Three new roads to the Planck scale

TL;DR

This paper presents three heuristic approaches rooted in relativistic gravity and quantum physics to derive the Planck scale, aligning closely with standard values and offering new perspectives on quantum-gravitational phenomena.

Contribution

It introduces three novel heuristic derivations of the Planck scale based on fundamental principles, expanding understanding of quantum-gravitational interactions.

Findings

Derived Planck scale quantities within one order of magnitude of standard values

Proposed pair creation of tiny causal bubbles as particles

Analyzed black hole evaporation at the Planck scale

Abstract

Three new heuristic derivations of the Planck scale are described. They are based on basic principles or phenomena of relativistic gravity and quantum physics. The Planck scale quantities thus obtained are within one order of magnitude of the "standard" ones. We contemplate the pair creation of causal bubbles so small that they can be treated as particles, the scattering of a matter wave off the background curvature of spacetime that it induces, and the Hawking evaporation of a black hole in a single burst at the Planck scale.

Equations

$$l_{\text{pl}}=\sqrt{\frac{G\hslash }{c^3}}=1.6\cdot 10^{-35}\text{mbox}m.$$

$$m_{\text{pl}}=\frac{l_{\text{pl}}{c}^2}{G}=\sqrt{\frac{\hslash c}{G}}=2.2\cdot 10^{-8}\text{mbox}kg,$$

$$E_{\text{pl}}=m_{\text{pl}}{c}^2=\sqrt{\frac{\hslash c^5}{G}}=1.3\cdot 10^{19}\text{mbox}GeV,$$

$${\rho }_{\text{pl}}=\frac{m_{\text{pl}}}{l_{\text{pl}}^3}=\frac{c^2}{l_{\text{pl}}^{2}G}=\frac{c^5}{\hslash G^2}=5.2\cdot 10^{96}\text{mbox}kg\cdot \text{mbox}{m}^{-3},$$

$$T_{\text{pl}}=\frac{E_{\text{pl}}}{k_{\text{B}}}=\frac{l_{\text{pl}}{c}^4}{k_{\text{B}}G}=\sqrt{\frac{\hslash c^5}{G{k}_{\text{B}}^2}}=1.4\cdot 10^{32}\text{mbox}K,$$

$$m=\sqrt{\frac{hc}{2G}}=\sqrt{\pi }{m}_{\text{pl}}\simeq 1.77m_{\text{pl}}.$$

$$d{s}^2=-d{t}^2+a^2(t)\left(d{x}^2+d{y}^2+d{z}^2\right),$$

$$m{c}^2=\frac{4\pi }{3}\rho {\left(H^{-1}c\right)}^3=\frac{H^{-1}{c}^5}{2G},$$

$$H^2=\frac{8\pi G}{3c^2}\rho$$

$$\frac{c}{H}\sim \lambda =\frac{h}{mc}.$$

$$H^2=\frac{c^5}{2Gh}.$$

$$\rho \sim \frac{3c^7}{16\pi G^{2}h}=\frac{3c^2}{32{\pi }^2}{\rho }_{\text{pl}}\simeq 10^{-2}{c}^2{\rho }_{\text{pl}},$$

$$l$$

$$m$$

$$E$$

$$T$$

$$R_{\text{S}}=\frac{2Gm}{c^2}=\frac{2G}{c^2}\left(\frac{4\pi R^3}{3}\frac{\rho }{c^2}\right)=\frac{2G}{c^2}\frac{4\pi }{3}\frac{\rho }{c^2}{\left(H^{-1}c\right)}^3=\frac{8\pi G}{3c}{H}^{-3}\rho .$$

$$\frac{4\pi }{3}\rho {\left(H^{-1}c\right)}^3{H}^{-1}\simeq \frac{\hslash }{2}$$

$$\frac{8\pi G}{3}\rho \frac{c^3}{G{H}^4}=\hslash .$$

$$\frac{\rho }{c^2}\simeq \frac{3c^5}{8\pi G^2\hslash }=\frac{3}{8\pi }{\rho }_{\text{pl}},$$

$$\frac{h}{\left(L{c}^2/G\right)c}\sim L$$

$$L=\sqrt{\frac{Gh}{c^3}}=\sqrt{2\pi }{l}_{\text{pl}}\simeq 2.51l_{\text{pl}}.$$

$$T_{\text{H}}=\frac{\hslash c^3}{8\pi G{k}_{\text{B}}m}.$$

$${\lambda }_{\text{max}}{T}_{\text{H}}=b=\frac{hc}{4.9651k_{\text{B}}}\simeq 2.8978\cdot 10^{-3}\text{mbox}m\cdot \text{mbox}K$$

$${\lambda }_{\text{max}}=\frac{b}{T_{\text{H}}}=\frac{8{\pi }^2}{4.9651}\frac{2Gm}{c^2}\simeq 15.90R_{\text{S}}.$$

$$E=\frac{hc}{{\lambda }_{\text{max}}}\sim \frac{hc}{16R_{\text{S}}}=\frac{h{c}^3}{32Gm}.$$

$$m\simeq \frac{NE}{c^2}\simeq \sqrt{\frac{hc}{16G}}=\sqrt{\frac{\pi }{8}}{m}_{\text{pl}}\sim 0.627m_{\text{pl}}.$$

Full text

I Introduction

General relativity and quantum mechanics are two great achievements of twentieth century physics. Gravity is completely classical in Einstein’s theory of general relativity, and quantum mechanics (broadly defined to include quantum field theory and particle physics) incorporates special relativity but excludes gravity. It is believed that these two completely separate theories should merge at the Planck scale, at which general-relativistic effects become comparable to quantum ones. No definitive theory of quantum gravity is available, although much work has gone into string theories, loop quantum gravity, and other approaches (e.g.,1 ; 2 ; 3 ; 4 see also 5 and see 6 for a popular exposition).

The Planck scale was introduced by Planck himself 7 in 1899, therefore predating the Planck law for blackbody radiation. The importance of the Planck units was realized by Eddington 8 and the idea that gravitation and quantum mechanics should be taken into account simultaneously at this scale was spread by Wheeler 9 ; 10 and has bounced around ever since. The themes that a fundamental system of units exists in nature and that the values of these units can perhaps be derived in a super-theory have been the subject of a large literature (see Ref. 11 for an excellent introduction).

All derivations of the Planck scale more or less correspond to taking various combinations of the fundamental constants $G$ (Newton’s constant) associated with gravity, $c$ (the speed of light in vacuo) characterizing relativity, and the Planck constant $h$ (or the reduced Planck constant $\hbar\equiv h/(2\pi)$) which signals quantum mechanics. Usually the Planck scale is deduced, following Planck, on a purely dimensional basis 7 or it is derived using the concept of a black hole in conjunction with that of a matter wave. The simplest derivation of the Planck scale notes that by combining the three fundamental constants $G,c$, and $\hbar$ one obtains a unique quantity with the dimensions of a length, the Planck length

[TABLE]

By combining $l_{\text{pl}}$ with $G$ and $c$ one then obtains the Planck mass

[TABLE]

the Planck energy

[TABLE]

the Planck mass density

[TABLE]

and the Planck temperature

[TABLE]

where $k_{\text{B}}$ is the Boltzmann constant. We denote with $x_{\text{pl}}$ the Planck scale value of a quantity $x$ determined by dimensional analysis as in the above. Two suggestive alternative derivations of the Planck scale appear in the literature and are reviewed in the next two subsections. At least six more roads to the Planck scale, which are slightly more complicated, are known and have been discussed in Ref. [15]. How many ways to obtain the Planck scale without a full quantum gravity theory are possible? The challenge of finding them can be fun and very creative. Other possibilities to heuristically derive the Planck scale certainly exist: in Secs. II-IV we propose three new ones based on pair creation of “particle-universes”, the propagation of matter waves on a curved spacetime, or the Hawking radiation from black holes.

I.1 A Planck size black hole

In what is probably the most popular derivation of the Planck scale, one postulates that a particle of mass $m$ and Compton wavelength $\lambda=h/(mc)$, which has Planck energy, collapses to a black hole of radius $R_{\text{S}}=2Gm/c^{2}$ (the Schwarzschild radius of a spherical static black hole of mass $m$ 12 ; 13 ). Like all orders of magnitude estimates, this procedure is not rigorous since it extrapolates the concepts of black hole and of Compton wavelength to a new regime in which both concepts would probably lose their accepted meanings and would, strictly speaking, cease being valid. However, this is how one gains intuition into a new physical regime.

Equating the Compton wavelength of this mass $m$ to its black hole radius gives

[TABLE]

I.2 A universe of size comparable with its Compton

wavelength

It is not compulsory to restrict to black holes in heuristic derivations of the Planck scale, although black holes certainly constitute some of the most characteristic phenomena predicted by relativistic gravity.12 ; 13 Why not use a relativistic universe instead of a black hole? This approach is followed in the following argument proposed in John Barrow’s Book of Universes 16 (but it does not appear in the technical literature and it definitely deserves to be included in the pedagogical literature).

Cosmology can only be described in a fully consistent and general way by a relativistic theory of gravity and one can rightly regard a description of the universe as phenomenology of relativistic gravity on par with the prediction of black holes. Consider a spatially homogeneous and isotropic universe which, for simplicity, will be taken to be a spatially flat Friedmann-Lemaître-Robertson-Walker spacetime with line element

[TABLE]

and with scale factor $a(t)$ and Hubble parameter $H(t)\equiv\dot{a}/a$. An overdot denotes differentiation with respect to the comoving time $t$ measured by observers who see the 3-space around them homogeneous and isotropic. The size of the observable universe is its Hubble radius $cH^{-1}$ which is also, in order of magnitude, the radius of curvature (in the sense of four-dimensional curvature) of this space. Consider the mass $m$ enclosed in a Hubble sphere, given by

[TABLE]

where $\rho$ is the cosmological energy density and in the last equality we used the Friedmann equation 12 ; 13

[TABLE]

(note that, following standard notation, $\rho_{\text{pl}}$ and $\rho$ denote a mass density and an energy density, respectively). The Planck scale is reached when the Compton wavelength of the mass $m$ is comparable with the Hubble radius, i.e., when

[TABLE]

This procedure implies that quantum effects (Compton wavelength) are of the same order of gravitational effects (cosmology described by the Friedmann equation). Clearly, we extrapolate Eq. (9) to a new quantum gravity regime from the realm of validity of general relativity and we extrapolate the concept of Compton wavelength from the realm of ordinary quantum mechanics. This extrapolation is necessary in order to learn something about the Planck scale, although it is not rigorous.

The expression (8) of $m$ then gives

[TABLE]

Using again Eq. (9) yields the energy density

[TABLE]

from which the other Planck quantities (1)-(5) can be deduced by dimensional analysis. One obtains

[TABLE]

At first sight the argument of a universe with size comparable with its Compton wavelength is not dissimilar in spirit from the popular argument comparing the Schwarzschild radius of a black hole with its Compton wavelength. In fact, it is commonly remarked that the universe is a relativistic system by showing that the size of the observable universe is the same as the Schwarzschild radius of the mass $m$ contained in it, for

[TABLE]

Equation (9) then yields $R_{\text{S}}\simeq cH^{-1}$, which is often reported in the popular science literature by saying that the universe is a giant black hole. This argument is definitely too naive because the Schwarzschild radius pertains to the Schwarzschild solution of the Einstein equations,12 ; 13 which is very different from the Friedmann-Lemaître-Robertson-Walker one. If one accepted this argument, then comparing the size of the visible universe $cH^{-1}$ with the Compton wavelength of the mass contained in it would be numerically similar to comparing its Schwarzschild radius with this wavelength. However, the step describing the visible universe as a black hole (which is extremely questionable if not altogether incorrect) is logically not needed in the procedure expressed by Eq. (10).

Turning things around but in keeping with the spirit of the derivation above, it has also been noted that equating the Planck density to the density of a sphere containing the mass of the observable universe produces the size of the nucleus (or the pion Compton wavelength) as the radius of this sphere.20

II Pair creation of particle-universes

Another approach to the Planck scale is the following. The idea of a universe which is quantum-mechanical in nature has been present in the literature for a long time and the use of the uncertainty principle to argue something about the universe goes back to Tryon’s 1973 proposal that the universe may have originated as a vacuum fluctuation.Tryon This notion of creation features prominently also in recent popular literature.Krauss Consider now universes so small that they are ruled by quantum mechanics and regard the mass-energies contained in them as elementary particles. At high energies there could be production of pairs of such “particle-antiparticle universes”. Again, one goes beyond known and explored regimes of general relativity and ordinary quantum mechanics by extrapolating facts well known in these regimes to the unknown Planck regime. The Heisenberg uncertainty principle $\Delta E\Delta t\geq\hbar/2$ can be used by assuming that $\Delta E$ is the energy contained in a Friedmann-Lemaître-Robertson-Walker causal bubble of radius $R\sim H^{-1}c$ containing the energy $\Delta E\simeq 4\pi\rho R^{3}/3$. Setting $\Delta t\sim H^{-1}$ (the age of this very young universe), $\Delta E\Delta t\simeq\hbar/2$ gives

[TABLE]

which can be rewritten as

[TABLE]

Equation (9) then yields the mass density

[TABLE]

one order of magnitude smaller than the “standard” Planck mass density (4). The other Planckian quantities can then be derived from $\rho$ and the fundamental constants $G,c$, and $h$.

III Scattering of a matter wave off the background

curvature of spacetime

The second alternative road to the Planck scale comes from the fact that, in general, waves propagating on a curved background spacetime scatter off it.DWB ; 17 ; 18 ; 19 This phenomenon is well known and can be interpreted as if these waves had an effective mass induced by the spacetime curvature. It is experienced by waves with wavelength $\lambda$ comparable with, or larger than, the radius of curvature $L$ of spacetime. High frequency waves do not “feel” the larger scale inhomogeneities of the spacetime curvature and, as is intuitive, essentially propagate as if they were in flat spacetime.17 ; 18 ; 12 ; 19 The phenomenon is not dissimilar from the scattering experienced by a wave propagating through an inhomogeneous medium when its wavelength is comparable with the typical size of the inhomogeneities. Again, we extrapolate the backscattering of a test-field wave by the (fixed) background curvature of spacetime to a new regime in which this wave packet gravitates, bends spacetime and, at the Planck scale, impedes its own propagation. Clearly, this extrapolation is not rigorous, like all order of magnitude estimates. However, we can gain some confidence in this procedure a posteriori by noting that it produces a Planck scale of the same order of magnitude as that obtained by the other methods exposed here.

Consider now a matter wave associated with a particle of mass $m$ and Compton wavelength $\lambda=h/(mc)$ scattering off the curvature of spacetime. The Planck scale can be pictured as that at which the spacetime curvature is caused by the mass $m$ itself and the radius of curvature of spacetime due to this mass is comparable with the Compton wavelength. Essentially, high frequency waves do not backscatter but, at the Planck scale, there can be no waves shorter than the background curvature radius. Dimensionally, the length scale $L$ associated with the mass $m$ (the radius of curvature of spacetime) is given by $m=Lc^{2}/G$ and quantum and gravitational effects become comparable when $\lambda\sim L$, which gives

[TABLE]

or

[TABLE]

In other words, if we pack enough energy into a matter wave so that it curves spacetime, the curvature induced by this wave will impede its own propagation when the Planck scale is reached. When the energy of this wave becomes too compact, the propagation of the matter wave is affected drastically.

IV Hawking evaporation of a black hole in a single

burst

Hawking’s discovery that, quantum mechanically, black holes emit a thermal spectrum of radiation allowed for the development of black hole thermodynamics by assigning a non-zero temperature to black holes.14 In the approximation of a fixed black hole background and of a test quantum field in this spacetime, a spherical static black hole of mass $m$ emits a thermal spectrum at the Hawking temperature

[TABLE]

As is well known, the emitted radiation peaks at a wavelength $\lambda_{\text{max}}$ larger than the horizon radius $R_{\text{S}}=2Gm/c^{2}$. In fact, using Wien’s law of displacement for blackbodies

[TABLE]

and Eq. (23), one obtains

[TABLE]

Therefore, most of the thermal radiation is emitted at wavelengths comparable to, or larger than, the black hole horizon, giving a fuzzy image of the black hole.

Heuristically, one can extrapolate Hawking’s prediction to a Planck regime in which the loss of energy is comparable with the black hole mass. Then the Planck scale is reached when the entire black hole mass $m$ is radiated in a single burst of $N$ particles of wavelength $\sim\lambda_{\text{max}}$ and energy

[TABLE]

Although certainly not rigorous, this procedure provides a Planck scale of the same order of magnitude as the other procedures considered (which is all that one can expect from an order of magnitude estimate). Assuming $N$ of order unity (say, $N=2$) and equating this energy with the black hole energy $mc^{2}$ yields

[TABLE]

V Discussion

Although black holes are a most striking prediction of Einstein’s theory of gravity,12 ; 13 they do not constitute the entire phenomenology of general relativity and there is no need to limit oneself to the black hole concept in heuristic derivations of the Planck scale. One can consider cosmology as well, which is appropriate since cosmology can only be discussed in the context of relativistic gravity. This approach leads to Barrow’s new heuristic derivation of the Planck scale 16 by considering, in a Friedmann-Lemaître-Robertson-Walker universe, a Hubble sphere with size comparable to the Compton wavelength of the mass it contains. Alternatively, one can consider the pair creation of causal bubbles so small that they can be treated as particles, or one can derive the Planck scale using the scattering of waves off the background curvature of spacetime which leads again, in order of magnitude, to the Planck scale when applied to matter waves. Alternatively, one can consider a black hole that evaporates completely in a single burst at the Planck scale. Of course, other approaches to the Planck scale are in principle conceivable. Although quantum gravity is certainly not a subject of undergraduate university courses, the exercice of imagining new heuristic avenues to the Planck scale can be fun and can stimulate the imagination of both undergraduate and graduate students, as well as being an exercise in dimensional analysis.

Acknowledgements.

The author is grateful to John Barrow for a discussion and for pointing out Ref. 8, and to two referees for helpful suggestions. This work is supported by Bishop’s University and by the Natural Sciences and Engineering Research Council of Canada.