Planck length, Planck time, and Planck mass 

Planck length, Planck time and Planck time will be defined from the fundamental constants of physics which are the velocity of light noted c, the gravitational constant noted G, and Planck’s constant noted h,  by using dimensional arguments. In the MKS system, their values ​​are:

c = 299 792 458 m s– 1,

G ≈ 6.674 30 × 10−11 m3 kg− 1 s− 2,

h ≈ 6.626 070 040 × 10−34 kg m2s− 1

h ≈ 1.054 571 800 × 10−34 kg m2 s− 1

For c, this is an exact value (by definition), the others are measured values ​​so are approximated values, the value of all these constants is not predicted by the theory, they are called free parameters. The dimensions of these constants are listed.

The Planck length lP will therefore be defined by the simplest product of these constants which has the dimension of a length, for the time tP and the Planck mass mP, it is the same principle but for a time and a mass. This gives:

mP = (hc /G)1/2 ≈ 2,177.10 -8 kg,

tP = (hG /c5)1/2 ≈5,391.10-44 s

lP = c.tP= (hG /c3)1/2 ≈1,616.10 -35m,

We can check that these values ​​have the correct dimension and with the values ​​of the constants c, G, h, in the MKS system, that their values are correct.

Planck force

By using the law f = m γ, where f is the force applied to mass m and γ the resulting acceleration, by using Planck’s values ​​for the operands, we get:

fP = mP γP, with γP = lP / (tP ²) →fP = c4/G ≈1.21. 1044 Newtons (m.kg.s– 2)

This huge value is independant of the value of the Planck constant! In other words, whatever the value of h is, we get this result! Planck force is an invariant of physics, which will be also found in other phenomena, such as:

Rough estimated mass of the universe

This c4/G factor seems to play a general role. Since this “Planck” force is independent of the value of Planck’s constant, let us consider the value of a modified Planck constant, denoted hu, (h universe) where “the modified Planck mass” would be equal to that of the universe.

Current data assigns a mass of roughly one hundred billion (1011) solar masses to our galaxy and the number of galaxies is estimated to be around 1000 billion, 1012 (these figures are recent, earlier figures have been revised upwards).

With 1012 galaxies of 1011 solar masses and a Solar mass of  ≈ 2.1030kg, we get:

Mass of the universe = Mu ≈2.1053 kg.

According to the Planck mass, (mp ≈ 2 .10-8 kg) it is necessary to multiply by a factor K ≈ 1061, for getting the mass of the universe. As in the definition of Planck’s mass, it is its square root that is involved, the hu constant to be used, instead of Planck’s constant, is such that:

hu ≈ 10122 h.

Vacuum energy and cosmological constant

For explaining the acceleration of the expansion of the universe, one introduced the concept of dark energy. One mathematical solution for dark energy is the cosmological constant. The vacuum energy was proposed as a physical solution of this cosmological constant.

This hypothesis was discarded because of a huge discrepancy (around 10122) between the calculated value of the cosmological constant resulting from vacuum energy and its current value, measured by cosmologists. There is an annex in my book explaining that, in more details.

Due to this huge discrepancy, there are issues about the physical representation of vacuum energy in cosmology, this being considered as a major problem!

But, per our dimensional analysis, the vacuum energy of the whole universe should be calculated not with the physical Planck constant h but with hu, which introduces a 10122 factor, between the calculated vacuum energies.

Estimate of the vacuum energy by this dimensional analysis

The dimension of the cosmological constant λ being [L]-2  (reverse of a squared length). This can be obtained by calculating (T.c)-2, where T is the age of the universe and c the celerity of light.

The calculation with an age of 13.7 billion years gives a value of λ =0.6 x 10-52  m-2, compared with the measured value which is  1.088 x 10-52.

This undervalued value is explained by the fact that, in our dimensional analysis, we do not take into account matter (baryonic and black) whose effect, contrary to that of the cosmological constant, is to slow the expansion.

The value obtained by dimensional analysis corresponds to a phenomenology that would be only due to the cosmological constant.

It is therefore undervalued, because it does not take into account the opposite effect of matter.

We find that the energy of the vacuum, such defined, reflects, in order of magnitude, the observed black energy and the value of the associated cosmological constant.

Calculation of the cosmological constant by using the Newton second law

By using the law:

fplanck = M.γ

where M is the mass of the universe, and γ the acceleration associated with the force of Planck we obtain:

γ = 0.6 x 10-9 m / sec²

By converting, by analysis dimensional, λ, the cosmological constant [L] -2 to obtain an acceleration [L] [T] -2, on the scale of the universe, it must be multiplied by c3 and by t = age of the universe ( in seconds).

γ = λ. c3 .t (age of the universe). –> λ =. c-3 .t-1 γ-1

This calculation will show that we obtain the same value for λ .

This result is obtained by two different methods, which is interesting.

Let us add, that if we are able to measure these values precisely that we will give a more realistic value of the mass of the universe, since they depend on it.

Does cosmological constant depend on time?

This last equation shows, on the other hand, that the value of the cosmological constant depends on the age of the universe, which seems contradictory with its definition.

But perhaps the physical phenomenon at work in the acceleration of the universe, which appears to us as a cosmological constant, is not one, especially as if it is a question of an energy of vaccuum, it is not inconceivable that it depends on time. This issue is under investigation curently.

An article listed on the main page “Hubble constant problem” reviews how the concept ofPlanck’s force and its related cosmological constant may solve this issue.

But above all, and fundamentally, let us remember that the age of the universe is linked to one specific description of cosmology and only reflects the value of a past in relation to our present, in a specific coordinate system (Robetson-Walker) where, moreover, time (called cosmological time) is not ours.

Remember overall that the universe, in relativity, is a space-time, where the notion of age of this space-time has no meaning …

From this, we should consider the dimensional analysis not only as as brain storming but a way aimed to stimulate our mind on this topic….

Justification of the physical relevance of the change of scale.

From a physical point of view this proposition can be supported by statistical considerations. You can divide the universe into microscopic cells. At the level of a microscopic cell, the vacuum state which undergoes fluctuations linked to the process of creation / annihilation of particle-antiparticle pairs can be represented by a random variable governed by a statistical law, binomial law or Poisson law when the probability of events is low. Anywhere in the vacuum throughout the universe the random variables associated with these micro-cells are independent.

Statistics tell us that the distribution law of the state of the universe, which is the combination of the states of the huge number of microscopic cells, at the scale of the universe, resulting from all these independent random variables at microscopic scale, is a random variable which tends, whatever the statistical law is at the microscopic level, towards a normal distribution law (Gaussian) whose parameters, likelihood and variance, are calculated from microscopic laws and the configuration of the universe in microscopic terms. Under these conditions, a vacuum energy at the scale of the universe is a physical hypothesis.

Planck length and Planck time at the universe’s scale.

To calculate the length L and the time T associated with the scale of the universe, with the same “Planck force”, the values ​​of the Planck scale must be multiplied by: K ≈ 1061.

By applying this, we get  :

L ≈ lp x 1061 ≈ 1.6 10-35 x1061 m ≈1, 6 x 1026 m ≈1.7 1010 al.

That is 17 billion light years.

T = tp x 1061 ≈ 5.4 10-44 x1061s ≈5.4 x 1017 s ≈17 x1010 years.

That’s 17 billion years.

Given the inaccuracies in the estimates of the mass of the universe, we see that we get figures that are of the order of magnitude of what is adopted today.

Conversely, to obtain the correct figures, we just need to correct the mass of the universe where, for example, the number of galaxies should be estimated at 800 billion, instead of 1000 billion.

 This topic follows an amazing proposal from Edouard Bassinot, to define a “force of Planck”, defined by the second law of Newton with the Planck parameters. I was intrigued by the proposal and tried to explore its possible consequences. One of them would be that the gravitational constant G and the celerity of light c would be not free parameters in the theory. They would be enforced, by the physical parameters (age, size, and mass) of the universe.

  h = h / 2π, in place of h, is often used, because angular pulsation θ/s is more convenient than frequency in physics. The factor 2π results from the fact that 1 Herz is equal to 2π radians/second.

  Let us stress that the value of the constants G and c are not predicted by the theory. They are what we call “free parameters”. The exact value of c results from a convention allowing to define the units of length and time! For the constant h, also unpredictable (therefore free), the relation E = h.f, where f is the frequency of a photon and E its energy allows to measure its value. The constant h, first introduced in physics for the radiation of the black body (see another item on the website), is ubiquitous in quantum mechanics.

 In a year there are 3600x 24x 365 ≈ 3.15 107 seconds and in a light year 9.45 1015 meters.