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Planck units
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In particle physics and physical cosmology, Planck units are a set of units of measurement defined exclusively in terms of five universal physical constants, in such a manner that these five 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 also known as 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 system of several systems of natural units, but Planck units are not based on properties of any prototype object or particle (that would be arbitrarily chosen), but rather on only the properties of free space. Planck units have significance for theoretical physics since they simplify several recurring algebraic expressions of physical law by nondimensionalization. They are relevant in research on unified theories such as quantum gravity.
The term "Planck scale" refers to magnitudes of space, time, energy and other units, beyond (or below) which the predictions of the Standard Model, quantum field theory and general relativity are no longer reconcilable, and quantum effects of gravity are expected to dominate. This region may be characterized by energies around 7009195465531414000♠1.22×1019 GeV (the Planck energy), time intervals around 6956538999999999999♠5.39×10−44 s (the Planck time) and lengths around 6965162000000000000♠1.62×10−35 m (the Planck length). At the Planck scale, current models are not expected to be a useful guide to the cosmos, and physicists no longer have any scientific model whatsoever to suggest how the physical universe behaves. 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 five universal constants that Planck units, by definition, normalize to 1 are:
the speed of light in a vacuum, c,
the gravitational constant, G,
the reduced Planck constant, ħ,
the Coulomb constant, ke = 1/4πε0
the Boltzmann constant, kB
Each of these constants can be associated with a fundamental physical theory or concept: c with special relativity, G with general relativity, ħ with quantum mechanics, ε0 with electromagnetism, and kB with the notion of temperature/energy (statistical mechanics and thermodynamics).
Contents
1 Base units
2 Derived units
3 Significance
3.1 Cosmology
4 History
5 List of physical equations
6 Alternative choices of normalization
6.1 Gravity
6.2 Electromagnetism
6.3 Temperature
7 Planck units and the invariant scaling of nature
8 See also
9 Notes
10 References
11 External links
Base units[edit]
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 may be selected, and the Planck base unit of length is then known simply as the Planck length, the base unit of time is the Planck time, and so on. These units are derived from the five dimensional universal physical constants of Table 1, in such a manner that these constants are eliminated from fundamental selected equations of physical law when physical quantities are expressed in terms of Planck units. For example, Newton's law of universal gravitation,
F = G m 1 m 2 r 2 = ( F P l P 2 m P 2 ) m 1 m 2 r 2 {\displaystyle {\begin{aligned}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}}}\\\end{aligned}}}
can be expressed as
F F P = ( m 1 m P ) ( m 2 m P ) ( r l P ) 2 . {\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 units, but the second equation, with G missing, 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 all physical quantities are expressed in terms of Planck units, the ratios above may be expressed simply with the symbols of physical quantity, without being scaled explicitly by their corresponding unit:
F = m 1 m 2 r 2 . {\displaystyle F={\frac {m_{1}m_{2}}{r^{2}}}\ .}
In order for this last equation to be valid (without G present), F, m1, m2, and r are understood to be the dimensionless numerical values of these quantities measured in terms of Planck units. This is why Planck units or any other use of 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."[1]
Table 1: Dimensional universal physical constants normalized with Planck units
Constant
Symbol
Dimension
Value (SI units)[2]
Speed of light in vacuum c L T −1 7008299792458000000♠2.99792458×108 m⋅s−1
(exact by definition of metre)
Gravitational constant G L3 M−1 T −2 6989667408000000000♠6.67408(31)×10−11 m3⋅kg−1⋅s−2[3]
Reduced Planck constant ħ = h/2π
where h is the Planck constant L2 M T −1 6966105457179999999♠1.054571800(13)×10−34 J⋅s[4]
Coulomb constant ke = 1/4πε0
where ε0 is the permittivity of free space L3 M T −2 Q−2 7009898755178736817♠8.9875517873681764×109 kg⋅m3⋅s−4⋅A−2
(exact by definitions of ampere and metre)
Boltzmann constant kB L2 M T −2 Θ−1 6977138064851999999♠1.38064852(79)×10−23 J⋅K−1[5]
Key: L = length, M = mass, T = time, Q = electric charge, Θ = temperature.
As can be seen above, the gravitational attractive force of two bodies of 1 Planck mass each, set apart by 1 Planck length is 1 Planck force. Likewise, the distance traveled by light during 1 Planck time is 1 Planck length. To determine, in terms of SI or another existing system of units, the quantitative values of the five base Planck units, those two equations and three others must be satisfied:
l P = c t P {\displaystyle l_{\text{P}}=c\ t_{\text{P}}}
F P = m P l P t P 2 = G m P 2 l P 2 {\displaystyle F_{\text{P}}={\frac {m_{\text{P}}l_{\text{P}}}{t_{\text{P}}^{2}}}=G\ {\frac {m_{\text{P}}^{2}}{l_{\text{P}}^{2}}}}
E P = m P l P 2 t P 2 = ℏ 1 t P {\displaystyle E_{\text{P}}={\frac {m_{\text{P}}l_{\text{P}}^{2}}{t_{\text{P}}^{2}}}=\hbar \ {\frac {1}{t_{\text{P}}}}}
F P = m P l P t P 2 = 1 4 π ε 0 q P 2 l P 2 {\displaystyle F_{\text{P}}={\frac {m_{\text{P}}l_{\text{P}}}{t_{\text{P}}^{2}}}={\frac {1}{4\pi \varepsilon _{0}}}\ {\frac {q_{\text{P}}^{2}}{l_{\text{P}}^{2}}}}
E P = m P l P 2 t P 2 = k B T P . {\displaystyle E_{\text{P}}={\frac {m_{\text{P}}l_{\text{P}}^{2}}{t_{\text{P}}^{2}}}=k_{\text{B}}\ T_{\text{P}}.}
Solving the five equations above for the five unknowns results in a unique set of values for the five base Planck units:
Table 2: Base Planck units
Name
Dimension
Expression
Value (SI unit)s[2]
Planck length Length (L) l P = ℏ G c 3 {\displaystyle l_{\text{P}}={\sqrt {\frac {\hbar G}{c^{3}}}}} 6965161622900000000♠1.616229(38)×10−35 m[6]
Planck mass Mass (M) m P = ℏ c G {\displaystyle m_{\text{P}}={\sqrt {\frac {\hbar c}{G}}}} 6992217647000000000♠2.176470(51)×10−8 kg[7]
Planck time Time (T) t P = l P c = ℏ m P c 2 = ℏ G c 5 {\displaystyle t_{\text{P}}={\frac {l_{\text{P}}}{c}}={\frac {\hbar }{m_{\text{P}}c^{2}}}={\sqrt {\frac {\hbar G}{c^{5}}}}} 6956539115999999999♠5.39116(13)×10−44 s[8]
Planck charge Electric charge (Q) q P = 4 π ε 0 ℏ c = e α {\displaystyle q_{\text{P}}={\sqrt {4\pi \varepsilon _{0}\hbar c}}={\frac {e}{\sqrt {\alpha }}}} 1.875 545 956(41) × 10−18 C[9][4][10]
Planck temperature Temperature (Θ) T P = m P c 2 k B = ℏ c 5 G k B 2 {\displaystyle T_{\text{P}}={\frac {m_{\text{P}}c^{2}}{k_{\text{B}}}}={\sqrt {\frac {\hbar c^{5}}{Gk_{\text{B}}^{2}}}}} 7032141680800000000♠1.416808(33)×1032 K[11]
Table 2 clearly defines Planck units in terms of the fundamental constants. Yet relative to other units of measurement such as SI, the values of the Planck units are only known approximately. This is mostly due to uncertainty in the value of the gravitational constant G.
Today the value of the speed of light c in SI units is not subject to measurement error, because the SI base unit of length, the metre, is now defined as the length of the path travelled by light in vacuum during a time interval of 1/299792458 of a second. Hence the value of c is now exact by definition, and contributes no uncertainty to the SI equivalents of the Planck units. The same is true of the value of the vacuum permittivity ε0, due to the definition of ampere which sets the vacuum permeability μ0 to 4π × 10−7 H/m and the fact that μ0ε0 = 1/c2. The numerical value of the reduced Planck constant ħ has been determined experimentally to 12 parts per billion, while that of G has been determined experimentally to no better than 1 part in 7004213000000000000♠21300 (or 7004470000000000000♠47000 parts per billion).[2] G appears in the definition of almost every Planck unit in Tables 2 and 3. Hence the uncertainty in the values of the Table 2 and 3 SI equivalents of the Planck units derives almost entirely from uncertainty in the value of G. (The propagation of the error in G is a function of the exponent of G in the algebraic expression for a unit. Since that exponent is ±1/2 for every base unit other than Planck charge, the relative uncertainty of each base unit is about one half that of G. This is indeed the case; according to CODATA, the experimental values of the SI equivalents of the base Planck units are known to about 1 part in 7004435000000000000♠43500, or 7004230000000000000♠23000 parts per billion.)
Derived units[edit]
In any system of measurement, units for many physical quantities can be derived from base units. Table 3 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 3: Derived Planck units
Name
Dimension
Expression
Approximate SI equivalent
Planck area Area (L2) l P 2 = ℏ G c 3 {\displaystyle l_{\text{P}}^{2}={\frac {\hbar G}{c^{3}}}} 2.6121 × 10−70 m2
Planck volume Volume (L3) l P 3 = ( ℏ G c 3 ) 3 2 = ( ℏ G ) 3 c 9 {\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
Planck momentum Momentum (LMT−1) m P c = ℏ l P = ℏ c 3 G {\displaystyle m_{\text{P}}c={\frac {\hbar }{l_{\text{P}}}}={\sqrt {\frac {\hbar c^{3}}{G}}}} 6.52485 kg⋅m/s
Planck energy Energy (L2MT−2) E P = m P c 2 = ℏ t P = ℏ c 5 G {\displaystyle E_{\text{P}}=m_{\text{P}}c^{2}={\frac {\hbar }{t_{\text{P}}}}={\sqrt {\frac {\hbar c^{5}}{G}}}} 1.9561 × 109 J
Planck force Force (LMT−2) F P = E P l P = ℏ l P t P = c 4 G {\displaystyle F_{\text{P}}={\frac {E_{\text{P}}}{l_{\text{P}}}}={\frac {\hbar }{l_{\text{P}}t_{\text{P}}}}={\frac {c^{4}}{G}}} 1.21027 × 1044 N
Planck power Power (L2MT−3) P P = E P t P = ℏ t P 2 = c 5 G {\displaystyle P_{\text{P}}={\frac {E_{\text{P}}}{t_{\text{P}}}}={\frac {\hbar }{t_{\text{P}}^{2}}}={\frac {c^{5}}{G}}} 3.62831 × 1052 W
Planck density Density (L−3M) ρ P = m P l P 3 = ℏ t P l P 5 = c 5 ℏ G 2 {\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}}}} 5.15500 × 1096 kg/m3
Planck energy density Energy density (L−1MT−2) ρ P E = E P l P 3 = c 7 ℏ G 2 {\displaystyle \rho _{\text{P}}^{E}={\frac {E_{\text{P}}}{l_{\text{P}}^{3}}}={\frac {c^{7}}{\hbar G^{2}}}} 4.633 × 10113 J/m3
Planck intensity Intensity (MT−3) I P = ρ P E c = P P l P 2 = c 8 ℏ G 2 {\displaystyle I_{\text{P}}=\rho _{\text{P}}^{E}c={\frac {P_{\text{P}}}{l_{\text{P}}^{2}}}={\frac {c^{8}}{\hbar G^{2}}}} 1.38893 × 10122 W/m2
Planck angular frequency Angular frequency (T−1) ω P = 1 t P = c 5 ℏ G {\displaystyle \omega _{\text{P}}={\frac {1}{t_{\text{P}}}}={\sqrt {\frac {c^{5}}{\hbar G}}}} 1.85487 × 1043 rad/s
Planck pressure Pressure (L−1MT−2) p P = F P l P 2 = ℏ l P 3 t P = c 7 ℏ G 2 {\displaystyle p_{\text{P}}={\frac {F_{\text{P}}}{l_{\text{P}}^{2}}}={\frac {\hbar }{l_{\text{P}}^{3}t_{\text{P}}}}={\frac {c^{7}}{\hbar G^{2}}}} 4.633 × 10113 Pa
Planck current Electric current (T−1Q) I P = q P t P = 4 π ϵ 0 c 6 G {\displaystyle I_{\text{P}}={\frac {q_{\text{P}}}{t_{\text{P}}}}={\sqrt {\frac {4\pi \epsilon _{0}c^{6}}{G}}}} 3.4789 × 1025 A
Planck voltage Voltage (L2MT−2Q−1) V P = E P q P = ℏ t P q P = c 4 4 π ϵ 0 G {\displaystyle V_{\text{P}}={\frac {E_{\text{P}}}{q_{\text{P}}}}={\frac {\hbar }{t_{\text{P}}q_{\text{P}}}}={\sqrt {\frac {c^{4}}{4\pi \epsilon _{0}G}}}} 1.04295 × 1027 V
Planck impedance Resistance (L2MT−1Q−2) Z P = V P I P = ℏ q P 2 = 1 4 π ϵ 0 c = Z 0 4 π {\displaystyle Z_{\text{P}}={\frac {V_{\text{P}}}{I_{\text{P}}}}={\frac {\hbar }{q_{\text{P}}^{2}}}={\frac {1}{4\pi \epsilon _{0}c}}={\frac {Z_{0}}{4\pi }}} 29.9792458 Ω
Planck magnetic inductance Magnetic induction (MT−1Q−1) B P = F P l P I P = c 5 ℏ G 2 4 π ϵ 0 {\displaystyle B_{\text{P}}={\frac {F_{\text{P}}}{l_{\text{P}}I_{\text{P}}}}={\sqrt {\frac {c^{5}}{\hbar G^{2}4\pi \epsilon _{0}}}}} 2.1526 x 1053 T
Planck electrical inductance Inductance (L2MQ−2) L P = E P I P = m P l P 2 q P = G ℏ c 7 ( 4 π ϵ 0 ) 2 {\displaystyle L_{\text{P}}={\frac {E_{\text{P}}}{I_{\text{P}}}}={\frac {m_{\text{P}}l_{\text{P}}^{2}}{q_{\text{P}}}}={\sqrt {\frac {G\hbar }{c^{7}(4\pi \epsilon _{0})^{2}}}}} 1.616 x 10−42 H
Planck volumetric flow rate Volumetric flow rate (L3T−1) Q P = l P 2 c = l P 3 t P = ℏ G c 2 {\displaystyle Q_{\text{P}}=l_{\text{P}}^{2}c={\frac {l_{\text{P}}^{3}}{t_{\text{P}}}}={\frac {\hbar G}{c^{2}}}} 7.83 x 10−62 m3/s
Planck viscosity Dynamic viscosity (L−1MT−1) μ P = p P t P = c 9 G 3 ℏ {\displaystyle \mu _{\text{P}}=p_{\text{P}}t_{\text{P}}={\sqrt {\frac {c^{9}}{G^{3}\hbar }}}} 2.498 x 1070 Pa s
Planck acceleration Acceleration (LT−2) a P = c t P = c 7 ℏ G {\displaystyle a_{\text{P}}={\frac {c}{t_{\text{P}}}}={\sqrt {\frac {c^{7}}{\hbar G}}}} 5.560815 × 1051 m/s2
The charge, as other Planck units, was not originally defined by Planck. It is a unit of charge that is a natural addition to the other units of Planck, and is used in some publications.[12][13] The elementary charge, measured in terms of the Planck charge, is
e = α ⋅ q P ≈ 0.085424543 ⋅ q P {\displaystyle e={\sqrt {\alpha }}\cdot q_{\text{P}}\approx 0.085424543\cdot q_{\text{P}}\,}
where α {\displaystyle {\alpha }\ } is the fine-structure constant
α = ( e q P ) 2 = e 2 4 π ε 0 ℏ c ≈ 1 137.03599911 {\displaystyle \alpha =\left({\frac {e}{q_{\text{P}}}}\right)^{2}={\frac {e^{2}}{4\pi \varepsilon _{0}\hbar c}}\approx {\frac {1}{137.03599911}}}
Some Planck units are suitable for measuring quantities that are familiar from daily experience. For example:
1 Planck mass is about 22 micrograms;
1 Planck momentum is about 6.5 kg⋅m/s;
1 Planck energy is about 540 kW⋅h;
1 Planck charge is close to 11.7 elementary charges;
1 Planck impedance is close to 30 ohms.
However, most Planck units are many orders of magnitude too large or too small to be of practical use, so that Planck units as a system are really only relevant to theoretical physics. In fact, 1 Planck unit is often the largest or smallest value of a physical quantity that makes sense according to our current understanding. For example:
A speed of 1 Planck length per Planck time is the speed of light in a vacuum, the maximum possible physical speed in special relativity;[14] 1 nano-(Planck length per Planck time) is about 1.079 km/h.
Our understanding of the Big Bang begins with the Planck epoch, when the universe was 1 Planck time old and 1 Planck length in diameter, and had a Planck temperature of 1. At that moment, quantum theory as presently understood becomes applicable. Understanding the universe when it was less than 1 Planck time old requires a theory of quantum gravity that would incorporate quantum effects into general relativity. Such a theory does not yet exist.
Significance[edit]
Planck units are free of anthropocentric arbitrariness. Some physicists argue that communication with extraterrestrial intelligence would have to employ such a system of units in order to be understood.[15] 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.
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)].[16]
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 to 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.
Cosmology[edit]
Main article: Chronology of the Universe
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, tP, or approximately 10−43 seconds.[17] 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. Inconceivably 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 tP).[18]
Relative to the Planck epoch, the observable universe today looks extreme when expressed in Planck units, as in this set of approximations:[19][20]
Table 5: 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 x 10−123 ρP 9.9 x 10−27 kg m-3
Temperature 1.9 × 10−32 TP 2.725 K
temperature of the cosmic microwave background radiation
Cosmological constant 5.6 × 10−122 t−2
P 1.9 × 10−35 s−2
Hubble constant 1.18 × 10−61 t−1
P 67.8 (km/s)/Mpc
Further information: Time-variation of physical constants and Dirac large numbers hypothesis
The recurrence of large numbers close or related to 1060 in the above table is a coincidence that intrigues some theorists. It is an example of the kind of large numbers coincidence that led theorists such as Eddington and Dirac to develop alternative physical theories (e.g. a variable speed of light or Dirac varying-G theory).[21] After the measurement of the cosmological constant 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 squared.[22] Barrow and Shaw (2011) 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.[23]
History[edit]
Natural units began 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.[24] In 1898, Max Planck discovered that action is quantized, and published the result in a paper presented to the Prussian Academy of Sciences in May 1899.[25][26] At the end of the paper, Planck introduced, as a consequence of his discovery, the base units later named in his honor. The Planck units are based on the quantum of action, now usually known as Planck's constant. Planck called the constant b in his paper, though h is now common. Planck underlined the universality of the new unit system, writing:
...ihre Bedeutung für alle Zeiten und für alle, auch außerirdische und außermenschliche Kulturen notwendig behalten und welche daher als »natürliche Maßeinheiten« bezeichnet werden können...
...These necessarily retain their meaning for all times and for all civilizations, even extraterrestrial and non-human ones, and can therefore be designated as "natural units"...
Planck considered only the units based on the universal constants G, ħ, c, and kB to arrive at natural units for length, time, mass, and temperature.[26] Planck did not adopt any electromagnetic units. However, since the non-rationalized gravitational constant, G, is set to 1, a natural extension of Planck units to a unit of electric charge is to also set the non-rationalized Coulomb constant, ke, to 1 as well.[27] Another convention is to use the elementary charge as the basic unit of electric charge in Planck system.[28] Such system is convenient for black hole physics. The two conventions for unit charge differ by a factor of the square root of the fine-structure constant. Planck's paper also gave numerical values for the base units that were close to modern values.
List of physical equations[edit]
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 6 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 6: How Planck units simplify the key equations of physics
SI form
Nondimensionalized form
Newton's law of universal gravitation F = G m 1 m 2 r 2 {\displaystyle F=G{\frac {m_{1}m_{2}}{r^{2}}}} F = m 1 m 2 r 2 {\displaystyle F={\frac {m_{1}m_{2}}{r^{2}}}}
Einstein field equations in general relativity G μ ν = 8 π G c 4 T μ ν {\displaystyle {G_{\mu \nu }=8\pi {G \over c^{4}}T_{\mu \nu }}\ } G μ ν = 8 π T μ ν {\displaystyle {G_{\mu \nu }=8\pi T_{\mu \nu }}\ }
Mass–energy equivalence in special relativity E = m c 2 {\displaystyle {E=mc^{2}}\ } E = m {\displaystyle {E=m}\ }
Energy–momentum relation E 2 = m 2 c 4 + p 2 c 2 {\displaystyle E^{2}=m^{2}c^{4}+p^{2}c^{2}\;} E 2 = m 2 + p 2 {\displaystyle E^{2}=m^{2}+p^{2}\;}
Thermal energy per particle per degree of freedom E = 1 2 k B T {\displaystyle {E={\tfrac {1}{2}}k_{\text{B}}T}\ } E = 1 2 T {\displaystyle {E={\tfrac {1}{2}}T}\ }
Boltzmann's entropy formula S = k B ln  Ω {\displaystyle {S=k_{\text{B}}\ln \Omega }\ } S = ln  Ω {\displaystyle {S=\ln \Omega }\ }
Planck–Einstein relation for energy and angular frequency E = ℏ ω {\displaystyle {E=\hbar \omega }\ } E = ω {\displaystyle {E=\omega }\ }
Planck's law (surface intensity per unit solid angle per unit angular frequency) for black body at temperature T. I ( ω , T ) = ℏ ω 3 4 π 3 c 2 1 e ℏ ω k B T − 1 {\displaystyle I(\omega ,T)={\frac {\hbar \omega ^{3}}{4\pi ^{3}c^{2}}}~{\frac {1}{e^{\frac {\hbar \omega }{k_{\text{B}}T}}-1}}} I ( ω , T ) = ω 3 4 π 3 1 e ω / T − 1 {\displaystyle I(\omega ,T)={\frac {\omega ^{3}}{4\pi ^{3}}}~{\frac {1}{e^{\omega /T}-1}}}
Stefan–Boltzmann constant σ defined σ = π 2 k B 4 60 ℏ 3 c 2 {\displaystyle \sigma ={\frac {\pi ^{2}k_{\text{B}}^{4}}{60\hbar ^{3}c^{2}}}} σ = π 2 60 {\displaystyle \ \sigma ={\frac {\pi ^{2}}{60}}}
Bekenstein–Hawking black hole entropy[29] S BH = A BH k B c 3 4 G ℏ = 4 π G k B m BH 2 ℏ c {\displaystyle S_{\text{BH}}={\frac {A_{\text{BH}}k_{\text{B}}c^{3}}{4G\hbar }}={\frac {4\pi Gk_{\text{B}}m_{\text{BH}}^{2}}{\hbar c}}} S BH = A BH 4 = 4 π m BH 2 {\displaystyle S_{\text{BH}}={\frac {A_{\text{BH}}}{4}}=4\pi m_{\text{BH}}^{2}}
Schrödinger's equation − ℏ 2 2 m ∇ 2 ψ ( r , t ) + V ( r , t ) ψ ( r , t ) = i ℏ ∂ ψ ( r , t ) ∂ t {\displaystyle -{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\psi (\mathbf {r} ,t)+V(\mathbf {r} ,t)\psi (\mathbf {r} ,t)=i\hbar {\frac {\partial \psi (\mathbf {r} ,t)}{\partial t}}} − 1 2 m ∇ 2 ψ ( r , t ) + V ( r , t ) ψ ( r , t ) = i ∂ ψ ( r , t ) ∂ t {\displaystyle -{\frac {1}{2m}}\nabla ^{2}\psi (\mathbf {r} ,t)+V(\mathbf {r} ,t)\psi (\mathbf {r} ,t)=i{\frac {\partial \psi (\mathbf {r} ,t)}{\partial t}}}
Hamiltonian form of Schrödinger's equation H | ψ t ⟩ = i ℏ ∂ ∂ t | ψ t ⟩ {\displaystyle H\left|\psi _{t}\right\rangle =i\hbar {\frac {\partial }{\partial t}}\left|\psi _{t}\right\rangle } H | ψ t ⟩ = i ∂ ∂ t | ψ t ⟩ {\displaystyle H\left|\psi _{t}\right\rangle =i{\frac {\partial }{\partial t}}\left|\psi _{t}\right\rangle }
Covariant form of the Dirac equation ( i ℏ γ μ ∂ μ − m c ) ψ = 0 {\displaystyle \ (i\hbar \gamma ^{\mu }\partial _{\mu }-mc)\psi =0} ( i γ μ ∂ μ − m ) ψ = 0 {\displaystyle \ (i\gamma ^{\mu }\partial _{\mu }-m)\psi =0}
Coulomb's law F = 1 4 π ϵ 0 q 1 q 2 r 2 {\displaystyle F={\frac {1}{4\pi \epsilon _{0}}}{\frac {q_{1}q_{2}}{r^{2}}}} F = q 1 q 2 r 2 {\displaystyle F={\frac {q_{1}q_{2}}{r^{2}}}}
Maxwell's equations ∇ ⋅ E = 1 ϵ 0 ρ {\displaystyle \nabla \cdot \mathbf {E} ={\frac {1}{\epsilon _{0}}}\rho }
∇ ⋅ B = 0 {\displaystyle \nabla \cdot \mathbf {B} =0\ }
∇ × E = − ∂ B ∂ t {\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}}
∇ × B = 1 c 2 ( 1 ϵ 0 J + ∂ E ∂ t ) {\displaystyle \nabla \times \mathbf {B} ={\frac {1}{c^{2}}}\left({\frac {1}{\epsilon _{0}}}\mathbf {J} +{\frac {\partial \mathbf {E} }{\partial t}}\right)}
∇ ⋅ E = 4 π ρ {\displaystyle \nabla \cdot \mathbf {E} =4\pi \rho \ }
∇ ⋅ B = 0 {\displaystyle \nabla \cdot \mathbf {B} =0\ }
∇ × E = − ∂ B ∂ t {\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}}
∇ × B = 4 π J + ∂ E ∂ t {\displaystyle \nabla \times \mathbf {B} =4\pi \mathbf {J} +{\frac {\partial \mathbf {E} }{\partial t}}}
Ideal gas law P V = n R T {\displaystyle PV=nRT} P V = N T {\displaystyle PV=NT}
Alternative choices of normalization[edit]
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. 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 have to 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 either 4πG (or 8πG or 16πG) to 1. Doing so would introduce a factor of 1/4π (or 1/8π or 1/16π) 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.
The rationalized Planck units are c = 4 π G = ℏ = ϵ 0 = k B = 1 {\displaystyle c=4\pi G=\hbar =\epsilon _{0}=k_{\text{B}}=1} . This is also the Planck units based on Lorentz–Heaviside units instead of Gaussian units) as depicted above.
There are several possible alternative normalizations.
Gravity[edit]
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:
Gauss's law for gravity becomes Φg = −M (rather than Φg = −4πM in Planck units).
The Bekenstein–Hawking formula for the entropy of a black hole in terms of its mass mBH and the area of its event horizon ABH simplifies to SBH = πABH = (mBH)2, where ABH and mBH are both measured in a slight modification of reduced Planck units, which are described below.
The characteristic impedance Z0 of gravitational radiation in free space becomes equal to 1. (It is normally expressed as 4πG/c)[30]
No factors of 4π appear in the gravitoelectromagnetic (GEM) equations, which hold in weak gravitational fields or locally flat space-time. 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.
Setting 8πG = 1. This would eliminate 8πG from the Einstein field equations, Einstein–Hilbert action, Friedmann equations, and the Poisson equation 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 SBH = (mBH)2/2 = 2πABH.
Setting 16πG = 1. This would eliminate the constant c4/16πG from the Einstein–Hilbert action. The form of the Einstein field equations with cosmological constant Λ becomes Rμν + Λgμν = 1/2(Rgμν + Tμν).
Electromagnetism[edit]
Planck units normalize to 1 the Coulomb force constant ke = 1/4πε0 (as does the cgs system of units). This sets the Planck impedance, ZP equal to Z0/4π, where Z0 is the characteristic impedance of free space.
Normalizing the permittivity of free space ε0 to 1:
Sets the permeability of free space μ0 = 1 (because c = 1).
Sets the unit impedance or unit resistance to the characteristic impedance of free space, ZP = Z0 (or sets the characteristic impedance of free space Z0 to 1).
Eliminates 4π from the nondimensionalized form of Maxwell's equations.
Eliminates ε0 from the nondimensionalized form of Coulomb's law, but has 4πr2 remaining in the denominator (which is the surface area of the enclosing sphere at radius r).
Equates the notions of flux density and field strength in free space.
In this case the elementary charge, measured in terms of this rationalized Planck charge, is
e = 4 π α ⋅ q (P) ≈ 0.30282212 ⋅ q (P) {\displaystyle e={\sqrt {4\pi \alpha }}\cdot q_{\text{(P)}}\approx 0.30282212\cdot q_{\text{(P)}}\,}
where α {\displaystyle {\alpha }\ } is the fine-structure constant. This convention is seen in high-energy physics.
Temperature[edit]
Planck normalized to 1 the Boltzmann constant kB.
Normalizing 1/2kB to 1:
Removes the factor of 1/2 in the nondimensionalized equation for the thermal energy per particle per degree of freedom.
Introduces a factor of 2 into the nondimensionalized form of Boltzmann's entropy formula.
Does not affect the value of any of the base or derived Planck units listed in Tables 3 and 4 other than the Planck temperature, which it doubles.
Planck units and the invariant scaling of nature[edit]
Some theorists (such as Dirac and Milne) have proposed cosmologies that conjecture that physical "constants" might actually change over time (e.g. a variable speed of light or Dirac varying-G theory). Such cosmologies have not gained mainstream acceptance and yet there is still considerable scientific interest in the possibility that physical "constants" might change, although such propositions introduce difficult questions. Perhaps the first question to address is: How would such a change make a noticeable operational difference in physical measurement or, more fundamentally, our perception of reality? If some particular physical constant had changed, how would we notice it, or how would physical reality be different? Which changed constants result in a meaningful and measurable difference in physical reality? If a physical constant that is not dimensionless, such as the speed of light, did in fact change, would we be able to notice it or measure it unambiguously? – a question examined by Michael Duff in his paper "Comment on time-variation of fundamental constants".[31]
George Gamow argued in his book Mr Tompkins in Wonderland that a sufficient change in a dimensionful physical constant, such as the speed of light in a vacuum, would result in obvious perceptible changes. But this idea is challenged:
[An] important lesson we learn from the way that pure numbers like α define the world is what it really means for worlds to be different. The pure number we call the fine structure constant and denote by α is a combination of the electron charge, e, the speed of light, c, and Planck's constant, h. At first we might be tempted to think that a world in which the speed of light was slower would be a different world. But this would be a mistake. If c, h, and e were all changed so that the values they have in metric (or any other) units were different when we looked them up in our tables of physical constants, but the value of α remained the same, this new world would be observationally indistinguishable from our world. The only thing that counts in the definition of worlds are the values of the dimensionless constants of Nature. If all masses were doubled in value [including the Planck mass mP ] you cannot tell because all the pure numbers defined by the ratios of any pair of masses are unchanged.
—  Barrow 2002[19]
Referring to Duff's "Comment on time-variation of fundamental constants"[31] and Duff, Okun, and Veneziano's paper "Trialogue on the number of fundamental constants",[32] particularly the section entitled "The operationally indistinguishable world of Mr. Tompkins", if all physical quantities (masses and other properties of particles) were expressed in terms of Planck units, those quantities would be dimensionless numbers (mass divided by the Planck mass, length divided by the Planck length, etc.) and the only quantities that we ultimately measure in physical experiments or in our perception of reality are dimensionless numbers. When one commonly measures a length with a ruler or tape-measure, that person is actually counting tick marks on a given standard or is measuring the length relative to that given standard, which is a dimensionless value. It is no different for physical experiments, as all physical quantities are measured relative to some other like-dimensioned quantity.
We can notice a difference if some dimensionless physical quantity such as fine-structure constant, α, changes or the proton-to-electron mass ratio, mp/me, changes (atomic structures would change) but if all dimensionless physical quantities remained unchanged (this includes all possible ratios of identically dimensioned physical quantity), we cannot tell if a dimensionful quantity, such as the speed of light, c, has changed. And, indeed, the Tompkins concept becomes meaningless in our perception of reality if a dimensional quantity such as c has changed, even drastically.
If the speed of light c, were somehow suddenly cut in half and changed to 1/2c (but with the axiom that all dimensionless physical quantities remain the same), then the Planck length would increase by a factor of 2√2 from the point of view of some unaffected observer on the outside.[citation needed] Measured by "mortal" observers in terms of Planck units, the new speed of light would remain as 1 new Planck length per 1 new Planck time – which is no different from the old measurement. But, since by axiom, the size of atoms (approximately the Bohr radius) are related to the Planck length by an unchanging dimensionless constant of proportionality:
a 0 = 4 π ϵ 0 ℏ 2 m e e 2 = m P m e α l P . {\displaystyle a_{0}={\frac {4\pi \epsilon _{0}\hbar ^{2}}{m_{e}e^{2}}}={\frac {m_{\text{P}}}{m_{e}\alpha }}l_{\text{P}}.}
Then atoms would be bigger (in one dimension) by 2√2, each of us would be taller by 2√2, and so would our metre sticks be taller (and wider and thicker) by a factor of 2√2. Our perception of distance and lengths relative to the Planck length is, by axiom, an unchanging dimensionless constant.
Our clocks would tick slower by a factor of 4√2 (from the point of view of this unaffected observer on the outside) because the Planck time has increased by 4√2 but we would not know the difference (our perception of durations of time relative to the Planck time is, by axiom, an unchanging dimensionless constant). This hypothetical unaffected observer on the outside might observe that light now propagates at half the speed that it previously did (as well as all other observed velocities) but it would still travel 7008299792458000000♠299792458 of our new metres in the time elapsed by one of our new seconds (1/2c × 4√2 ÷ 2√2 continues to equal 7008299792458000000♠299792458 m/s). We would not notice any difference.
This contradicts what George Gamow writes in his book Mr. Tompkins; there, Gamow suggests that if a dimension-dependent universal constant such as c changed significantly, we would easily notice the difference. The disagreement is better thought of as the ambiguity in the phrase "changing a physical constant"; what would happen depends on whether (1) all other dimensionless constants were kept the same, or whether (2) all other dimension-dependent constants are kept the same. The second choice is a somewhat confusing possibility, since most of our units of measurement are defined in relation to the outcomes of physical experiments, and the experimental results depend on the constants. (The only exception is the kilogram.) Gamow does not address this subtlety; the thought experiments he conducts in his popular works assume the second choice for "changing a physical constant". And Duff or Barrow would point out that ascribing a change in measurable reality, i.e. α, to a specific dimensional component quantity, such as c, is unjustified. The very same operational difference in measurement or perceived reality could just as well be caused by a change in h or e if α is changed and no other dimensionless constants are changed. It is only the dimensionless physical constants that ultimately matter in the definition of worlds.[31][33]
This unvarying aspect of the Planck-relative scale, or that of any other system of natural units, leads many theorists to conclude that a hypothetical change in dimensionful physical constants can only be manifest as a change in dimensionless physical constants. One such dimensionless physical constant is the fine-structure constant. There are some experimental physicists who assert they have in fact measured a change in the fine structure constant[34] and this has intensified the debate about the measurement of physical constants. According to some theorists[35] there are some very special circumstances in which changes in the fine-structure constant can be measured as a change in dimensionful physical constants. Others however reject the possibility of measuring a change in dimensionful physical constants under any circumstance.[31] The difficulty or even the impossibility of measuring changes in dimensionful physical constants has led some theorists to debate with each other whether or not a dimensionful physical constant has any practical significance at all and that in turn leads to questions about which dimensionful physical constants are meaningful.[32]
See also[edit]
cGh physics
Dimensional analysis
Doubly special relativity
Planck particle
Planck scale
Planck time
Zero-point energy
Notes[edit]
Jump up ^ Wesson, P. S. (1980). "The application of dimensional analysis to cosmology". Space Science Reviews. 27 (2): 117. Bibcode:1980SSRv...27..109W. doi:10.1007/bf00212237.
^ Jump up to: a b c Fundamental Physical Constants from NIST
Jump up ^ "CODATA Value: Newtonian constant of gravitation". The NIST Reference on Constants, Units, and Uncertainty. US National Institute of Standards and Technology. June 2015. Retrieved 2015-09-25. 2014 CODATA recommended values
^ Jump up to: a b "CODATA Value: Planck constant over 2 pi". The NIST Reference on Constants, Units, and Uncertainty. US National Institute of Standards and Technology. June 2015. Retrieved 2015-09-25. 2014 CODATA recommended values
Jump up ^ "CODATA Value: Boltzmann constant". The NIST Reference on Constants, Units, and Uncertainty. US National Institute of Standards and Technology. June 2015. Retrieved 2015-09-25. 2014 CODATA recommended values
Jump up ^ "CODATA Value: Planck length". The NIST Reference on Constants, Units, and Uncertainty. US National Institute of Standards and Technology. June 2015. Retrieved 2017-06-22. 2014 CODATA recommended values
Jump up ^ "CODATA Value: Planck mass". The NIST Reference on Constants, Units, and Uncertainty. US National Institute of Standards and Technology. June 2015. Retrieved 2017-06-22. 2014 CODATA recommended values
Jump up ^ "CODATA Value: Planck time". The NIST Reference on Constants, Units, and Uncertainty. US National Institute of Standards and Technology. June 2015. Retrieved 2017-06-22. 2014 CODATA recommended values
Jump up ^ "CODATA Value: electric constant". The NIST Reference on Constants, Units, and Uncertainty. US National Institute of Standards and Technology. June 2015. Retrieved 2015-09-25. 2014 CODATA recommended values
Jump up ^ "CODATA Value: speed of light in vacuum". The NIST Reference on Constants, Units, and Uncertainty. US National Institute of Standards and Technology. June 2015. Retrieved 2015-09-25. 2014 CODATA recommended values
Jump up ^ "CODATA Value: Planck temperature". The NIST Reference on Constants, Units, and Uncertainty. US National Institute of Standards and Technology. June 2015. Retrieved 2017-06-22. 2014 CODATA recommended values
Jump up ^ [Theory of Quantized Space – Date of registration 21/9/1994 N. 344146 protocol 4646 Presidency of the Council of Ministers – Italy – Dep. Information and Publishing, literary, artistic and scientific property]
Jump up ^ Electromagnetic Unification Electronic Conception of the Space, the Energy and the Matter
Jump up ^ Feynman, R. P.; Leighton, R. B.; Sands, M. (1963). "The Special Theory of Relativity". The Feynman Lectures on Physics. 1 "Mainly mechanics, radiation, and heat". Addison-Wesley. pp. 15–9. ISBN 978-0-7382-0008-8. LCCN 63020717.
Jump up ^ Michael W. Busch, Rachel M. Reddick (2010) "Testing SETI Message Designs," Astrobiology Science Conference 2010, April 26–29, 2010, League City, Texas.
Jump up ^ June 2001 Physics Today
Jump up ^ Staff. "Birth of the Universe". University of Oregon. Retrieved September 24, 2016. - discusses "Planck time" and "Planck era" at the very beginning of the Universe
Jump up ^ Edward W. Kolb; Michael S. Turner (1994). The Early Universe. Basic Books. p. 447. ISBN 978-0-201-62674-2. Retrieved 10 April 2010.
^ Jump up to: a b John D. Barrow, 2002. The Constants of Nature; From Alpha to Omega – The Numbers that Encode the Deepest Secrets of the Universe. Pantheon Books. ISBN 0-375-42221-8.
Jump up ^ Barrow, John D.; Tipler, Frank J. (1988). The Anthropic Cosmological Principle. Oxford University Press. ISBN 978-0-19-282147-8. LCCN 87028148.
Jump up ^ P.A.M. Dirac (1938). "A New Basis for Cosmology". Proceedings of the Royal Society A. 165 (921): 199–208. Bibcode:1938RSPSA.165..199D. doi:10.1098/rspa.1938.0053.
Jump up ^ J.D. Barrow and F.J. Tipler, The Anthropic Cosmological Principle, Oxford UP, Oxford (1986), chapter 6.9.
Jump up ^ Barrow, John D.; Shaw, Douglas J. (2011). "The value of the cosmological constant". General Relativity and Gravitation. 43 (10): 2555–2560. arXiv:1105.3105. Bibcode:2011GReGr..43.2555B. doi:10.1007/s10714-011-1199-1.
Jump up ^ John Barrow, The Constants of Nature: The Numbers That Encode the Deepest Secrets of the Universe, 2003, chapter 1
Jump up ^ Planck (1899), p. 479.
^ Jump up to: a b *Tomilin, K. A., 1999, "Natural Systems of Units: To the Centenary Anniversary of the Planck System", 287–296.
Jump up ^ Pavšic, Matej (2001). The Landscape of Theoretical Physics: A Global View. Dordrecht: Kluwer Academic. pp. 347–352. ISBN 978-0-7923-7006-2. https://arxiv.org/abs/gr-qc/0610061
Jump up ^ Tomilin, K. (1999). "Fine-structure constant and dimension analysis". Eur. J. Phys. 20 (5): L39–L40. Bibcode:1999EJPh...20L..39T. doi:10.1088/0143-0807/20/5/404.
Jump up ^ Also see Roger Penrose (1989) The Road to Reality. Oxford Univ. Press: 714-17. Knopf.
Jump up ^ General relativity predicts that gravitational radiation propagates at the same speed as electromagnetic radiation.
^ Jump up to: a b c d Michael Duff (2002). "Comment on time-variation of fundamental constants". arXiv:hep-th/0208093.
^ Jump up to: a b Duff, Michael; Okun, Lev; Veneziano, Gabriele (2002). "Trialogue on the number of fundamental constants". Journal of High Energy Physics. 3 (3): 023. arXiv:physics/0110060. Bibcode:2002JHEP...03..023D. doi:10.1088/1126-6708/2002/03/023.
Jump up ^ John Baez How Many Fundamental Constants Are There?
Jump up ^ Webb, J. K.; et al. (1999). "Further evidence for cosmological evolution of the fine structure constant". Phys. Rev. Lett. 87 (9): 884. arXiv:astro-ph/0012539v3. Bibcode:2001PhRvL..87i1301W. doi:10.1103/PhysRevLett.87.091301. PMID 11531558.
Jump up ^ Davies, Paul C.; Davis, T. M.; Lineweaver, C. H. (2002). "Cosmology: Black Holes Constrain Varying Constants". Nature. 418 (6898): 602–3. Bibcode:2002Natur.418..602D. doi:10.1038/418602a. PMID 12167848.
References[edit]
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). The Road to Reality. New York: Alfred A. Knopf. Section 31.1. ISBN 978-0-679-45443-4.
Planck, Max (1899). "Über irreversible Strahlungsvorgänge". Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin. 5: 440–480. 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. Archived from the original (PDF) on 2006-06-17.
External links[edit]
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" (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.
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Plantation economy
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The examples and perspective in this article deal primarily with the United States and do not represent a worldwide view of the subject. You may improve this article, discuss the issue on the talk page, or create a new article, as appropriate. (December 2010) (Learn how and when to remove this template message)
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A plantation economy is an economy based on agricultural mass production, usually of a few commodity crops grown on large farms called plantations. Plantation economies rely on the export of cash crops as a source of income. Prominent crops included cotton, rubber, sugar cane, tobacco, figs, rice, kapok, sisal, and species in the genus Indigofera, used to produce indigo dye.
The longer a crop's harvest period, the more efficient plantations become. Economies of scale are also achieved when the distance to market is long. Plantation crops usually need processing immediately after harvesting. Sugarcane, tea, sisal, and palm oil are most suited to plantations, while coconuts, rubber, and cotton are suitable to a lesser extent.[1]
Contents
[hide]
1 North American plantations
1.1 Slavery
2 Atlantic slave trade
3 Industrial Revolution in Europe
4 Sugar plantations
5 Indigo plantations
6 Currencies
7 See also
8 References
North American plantations[edit]
Main article: Plantations in the American South
In the Thirteen Colonies, plantations were concentrated in the South. These colonies included Maryland, Virginia, North Carolina, South Carolina, and Georgia. They had good soil and almost year-round growing seasons, ideal for crops such as rice and tobacco. The existence of many waterways in the region made transportation easier. Each colony specialized in one or two crops, with Virginia standing out in tobacco production[2]
Slavery[edit]
See also: Slavery in the United States and Slavery in the colonial United States
Sugar plantation in the British colony of Antigua, 1823
Planters embraced the use of slaves mainly because indentured labor became expensive. Some indentured servants were also leaving to start their own farms as land was widely available. Colonists tried to use Native Americans for labor, but they were susceptible to European diseases and died in large numbers. The plantation owners then turned to enslaved Africans for labor. In 1665, there were fewer than 500 Africans in Virginia but by 1750, 85 percent of the 235,000 slaves lived in the Southern colonies, Virginia included. Africans made up 40 percent of the South’s population.[3]
According to the 1840 United States Census, one out of every four families in Virginia owned slaves. There were over 100 plantation owners who owned over 100 slaves.[4]
Number of slaves in the Lower South: 2,312,352 (47% of total population).
Number of slaves in the Upper South: 1,208,758 (9% of total population).
Number of slaves in the Border States: 432,586 (13% of total population).
Fewer than one-third of Southern families owned slaves at the peak of slavery in 1860 prior to the Civil War. In Mississippi and South Carolina the figure was 49% and 46% respectively. The total number of slave owners was 385,000 (including, in Louisiana, a small percentage of free African Americans), amounting to approximately 3.8% of the 15 slave states population. By counting only slave owners, this does not acknowledge people who benefited from slavery by being in a slave owning family of an owner and thus the true extent of slavery in 1860 on the eve of the Civil War.
Tobacco field
On a plantation with more than 100 slaves, the capital value of the slaves was greater than the capital value of the land and farming implements.
Per the 1860 census, at the peak of Slavery and the Plantation Economy in the US, the % of slaveholding families was as follows:[5]
Slaveholding Families %
Groups of Slave States relating to the Civil War
States in the Group
26% 15 Slave States where slavery was legal Alabama, Arkansas, Delaware, Florida, Georgia, Kentucky, Louisiana, Maryland, Mississippi, Missouri, North Carolina, South Carolina, Tennessee, Texas, Virginia
31% 11 Confederate States that seceded Alabama, Arkansas, Florida, Georgia, Louisiana, Mississippi, North Carolina, South Carolina, Tennessee, Texas, Virginia
37% 1st 7 states seceding before Lincoln's Inauguration Alabama, Florida, Georgia, Louisiana, Mississippi, South Carolina, Texas
25% 2nd 4 states seceding after Fort Sumter Arkansas, North Carolina, Tennessee, Virginia
16% 4 Border States that did not secede Delaware, Kentucky, Maryland, Missouri
Atlantic slave trade[edit]
Main article: Atlantic slave trade
Enslaved Africans were brought from Africa by the English and other European powers, for their Western Hemisphere colonies. They were shipped from ports in West Africa to the New World. The journey from Africa across the Atlantic Ocean was called "the middle passage", and was one of the three legs which comprised the [trade] among the continents of Europe, the Americas, and Africa.
By some estimates, it is said that some ten million Africans were brought to the Americas. Only about 6% ended up in the North American colonies, while the majority were taken to the Caribbean colonies and South America. A reason many did not make it to the colonies at all was disease and illness. Underneath the slave ship's decks, Africans were held chest-to-chest and could not do much moving. There was waste and urine throughout the hold; this caused the captives to get sick and to die from illnesses that could not be cured.[6]
As the plantation economy expanded, the slave trade grew to meet the growing demand for labor.
Industrial Revolution in Europe[edit]
Main article: Industrial Revolution
Western Europe was the final destination for the plantation produce. At this time, Europe was starting to industrialize, and it needed a lot of materials to manufacture goods. Being the power center of the world at the time, they exploited the New World and Africa to industrialize. Africa supplied slaves for the plantations; the New World produced raw material for industries in Europe. Manufactured goods, of higher value, were then sold both to Africa and the New World. The system was largely run by European merchants[7]
Sugar plantations[edit]
Sugar cane workers in Puerto Rico, 1941
Sugar has a long history as a plantation crop. Cultivation of sugar had to follow a precise scientific system to profit from the production. Sugar plantations everywhere were disproportionate consumers of labor, often enslaved, because of the high mortality of the plantation laborers. In Brazil, plantations were called casas grandes and suffered from similar issues.
The slaves working the sugar plantation were caught in an unceasing rhythm of arduous labor year after year. Sugarcane is harvested about 18 months after planting and the plantations usually divided their land for efficiency. One plot was lying fallow, one plot was growing cane, and the final plot was being harvested. During the December–May rainy season, slaves planted, fertilized with animal dung, and weeded. From January to June, they harvested the cane by chopping the plants off close to the ground, stripping the leaves and then cutting them into shorter strips to be bundled off to be sent to the sugar cane mill.
In the mill, the cane was crushed using a three-roller mill. The juice from the crushing of the cane was then boiled or clarified until it crystallized into sugar. Some plantations also went a step further and distilled the molasses, the liquid left after the sugar is boiled or clarified, to make rum. The sugar was then shipped back to Europe. For the slave laborer, the routine started all over again.
With the 19th-century abolition of slavery, plantations continued to grow sugar cane, but sugar beets, which can be grown in temperate climates, increased their share of the sugar market.
Indigo plantations[edit]
Indigofera was a major crop cultivated during the 18th century, in Venezuela, Guatemala—and Haiti until the slave rebellion against France that left them embargoed by Europe and India in the 19th and 20th centuries. The indigo crop was grown for making blue indigo dye in the pre-industrial age.
Mahatma Gandhi's investigation of indigo workers' claims of exploitation led to the passage of the Champaran Agrarian Bill in 1917 by the British colonial government.
Currencies[edit]
This section needs expansion. You can help by adding to it. (December 2017)
The currency for the Georgia colony was the Pound sterling, a gold coin or very light green bill with the national pound symbol. The currency is worth about one and one half U.S. dollar. The pound sterling is actually still the currency in the United Kingdom, which isn't completely surprising for a country that continues traditions as much as it does.
The currency for the New York was at the time of the colony the New York pound, a fitting name for the colony's currency. This currency was primarily used and made in the 1700 hundreds.
See also[edit]
Banana republic
History of commercial tobacco in the United States
History of sugar
King Cotton
Sugar plantations in the Caribbean
Tropical agriculture
References[edit]
Jump up ^ Jeffery Paige, Agrarian Revolution, 1975.
Jump up ^ The Southern Colonies:Plantations and Slavery, by Kalpesh Khanna Kapurtalawla
Jump up ^ The Southern Colonies: Plantations and Slavery
Jump up ^ "PBS The Slaves' Story". Retrieved 2006-03-24.
Jump up ^ Bonekemper III, Edward H. (2015) The Myth of the Lost Cause: why the South fought the Civil War and why the North won. Regnery Publishing pg 39
Jump up ^ Stephen Behrendt (1999). "Transatlantic Slave Trade". Africana: The Encyclopedia of the African and African American Experience
Jump up ^ The Abolition Project, http://abolition.e2bn.org/slavery_42.html, accessed 3-26-2013
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Plantation agriculture in the Southeastern United States
The plantation
Plantations in the American South
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