Plankovo vreme

U kvantnoj mehanici, Plankovo vreme (t_P) je jedinica vremena u sistemu prirodnih jedinica poznatih kao Plankove jedinice. Plankova vremenska jedinica je vreme potrebno za putovanje svetlosti na udaljenost od 1 Plankove dužine u vakuumu, što je vremenski interval od približno 5.39 × 10 ⁻⁴⁴ s. Ova jedinica je nazvana po Maksu Planku, koji je prvi predložio njeno postojanje.

Plankovo vreme je definisano kao:

$t_{\mathrm {P} }\equiv {\sqrt {\frac {\hbar G}{c^{5}}}}$

gde je: ħ = ^h⁄_2π redukovana Plankova konstant (ponekad se h koristi umesto ħ u definiciji), G = gravitaciona konstanta, и c = brzina svetlosti u vakuumu. Koristeći poznate vrednosti konstanti, približna ekvivalentna vrednost u smislu SI jedinice, sekunde, je $1\ t_{\mathrm {P} }\approx 5.391\,245(60)\times 10^{-44}\ \mathrm {s} ,$ pri čemu cifre u zagradi označavaju standardnu grešku aproksimirane vrednosti.

Uvod

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,

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

can be expressed as:

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

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 corresponding 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:

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

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."

Istorija

The concept of natural units was introduced in 1874, when George Johnstone Stoney, noting that electric charge is quantized, derived units of length, time, and mass, now named Stoney units in his honor. Stoney chose his units so that G, c, and the electron charge e would be numerically equal 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 that were 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 black-body 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 nothwendig behalten und welche daher als »natürliche Maasseinheiten« 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 $G$, $h$, $c$, and $k_{\rm {B}}$ to arrive at natural units for length, time, mass, and temperature. His definitions differ from the modern ones by a factor of ${\sqrt {2\pi }}$, because the modern definitions use $\hbar$ rather than $h$.

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

Name	Dimension	Expression	Value (SI units)
Planck length	length (L)	$l_{\text{P}}={\sqrt {\frac {\hbar G}{c^{3}}}}$	6965161625500000000♠1,616255(18)×10−35 m
Planck mass	mass (M)	$m_{\text{P}}={\sqrt {\frac {\hbar c}{G}}}$	6992217643400000000♠2,176434(24)×10−8 kg
Planck time	time (T)	$t_{\text{P}}={\sqrt {\frac {\hbar G}{c^{5}}}}$	6956539124699999999♠5,391247(60)×10−44 s
Planck temperature	temperature (Θ)	$T_{\text{P}}={\sqrt {\frac {\hbar c^{5}}{Gk_{\text{B}}^{2}}}}$	7032141678400000000♠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. Some authors 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, so that either the Coulomb constant $k_{e}$ or the vacuum permittivity $\epsilon _{0}$ is normalized to 1. Thus, depending on the author's choice, this charge unit is given by

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

for $k_{\text{e}}=1$, or

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

for $\varepsilon _{0}=1$. Some of these tabulations also replace mass with energy when doing so.

Fizički značaj

Plankovo vreme je jedinstvena kombinacija gravitacione konstante G, specijalno-relativističke konstante c, i kvantne konstante ħ, kojom je proizvedena konstanta sa dimenzijom vremena. Pošto Plankovo vreme proističe iz dimenzione analize, koja ignoriše konstantne faktore, nema razloga da se veruje da tačno jedna jedinica Planckovog vremena ima neki poseban fizički značaj. Naprotiv, Plankovo vreme predstavlja grubu vremensku skalu na kojoj je verovatno da će kvantni gravitacioni efekti postati značajni. To esencijalno znači da, iako manje jedinice vremena mogu postojati, one su toliko male da je njihov uticaj na naše postojanje zanemarljiv. Priroda tih efekata, kao i tačna vremenska skala na kojoj će se pojaviti, bi trebalo da budu izvedeni iz teorije kvantne gravitacije.

Recipročna vrednost Plankovog vremena, koja je Plankova frekvencija, može se interpretirati kao gornja granica frekvencije talasa. Ovo sledi iz tumačenja Plankove dužine kao minimalne dužine, a time i donje granice na talasnoj dužini.

Svi naučni eksperimenti i ljudska iskustva se dešavaju tokom vremenskih skala koje su mnogo redova veličine duže od Plankovog vremena, čineći događaje koji se dešavaju na Plankovoj skali nedetektivim putem trenutnih naučnih saznanja. Prema podacima iz novembra 2016. godine, najmanja neizvesnost vremenskog intervala u direktnim merenjima je reda 850 zeptosekundi (8,50 × 10⁻¹⁹ sekundi).

Vidi još

• Atosekund
• Hronon
• Redovi veličine (vreme)
• Plankova energija
• Prirodne jedinice
• Kvantni časovnik

Literatura

• Barrow, John D. (2002). The Constants of Nature; From Alpha to Omega – The Numbers that Encode the Deepest Secrets of the Universe. New York: Pantheon Books. ISBN 978-0-375-42221-8.  Easier.
• Barrow, John D.; Tipler,, Frank J. (1986). The Anthropic Cosmological Principle. Oxford: Claredon Press. ISBN 978-0-19-851949-2.  Harder.
• Penrose, Roger (2005). „Section 31.1”. The Road to Reality. New York: Alfred A. Knopf. ISBN 978-0-679-45443-4. 
• Planck, Max (1899). „Über irreversible Strahlungsvorgänge”. Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin (на језику: немачки). 5: 440—480. Архивирано из оригинала 03. 04. 2009. г. Приступљено 28. 06. 2019.  pp. 478–80 contain the first appearance of the Planck base units other than the Planck charge, and of Planck's constant, which Planck denoted by b. a and f in this paper correspond to k and G in this entry.
• Tomilin, K. A. (1999). „Natural Systems of Units: To the Centenary Anniversary of the Planck System” (PDF). Proceedings Of The XXII Workshop On High Energy Physics And Field Theory: 287—296. Архивирано из оригинала (PDF) 17. 6. 2006. г. 

Spoljašnje veze

• "Planck Era" and "Planck Time" Архивирано на сајту Wayback Machine (28. новембар 2018) (up to 10⁻⁴³ seconds after birth of Universe) (University of Oregon).
• Value of the fundamental constants, including the Planck base units, as reported by the National Institute of Standards and Technology (NIST).
• Sections C-E of collection of resources bear on Planck units. As of 2011, those pages had been removed from the planck.org web site. Use the Wayback Machine to access pre-2011 versions of the website. Good discussion of why 8πG should be normalized to 1 when doing general relativity and quantum gravity. Many links.
• The universe and the parameters that describe it in Planck units Pulls together various physics concepts into one unifying picture.
• "Planck Era" and "Planck Time" Архивирано на сајту Wayback Machine (28. новембар 2018) (up to 10⁻⁴³ seconds after birth of Universe) (University of Oregon).
• Constants of nature: Quantum Space Theory offers a different set of Planck units and defines 31 physical constants in terms of them.