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The and the Planck’s Units A Simplification of the Quantum Realm⇤

Espen Gaarder Haug† Norwegian University of Life Sciences

(Dated: Preprint where final version published Physics Essays 2016) Abstract: In this paper, I suggest a new way to write the gravitational constant that makes all of the : Planck , Planck , Planck , and Planck more intuitive and simpler to understand. By writing the gravitational constant in a Planck functional form, we can rewrite all of the Planck units (without changing their values). Hopefully this can be a small step on the way to a better understanding of the quantum realm.

R´esum´e: Dans cet article, je propose une nouvelle faon d’´ecrire la constante gravitationnelle qui rend toutes les unit´es de Planck, la longueur de Planck, le temps de Planck, la masse de Planck, et l’´energie de Planck, plus intuitives et plus simples comprendre. En crivant la constante gravitationnelle sous une forme fonctionnelle de Planck, nous pouvons r´e´ecrire toutes les unit´es de Planck (sans changer leur valeur). Nous esp´erons que cela puisse tre un petit pas vers une comprhension encore meilleure du domaine quantique.

Key words: Gravitational constant, , Planck units: length, time, mass, energy, quan- tum physics.

I. A NEW PERSPECTIVE ON THE PLANCK UNITS Next the Planck mass in this context results in We suggest that ’s gravitational constant[1] can be written as a function of Planck’s reduced constant ~c ~c ~ 1 mp = = 2 3 = (4) G lpc l c 2 3 s p s p lpc ~ Gp = (1) ~ Based on the gravitational constant, the Planck energy where ~ is the reduced Planck’s constant [2], c is the can be simplified to well tested round-trip speed of , and lp is the [3]. We could call this Planck’s form of the grav- itational constant. The Planck length lp is calibrated so 2 ~c 2 ~ 1 2 ~ Ep = mpc = c = c = c (5) that Gp matches our best estimate for the gravitational sGp lp c lp constant. We can use the gravitational constant to find the Planck length, or the Planck length to set the grav- And finally we can also rewrite the reduced Compton itational constant. In our view, the Planck form of the wavelength: gravitational constant enables us to rewrite Planck’s con- stants in a way that simplifies and gives deeper insight, potentially opening up the path for new interpretations ~ ~ 1 = 1 = 1 = lp (6) mpc ~ c in physics. lp c lp Based on this, the Planck length is given by I summarize a series of rewritten Planck units in Table 1. l2 c3 G p One interesting thing to note from the table is that in ~ p ~ ~ lp = = = lp (2) the Planck form of the Planck units, one has c1.5, c2.5, c3 s c3 r c3.5 and c4.5 as well as c4, c5, c7, c8 and it is very hard to Here the Planck length is simply our constant lp.Fur- find any intuition in c powered to such numbers. In the ther, the Planck time in this context is rewritten forms introduced in this paper, we only have c in most of the units, and c2 for only the Planck power and Planck . We have gotten rid of the square 2 3 lpc root as well as the high-powered, non-intuitive notation ~Gp ~ lp t = = ~ = (3) in the Planck units. p 5 s 5 One could arguer thatc rewritingc the gravitationalc con- stant in this way creates a circular argument, since the 2

TABLE I. The table shows the standard Planck units and the units rewritten in the simpler and more intuitive form. Units: “Normal”-form: Simplified-form: 2 3 11 lpc Gravitational constant G 6.67408 10 Gp = ⇡ ⇥ ~ ~G Planck length lp = c3 lp = lp q ~G lp Planck time tp = c5 tp = c q ~c ~ 1 Planck mass mp = mp = G lp c q~c5 ~ Planck energy Ep = Ep = c G lp 2 ~ ~ 1 2 Relationship mass and energy Ep = qmpc c = c lp lp c ~ Reduced lp mpc 2 ~G 2 2 Planck lp = c3 lp = lp 3 ~3G3 3 3 Planck lp = c9 lp = lp 4 Ep ~ c ~ c Planck Fp = =q = Fp = lp lptp G lp lp 5 2 Ep c ~ c Planck power Pp = = Pp = tp G lp lp 1 5 ~ mp c lp c ~ 1 ~ 1 Planck mass ⇢p = 3 = 2 ⇢p = 3 = 4 = 3 lp ~G lp lp c lp clp 7 ~ c E Ep c E lp ~ ~ c Planck energy density ⇢p = 3 = 2 ⇢p = 3 = 4 c = 3 lp ~G lp lp lp lp 8 2 c ~ c Planck intensity Ip = ⇢pc = 2 Ip = 3 ~G lp lp 1 c5 1 c Planck !p = = !p = = tp ~G lp lp c 7 Fp ~ q c ~ c Planck pressure pp = 2 = 3 = 2 pp = 3 lp lptp ~G lp lp

Planck length is derived from Newton’s gravitational con- adds support to the view that the Planck length could be stant. We should consider this in a historical perspective. just as fundamental as big G, if not more so. Although Newton’s gravitational constant was discovered long be- our technology is not advanced enough yet for precision fore the Planck length was even considered (in 1906). in such analysis, it may be in the not too distant future. The Planck length was derived from the gravitational One could argue that the equation is biting its own constant, the , and the . tail as the maximum velocity is a function of the Planck However, it could have taken place the other way around length, e.g. we have simply invented a circular solution if the Planck length had been introduced as a “hypothet- to the Planck length with no real solution. However, this ical” fundamental entity first. The fact that Newton’s is a misconception. The important point is that vmax gravitational constant was discovered before the Planck can be measured experimentally (At a minimum within length was established does not necessarily make it more a thought experiment, that is until our technology for fundamental than the Planck length. Newton’s gravita- accelerating particles get more advanced.) and we know tional constant was likely discovered first because it was that vmax is the composite structure, thus we can use this easier to measure; this is true even if Big G is hard to to extract lp. We typically know the reduced Compton measure accurately; see [4–8]. Even so, Big G is easier wavelength (of an electron, for example) and we know c to measure (indirectly) than the Planck length. per definition; based on this, we can extract lp. Remark- Recently Haug [9, 10] has given a new theoretical in- ably, we need no knowledge of G or ~ to find the find the sight strongly suggesting that fundamental particles in Planck length and even the Planck mass. Einstein’s relativistic mass equation actually have a max- The maximum velocity derived by Haug can also be 1 imum velocity just below that of the speed of light. This written as a function of G, c, ~ and me can be seen as an additional “boundary condition” that a↵ects the interpretation of Einstein’s relativistic mass Gm2 l2 energy formula without actually changing the formula it- v = c 1 e = c 1 p c 1 1.7517 10 45 max ¯2 self. The maximum velocity can be estimated accurately r ~c s e ⇡ ⇥ and is far above the velocity currently attained at the p (7) LHC. Still, what is most interesting in this context is Since vmax here is a function of the universal constants that the Planck length can be found experimentally to G, ~, and c one could try to argue that this is evidence be only a function of the speed of light c and the reduced Compton wavelength of the mass in question; the Haug v2 formula for the Planck length is: l = ¯ 1 max .This 1 Thanks to an anonymous referee for pointing this out. p c2 q 3 that lp must be a function of G and ~ and c and not From the rewriting above, we see that that fine struc- that G is a function of lp. In other words, that G must ture constants appear in several places and cancel each be a universal constant and lp is just a derived constant. other out, basically illustrating the Planck relationship However, the beauty of equation 7 is that G and ~ cancel described in 1906. In other words, the Planck length out and that we are left with vmax as a function of c, lp (and thereby the gravitational constant) is not directly only and the reduced Compton wavelength of the particle dependent on electromagnetic constants and we do not in question, ¯, and not of G and ~. It is worth pointing seem to lose any information by writing Newton’s grav- l2 c3 out that, for example, the reduced Compton wavelength itational constant in the form G = p . Naturally this ~ of an electron can experimentally be found completely does not exclude the possibility of other relationships ex- independent of any knowledge of G,see[11]. To find lp isting between and , but an in- one need the reduced Compton wavelength that can be depth discussion of such ideas is outside the scope of this found totally independent on G as well at the maximum paper. velocity for an electron, vmax. This maximum velocity In the Appendix we have derived the same relationship has to be found experimentally. This maximum veloc- for big G based on . Since dimen- ity for an electron are very close to c, but still higher sional analysis has certain limitations and weaknesses, than velocities one operate with at LHC. However, the it should not be used alone as a “proof” that this is fact that something is predicted and not found yet is not an important relationship. However, it is an additional a sucient argument for rejecting a theory, this should tool that can support the idea that big G written in this simply encorrage further investigation. form could be highly relevant, particularly for simplifying There exists an alternative way to find lp that is not many of the Planck units. dependent on G or ~.Further[9, 10] shows two other The approach to writing the gravitational constant as ways to derive vmax totally independent on G and ~. shown here could have important benefits for physics We are not questioning if G is a universal constant, we because it can be used to simplify Planck units and to are asking if G could be a universal composite constant quantize many gravitational formulas. This could lead consisting of even more fundamental constants, and we to new intuition and interpretations about the depth have based on this report reason to think these are c, ~, of reality. Haug [13] has recently shown how the same and lp. rewriting of Newton’s gravitational constant can be We may never be able to measure the Planck length used to simplify and quantize Newton and Einstein’s directly, but only indirectly through G or hopefully also gravitation theories without changing their output through some other such as recently sug- values. gested by Haug. That we today can measure G and not lp independently of G yet does not necessarily mean that lp is less fundamental than the gravitational constant. It is interesting to note that the Planck length can also II. SUMMARY be obtained by the modification of Stoney’s [12] relating Newton’s constant G to electromagnetic con- By making the gravitational constant a functional form stants: of the reduced Planck constant, we can rewrite the Planck units into simpler and more intuitive forms. As a mini- mum these should be somewhat easier to remember and 2 Gkee work with. It should be easier to interpret c than, for lp = 4 (8) 4.5 r ↵c example, c . Hopefully the rewritten and simplified Planck units can, over time, be a step in the right di- ~ p 7 rection in helping us to better understand the quantum Since we have: e = c p↵ 10 , and Coulomb’s con- 2 7 realm. stant, ke = c 10 , weq get

APPENDIX: DIMENSIONAL ANALYSIS 2 Gkee lp = 4 r ↵c If we assume the Planck length could be an even more 2 fundamental constant than G, then we can also find G Gc2 10 7 ~ p↵p107 through “traditional” dimensional analysis. Here we will v ⇥ c l = u ✓ ◆ assume that the speed of light c, the Planck length lp, and p u q4 u ↵c the reduced Planck constant ~ are the three fundamental t Universal constants. The dimensions of G and the three Gc2 10 7 ~ ↵ 107 l = ⇥ c ⇥ fundamental constants are p s ↵c4 G~ L3 l = (9) 3 1 2 p 3 [G]= 2 = L M T (10) r c MT 4

2 Lenght : 3 = ↵ + +2 (15) L 2 1 [~]=M = ML T (11) Mass : 1= (16) T Time : 2= (17) This gives us

L 1 [c]= = LT (12) T ↵ =2

=3 [lp]=L (13)

Based on this, we have = 1 which means

↵ G = lp c ~ 2 3 ↵ lpc G = lp c ~ = (18) L3 L L2 ~ = L↵ M (14) MT2 T T ✓ ◆ ✓ ◆ As stated previously, dimensional analysis should be used with great care. Still, we think as a tool it adds support to our theory that it could be useful to write Based on this, we obtain the following three equations Newton’s gravitational constant in this form.

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