma^2\left(c^2-v^2\right)$$||$$V_\mu V^\mu=c^2$$||$$\gamma^2\left(c^2-v^2\right)=c^2$$
 * $$V_\mu$$||$$V_\mu V^\mu=\gamma^2\left(c^2-v^2\right)$$||$$V_\mu V^\mu=c^2$$||$$\gamma^2\left(c^2-v^2\right)=c^2$$
 * $$V_\mu$$||$$V_\mu V^\mu=\gamma^2\left(c^2-v^2\right)$$||$$V_\mu V^\mu=c^2$$||$$\gamma^2\left(c^2-v^2\right)=c^2$$

reduces to definition of $$\gamma$$
 * Luminal speed c
 * $$P_\mu$$||$$P_\mu P^\mu=\left(\dfrac{E}{c}\right)^2-p^2$$||$$P_\mu P^\mu=\left(mc\right)^2$$||$$\left(\dfrac{E}{c}\right)^2-p^2=\left(mc\right)^2$$||Invariant mass m
 * }
 * }
 * }

Lorentz transformation (inertial frames)
<ol> <li>Transformation: For a boost in an arbitary direction (no rotations)


 * $$ {A'}_\mu = \Lambda_{\mu}{}^{\lambda}A_\lambda \,\!$$

Components are:


 * $$ \begin{align} \Lambda_{00} & = \gamma, \\

\Lambda_{0i} & = \Lambda_{i0} = - \gamma \beta_{i}, \\ \Lambda_{ij} & = \Lambda_{ji} = ( \gamma - 1 )\dfrac{\beta_{i}\beta_{j}}{\beta^{2}} + \delta_{ij}= ( \gamma - 1 )\dfrac{v_i v_j}{v^2} + \delta_{ij}, \\ \end{align} \,\!$$

in matrix representation:


 * $$\Lambda_{\mu\lambda} = \begin{bmatrix}

\gamma & - \gamma \boldsymbol{\beta}^\mathrm{T} \\ -\gamma\boldsymbol{\beta} & \mathbf{I} + (\gamma-1) \boldsymbol{\beta}\boldsymbol{\beta}^\mathrm{T}/\beta^2 \\ \end{bmatrix}\,\!$$

Most physical quantities are best described as (components of) tensors (also spinors). All physical 4-vetcors and tensors transform by the rule:


 * $$T^{\alpha' \beta' \cdots \zeta'}_{\theta' \iota' \cdots \kappa'} =

\Lambda^{\alpha'}{}_{\mu} \Lambda^{\beta'}{}_{\nu} \cdots \Lambda^{\zeta'}{}_{\rho} \Lambda_{\theta'}{}^{\sigma} \Lambda_{\iota'}{}^{\upsilon} \cdots \Lambda_{\kappa'}{}^{\phi} T^{\mu \nu \cdots \rho}_{\sigma \upsilon \cdots \phi}$$</li>

<li> </li>Compositions of two transformations:


 * $$B(\mathbf{u})B(\mathbf{v})=B\left ( \mathbf{u}\oplus\mathbf{v} \right )\mathrm{Gyr}\left [ \mathbf{u},\mathbf{v}\right ]=\mathrm{Gyr}\left [\mathbf{u},\mathbf{v} \right ]B \left ( \mathbf{v}\oplus\mathbf{u} \right )$$,

including rotations U and V


 * $$L(\mathbf{u},U)L(\mathbf{u},V)=L(\mathbf{u}\oplus U\mathbf{v}, \mathrm{gyr}[\mathbf{u},U\mathbf{v}]UV)$$

(cf Gyrovector space) </ol>

Summary

 * 1) Analyze system and impose boundary conditions
 * 2) Solve Einsteins field equatios for the metric,
 * 3) Use the Geodesic equation and solve for the geodesic deviation to calculate the motion of masses in curved spacetime, equivalently the gravitational field.

Definitions

 * 1) Some derivative to be named: $$\dfrac{\mathrm{D}A^{\mu}}{\mathrm{D}\lambda}=\dfrac{\mathrm{d}A^{\mu}}{\mathrm{d}\lambda}+\Gamma_{\alpha\beta}^{\mu}A^{\alpha}t^{\beta}$$.
 * 2) Any contravariant 4-vector $$A^\mu$$
 * 3) Tangent vector $$t^{\mu}=\dfrac{\mathrm{d}x^{\mu}}{\mathrm{d}\lambda}$$
 * 4) Affine parameter λ.

Formalism

 * {| class="wikitable"

! Formalism !! Einstein's field equations !! Geodesic equations
 * Tensor form
 * $$G^{\alpha\beta}=\dfrac{8\pi G}{c^{4}}T^{\alpha\beta}$$
 * Geodesic equation
 * Geodesic equation

$$\dfrac{\mathrm{D}A^{\mu}}{\mathrm{D}\lambda}=0$$

Geodesic deviation $$\xi^{\mu}$$:

$$\dfrac{\mathrm{D^{2}}\xi^{\mu}}{\mathrm{D}\lambda^{2}}=-R_{\alpha\beta\gamma}^{\mu}t^{\alpha}\xi^{\beta}t^{\gamma}$$
 * Differential forms || ||
 * Geometric algebra || ||
 * }
 * Geometric algebra || ||
 * }