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.

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., Refs. 1–4, see also Ref. 5, and see Ref. 6 for a popular exposition).

The Planck scale was introduced by Planck himself7 in 1899, therefore predating the Planck law for blackbody radiation. The importance of the Planck units was realized by Eddington8 and the idea that gravitation and quantum mechanics should be taken into account simultaneously at this scale was spread by Wheeler9,10 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 h/(2π)] which signals quantum mechanics. Usually the Planck scale is deduced, following Planck, on a purely dimensional basis7 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 one obtains a unique quantity with the dimensions of a length, the Planck length

lpl=Gc3=1.6×1035m.
(1)

By combining lpl with G and c one then obtains the Planck mass

mpl=lplc2G=cG=2.2×108kg,
(2)

the Planck energy

Epl=mplc2=c5G=1.3×1019GeV,
(3)

the Planck mass density

ρpl=mpllpl3=c2lpl2G=c5G2=5.2×1096kgm3,
(4)

and the Planck temperature

Tpl=EplkB=lplc4kBG=c5GkB2=1.4×1032K,
(5)

where kB is the Boltzmann constant. We denote with xpl 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 Secs. I A and I B. 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.

In what is probably the most popular derivation of the Planck scale, one postulates that a particle of mass m and Compton wavelength λ=h/(mc), which has Planck energy, collapses to a black hole of radius RS=2Gm/c2 (the Schwarzschild radius of a spherical static black hole of mass m (Refs. 12 and 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

m=hc2G=πmpl1.77mpl.
(6)

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 Universes16 (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

ds2=dt2+a2(t)(dx2+dy2+dz2),
(7)

and with scale factor a(t) and Hubble parameter H(t)ȧ/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 cH1 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

mc2=4π3ρ(H1c)3=H1c52G,
(8)

where ρ is the cosmological energy density and in the last equality we used the Friedmann equation12,13

H2=8πG3c2ρ
(9)

(note that, following standard notation, ρpl and ρ 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

cHλ=hmc.
(10)

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

H2=c52Gh.
(11)

Using again Eq. (9) yields the energy density

ρ3c716πG2h=3c232π2ρpl102c2ρpl,
(12)

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

l=cGρ10lpl,
(13)
m=lc2G10mpl,
(14)
E=mc210Epl,
(15)
T=EkB10Tpl.
(16)

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

RS=2Gmc2=2Gc2(4πR33ρc2)=2Gc24π3ρc2(H1c)3=8πG3cH3ρ.
(17)

Equation (9) then yields RScH1, 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 cH1 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 in 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.17 

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.18 This notion of creation features prominently also in recent popular literature.19 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 ΔEΔt/2 can be used by assuming that ΔE is the energy contained in a Friedmann-Lemaître-Robertson-Walker causal bubble of radius RH1c containing the energy ΔE4πρR3/3. Setting ΔtH1 (the age of this very young universe), ΔEΔt/2 gives

4π3ρ(H1c)3H12
(18)

which can be rewritten as

8πG3ρc3GH4=.
(19)

Equation (9) then yields the mass density

ρc23c58πG2=38πρpl,
(20)

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

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.20–23 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 λ 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.12,21–23 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 λ=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=Lc2/G and quantum and gravitational effects become comparable when λL, which gives

h(Lc2/G)cL
(21)

or

L=Ghc3=2πlpl2.51lpl.
(22)

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.

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

TH=c38πGkBm.
(23)

As is well known, the emitted radiation peaks at a wavelength λmax larger than the horizon radius RS=2Gm/c2. In fact, using Wien's law of displacement for blackbodies

λmaxTH=b=hc4.9651kB2.8978×103mK
(24)

and Eq. (23), one obtains

λmax=bTH=8π24.96512Gmc215.90RS.
(25)

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 λmax and energy

E=hcλmaxhc16RS=hc332Gm.
(26)

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 mc2 yields

mNEc2hc16G=π8mpl0.627mpl.
(27)

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 scale16 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 exercise 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.

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 was supported by Bishop's University and by the Natural Sciences and Engineering Research Council of Canada.

1.
M. B.
Green
,
G. H.
Schwarz
, and
E.
Witten
,
Superstring Theory
(
Cambridge U.P.
,
Cambridge, UK
,
1987
).
2.
J.
Polchinski
,
String Theory
(
Cambridge U.P.
,
Cambridge, UK
,
2005
).
3.
C.
Rovelli
,
Quantum Gravity
(
Cambridge U.P.
,
Cambridge, UK
,
2007
).
4.
C.
Kiefer
,
Quantum Gravity
(
Oxford U.P.
,
Oxford, UK
,
2004
).
5.
D.
Oriti
,
Approaches to Quantum Gravity
(
Cambridge U.P.
,
Cambridge, UK
,
2009
).
6.
L.
Smolin
,
Three Roads to Quantum Gravity
(
Weidenfeld & Nicolson
,
London, UK
,
2000
).
7.
M.
Planck
, “
Ueber irreversible Strahlungsvorgänge
,”
Sitzungsberichte der Koniglich Preussischen Akademie der Wissenschaften zu Berlin
5
,
440
480
(
1899
).
Also as
M.
Planck
,
Ann. Phys.
11
,
69
122
(
1900
), translated in M. Planck, The Theory of Heat Radiation, translated by M. Masius (Dover, New York, 1959).
8.
A. S.
Eddington
, “
Report on the relativity theory of gravitation
,”
Physical Society of London
(
Fleetway Press
,
London
,
1918
).
9.
J. A.
Wheeler
, “
Geons
,”
Phys. Rev.
97
,
511
536
(
1955
).
10.
J. A.
Wheeler
,
Geometrodynamics
(
Academic Press
,
New York and London
,
1962
).
11.
J. D.
Barrow
,
The Constants of Nature
(
Pantheon Books
,
New York
,
2002
).
12.
C. W.
Misner
,
K. S.
Thorne
, and
J. A.
Wheeler
,
Gravitation
(
Freeman
,
San Francisco
,
1973
).
13.
R. M.
Wald
,
General Relativity
(
Chicago U.P.
,
Chicago
,
1984
).
14.
S. W.
Hawking
, “
Particle creation by black holes
,”
Comm. Math. Phys.
43
,
199
220
(
1975
);
S. W.
Hawking
,
Erratum
46
,
206
206
(
1976
).
15.
R. J.
Adler
, “
Six easy roads to the Planck scale
,”
Am. J. Phys.
78
,
925
932
(
2010
).
16.
J. D.
Barrow
,
The Book of Universes
(
W.W. Norton & C.
,
New York
,
2011
), p.
185
and p. 260.
17.
N. A.
Misnikova
and
B. N.
Shvilkin
, “
A possible relation of the mass of the Universe with the characteristic sizes of elementary particles
,” preprint arXiv:1208.0824 (
2012
).
18.
E. P.
Tryon
, “
Is the universe a vacuum fluctuation?
,”
Nature
246
,
396
397
(
1973
).
19.
L.
Krauss
,
A Universe from Nothing
(
Free Press, Simon & Schuster
,
New York
,
2012
).
20.
B. S.
DeWitt
and
R. W.
Brehme
, “
Radiation damping in a gravitational field
,”
Ann. Phys.
9
,
220
259
(
1960
).
21.
J.
Hadamard
,
Lectures on Cauchy's Problem in Linear Partial Differential Equations
(
Dover
,
New York
,
1952
).
22.
W.
Kundt
and
E. T.
Newman
, “
Hyperbolic differential equations in two dimensions
,”
J. Math. Phys.
9
,
2193
2210
(
1968
).
23.
F. G.
Friedlander
,
The Wave Equation on a Curved Spacetime
(
Cambridge U.P.
,
Cambridge, UK
,
1975
).