Table of contents
Planck length, Planck time, and Planck mass . Planck force. Planck force and the Einstein equation. Is Planck’s force a constant of the universe in relativity? Definition by the Einstein equation. Definition with Newton second law..Gravitational waves. Gravitational force of 2 black holes in contact (Maximum gravitational force?). Planck Force invariance applied to the universe. Vacuum energy and cosmological constant. Estimate of the vacuum energy by this dimensional analysis. Justification of the physical relevance of the change of scale. Planck length and Planck time at the universe’s scale. Estimate of the Hubble constant by dimensional analysis. Numerology or structural property?.Numerology?.Structural property: G and c are not free parameters, they are enforced by the parameters of the universe! Some comments on this dimensional analysis. Gravitational mass, declined in 2 categories. Active gravitational mass. Passive gravitational mass. Inertial mass. Can we define an inertia of the universe? Higgs field and Mach principle. Critics about this approach. A heuristic approach. What about the concept of force in general relativity? Notes.
Planck length, Planck time, and Planck mass 
Planck length, Planck time and Planck mass 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 x 10 -8 kg,
tP = (hG /c5)1/2 ≈5,391 x 10-44 s
lP = c.tP= (hG /c3)1/2 ≈1,616 x 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.
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 x 1044 Newtons (m.kg.s– 2)
This huge value is independant of the value of the Planck constant h!
Planck force and the Einstein equation
To ensure the homogeneity of the Einstein equation, we must multiply the Einstein tensor by a constant dimensioned as a force. But, what is the value of this dimensioned constant? It is remarkable that, as the Einstein equation below shows, that it is the Planck force, multiplied by a factor of 8π, that appears in this equation.
Gµν= (8πG.Tµν)/c4 → (c4/G)Gµν= (8π.Tµν) → (fp)Gµν= (8π.Tµν)
As such, it will not be surprising, as we are going to show, to find it in general relativity in its presentation of many physical domains.
Whatever the value of h is, we get this result! This invariance of the Planck force allow to use other values of length and time just by modifying the value of h and, by this property, exploring the phenomenology resulting from the Planck force at others scales.
Is Planck’s force a constant of the universe in relativity?
General relativity is a geometric theory of gravitation.
Definition by the Einstein equation
In the Einstein equation, the theoretical model is therefore represented by the left hand side, which is the geometric part, of the equation. This must be associated with the mind of the physicist who conceived the theory.
The member on the right side in the Einstein equation who expresses physical constraints will anchor this beautiful intellectual construction to the “real” world, described by physical phenomena, the only ones that are accessible to us and which will be the subject of experiments to validate the theoretical model.
This anchoring is done via this Planck force which plays the role of a mediator between the theoretical world and the physical world.
Let us remember that Einstein’s credo for qualifying relativistic cosmology as scientific cosmology is that, unlike the classical approach there is not a container (the universe) and a content (the astrophysical objects that inhabit it), but that this makes an inseparable whole (a space-time) and that therefore, the sensitive world (the phenomena) gives us access to the structure and characteristics of the universe which is a space-time.
Let us also point out that in this approach, time and space no longer exist as fundamental entity and are only “shadows” (as in Plato’s cave allegory) of space-time, the only entity having a status of a fundamental physical entity.
Anyway, whatever the geometry is, it is always this constant that applies, so as such, as we must find it everywhere in all the solutions, we are then justified in considering it as a constant of the theory .
It was built from Planck’s quantities but does not depend on Planck’s constant, which allows it to be valid at all scales.
Since it does not depend on the value of Planck’s constant, can we define it otherwise?
From its nature, we see that it would take a definition from G and c, which dimensional analysis would give us c4/G.
Definition with Newton second law
But would its epistemological justification also be Newton’s second law f = m γ, with any value of m, which could, for instance, be the mass of the universe, the largest mass as opposed to Planck’s mass which is the smallest.
This would then give us a value of an acceleration γ, the nature of which would be linked to the universe.
More details in the article: « Is dark energy vacuum energy at scale of universe? » in this website which introduced the idea that this may solve the problem of discrepancy of the measure of the Hubble law.
We will study these possibilities in the course of the document.
As Planck force is an invariant of physics, it will be found in phenomena, such as:
The amplitude “a” of the stretch and squeeze of space is given by the formula:
a ≈ [2G / (c4r) ]Q’’(t-r /c)
We notice the constant:
2G / c4 ≈ 1.65 x 10-44 m-1kg-1 s²
This shows that the “elasticity  ” of the space is proportional to the inverse of Planck’s force.
In this formula, Q is the quadrupole of the system emitting gravitational waves, Q” is its second derivative, with respect to time.
Gravitational force of 2 black holes in contact (Maximum gravitational force?)
A black hole of mass M has a Schwarzschild radius (defining a horizon). r ≈ 2GM / c². If we use  the law:
f = G m1 m2 / R²
With m1 = m2 = M. When 2 black holes (identical for simplifying) are in contact, their distance to their centers is R = 2r. Inserting this in the previous equation will give:
f = G M ²/(2r) ² = (c4GM ²) / (4GM) ² = c4/ 16G
We still find this same factor : c4/G
Note that one may consider this force as the maximum gravitational force between two distinct bodies since beyond, the bodies will interpenetrate and lose their identity.
We will notice that this is bound to the « rigidity » of space as defined by the gravitational waves phenomenology. One would say that a force greater than the Planck force would tear the space continuum.
In addition, and maybe overall, the importance of the Planck force in relativity is the dimensional constant allowing the homogeneity of the Einstein equation, whose left member is a geometrical parameter and the right member a physical parameter. Planck force links the geometrical parameter of the general relativity which is a geometrical theory of gravitation to that of the physical world (energy, momentum, masses,..).
Planck Force invariance applied to the universe.
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 x 1030kg, we get:
Estimated mass of the universe = Mu ≈2x 1053 kg.
According to the Planck mass, (mp ≈ 2 x 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 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 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.
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 x 10-35 x 1061 m ≈1, 6 x 1026 m ≈1.7 x 1010 al.
That is 17 billion light years.
T = tp x 1061 ≈ 5.4 x 10-44 x 1061s ≈5.4 x 1017 s ≈1.7 x1010 years.
That’s 17 billion years.
As the Planck force would account for the dark energy (cosmological constant), in this model does this mean that the values obtained account only for the phenomenology driven by the cosmological constant eventhough we used the estimated total baryonic mass of the universe for getting the scale?
Anyway, 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.
Estimate of the Hubble constant by dimensional analysis
Note that we could also calculate the Hubble H0 constant (let us take H0 = 70km/s/ Megaparsec,) considering the acceleration of Planck, aP for the universe where FP is the force of Planck and MU the mass of the universe:
aP=FP/MU ≈ 1.21 x 1044 / (1.6 x 1053) ≈7.5 x 10-10 m.s-2 and comparing this value to:
c.H0 ≈ 3 x 108 x 2.27 x 10-18 ≈ 6.8 x 10-10.m.s-2
Which has also the dimension of an acceleration.
The value of H0 deduced from the dimensional analysis is H0 = 77 km/s/Mpc. This is over-estimated by about 10%.
Numerology or structural property?
It seems quite surprising that by using the total mass of the universe, the value gravitational constant G, the speed of light c, with the notion of force (associated with the concept of inertia) as a unifying link, between the scales since it does not depend on it, but notwithstanding with the cosmological model of the universe and with its content (different kind of fluids), we obtain the orders of magnitude of the universe, as far as we know it.
Structural property: G and c are not free parameters, they are enforced by the parameters of the universe!
A reasonable explanation would be, that, in fact, that these free parameters, are not free at all, but are determined by the universe which reveals the value of the constants of physics linked to its content and its structure!
According to that, if we have measured the age T, the size L, and the mass M of the universe we can deduce the values of c and G per the two following equations:
L = c. T → c = L/T ≈ (1, 6 x 1026 m)/(5.4 x 1017 s)≈3x 108m/s
fP = c4/G= M.L/T²→ G = c4 T²/ML = L4T²/T4ML =
G= L3/(M T²)≈(1, 6 x 1026 m)3/[(2 x 1053 kg)(5.4 x 1017 s)²] ≈ 7 x 10-11m3 kg− 1 s− 2
The results are approximate because, in the example, we kept only two digits in numbers for the calculation. With all significative digits in the numbers, per construction of the value T, L, and M, obviously, you would recover the right values.
Let us point out that we needed these two equations (two constraints) , including that of the “Planck Force”, for calculating the unknown values of G and c.
Therefore, in this approach, the “Planck force”, an unexpected constraint, the interpretation of which would be the “elasticity of space” a concept used in cosmology for gravitational waves and bound to the inertia of the whole universe, is necessary. This would lead us to explore an analysis where it would play a major role. We expect that this new perspective may enlighten the understanding of the theory.
Some comments on this dimensional analysis
About mass, we must specify what kind of mass is involved.
Gravitational mass, declined in 2 categories
Active gravitational mass
Usually, when we consider the mass of the universe, we refer to its active gravitational mass. This mass generates the gravitational field. In general relativity, all the masses contribute to the geometry of the spacetime and, in turn, each mass couples with this spacetime which is defined by all. The spacetime is represented by a manifold in general relativity. But in our analysis, this is not this mass which is addressed. Would this discard our analysis, as we do not address the right concept of mass?
Passive gravitational mass
This passive gravitational mass is the coupling coefficient to the gravitational field generated by the active mass. In relativity, all masses follow geodesics of the spacetime that they contribute to define. This is the coupling process in general relativity.
It is defined by Newton’s second law f = mγ, and it is this that we used in our analysis. We can start by looking at the relations between the masses. In relativity, the inertial mass of any entity is equal to its passive gravitational mass, this is the principle of equivalence which is also valid in Newtonian physics but only for massive bodies.
Likely, this inspired E. Mach when under a philosophical approach he claimed that the inertia was from gravitational origin, this explaining the equivalence principle, as in fact, it is not two phenomenology’s, but the same.
In classical mechanics the active gravitational mass of a body is equal to its passive gravitational mass, according to the action-reaction principle. So in Newtonian physics all masses are equal. What is true for one, is true for the others.
In relativity, a priori, this is less obvious, because the active mass can involve integrations in space-time that are not always well defined.
But we can try to define what could be the inertial mass of the universe by the following arguments:
Einstein’s equation provides a solution, according to the principle of least action, which is a geometry for the universe, where all bodies contribute to its definition and where, in return, all bodies follow the geodesics of the geometry that they define.
Can we define an inertia of the universe?
In general relativity the universe is an “isolated” system. He is the “whole thing”, as its mathematical description by a manifold attest. He does not need to refer to anything else in order to exist and be described. Under these conditions, defining inertia seems impossible.
But, let us not forget that general relativity is a theory of gravitation only and that in the universe other interactions exist and may interact with the masses and the energy ruling the structure of the universe. This coupling with these other interactions is generally described by local spaces, which one calls “internal spaces”, at each point of relativistic spacetime. These internal spaces, which are therefore in local contact with relativistic space, can interact with it.
In these internal spaces, the laws of interactions other than gravitation and their couplings with relativistic spacetime are described. These internal spaces are tangent at each point of the manifold that describes the spacetime of general relativity.
The whole system constitutes a “bundle” where the manifold describing the relativistic spacetime is called the “base”, and where the internal spaces the “fibers”.
As a result, the spacetime of general relativity, defined by the least action principle by Einstein’s equation which describes an “equilibrium” (stable) universe, is not an isolated system.
It can be “disturbed” by other interactions. Therefore, a concept of inertia can be introduced as follows.
The inertia of relativistic spacetime will be its resistance to a change in this “stable” gravitational state since it is defined by the principle of least action; If there had been only gravity there would have been no reason to get out of this state.
So if, for some reason unrelated to gravity,  locally, a body deviates from the geodesic defined by the solution of Einstein’s equation, it will be an upheaval.
Indeed, this will not result only in a local change because, in accordance with the Einstein equation which defines a global universe, with these new data, it is all the geometry of the whole universe, which is modified, even if this modification is infinitesimal!
Therefore, the inertia can be defined as the resistance to a change from this “perfect and stable” state. 
But this is not instantaneous, per an inertial reaction to this change , gravitational waves will be emitted  and the new stable state of the universe will be only completed when gravitational waves reached the limit of the universe, these limits depending on the size and on the geometry of the universe. This may take an infinite time.
A new stable solution will emerge from this perturbated previous solution and so on. In this way, we see that we can define the inertial mass of the universe.
In other words, the inertia of the whole universe emerges from this phenomenology. The gravitational waves propagate in this spacetime as an inertial reaction to this perturbation. The physical parameters, especially the G/c4 factor associated to the “elasticity” of the universe, which rules the amplitude of these waves disturbing the spacetime geometry in this transient process up to the final stable state, of this propagation will characterize the inertial properties of the whole universe.
In modern physics the inertial mass is asserted to come from the coupling with the Higgs field , which is filling the space, but we do not know where this field come from.
Higgs field and Mach principle
In modern physics, the inertial mass of particles (bosons and fermions) arises from their coupling with the Higgs field,  which fills the whole space. To explain the equivalence of passive gravitational mass and inertial mass of bodies, in the nineteenth century E. Mach, on philosophical and mechanical considerations, claimed that it was because the inertial mass results from the gravitational interaction between all the masses in the universe.
Whether inertia is conferred by gravity this would explain why the passive gravitational mass is equal to the inertial mass. In this phenomenology the whole universe is involved, something similar to the phenomenology of the Higgs field which fills the whole universe.
Mach’s assumption is compatible with the inertia of the universe as described by general relativity, which is a universe “in equilibrium, satisfying the principle of least action, where a local disturbance causes a global inertial reaction involving gravitational waves. Apart from this global character, Mach’s assumption appears to be quite different from the phenomenology associated to the Higgs boson.
The Higgs field was introduced for explaining the mass of some bosons and that of matter (fermions) since quantum formalism did not allow it. This looks like a desperate trick. Nevertheless, it is possible that nature is like this.
Critics about this approach
A heuristic approach
The main critic is that it is a dimensional approach which, even though its heuristic contribution is valuable, is not a rigorous physical theory. This is true, but nevertheless, such approach brought to us interesting information, which was likely hidden in the parameters that we used. At least, it allowed to extract this information from these parameters, and it is not hopeless that it may provide a guideline to a new theory.
In addition, in this document, we did not always comply with the relativistic approach. But being a study relating orders of magnitude, where we make bold assumptions, for example, the estimation of the total mass of the universe, which has evolved significantly over the past 20 years, the unexpected results of these approximations are, at least, instructive.
What about the concept of force in general relativity?
We used the concept of “gravitational force” while in general relativity, which is the current theory, using a geometrical formalism for describing the gravitation, this concept is replaced by the curvature of the geometry of space-time
Indeed whether, in contrast to classical mechanics, in general relativity gravitation is not considered as a force, for the universe, as a whole entity, we defined its inertia which is related to a concept of force, induced in the representation of the amplitude of the gravitational waves  by the concept of elasticity of space that we had related to the inverse of Planck’s force, as described in a previous chapter.
To be continued….
 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.
 Elasticity is related to elastic stretch of a body (eg a spring) submitted to a stretching constraint. A low elasticity, such that of the space in the gravitational waves phenomenology, provides a huge resistance to stretch force resulting in a small stretch. Elasticity has the dimension of a length (in meters) divided by a force , which is here the Planck force.
 Comments on the approach: The first objection that comes to mind is that this approach is Newtonian-like, we used the Newtonian law, F = G.m1.m2 / r², giving the force F of attraction between 2 bodies, one of mass m1 and the other of mass m2, separated by a distance noted r.
 In a year there are approximatively 3600x 24x 365 = 3.15 x 107 seconds and in a light year 9.45 x 1015 meters.
 If, gravitation was only involved, this should not occur, because the Einstein equation describes the more stable universe, that with the lower possible energy. But they are other interactions, such as electromagnetism which may disturb this quiet and perfect situation.
 These waves are the inertial reaction to a change of the geometry of the universe under a disturbance of the established stable state of the system, where the motion of any body was geodesic. Aristote would have called geodesic motion natural motion unlike that resulting from a disturbance that he called violent movements! This emission of waves, as an inertial reaction to a disturbance of a “natural” state of a system is a general process in physics. For instance, electromagnetic waves are emitted, as an inertial reaction, when one accelerates a charged particle. This radiation induces a loss of energy of the particle. It is the same for gravitational radiation. Likely the internal potential energy would decrease.
 This field is scalar. The relativistic formalism is tensorial. How compatibility can be defined?
 To be rigorous, we must remember that the equation giving the amplitude of gravitational waves, cited in a previous chapter, results from a “linearization” of the Riemann tensor in a weak field where the relativistic metric gμν is approximated by the metric of Minkowski ημν to which we add a perturbation hμν assumed to be small with respect to 1 (gμν ≈ ημν + hμν). This is not relativity in the strict sense but a “post-Newtonian” theory. Would this invalidating completely the conceptual remarks that we made or rather this would amend them, the latter proposal being credible because of their validity in weak field.