Stationary-action principle

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https://en.wikipedia.org/wiki/Stationary-action_principle 

The stationary-action principle – also known as the principle of least action – is a variational principle that, when applied to the action of a mechanical system, yields the equations of motion for that system. The principle states that the trajectories (i.e. the solutions of the equations of motion) are stationary points of the system's action functional. The term "least action" is a historical misnomer since the principle has no minimality requirement: the value of the action functional need not be minimal (even locally) on the trajectories.

The principle can be used to derive Newtonian, Lagrangian and Hamiltonian equations of motion, and even general relativity (see Einstein–Hilbert action). In relativity, a different action must be minimized or maximized.

The classical mechanics and electromagnetic expressions are a consequence of quantum mechanics. The stationary action method helped in the development of quantum mechanics. In 1933, the physicist Paul Dirac demonstrated how this principle can be used in quantum calculations by discerning the quantum mechanical underpinning of the principle in the quantum interference of amplitudes. Subsequently Julian Schwinger and Richard Feynman independently applied this principle in quantum electrodynamics.

The principle remains central in modern physics and mathematics, being applied in thermodynamics, fluid mechanics, the theory of relativity, quantum mechanics, particle physics, and string theory and is a focus of modern mathematical investigation in Morse theory. Maupertuis' principle and Hamilton's principle exemplify the principle of stationary action.

The action principle is preceded by earlier ideas in optics. In ancient Greece, Euclid wrote in his Catoptrica that, for the path of light reflecting from a mirror, the angle of incidence equals the angle of reflection. Hero of Alexandria later showed that this path was the shortest length and least time.

Scholars often credit Pierre Louis Maupertuis for formulating the principle of least action because he wrote about it in 1744 and 1746. However, Leonhard Euler discussed the principle in 1744, and evidence shows that Gottfried Leibniz preceded both by 39 years.

General statement

As the system evolves, q traces a path through configuration space (only some are shown). The path taken by the system (red) has a stationary action (δS = 0) under small changes in the configuration of the system (δq).

The starting point is the action, denoted ${\displaystyle {\mathcal {S}}}$ (calligraphic S), of a physical system. It is defined as the integral of the Lagrangian L between two instants of time t₁ and t₂ – technically a functional of the N generalized coordinates q = (q₁, q₂, ... , q_N) which are functions of time and define the configuration of the system:

${\displaystyle \mathbf {q} :\mathbf {R} \to \mathbf {R} ^{N}}$

${\displaystyle {\mathcal {S}}[\mathbf {q} ,t_{1},t_{2}]=\int _{t_{1}}^{t_{2}}L(\mathbf {q} (t),\mathbf {\dot {q}} (t),t)dt}$

where the dot denotes the time derivative, and t is time.

Mathematically the principle is

${\displaystyle \delta {\mathcal {S}}=0,}$

where δ (lowercase Greek delta) means a small change. In words this reads:

The path taken by the system between times t₁ and t₂ and configurations q₁ and q₂ is the one for which the action is stationary (no change) to first order.

Stationary action is not always a minimum, despite the historical name of least action. It is a minimum principle for sufficiently short, finite segments in the path.

In applications the statement and definition of action are taken together:

${\displaystyle \delta \int _{t_{1}}^{t_{2}}L(\mathbf {q} ,\mathbf {\dot {q}} ,t)dt=0.}$

The action and Lagrangian both contain the dynamics of the system for all times. The term "path" simply refers to a curve traced out by the system in terms of the coordinates in the configuration space, i.e. the curve q(t), parameterized by time (see also parametric equation for this concept).

Origins, statements, and controversy

Fermat

Main article: Fermat's principle

In the 1600s, Pierre de Fermat postulated that "light travels between two given points along the path of shortest time," which is known as the principle of least time or Fermat's principle.

Maupertuis

Main article: Maupertuis principle

Credit for the formulation of the principle of least action is commonly given to Pierre Louis Maupertuis, who felt that "Nature is thrifty in all its actions", and applied the principle broadly:

The laws of movement and of rest deduced from this principle being precisely the same as those observed in nature, we can admire the application of it to all phenomena. The movement of animals, the vegetative growth of plants ... are only its consequences; and the spectacle of the universe becomes so much the grander, so much more beautiful, the worthier of its Author, when one knows that a small number of laws, most wisely established, suffice for all movements.

— Pierre Louis Maupertuis

This notion of Maupertuis, although somewhat deterministic today, does capture much of the essence of mechanics.

In application to physics, Maupertuis suggested that the quantity to be minimized was the product of the duration (time) of movement within a system by the "vis viva",

Maupertuis' principle

${\displaystyle \delta \int 2T(t)dt=0}$

which is the integral of twice what we now call the kinetic energy T of the system.

Euler

Leonhard Euler gave a formulation of the action principle in 1744, in very recognizable terms, in the Additamentum 2 to his Methodus Inveniendi Lineas Curvas Maximi Minive Proprietate Gaudentes. Beginning with the second paragraph:

Let the mass of the projectile be M, and let its speed be v while being moved over an infinitesimal distance ds. The body will have a momentum Mv that, when multiplied by the distance ds, will give Mv ds, the momentum of the body integrated over the distance ds. Now I assert that the curve thus described by the body to be the curve (from among all other curves connecting the same endpoints) that minimizes

${\displaystyle \int Mv\,ds}$

or, provided that M is constant along the path,

${\displaystyle M\int v\,ds}$.

— Leonhard Euler

As Euler states, ∫Mvds is the integral of the momentum over distance travelled, which, in modern notation, equals the abbreviated or reduced action

Euler's principle

${\displaystyle \delta \int p\,dq=0}$

Thus, Euler made an equivalent and (apparently) independent statement of the variational principle in the same year as Maupertuis, albeit slightly later. Curiously, Euler did not claim any priority, as the following episode shows.

Disputed priority

Maupertuis' priority was disputed in 1751 by the mathematician Samuel König, who claimed that it had been invented by Gottfried Leibniz in 1707. Although similar to many of Leibniz's arguments, the principle itself has not been documented in Leibniz's works. König himself showed a copy of a 1707 letter from Leibniz to Jacob Hermann with the principle, but the original letter has been lost. In contentious proceedings, König was accused of forgery, and even the King of Prussia entered the debate, defending Maupertuis (the head of his Academy), while Voltaire defended König.

Euler, rather than claiming priority, was a staunch defender of Maupertuis, and Euler himself prosecuted König for forgery before the Berlin Academy on 13 April 1752. The claims of forgery were re-examined 150 years later, and archival work by C.I. Gerhardt in 1898 and W. Kabitz in 1913 uncovered other copies of the letter, and three others cited by König, in the Bernoulli archives.

Further development

Euler continued to write on the topic; in his Réflexions sur quelques loix générales de la nature (1748), he called action "effort". His expression corresponds to modern potential energy, and his statement of least action says that the total potential energy of a system of bodies at rest is minimized, a principle of modern statics.

Lagrange and Hamilton

Main article: Hamilton's principle

Much of the calculus of variations was stated by Joseph-Louis Lagrange in 1760 and he proceeded to apply this to problems in dynamics. In Mécanique analytique (1788) Lagrange derived the general equations of motion of a mechanical body. William Rowan Hamilton in 1834 and 1835 applied the variational principle to the classical Lagrangian function

${\displaystyle L=T-V}$

to obtain the Euler–Lagrange equations in their present form.

Jacobi, Morse and Caratheodory

In 1842, Carl Gustav Jacobi tackled the problem of whether the variational principle always found minima as opposed to other stationary points (maxima or stationary saddle points); most of his work focused on geodesics on two-dimensional surfaces. The first clear general statements were given by Marston Morse in the 1920s and 1930s, leading to what is now known as Morse theory. For example, Morse showed that the number of conjugate points in a trajectory equalled the number of negative eigenvalues in the second variation of the Lagrangian. A particularly elegant derivation of the Euler-Lagrange equation was formulated by Constantin Caratheodory and published by him in 1935.

Gauss and Hertz

Other extremal principles of classical mechanics have been formulated, such as Gauss's principle of least constraint and its corollary, Hertz's principle of least curvature.

Disputes about possible teleological aspects

The mathematical equivalence of the differential equations of motion and their integral counterpart has important philosophical implications. The differential equations are statements about quantities localized to a single point in space or single moment of time. For example, Newton's second law

${\displaystyle \mathbf {F} =m\mathbf {a} }$

states that the instantaneous force F applied to a mass m produces an acceleration a at the same instant. By contrast, the action principle is not localized to a point; rather, it involves integrals over an interval of time and (for fields) an extended region of space. Moreover, in the usual formulation of classical action principles, the initial and final states of the system are fixed, e.g.,

Given that the particle begins at position x₁ at time t₁ and ends at position x₂ at time t₂, the physical trajectory that connects these two endpoints is an extremum of the action integral.

In particular, the fixing of the final state has been interpreted as giving the action principle a teleological character which has been controversial historically. However, according to W. Yourgrau and S. Mandelstam, the teleological approach... presupposes that the variational principles themselves have mathematical characteristics which they de facto do not possess In addition, some critics maintain this apparent teleology occurs because of the way in which the question was asked. By specifying some but not all aspects of both the initial and final conditions (the positions but not the velocities) we are making some inferences about the initial conditions from the final conditions, and it is this "backward" inference that can be seen as a teleological explanation. Teleology can also be overcome if we consider the classical description as a limiting case of the quantum formalism of path integration, in which stationary paths are obtained as a result of interference of amplitudes along all possible paths.

The short story Story of Your Life by the speculative fiction writer Ted Chiang contains visual depictions of Fermat's Principle along with a discussion of its teleological dimension. Keith Devlin's The Math Instinct contains a chapter, "Elvis the Welsh Corgi Who Can Do Calculus" that discusses the calculus "embedded" in some animals as they solve the "least time" problem in actual situations.

Action (physics)

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Action_(physics)

In physics, action is a numerical value describing how a physical system has changed over time. Action is significant because the equations of motion of the system can be derived through the principle of stationary action. In the simple case of a single particle moving with a specified velocity, the action is the momentum of the particle times the distance it moves, added up along its path, or equivalently, twice its kinetic energy times the length of time for which it has that amount of energy, added up over the period of time under consideration. For more complicated systems, all such quantities are added together. More formally, action is a mathematical functional which takes the trajectory, also called path or history, of the system as its argument and has a real number as its result. Generally, the action takes different values for different paths. Action has dimensions of energy × time or momentum × length, and its SI unit is joule-second (like the Planck constant h).

Introduction

Main article: Hamilton's principle

Hamilton's principle states that the differential equations of motion for any physical system can be re-formulated as an equivalent integral equation. Thus, there are two distinct approaches for formulating dynamical models.

It applies not only to the classical mechanics of a single particle, but also to classical fields such as the electromagnetic and gravitational fields. Hamilton's principle has also been extended to quantum mechanics and quantum field theory—in particular the path integral formulation of quantum mechanics makes use of the concept—where a physical system randomly follows one of the possible paths, with the phase of the probability amplitude for each path being determined by the action for the path.

Solution of differential equation

Empirical laws are frequently expressed as differential equations, which describe how physical quantities such as position and momentum change continuously with time, space or a generalization thereof. Given the initial and boundary conditions for the situation, the "solution" to these empirical equations is one or more functions that describe the behavior of the system and are called equations of motion.

Minimization of action integral

Action is a part of an alternative approach to finding such equations of motion. Classical mechanics postulates that the path actually followed by a physical system is that for which the action is minimized, or more generally, is stationary. In other words, the action satisfies a variational principle: the principle of stationary action (see also below). The action is defined by an integral, and the classical equations of motion of a system can be derived by minimizing the value of that integral.

This simple principle provides deep insights into physics, and is an important concept in modern theoretical physics.

History

Action was defined in several now obsolete ways during the development of the concept.

• Gottfried Leibniz, Johann Bernoulli and Pierre Louis Maupertuis defined the action for light as the integral of its speed or inverse speed along its path length.
• Leonhard Euler (and, possibly, Leibniz) defined action for a material particle as the integral of the particle's speed along its path through space.
• Pierre Louis Maupertuis introduced several ad hoc and contradictory definitions of action within a single article, defining action as potential energy, as virtual kinetic energy, and as a hybrid that ensured conservation of momentum in collisions.

Mathematical definition

Expressed in mathematical language, using the calculus of variations, the evolution of a physical system (i.e., how the system actually progresses from one state to another) corresponds to a stationary point (usually, a minimum) of the action.

Several different definitions of "the action" are in common use in physics. The action is usually an integral over time. However, when the action pertains to fields, it may be integrated over spatial variables as well. In some cases, the action is integrated along the path followed by the physical system.

The action is typically represented as an integral over time, taken along the path of the system between the initial time and the final time of the development of the system:

${\displaystyle {\mathcal {S}}=\int _{t_{1}}^{t_{2}}L\,dt,}$

where the integrand L is called the Lagrangian. For the action integral to be well-defined, the trajectory has to be bounded in time and space.

Action has the dimensions of [energy] × [time], and its SI unit is joule-second, which is identical to the unit of angular momentum.

Action in classical physics

In classical physics, the term "action" has a number of meanings.

Action (functional)

Most commonly, the term is used for a functional ${\displaystyle {\mathcal {S}}}$ which takes a function of time and (for fields) space as input and returns a scalar. In classical mechanics, the input function is the evolution q(t) of the system between two times t₁ and t₂, where q represents the generalized coordinates. The action ${\displaystyle {\mathcal {S}}[\mathbf {q} (t)]}$ is defined as the integral of the Lagrangian L for an input evolution between the two times:

${\displaystyle {\mathcal {S}}[\mathbf {q} (t)]=\int _{t_{1}}^{t_{2}}L(\mathbf {q} (t),{\dot {\mathbf {q} }}(t),t)\,dt,}$

where the endpoints of the evolution are fixed and defined as ${\displaystyle \mathbf {q} _{1}=\mathbf {q} (t_{1})}$ and ${\displaystyle \mathbf {q} _{2}=\mathbf {q} (t_{2})}$. According to Hamilton's principle, the true evolution q_true(t) is an evolution for which the action ${\displaystyle {\mathcal {S}}[\mathbf {q} (t)]}$ is stationary (a minimum, maximum, or a saddle point). This principle results in the equations of motion in Lagrangian mechanics.

Abbreviated action (functional)

Usually denoted as ${\displaystyle {\mathcal {S}}_{0}}$, this is also a functional. Here the input function is the path followed by the physical system without regard to its parameterization by time. For example, the path of a planetary orbit is an ellipse, and the path of a particle in a uniform gravitational field is a parabola; in both cases, the path does not depend on how fast the particle traverses the path. The abbreviated action ${\displaystyle {\mathcal {S}}_{0}}$ is defined as the integral of the generalized momenta along a path in the generalized coordinates:

${\displaystyle {\mathcal {S}}_{0}=\int \mathbf {p} \cdot d\mathbf {q} =\int p_{i}\,dq_{i}.}$

According to Maupertuis' principle, the true path is a path for which the abbreviated action ${\displaystyle {\mathcal {S}}_{0}}$ is stationary.

Hamilton's principal function

Main article: Hamilton–Jacobi equation

The Hamilton's principal function ${\displaystyle S=S(q,t;q_{0},t_{0})}$ is obtained from the action functional ${\displaystyle {\mathcal {S}}}$ by fixing the initial time ${\displaystyle t_{0}}$ and the initial endpoint ${\displaystyle q_{0},}$ while allowing the upper time limit ${\displaystyle t}$ and the second endpoint ${\displaystyle q}$ to vary. The Hamilton's principal function satisfies the Hamilton–Jacobi equation, a formulation of classical mechanics. Due to a similarity with the Schrödinger equation, the Hamilton–Jacobi equation provides, arguably, the most direct link with quantum mechanics.

Hamilton's characteristic function

When the total energy E is conserved, the Hamilton–Jacobi equation can be solved with the additive separation of variables:

${\displaystyle S(q_{1},\dots ,q_{N},t)=W(q_{1},\dots ,q_{N})-E\cdot t,}$

where the time-independent function W(q₁, q₂, … q_N) is called Hamilton's characteristic function. The physical significance of this function is understood by taking its total time derivative

${\displaystyle {\frac {dW}{dt}}={\frac {\partial W}{\partial q_{i}}}{\dot {q}}_{i}=p_{i}{\dot {q}}_{i}.}$

This can be integrated to give

${\displaystyle W(q_{1},\dots ,q_{N})=\int p_{i}{\dot {q}}_{i}\,dt=\int p_{i}\,dq_{i},}$

which is just the abbreviated action.

Other solutions of Hamilton–Jacobi equations

The Hamilton–Jacobi equations are often solved by additive separability; in some cases, the individual terms of the solution, e.g., S_k(q_k), are also called an "action".

Action of a generalized coordinate

This is a single variable J_k in the action-angle coordinates, defined by integrating a single generalized momentum around a closed path in phase space, corresponding to rotating or oscillating motion:

${\displaystyle J_{k}=\oint p_{k}\,dq_{k}}$

The variable J_k is called the "action" of the generalized coordinate q_k; the corresponding canonical variable conjugate to J_k is its "angle" w_k, for reasons described more fully under action-angle coordinates. The integration is only over a single variable q_k and, therefore, unlike the integrated dot product in the abbreviated action integral above. The J_k variable equals the change in S_k(q_k) as q_k is varied around the closed path. For several physical systems of interest, J_k is either a constant or varies very slowly; hence, the variable J_k is often used in perturbation calculations and in determining adiabatic invariants.

Action for a Hamiltonian flow

See tautological one-form.

Euler–Lagrange equations

Main article: Euler–Lagrange equations

In Lagrangian mechanics, the requirement that the action integral be stationary under small perturbations is equivalent to a set of differential equations (called the Euler–Lagrange equations) that may be obtained using the calculus of variations.

The action principle

Main article: principle of stationary action

Classical fields

See also: Einstein–Hilbert action

The action principle can be extended to obtain the equations of motion for fields, such as the electromagnetic field or gravitational field.

The Einstein equation utilizes the Einstein–Hilbert action as constrained by a variational principle.

The trajectory (path in spacetime) of a body in a gravitational field can be found using the action principle. For a free falling body, this trajectory is a geodesic.

Conservation laws

Main article: Conservation laws

Implications of symmetries in a physical situation can be found with the action principle, together with the Euler–Lagrange equations, which are derived from the action principle. An example is Noether's theorem, which states that to every continuous symmetry in a physical situation there corresponds a conservation law (and conversely). This deep connection requires that the action principle be assumed.

Quantum mechanics and quantum field theory

Main articles: Quantum mechanics and quantum field theory

In quantum mechanics, the system does not follow a single path whose action is stationary, but the behavior of the system depends on all permitted paths and the value of their action. The action corresponding to the various paths is used to calculate the path integral, which gives the probability amplitudes of the various outcomes.

Although equivalent in classical mechanics with Newton's laws, the action principle is better suited for generalizations and plays an important role in modern physics. Indeed, this principle is one of the great generalizations in physical science. It is best understood within quantum mechanics, particularly in Richard Feynman's path integral formulation, where it arises out of destructive interference of quantum amplitudes.

Maxwell's equations can also be derived as conditions of stationary action.

Single relativistic particle

Main article: Theory of relativity

When relativistic effects are significant, the action of a point particle of mass m travelling a world line C parametrized by the proper time ${\displaystyle \tau }$ is

${\displaystyle S=-mc^{2}\int _{C}\,d\tau .}$

If instead, the particle is parametrized by the coordinate time t of the particle and the coordinate time ranges from t₁ to t₂, then the action becomes

${\displaystyle S=\int _{t1}^{t2}L\,dt,}$

where the Lagrangian is

${\displaystyle L=-mc^{2}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}.}$

Modern extensions

The action principle can be generalized still further. For example, the action need not be an integral, because nonlocal actions are possible. The configuration space need not even be a functional space, given certain features such as noncommutative geometry. However, a physical basis for these mathematical extensions remains to be established experimentally.

 

Planck units

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Planck_units 

In particle physics and physical cosmology, Planck units are a set of units of measurement defined exclusively in terms of four universal physical constants, in such a manner that these physical constants take on the numerical value of 1 when expressed in terms of these units.

Originally proposed in 1899 by German physicist Max Planck, these units are a system of natural units because the origin of their definition comes only from properties of nature and not from any human construct. Planck units are only one of several systems of natural units, but Planck units are not based on properties of any prototype object or particle (the choice of which is inherently arbitrary), but rather on only the properties of free space. They are relevant in research on unified theories such as quantum gravity.

The term Planck scale refers to quantities of space, time, energy and other units that are similar in magnitude to corresponding Planck units. This region may be characterized by energies of around 10¹⁹ GeV, time intervals of around 10⁻⁴³ s and lengths of around 10⁻³⁵ m (approximately respectively the energy-equivalent of the Planck mass, the Planck time and the Planck length). At the Planck scale, the predictions of the Standard Model, quantum field theory and general relativity are not expected to apply, and quantum effects of gravity are expected to dominate. The best-known example is represented by the conditions in the first 10⁻⁴³ seconds of our universe after the Big Bang, approximately 13.8 billion years ago.

The four universal constants that, by definition, have a numeric value 1 when expressed in these units are:

• the speed of light in a vacuum, c,
• the gravitational constant, G,
• the reduced Planck constant, ħ,
• the Boltzmann constant, k_B.

Planck units do not incorporate an electromagnetic dimension. Some authors choose to extend the system to electromagnetism by, for example, adding either the electric constant ε₀ or 4πε₀ to this list. Similarly, authors choose to use variants of the system that give other numeric values to one or more of the four constants above.

Introduction

Any system of measurement may be assigned a mutually independent set of base quantities and associated base units, from which all other quantities and units may be derived. In the International System of Units, for example, the SI base quantities include length with the associated unit of the metre. In the system of Planck units, a similar set of base quantities and associated units may be selected, in terms of which other quantities and coherent units may be expressed. The Planck unit of length has become known as the Planck length, and the Planck unit of time is known as the Planck time, but this nomenclature has not been established as extending to all quantities.

All Planck units are derived from the dimensional universal physical constants that define the system, and in a convention in which these units are omitted (i.e. treated as having the dimensionless value 1), these constants are then eliminated from equations of physics in which they appear. For example, Newton's law of universal gravitation,

${\displaystyle F=G{\frac {m_{1}m_{2}}{r^{2}}}=\left({\frac {F_{\text{P}}l_{\text{P}}^{2}}{m_{\text{P}}^{2}}}\right){\frac {m_{1}m_{2}}{r^{2}}}}$

can be expressed as:

${\displaystyle {\frac {F}{F_{\text{P}}}}={\frac {\left({\dfrac {m_{1}}{m_{\text{P}}}}\right)\left({\dfrac {m_{2}}{m_{\text{P}}}}\right)}{\left({\dfrac {r}{l_{\text{P}}}}\right)^{2}}}.}$

Both equations are dimensionally consistent and equally valid in any system of quantities, but the second equation, with G absent, is relating only dimensionless quantities since any ratio of two like-dimensioned quantities is a dimensionless quantity. If, by a shorthand convention, it is understood that each physical quantity is the corresponing ratio with a coherent Planck unit (or "expressed in Planck units"), the ratios above may be expressed simply with the symbols of physical quantity, without being scaled explicitly by their corresponding unit:

${\displaystyle F'={\frac {m_{1}'m_{2}'}{r'^{2}}}.}$

This last equation (without G) is valid with F′, m₁′, m₂′, and r′ being the dimensionless ratio quantities corresponding to the standard quantities, written e.g. F′ ≘ F or F′ = F/F_P, but not as a direct equality of quantities. This may seem to be "setting the constants c, G, etc., to 1" if the correspondence of the quantities is thought of as equality. For this reason, Planck or other natural units should be employed with care. Referring to "G = c = 1", Paul S. Wesson wrote that, "Mathematically it is an acceptable trick which saves labour. Physically it represents a loss of information and can lead to confusion."

History and definition

The concept of natural units was introduced in 1881, when George Johnstone Stoney, noting that electric charge is quantized, derived units of length, time, and mass, now named Stoney units in his honor, by normalizing G, c, and the electron charge, e, to 1. In 1899, one year before the advent of quantum theory, Max Planck introduced what became later known as the Planck constant. At the end of the paper, he proposed the base units later named in his honor. The Planck units are based on the quantum of action, now usually known as the Planck constant, which appeared in the Wien approximation for blackbody radiation. Planck underlined the universality of the new unit system, writing:

... die Möglichkeit gegeben ist, Einheiten für Länge, Masse, Zeit und Temperatur aufzustellen, welche, unabhängig von speciellen Körpern oder Substanzen, ihre Bedeutung für alle Zeiten und für alle, auch außerirdische und außermenschliche Culturen notwendig behalten und welche daher als »natürliche Maßeinheiten« bezeichnet werden können.

... it is possible to set up units for length, mass, time and temperature, which are independent of special bodies or substances, necessarily retaining their meaning for all times and for all civilizations, including extraterrestrial and non-human ones, which can be called "natural units of measure".

Planck considered only the units based on the universal constants ${\displaystyle G}$, ${\displaystyle h}$, ${\displaystyle c}$, and ${\displaystyle k_{\rm {B}}}$ to arrive at natural units for length, time, mass, and temperature. His definitions differ from the modern ones by a factor of ${\displaystyle {\sqrt {2\pi }}}$, because the modern definitions use ${\displaystyle \hbar }$ rather than ${\displaystyle h}$.

Table 1: Modern values for Planck's original choice of quantities

Name	Dimension	Expression	Value (SI units)
Planck length	length (L)	${\displaystyle l_{\text{P}}={\sqrt {\frac {\hbar G}{c^{3}}}}}$	1.616255(18)×10−35 m
Planck mass	mass (M)	${\displaystyle m_{\text{P}}={\sqrt {\frac {\hbar c}{G}}}}$	2.176434(24)×10−8 kg
Planck time	time (T)	${\displaystyle t_{\text{P}}={\sqrt {\frac {\hbar G}{c^{5}}}}}$	5.391247(60)×10−44 s
Planck temperature	temperature (Θ)	${\displaystyle T_{\text{P}}={\sqrt {\frac {\hbar c^{5}}{Gk_{\text{B}}^{2}}}}}$	1.416784(16)×1032 K

Unlike the case with the International System of Units, there is no official entity that establishes a definition of a Planck unit system. Frank Wilczek and Barton Zwiebach both define the base Planck units to be those of mass, length and time, regarding an additional unit for temperature to be redundant. Other tabulations add, in addition to a unit for temperature, a unit for electric charge, sometimes also replacing mass with energy when doing so. Depending on the author's choice, this charge unit is given by

${\displaystyle q_{\text{P}}={\sqrt {4\pi \epsilon _{0}\hbar c}}\approx 1.875546\times 10^{-18}{\text{ C}}\approx 11.7\ e}$

or

${\displaystyle q_{\text{P}}={\sqrt {\epsilon _{0}\hbar c}}\approx 5.290818\times 10^{-19}{\text{ C}}\approx 3.3\ e.}$

The Planck charge, as well as other electromagnetic units that can be defined like resistance and magnetic flux, are more difficult to interpret than Planck's original units and are used less frequently.

In SI units, the values of c, h, e and k_B are exact and the values of ε₀ and G in SI units respectively have relative uncertainties of 1.5×10⁻¹⁰ and 2.2×10⁻⁵. Hence, the uncertainties in the SI values of the Planck units derive almost entirely from uncertainty in the SI value of G.

Derived units

In any system of measurement, units for many physical quantities can be derived from base units. Table 2 offers a sample of derived Planck units, some of which in fact are seldom used. As with the base units, their use is mostly confined to theoretical physics because most of them are too large or too small for empirical or practical use and there are large uncertainties in their values.

Table 2: Coherent derived units of Planck units

Derived unit of	Expression	Approximate SI equivalent
area (L2)	${\displaystyle l_{\text{P}}^{2}={\frac {\hbar G}{c^{3}}}}$	2.6121×10−70 m2
volume (L3)	${\displaystyle l_{\text{P}}^{3}=\left({\frac {\hbar G}{c^{3}}}\right)^{\frac {3}{2}}={\sqrt {\frac {(\hbar G)^{3}}{c^{9}}}}}$	4.2217×10−105 m3
momentum (LMT−1)	${\displaystyle m_{\text{P}}c={\frac {\hbar }{l_{\text{P}}}}={\sqrt {\frac {\hbar c^{3}}{G}}}}$	6.5249 kg⋅m/s
energy (L2MT−2)	${\displaystyle E_{\text{P}}=m_{\text{P}}c^{2}={\frac {\hbar }{t_{\text{P}}}}={\sqrt {\frac {\hbar c^{5}}{G}}}}$	1.9561×109 J
force (LMT−2)	${\displaystyle F_{\text{P}}={\frac {E_{\text{P}}}{l_{\text{P}}}}={\frac {\hbar }{l_{\text{P}}t_{\text{P}}}}={\frac {c^{4}}{G}}}$	1.2103×1044 N
density (L−3M)	${\displaystyle \rho _{\text{P}}={\frac {m_{\text{P}}}{l_{\text{P}}^{3}}}={\frac {\hbar t_{\text{P}}}{l_{\text{P}}^{5}}}={\frac {c^{5}}{\hbar G^{2}}}}$	5.1550×1096 kg/m3
acceleration (LT−2)	${\displaystyle a_{\text{P}}={\frac {c}{t_{\text{P}}}}={\sqrt {\frac {c^{7}}{\hbar G}}}}$	5.5608×1051 m/s2
frequency (T−1)	${\displaystyle f_{p}={\frac {c}{l_{\text{P}}}}={\sqrt {\frac {c^{5}}{\hbar G}}}}$	1.8549×1043 s−1

Some Planck units, such as of time and length, are many orders of magnitude too large or too small to be of practical use, so that Planck units as a system are typically only relevant to theoretical physics. In some cases, a Planck unit may suggest a limit to a range of a physical quantity where present-day theories of physics apply. For example, our understanding of the Big Bang does not extend to the Planck epoch, i.e., when the universe was less than one Planck time old. Describing the universe during the Planck epoch requires a theory of quantum gravity that would incorporate quantum effects into general relativity. Such a theory does not yet exist.

Several quantities are not "extreme" in magnitude, such as the Planck mass, which is about 22 micrograms: very large in comparison with subatomic particles, and within the mass range of living organisms. It has been speculated that it may be an approximate lower limit at which a black hole could be formed by collapse. Similarly, the related units of energy and of momentum are in the range of some everyday phenomena.

Significance

Planck units have little anthropocentric arbitrariness, but do still involve some arbitrary choices in terms of the defining constants. Unlike the metre and second, which exist as base units in the SI system for historical reasons, the Planck length and Planck time are conceptually linked at a fundamental physical level. Consequently, natural units help physicists to reframe questions. Frank Wilczek puts it succinctly:

We see that the question [posed] is not, "Why is gravity so feeble?" but rather, "Why is the proton's mass so small?" For in natural (Planck) units, the strength of gravity simply is what it is, a primary quantity, while the proton's mass is the tiny number [1/(13 quintillion)].

While it is true that the electrostatic repulsive force between two protons (alone in free space) greatly exceeds the gravitational attractive force between the same two protons, this is not about the relative strengths of the two fundamental forces. From the point of view of Planck units, this is comparing apples with oranges, because mass and electric charge are incommensurable quantities. Rather, the disparity of magnitude of force is a manifestation of the fact that the charge on the protons is approximately the unit charge but the mass of the protons is far less than the unit mass.

Planck scale

In particle physics and physical cosmology, the Planck scale is an energy scale around 1.22×10¹⁹ GeV (the Planck energy, corresponding to the energy equivalent of the Planck mass, 2.17645×10⁻⁸ kg) at which quantum effects of gravity become strong. At this scale, present descriptions and theories of sub-atomic particle interactions in terms of quantum field theory break down and become inadequate, due to the impact of the apparent non-renormalizability of gravity within current theories.

Relationship to gravity

At the Planck length scale, the strength of gravity is expected to become comparable with the other forces, and it is theorized that all the fundamental forces are unified at that scale, but the exact mechanism of this unification remains unknown. The Planck scale is therefore the point where the effects of quantum gravity can no longer be ignored in other fundamental interactions, where current calculations and approaches begin to break down, and a means to take account of its impact is necessary.

While physicists have a fairly good understanding of the other fundamental interactions of forces on the quantum level, gravity is problematic, and cannot be integrated with quantum mechanics at very high energies using the usual framework of quantum field theory. At lesser energy levels it is usually ignored, while for energies approaching or exceeding the Planck scale, a new theory of quantum gravity is necessary. Approaches to this problem include string theory and M-theory, loop quantum gravity, noncommutative geometry, and causal set theory.

In cosmology

Main article: Chronology of the universe

Further information: Time-variation of fundamental constants

In Big Bang cosmology, the Planck epoch or Planck era is the earliest stage of the Big Bang, before the time passed was equal to the Planck time, t_P, or approximately 10⁻⁴³ seconds. There is no currently available physical theory to describe such short times, and it is not clear in what sense the concept of time is meaningful for values smaller than the Planck time. It is generally assumed that quantum effects of gravity dominate physical interactions at this time scale. At this scale, the unified force of the Standard Model is assumed to be unified with gravitation. Immeasurably hot and dense, the state of the Planck epoch was succeeded by the grand unification epoch, where gravitation is separated from the unified force of the Standard Model, in turn followed by the inflationary epoch, which ended after about 10⁻³² seconds (or about 10¹⁰ t_P).

Properties of the observable universe today expressed in Planck units:

Table 2: Today's universe in Planck units

Property of present-day observable universe	Approximate number of Planck units	Equivalents
Age	8.08 × 1060 tP	4.35 × 1017 s, or 13.8 × 109 years
Diameter	5.4 × 1061 lP	8.7 × 1026 m or 9.2 × 1010 light-years
Mass	approx. 1060 mP	3 × 1052 kg or 1.5 × 1022 solar masses (only counting stars) 1080 protons (sometimes known as the Eddington number)
Density	1.8 × 10−123 mP⋅lP−3	9.9 × 10−27 kg⋅m−3
Temperature	1.9 × 10−32 TP	2.725 K temperature of the cosmic microwave background radiation
Cosmological constant	2.9 × 10−122 l −2 P	1.1 × 10−52 m−2
Hubble constant	1.18 × 10−61 t −1 P	2.2 × 10−18 s−1 or 67.8 (km/s)/Mpc

After the measurement of the cosmological constant (Λ) in 1998, estimated at 10⁻¹²² in Planck units, it was noted that this is suggestively close to the reciprocal of the age of the universe (T) squared. Barrow and Shaw proposed a modified theory in which Λ is a field evolving in such a way that its value remains Λ ~ T⁻² throughout the history of the universe.

Analysis of the units

Planck length

Main article: Planck length

The Planck length, denoted ℓ_P, is a unit of length defined as:

${\displaystyle \ell _{\mathrm {P} }={\sqrt {\frac {\hbar G}{c^{3}}}}}$

It is equal to 1.616255(18)×10⁻³⁵ m, where the two digits enclosed by parentheses are the estimated standard error associated with the reported numerical value, or about 10⁻²⁰ times the diameter of a proton.

Planck time

The Planck time t_P is the time required for light to travel a distance of 1 Planck length in a vacuum, which is a time interval of approximately 5.39×10⁻⁴⁴ s. All scientific experiments and human experiences occur over time scales that are many orders of magnitude longer than the Planck time, making any events happening at the Planck scale undetectable with current scientific technology. As of October 2020, the smallest time interval uncertainty in direct measurements was on the order of 247 zeptoseconds (2.47×10⁻¹⁹ s).

While there is currently no known way to measure time intervals on the scale of the Planck time, researchers in 2020 proposed a theoretical apparatus and experiment that, if ever realized, could be capable of being influenced by effects of time as short as 10⁻³³ seconds, thus establishing an upper detectable limit for the quantization of a time that is roughly 20 billion times longer than the Planck time.

Planck energy

Most Planck units are extremely small, as in the case of Planck length or Planck time, or extremely large, as in the case of Planck temperature or Planck acceleration. For comparison, the Planck energy E_P is approximately equal to the energy stored in an automobile gas tank (57.2 L of gasoline at 34.2 MJ/L of chemical energy). The ultra-high-energy cosmic ray observed in 1991 had a measured energy of about 50 J, equivalent to about 2.5×10⁻⁸ E_P.

Planck unit of force

The Planck unit of force may be thought of as the derived unit of force in the Planck system if the Planck units of time, length, and mass are considered to be base units.

${\displaystyle F_{\text{P}}={\frac {m_{\text{P}}c}{t_{\text{P}}}}={\frac {c^{4}}{G}}=1.210295\times 10^{44}{\text{ N.}}}$

It is the gravitational attractive force of two bodies of 1 Planck mass each that are held 1 Planck length apart; equivalently, it is the electrostatic attractive or repulsive force of two Planck units of charges that are held 1 Planck length apart.

Various authors have argued that the Planck force is on the order of the maximum force that can be observed in nature. However, the validity of these conjectures has been disputed.

Planck temperature

The Planck temperature T_P is 1.416784(16)×10³² K. There are no known physical models able to describe temperatures greater than T_P; a quantum theory of gravity would be required to model the extreme energies attained.

List of physical equations

Physical quantities that have different dimensions (such as time and length) cannot be equated even if they are numerically equal (1 second is not the same as 1 metre). In theoretical physics, however, this scruple can be set aside, by a process called nondimensionalization. Table 3 shows how the use of Planck units simplifies many fundamental equations of physics, because this gives each of the five fundamental constants, and products of them, a simple numeric value of 1. In the SI form, the units should be accounted for. In the nondimensionalized form, the units, which are now Planck units, need not be written if their use is understood.

Table 3: How Planck units simplify the key equations of physics

Alternative choices of normalization

As already stated above, Planck units are derived by "normalizing" the numerical values of certain fundamental constants to 1. These normalizations are neither the only ones possible nor necessarily the best. Moreover, the choice of what factors to normalize, among the factors appearing in the fundamental equations of physics, is not evident, and the values of the Planck units are sensitive to this choice.

The factor 4π is ubiquitous in theoretical physics because the surface area of a sphere of radius r is 4πr² in contexts having spherical symmetry in three dimensions. This, along with the concept of flux, are the basis for the inverse-square law, Gauss's law, and the divergence operator applied to flux density. For example, gravitational and electrostatic fields produced by point charges have spherical symmetry (Barrow 2002: 214–15). The 4πr² appearing in the denominator of Coulomb's law in rationalized form, for example, follows from the flux of an electrostatic field being distributed uniformly on the surface of a sphere. Likewise for Newton's law of universal gravitation. (If space had more than three spatial dimensions, the factor 4π would be changed according to the geometry of the sphere in higher dimensions.)

Hence a substantial body of physical theory developed since Planck (1899) suggests normalizing not G but 4πG (or 8πG) to 1. Doing so would introduce a factor of 1/4π (or 1/8π) into the nondimensionalized form of the law of universal gravitation, consistent with the modern rationalized formulation of Coulomb's law in terms of the vacuum permittivity. In fact, alternative normalizations frequently preserve the factor of 1/4π in the nondimensionalized form of Coulomb's law as well, so that the nondimensionalized Maxwell's equations for electromagnetism and gravitoelectromagnetism both take the same form as those for electromagnetism in SI, which do not have any factors of 4π. When this is applied to electromagnetic constants, ε₀, this unit system is called "rationalized". When applied additionally to gravitation and Planck units, these are called rationalized Planck units and are seen in high-energy physics.

The rationalized Planck units are defined so that ${\displaystyle c=4\pi G=\hbar =\varepsilon _{0}=k_{\text{B}}=1}$.

There are several possible alternative normalizations.

Gravitational constant

In 1899, Newton's law of universal gravitation was still seen as exact, rather than as a convenient approximation holding for "small" velocities and masses (the approximate nature of Newton's law was shown following the development of general relativity in 1915). Hence Planck normalized to 1 the gravitational constant G in Newton's law. In theories emerging after 1899, G nearly always appears in formulae multiplied by 4π or a small integer multiple thereof. Hence, a choice to be made when designing a system of natural units is which, if any, instances of 4π appearing in the equations of physics are to be eliminated via the normalization.

• Normalizing 4πG to 1 (and therefore setting G = 1/4π):

• Gauss's law for gravity becomes Φ_g = −M (rather than Φ_g = −4πM in Planck units).
• Eliminates 4πG from the Poisson equation.
• Eliminates 4πG in the gravitoelectromagnetic (GEM) equations, which hold in weak gravitational fields or locally flat spacetime. These equations have the same form as Maxwell's equations (and the Lorentz force equation) of electromagnetism, with mass density replacing charge density, and with 1/4πG replacing ε₀.
• Normalizes the characteristic impedance Z_g of gravitational radiation in free space to 1 (normally expressed as 4πG/c).
• Eliminates 4πG from the Bekenstein–Hawking formula (for the entropy of a black hole in terms of its mass m_BH and the area of its event horizon A_BH) which is simplified to S_BH = πA_BH = (m_BH)².

• Setting 8πG = 1 (and therefore setting G = 1/8π). This would eliminate 8πG from the Einstein field equations, Einstein–Hilbert action, and the Friedmann equations, for gravitation. Planck units modified so that 8πG = 1 are known as reduced Planck units, because the Planck mass is divided by √8π. Also, the Bekenstein–Hawking formula for the entropy of a black hole simplifies to S_BH = (m_BH)²/2 = 2πA_BH.