Unitates Planckianae

O R H Unitates Planckianae
Longitudo	1.616255 × 10−35 m
Massa	2.176434 × 10−8 kg
Tempus	5.391247 × 10−44 s
Onus electricum	1.875546 × 10−18 C
Temperatura	1.416784 × 1032 K

Unitates Planckianae in cosmologia physica et in physica particularum elementarium sunt systema unitatum naturalium^(en)(d) a Maximiliano Planck physico Germanico anno 1899 propositum (undevicensimo anno ab unitatibus Stoneyanis creatis), sic excogitatum ut ex quinque tantum quantitatibus constantibus primariis dependeret: c, G, ħ, k_B et k_e.

... 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.

... possibile est quasdam unitates longitudinis, massae, temporis temperaturaeque sic definire ut e specialibus corporibus vel materie non dependeant, ut semper et per omnes civitates, extraterrestribus vel haud humanis inclusis, significationes suas servent, ut igitur "unitates naturales mensurae" appellari possint.

― Maximilianus Planck, "Über irreversible Strahlungsvorgänge"

Unitates

Ad systema mensurae Planckianum definiendum, hae quantitates constantes adhibentur:

• c, celeritas lucis in vacuo (L T⁻¹)
• G, constans gravitatis (L³ M⁻¹ T⁻²)
• ħ, constans Planckiana minuta (L² M T⁻¹)
• k_e, constans Coulombiana (L³ M T⁻² Q⁻²)
• k_B, constans Boltzmanniana (L² M T⁻² Θ⁻¹)

Unitates Planckianae emergunt numerum unius parti numerali talium constantium imponendo et systema quinque aequationum linearium solvendo.

Nomen	Dimensio	Aequatio	Mensura ad SI versa
Longitudo Planckiana	longitudo (L)	$l_{\text{P}}={\sqrt {\frac {\hbar G}{c^{3}}}}$	1.616255 × 10−35 m
Massa Planckiana	massa (M)	$m_{\text{P}}={\sqrt {\frac {\hbar c}{G}}}$	2.176434 × 10−8 kg
Tempus Planckianum	tempus (T)	$t_{\text{P}}={\sqrt {\frac {\hbar G}{c^{5}}}}$	5.391247 × 10−44 s
Onus electricum Planckianum	onus electricum (Q)	$q_{\text{P}}={\sqrt {\frac {\hbar c}{k_{\text{e}}}}}={\sqrt {4\pi \varepsilon _{0}\hbar c}}={\frac {e}{\sqrt {\alpha }}}$	1.875545956 × 10−18 C
Temperatura Planckiana	temperatura (Θ)	$T_{\text{P}}={\sqrt {\frac {\hbar c^{5}}{Gk_{\text{B}}^{2}}}}$	1.416784 × 1032 K

Forma originalis systematis a Maximiliano Planck proposita constantem Planckianam (h) ad vicem constantis Planckianae minutae (ħ) adhibebat et constante Coulombiana carebat. Idcirco unitates originales erant factoris ${\sqrt {2\pi }}\approx 2.5066$ maiores hodiernis unitatibus et sine ulla unitate oneris electrici.

Constantes quae systema Planckianum definiunt, cum per ipsas unitates Planckianas exprimantur, partem numeralem habent unius:

${\begin{aligned}c&=1\ l_{\text{P}}\cdot t_{\text{P}}^{-1}\\G&=1\ l_{\text{P}}^{3}\cdot t_{\text{P}}^{-2}\cdot m_{\text{P}}^{-1}\\\hbar &=1\ l_{\text{P}}^{3}\cdot t_{\text{P}}^{-2}\cdot m_{\text{P}}^{-1}\\k_{\text{B}}&=1\ l_{\text{P}}^{2}\cdot t_{\text{P}}^{-2}\cdot m_{\text{P}}\cdot T_{\text{P}}^{-1}\\k_{\text{e}}&=1\ l_{\text{P}}^{3}\cdot t_{\text{P}}^{-2}\cdot m_{\text{P}}\cdot q_{\text{P}}^{-2}\end{aligned}}$

Proportio inter unitates Planckianae et unitates Stoneyanas est ${\sqrt {{1}/{\alpha }}}\approx 11.7$. Idcirco: $l_{\text{S}}={\sqrt {\alpha }}\,l_{\text{P}};$ $m_{\text{S}}={\sqrt {\alpha }}\,m_{\text{P}};$ $t_{\text{S}}={\sqrt {\alpha }}\,t_{\text{P}};$ $q_{\text{S}}={\sqrt {\alpha }}\,q_{\text{P}}.$

Ad vicem constantis gravitatis (G) et constans Coulombiana (k_e), nonnulli auctores, rationem Oliverii Heaviside sequentes, praeferunt 4𝜋G et 4𝜋k_e uni aequare (c = ħ = 4𝜋G = 4𝜋k_e = k_B = 1). Talis emendatio, “rationalizatio” appellata, aliquando in physica altarum energiarum adhibetur et hoc modo unitates Planckianas transformat:

${\begin{aligned}l_{\text{P}}^{\prime }={\sqrt {\frac {4\pi G\hbar }{c^{3}}}}\approx 5.7295\times 10^{-35}\,{\text{m}}\\m_{\text{P}}^{\prime }={\sqrt {\frac {\hbar c}{4\pi G}}}\approx 6.1396\times 10^{-9}\,{\text{kg}}\\t_{\text{P}}^{\prime }={\sqrt {\frac {4\pi G\hbar }{c^{5}}}}\approx 1.9111\times 10^{-43}\,{\text{s}}\\q_{\text{P}}^{\prime }={\sqrt {\frac {\hbar c}{4\pi k_{e}}}}\approx 5.2908\times 10^{-19}\,{\text{C}}\\T_{\text{P}}^{\prime }={\sqrt {\frac {\hbar c^{5}}{4\pi Gk_{\text{B}}^{2}}}}\approx 3.9967\times 10^{31}\,{\text{K}}\end{aligned}}$

Nexus interni

• Longitudo Planckiana
• Massa Planckiana
• Celeritas lucis
• Constans Planckiana
• Constans Coulombiana
• Constans gravitatis
• Onus electricum Planckianum
• Physica particularum elementarium
• Radius Schwarzschildianus
• Systema internationale unitatum
• Temperatura Planckiana
• Tempus Planckianum
• Unitates Stoneyanae

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Unitates Planckianae