# Planck scale: Wikis

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# Encyclopedia

In particle physics and physical cosmology, the Planck scale is an energy scale around 1.22 × 1028 eV (which corresponds by the mass–energy equivalence to the Planck mass 2.17645 × 10−8 kg) at which quantum effects of gravity become strong. At this scale, the description of sub-atomic particle interactions in terms of quantum field theory breaks down (due to the non-renormalizability of gravity). That is; although physicists have a fairly good understanding of the other fundamental interactions or forces on the quantum level, gravity is problematic, and cannot be integrated with quantum mechanics (at high energies) using the usual framework of quantum field theory. For energies approaching the Planck scale, an exact theory of quantum gravity is required, and the current leading candidate is string theory, or its modernized form M-theory. Other approaches to this problem include Loop quantum gravity and Noncommutative geometry. At the Planck scale, the strength of gravity is expected to become comparable to 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 term Planck scale can also refer to a length scale or time scale.

Quantity SI equivalent
Planck time 5.39121 × 10−44 s
Planck mass 2.17645 × 10−8 kg

The Planck length is related to Planck energy by the uncertainty principle. At this scale, the concepts of size and distance break down, as quantum indeterminacy becomes virtually absolute. Because the Compton wavelength is roughly equal to the Schwarzschild radius of a black hole at the Planck scale, a photon with sufficient energy to probe this realm would yield no information whatsoever. Any photon energetic enough to precisely measure a Planck-sized object could actually create a particle of that dimension, but it would be massive enough to immediately become a black hole (a.k.a Planck particle), thus completely distorting that region of space, and swallowing the photon. This is the most extreme example possible of the uncertainty principle, and explains why only a quantum gravity theory reconciling general relativity with quantum mechanics will allow us to understand the dynamics of space-time at this scale. Planck scale dynamics is important for cosmology because if we trace the evolution of the cosmos back to the very beginning, at some very early stage the universe should have been so hot that processes involving energies as high as the Planck energy (corresponding to distances as short as the Planck length) may have occurred. This period is therefore called the Planck era or Planck epoch.

## Theoretical ideas

The nature of reality at the Planck scale is the subject of much debate in the world of physics, as it relates to a surprisingly broad range of topics. It may, in fact, be a fundamental aspect of the universe. In terms of size, the Planck scale is unimaginably small (many orders of magnitude smaller than a proton). In terms of energy, it is unimaginably 'hot' and energetic. The wavelength of a photon (and therefore its size) decreases as its frequency or energy increases. The fundamental limit for a photon's energy is the Planck energy, for the reasons cited above. This makes the Planck scale a fascinating realm for speculation by theoretical physicists from various schools of thought. Is the Planck scale domain a seething mass of virtual black holes? Is it a fabric of unimaginably fine loops or a spin foam network? Is it interpenetrated by innumerable Calabi-Yau manifolds,[1] which connect our 3-dimensional universe with a higher dimensional space? Perhaps our 3-D universe is 'sitting' on a 'brane'[2] which separates it from a 2, 5, or 10-dimensional universe and this accounts for the apparent 'weakness' of gravity in ours. These approaches, among several others, are being considered to gain insight into Planck scale dynamics. This would allow physicists to create a unified description of all the fundamental forces.

## Experiments probing the Planck Scale

Experimental evidence of Planck scale dynamics is difficult to obtain, and until quite recently was scant to non-existent. Although it remains impossible to probe this realm directly, as those energies are well beyond the capability of any current or planned particle accelerator, there possibly was a time when the universe itself achieved Planck scale energies, and we have measured the afterglow of that era with instruments such as the WMAP probe, which recently accumulated sufficient data to allow scientists to probe back to the first trillionth of a second after the Big Bang, near the electroweak phase transition. This is still several orders of magnitude away from the Planck epoch, when the universe was at the Planck scale, but planned probes such as Planck Surveyor and related experiments such as IceCube expect to greatly improve on current astrophysical measurements.

Results from the Relativistic Heavy Ion Collider have pushed back the particle physics frontier to discover the fluid nature of the quark-gluon plasma, and this process will be augmented by the Large Hadron Collider coming online soon at CERN, pushing back the 'cosmic clock' for particle physics still further. This is likely to add to the understanding of Planck scale dynamics, and sharpen the knowledge of what evolves from that state. No experiment current or planned will allow the precise probing or complete understanding of the Planck scale. Nonetheless, enough data have already been accumulated to narrow the field of workable inflationary universe theories, and to eliminate some theorized extensions to the Standard Model.

## Sub-Planck physics

Sub-Planck refers to conjectural physics beyond or smaller than the Planck scale.

The Elegant Universe[1] by Brian Greene discusses briefly the strange world of the sub-Planck and how it "creates" the quantum universe by its averages. In his later work, The Fabric of the Cosmos, Greene states that "the familiar notion of space and time do not extend into the sub-Planckian realm, which suggests that space and time as we currently understand them may be mere approximations to more fundamental concepts that still await our discovery.”

## References

1. ^ a b Greene, Brian. "The Elegant Universe". pp. 207–208. ISBN 0-375-70811-1.
2. ^ Arkani-Hamed, Nima; Savas Dimopoulos, Gia Dvali, Nemanja Kaloper (1999-11-17). Manyfold Universe. Retrieved 2007-07-20.

# Study guide

Up to date as of January 14, 2010

### From Wikiversity

In physics, Planck scale is the fundamental scale of matter, named after the German physicist Max Planck, who first proposed the ‘’Planck mass’’ in 1899. It defines the electric Planck coupling constant equal to the gravitational one:

$\alpha_{P} = \frac{q_P^2}{2hc\,\varepsilon_E} = \alpha_S/\alpha_S = 1, \$

where

$\alpha_{S} = \frac{e^2}{2hc\,\varepsilon_E} = \alpha, \$

is the Stoney scale fine structure constant;

• $q_P = \frac{e}{\sqrt{\alpha_S}}$ is the Planck charge;
• $\!e$ is the elementary charge;
• $\!h$ is the Planck constant;
• $\!c$ is the speed of light in vacuum;
• $\!\varepsilon_E$ is the electric constant, or permittivity of free space. Thus, the Stoney scale defines the force interaction dimensionless parameter, known as fine structure constant.

## History

Natural units began in 1881, when George Johnstone Stoney 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. (Stoney was also the first to hypothesize that electric charge is quantized and hence to see the fundamental character of e.) Max Planck first set out the base units (qP excepted) later named in his honor, in a paper presented to the Prussian Academy of Sciences in May 1899.[1][2] That paper also includes the first appearance of a constant named b, and later called h and named after him. The paper gave numerical values for the base units, in terms of the metric system of his day, that were remarkably close to those in Table 2. We are not sure just how Planck came to discover these units because his paper gave no algebraic details. But he did explain why he valued these units as follows:

...ihre Bedeutung fur alle Zeiten und fur alle, auch au?erirdische und au?ermenschliche Kulturen notwendig behalten und welche daher als »naturliche Ma?einheiten« bezeichnet werden konnen... ...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"...

## Fundamental units of vacuum

Dielectric constant[3]:

$\varepsilon_E = \varepsilon_0 = 8.854187817\cdot 10^{-12} \$ F m−1

Magnetic constant:

$\mu_E = \mu_0 = \frac{1}{\varepsilon_0 c^2} = 1.2566370614\cdot 10^{-6} \$ H m−1

Electrodynamic velocity of light:

$c_E = \frac{1}{\sqrt{\varepsilon_E\mu_E}} = 2.99792458\cdot 10^8 \$ m s−1

Electrodynamic vacuum impedance:

$\rho_{E0} = \sqrt{\frac{\mu_E}{\varepsilon_E}} = 376.730313461 \$ Ohm

Dielectric-like gravitational constant:

$\varepsilon_G = \frac{1}{4\pi G} = 1.192708\cdot 10^9 \$ kg s2 m−3

Magnetic-like gravitational constant:

$\mu_G = \frac{4\pi G}{c^2} = 9.328772\cdot 10^{-27} \$ m kg−1

Gravidynamic velocity of light:

$c_G = \frac{1}{\sqrt{\varepsilon_G\mu_G}} = 2.9979246\cdot 10^8 \$ m s−1

Gravidynamic vacuum impedance:

$\rho_{G0} = \sqrt{\frac{\mu_G}{\varepsilon_G}} = 2.7966954\cdot 10^{-18} \$ m2 kg−1 s−1

Considering that all Stoney and Planck units are derivatives from the ‘’vacuum units’’, therefore the last are more fundamental that units of any scale.

The above fundamental constants define naturally the following relationship between mass and electric charge:

$m_P = \sqrt{2hc\varepsilon_G} = e\sqrt{\frac{\varepsilon_G}{\alpha_S\varepsilon_E}} \$

and these values are the base units of the Planck scale.

## Primary Planck units

### Gravitational Planck units

$m_P = \sqrt{2hc\varepsilon_G} = m_S\ \sqrt{\alpha} = 2.17644(11)\cdot 10^{-8} \$ kg,

where $m_S \$ is the Stoney mass. Planck gravitational fine structure constant:

$\alpha_{GP} = \frac{m_P^2}{2hc\varepsilon_G} = \alpha_P = 1. \$

Planck "dynamic mass", or gravitational magnetic-like flux:

$\varphi_{GP} = \frac{h}{m_P} = 3.04396\cdot 10^{-26} \$ J s kg−1

Planck scale gravitational magnetic-like fine structure constant[4]

$\beta_{GP} = \frac{\varphi_{GP}^2}{2hc\mu_G} = 1/4 = 0.25. \$

Planck gravitational impedance quantum:

$R_{GP} = \frac{\varphi_{GS}}{m_P} = \frac{h}{m_P^2} = 1.39835\cdot 10^{-18} \$ J s kg−1

### Electromagnetic Planck units

Planck charge:

$q_P = \frac{e}{\sqrt{\alpha_S}} = 1.875545870(47)\cdot 10^{-18} \$ C

Planck electric fine structure constant:

$\alpha_{EP} = \frac{q_P^2}{2hc\,\varepsilon_E} = 1. \$

Planck magnetic charge, or flux:

$\varphi_{MP} = \frac{h}{q_P} = \sqrt{\alpha_S}\varphi_0 = 3.53290\cdot 10^{-16} \$ Wb

Planck scale magnetic fine structure constant[4]

$\beta_{MP} = \frac{\varphi_{MP}^2}{2hc\mu_E} = 1/4 = 0.25. \$

Planck electrodynamic impedance quantum:

$R_{EP} = \frac{\varphi_{MP}}{q_P} = \alpha_S\frac{h}{e^2} = 1.88365\cdot 10^2 \$ Ohm

is the s.c. von Klitzing constant for Planck scale.

## Secondary Planck scale units

All systems of measurement feature base units: in the International System of Units (SI), for example, the base unit of length is the meter. In the system of Stoney units, the Stoney base unit of length is known simply as the ‘’Stoney length’’, the base unit of time is the ‘’Stoney time’’, and so on. These units are derived from the presented above primary Stoney units, which are arranged in Table 1 so as to cancel out the unwanted dimensions, leaving only the dimension appropriate to each unit. (Like all systems of natural units, Stoney units are an instance of dimensional analysis.)

Used keys in the tables below: L = length, T = time, M = mass, Q = electric charge, ? = temperature. The values given without uncertainties are exact due to the definitions of the metre and the ampere.

Table 1: Secondary Planck units
Name Dimension Expressions SI equivalent with uncertainties[3]
Planck wavelength Length (L) $\lambda_P = \frac{h}{m_Pc}$ $1.01356\cdot 10^{-34}$ m
Planck time Time (T) $t_P = \frac{\lambda_P}{c}$ $3.96474\cdot 10^{-42}$ s
Planck classical radius Length (L) $r_{Pc} = \frac{\alpha_P \lambda_P}{2\pi}$ $1.61313\cdot 10^{-35}$ m
Planck Schwarzschild radius Length (L) $r_{SP} = 2r_{Pc} \$ $3.22626\cdot 10^{-35}$ m
Planck temperature Temperature (Θ) $T_P = \frac{m_P c^2}{k_B}$ $1.416785(71)\cdot 10^{32}$ K

## Planck scale forces

### Planck scale static forces

Electric Planck scale force:

$F_P(q_P\cdot q_P) = \frac{1}{4\pi \varepsilon_E}\cdot \frac{q_P^2}{r^2} = \frac{\alpha_{PE}\hbar c}{r^2}, \$

where $\alpha_{PE} = \frac{q_P^2}{2hc\varepsilon_E} = 1 \$ is the Planck electric fine structure constant. Gravity Planck scale force:

$F_P(m_P\cdot m_P) = \frac{1}{4\pi \varepsilon_G}\cdot \frac{m_P^2}{r^2} = \frac{\alpha_{PG}\hbar c}{r^2}, \$

where $\alpha_{PG} = \frac{m_P^2}{2hc\varepsilon_G} = 1 \$ is the gravity fine structure constant. Mixed (charge-mass interaction) Planck force:

$F_P(m_P\cdot q_P) = \frac{1}{4\pi \sqrt{\varepsilon_G\varepsilon_E}}\cdot \frac{m_P\cdot q_P}{r^2} = \frac{1}{\alpha_S} \sqrt{\alpha_G\alpha_E}\frac{\hbar c}{r^2} = \frac{\hbar c}{r^2}, \$

where $\sqrt{\alpha_{SG} \alpha_{SE}} = \alpha_S \$ is the mixed Stoney fine structure constant.

So, at the Planck scale we have the equality of all static forces which describes interactions between charges and masses:

$F_P(q_P\cdot q_P) = F_P(m_P\cdot m_P) = F_P(m_P\cdot q_P) = \frac{\hbar c}{r^2}. \$

### Planck scale dynamic forces

Magnetic Planck scale force:

$F_P(\varphi_{MP}\cdot \varphi_{MP}) = \frac{1}{4\pi \mu_E}\cdot \frac{\varphi_{MP}^2}{r^2} = \frac{\beta_{PE}\hbar c}{r^2}, \$

where $\beta_{PE} = \frac{\varphi_{MP}^2}{2hc\mu_E} = 1/4 \$ is the magnetic Planck ‘’fine structure constant’’. Gravitational magnetic-like force:

$F_P(\varphi_{GP}\cdot \varphi_{GP}) = \frac{1}{4\pi \mu_G}\cdot \frac{\varphi_{GP}^2}{r^2} = \frac{\beta_{PG}\hbar c}{r^2}, \$

where $\beta_{PG} = \frac{\varphi_{GP}^2}{2hc\mu_G} = 1/4 \$ is the magnetic-like Planck gravitational fine structure constant. Mixed dynamic (charge-mass interaction) force:

$F_P(\varphi_{MP}\cdot \varphi_{GP}) = \frac{1}{4\pi \sqrt{\mu_G\mu_E}}\cdot \frac{\varphi_{MP}\cdot \varphi_{GP}}{r^2} = \alpha_S\sqrt{\beta_{PG}\beta_{PE}}\frac{\hbar c}{r^2} = \frac{\alpha_S\beta_S \hbar c}{r^2}, \$

where $\sqrt{\beta_{PG} \beta_{PE}} = \beta_S = \frac{1}{4\alpha_S}. \$

So, at the Planck scale we have the equality of all dynamic forces which describes interactions between dynamic charges and masses:

$F_P(\varphi_{MP}\cdot \varphi_{MP}) = F_P(\varphi_{GP}\cdot \varphi_{GP}) = F_P(\varphi_{MP}\cdot \varphi_{GP}) = \frac{\alpha_S\beta_S \hbar c}{r^2} = \frac{\hbar c}{4r^2}. \$

## References

1. Planck (1899), p. 479.
2. *Tomilin, K. A., 1999, "Natural Systems of Units: To the Centenary Anniversary of the Planck System," 287-96.
3. 3.0 3.1 Latest (2006) values of the constants [1]
4. 4.0 4.1 Yakymakha O.L.(1989). High Temperature Quantum Galvanomagnetic Effects in the Two- Dimensional Inversion Layers of MOSFET's (In Russian). Kyiv: Vyscha Shkola. p.91. ISBN 5-11-002309-3. djvu

## Sources

• Planck, Max (1899). "Uber irreversible Strahlungsvorgange". Sitzungsberichte der Koniglich Preu?ischen 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.
• Template:Cite paper