LOG#155. Maximal tension, Λ and quantum gravity.

Hi, folks! Before the upcoming series of thematic blog posts (they are on their way, eager readers), I found a very interesting paper to discuss. The paper is this http://arxiv.org/abs/1408.1820 and the pdf file can be found here http://arxiv.org/pdf/1408.1820v1.pdf .

The author of the above paper are two old known theoretical physicists, Gary Gibbons (G.W.Gibbons) and John D. Barrow. The paper is titled quite gorgeously as

Maximum Tension: with and without a cosmological constant

Maximum force/tension is not new even to normal world (let me joke a little bit). You can easily find maximum force out there

force-01 maximalForceplay Maximum_force

Beyond marketing strategies in Theoretical Physics, why did that paper recall my attention? There are several reasons. Firstly, I am involved (as free researcher yet) in the study of physical principles for New Physics and Unification. Indeed, the principle of maximal force (tension) is known for me in different approaches I found these years. Maximal force can be linked to maximal acceleration principles, and they have been studied in Finsler geometries, C-space relativity, Born reciprocal relativity, and even from the Quantum Mechanics realm by Papini and Caianiello. Secondly, I am studying higher dimensional physics and mathematics at the moment, as well as dimensional analysis due to some MOOC I am currently doing. Thirdly, I had also read about the maximal force principle in Schiller’s free online book Motion Mountain (a great free e-book despite some of its bias towards non-standard approaches in the speculative final part of the mammoth reading). Fourthly, It discusses (more interestingly) the effect of a non null and positive cosmological constant (and de Sitter scale/units!) over the principle of maximal force. Finally, it is fun when you can complete/correct the paper and realize the typos AND when you disagree about some of the claims.

What about the content of the paper? It is interesting (but it has some typos and some ideas I don’t agree with, and I will tell you why right soon).

Using the Planck systems of units (I have discussed this and other interesting systems of units previously in this blog, here, here, here, herehere, and here, and no more “here” until now). The maximal force is connected to the Planck system of units, according to the authors, in a classical world only, through the equation

(1)    \boxed{F_m=\dfrac{c^4}{G}}

in a 4D spacetime (3d space) world. I don’t like that claim too much. Gravity is a fundamental interaction, and we suspect it should have a purely quantum origin (unless of course you find some classical theory reproducing all the Standard Model, Quantum Mechanics rules). Furthermore, you can rewrite the above maximal force in the following way

(2)    \boxed{F_m=\dfrac{c^4}{G}=\dfrac{\hbar c}{4L_p^2}}

where L_p is the Planck length

(3)    \boxed{L_p^2=\dfrac{G\hbar}{c^3}}

so

(4)    \boxed{\dfrac{1}{G}=\dfrac{\hbar}{L_p^2c^3}}

There, \hbar, c are the reduced Planck constant and the speed of light in vacuum, respectively. Remember that the Planck length is a really tiny distance, about 10^{-35} meters!

Thus, the claim that the maximal force is “classical” should be taken with care. If you take the gravitational constant as non-fundamental, ant you choose a fundamental microscopic length such as the Planck length as “truly fundamental”, you see that the maximal force is “quantum” in origin from the beginning, non classical at all! It results from the disguise as G that the maximal force seems to be classical in Nature. In my opinion, that a maximal/minimal magnitude be classical or quantum is depends on how you think about it. Of course, that is not the plan as I understand of the whole paper, but I think it should be said that some magnitudes are macroscopic (classical) or microscopic (quantum) depending on how you select your “fundamental physical dimensions”. Moreover, the existence of a maximal force implies a maximal power if you believe in special relativity as it is commonly explained, as v_m=c for known physical objects (matter fields, being massive, travel to less than that maximal speed, and massless objects travel at the speed of light, whenever you accept the postulates and tests of special relativity; no escape of that restriction is possible in the normal special relativity). Therefore,

(5) \boxed{P_m=F_mv_m=F_mc=\dfrac{c^4c}{4G}=\dfrac{c^5}{4G}=\dfrac{\hbar c^2}{4L_p^2}}

How is the maximal force principle justified in the introduction of the mentioned paper? It uses 2 arguments (well it uses 3, but as Stoney units are just a rescaled Planck units, they are 2 only):

1st. Maximal force (a conjecture in physics yet) is connected to the structure of static cosmic strings. From a static source, you get

\rho+\displaystyle{\sum_{i=1}^3p_i}=0

where \rho is the density and p_i is the pressure in the ith direction. Defining the mass per length as

\dfrac{M}{L}=\mu

the tension of the string, and since there is no newtonian source for the cosmic string so

\nabla^2\phi_N=0

and as the cosmic string supports a conical metric flat spacetime, it provides an angular deviation

\Delta \theta =8\pi G\mu c^{-2}>2\pi

only if F>F_m

2nd. Natural units. Natural Planck units (length, time and mass) are introduced according to

L_p^2=\dfrac{G\hbar}{c^3}

T_p^2=\dfrac{G\hbar}{c^5}=\dfrac{L_p^2}{c^2}

M_p^2=\dfrac{\hbar c}{G}=\dfrac{\hbar^2}{c^2L_p^2}

Remark: keeping the idea that G is fundamental, the squared Planck length, time and mass are LINEAR in \hbar. However, if you give up G as fundamental, the squared Planck length is LINEAR in \hbar but remarkably the Planck time is quantum due to L_p and the aquared Planck mass is QUADRATIC in \hbar. Then, it is claimed that maximal force/tension and power are classical, but that it can only be true if we keep the idea of the gravitational constant as a fundamental “constant”, something we suspect from different arguments that can not be completely true. Maximal force/tension and power are “classical” in the sense they do NOT contain the Planck constant in their expressions. But as we have remarked here, that is only due to the fact we conserve G as a fundamental non-varying constant. Several approaches to quantum gravity strongly suggest that G is not ultimately fundamental, but it would be similar to the Fermi constant G_F of nuclear weak interactions, secretly linked to the mass of the intermediate W gauge boson. Furthermore, without entering into the details you can find in the paper or reading the posts in my blog about different systems of “natural” units, Barrow and Gibbons remember us the first system of natural units, the Stoney units. I will write the equivalence with Planck units for clarification purposes here:

L_S^2=\dfrac{Ge^2}{c^4}=\dfrac{L_p^2e^2}{\hbar c}=L_p^2\alpha

T_S^2=\dfrac{L_p^2e^2}{\hbar c^3}=\dfrac{L_p^2\alpha}{c^2}=T_p^2\alpha

M_S^2=\dfrac{e^2}{G}=\dfrac{e^2\hbar}{L_p^2c^3}=\alpha\dfrac{\hbar^2}{L_p^2c^2}=\alpha M_P^2

Now we go into higher dimensions of space (or spacetime). In N space dimensions (N=D-1 lorentzian spacetime) we have

\left[G\right]=M^{-1}L^NT^{-2}

\left[e^2\right]=ML^NT^{-2}

\left[c\right]=LT^{-1}

\left[\hbar\right]=ML^2T^{-1}

and the equivalent to the electromagnetic fine structure constant in N dimensions read

\boxed{\alpha(N)=\hbar^{2-N}e^{N-1}G^{\frac{3-N}{2}}c^{N-4}}

Note that only when N=3 is gravity excluded from the fine structure! In general N space dimensions, we can get (according to the paper) a “non quantum” (I disagree with that terminology) maximal quantity

\boxed{Q=\mbox{mass}\cdot (\mbox{acceleration})^{N-2}=MA^{N-2}}

Remark: when N=3, you will recover the maximal force Q_m=F_m=MA. The main conjecture of the paper is the following statement:

Conjecture. In general relativity, with N spacetime dimensions (N=D-1 lorentzian spacetime), there exists a maximal upper bound to the magnitude Q, heuristically given by

\boxed{Q=\dfrac{c^{2(N-1)}}{G}}

up to (possibly) a multiplicative (dimensionless) constant.

The paper suggest using the N space Schwarzschild metric to guess the dimensionless factor. I will work the details for you here (omitted in the paper in its current version). Firstly, use the N space Schwarzschild horizon radius

R_s=\left(\dfrac{16\pi GM}{(N-1)\Omega_{N-1}c^2}\right)^{1/N-2}

with the N sphere area being (\Omega_2=4\pi)

\Omega_{N-1}=\dfrac{2\pi^{N/2}}{\Gamma (N/2)}

the expression for Q (N>3) is then given by

\boxed{Q=c^{2(N-1)}\left[\dfrac{(N-2)8\pi G}{(N-1)\Omega_{N-1}}\right]^{N-2}\left[\dfrac{(N-1)\Omega_{N-1}}{16\pi G}\right]^{N-1}=MA^{N-2}}

or

\boxed{Q=\dfrac{\Omega_{N-1}(N-1)(N-2)^{N-2}}{2^{N-2}\cdot 16\pi}\left(\dfrac{c^{2(N-1)}}{G}\right)=\dfrac{\Omega_{N-1}(N-1)(N-2)^{N-2}}{2^{N+2}\cdot \pi}\left(\dfrac{c^{2(N-1)}}{G}\right)=K(N)\left(\dfrac{c^{2(N-1)}}{G}\right)}

Check: 

N=3\rightarrow Q_3=MA=F=\left(\dfrac{4\pi G}{\Omega_2}\right)\left(\dfrac{\Omega_2}{8\pi G}\right)^2c^4=c^4G\dfrac{1}{G^2}=\dfrac{c^4}{4G}=F_m

Proof (N space dimensions):

From the N space Schwarzschild radius we get the following relationships

A)       R_s^{N-2}=\dfrac{16\pi G M}{(N-1)\Omega_{N-1}c^2}

B)       \dfrac{c^2R_s^{N-2}}{2M}=\dfrac{8\pi G}{(N-1)\Omega_{N-1}}

C)       \dfrac{(N-2)R_s^{N-2}c^2}{2M}=\left(\dfrac{N-2}{N-1}\right)\dfrac{8\pi G}{\Omega_{N-1}}

Introducing B) and C) into the formula

Q=c^{2(N-1)}\left[\dfrac{(N-2)c^2R_s^{N-2}}{(N-1)\Omega_{N-1}}\right]^{N-2}\left[\dfrac{(N-1)\Omega_{N-1}}{16\pi G}\right]^{N-1}

we obtain

Q=c^{2(N-1)}\left[\dfrac{(N-2)8\pi G}{2M}\right]^{N-2}\left[\dfrac{M}{R_s^{N-2}c^2}\right]^{N-1}

Simple algebra provides the result we are searching for

\boxed{Q=\dfrac{(N-2)^{N-2}}{2^{N-2}}M\dfrac{c^{2(N-2)}}{R_s^{(N-2)}}=\kappa_N\dfrac{Mc^{2(N-2)}}{R_s^{-(2-N)}}}

where

\kappa_N=\dfrac{(N-2)^{N-2}}{2^{N-2}}

is the dimensionless constant we were seeking before. Moreover, dimensional analysis prove that the last N space dimensionful formula has the right dimensions (mass times the acceleration to the N-2 power) since

\left[R_s^{-N+2}c^{2(N-2)}\right]=L^{-(N-2)}L^{2(N-2)}T^{-2(N-2)}=L^{N-2}T^{-2(N-2)}=\left(LT^{-2}\right)^{N-2}

and thus

\left[R_s^{-N+2}c^{2(N-2)}\right]=A^{N-2}

so

\boxed{Q=\kappa_NMA^{N-2}} q.e.d.

Remark: In Cosmology, the scale factor a(t) is proportional to t^n, the cosmic acceleration \ddot{a} is proportional to t^{n-2}, and thus, the force acting on the Universe grows as

F=F_p\left(\dfrac{t}{t_p}\right)^{n-2}

for t>t_p if n>2. By the other hand, the power is

P=P_p\left(\dfrac{t}{t_p}\right)^{2n-3}

for times t>t_p if n<3/2. Then, force and power decouple after Planck time (likely before that time).

The second part of the paper tries to generalize the above considerations when we have a positive non zero cosmological constant. This is important in physics because our Universe is currently expanding under dark energy on very large scales. What happens when we add a cosmological constant to these arguments? We will study only de N=3 case as the paper does, but similar arguments can be done for general N. Firstly, adding a \Lambda term into the Einstein Field Equations

R_{\mu\nu}- \dfrac{1}{2}g^{\alpha\beta}R_{\alpha\beta}g_{\mu\nu}=G_{\mu\nu}=\dfrac{8\pi G}{c^4}T_{\mu\nu}-\Lambda g_{\mu\nu}

or

G_{\mu\nu}=R_{\mu\nu}-\dfrac{1}{2}Rg_{\mu\nu}=2\pi\dfrac{T_{\mu\nu}}{F_m}-\Lambda g_{\mu\nu}

where F_m is the maximal force and dimensionally

\left[x^\mu\right]=L

\left[R_{\mu\nu}\right]=L^{-2}

\left[T_{\mu\nu}\right]=\dfrac{\mbox{Force}}{\mbox{Area}}=\mbox{Pressure}=ML^{-1}T^{-2}

\left[\Lambda\right]=L^{-2}

\left[g_{\mu\nu}\right]=1

The cosmological constant is certain kind of Hooke’s law. A positive cosmological constant implies a cosmic universal force at very large scales of magnitude

\mathbf{F}_\Lambda=M\dfrac{\Lambda c^2}{3}\mathbf{r}=M\dfrac{\Lambda c^2}{3}R\mathbf{u}_R

It is obvious than for a very tiny cosmological constant, the effect of that repulsive force is negligible for short distances (R<<1). Only when R>>1 or similar to the length scale of the cosmological constant this force will be detected (as it does).

Thus. the existence of the cosmological constant allows us to introduce a new set of “natural” units of length, time and mass related to it. They are called de Sitter units/dS units or “cosmological units” sometimes and they can be found in 2 standard forms:

1st. De Sitter units

L_{dS}=\sqrt{\dfrac{1}{\Lambda}}

T_{dS}=\dfrac{1}{c}\sqrt{\dfrac{1}{\Lambda}}=\dfrac{L_{dS}}{c}

M_{dS}=\dfrac{\hbar}{c}\sqrt{\Lambda}

2nd. Cosmological units (rescaled dS units)

L_{\Lambda}=\sqrt{\dfrac{3}{\Lambda}}=\sqrt{3}L_{dS}

T_{dS}=\dfrac{1}{c}\sqrt{\dfrac{3}{\Lambda}}=\sqrt{3}\dfrac{L_{dS}}{c}=\sqrt{3}T_{dS}

M_{dS}=\dfrac{\hbar}{c}\sqrt{\dfrac{\Lambda}{3}}=\dfrac{M_{dS}}{\sqrt{3}}=\dfrac{\sqrt{3}}{3}M_{dS}

Next, the paper says that we can NOT get CLASSICAL quantities from dS units that are INDEPENDENT of the Planck constant if they include the dS mass (energy). That claim is correct. Indeed, there are two arguments providing mass/force bounds

Argument 1. From Heisenberg Uncertainty Principle (HUP), we get

MR>\dfrac{\hbar}{c}

Argument 2. Black hole (BH) solutions, their existence, provide the horizon bound

R>R_s\rightarrow \dfrac{R}{M}>\dfrac{2G}{c^2}

In fact, for isotropic fluid spheres, the known Buchdahl’s bound holds

R>\dfrac{9}{4}\dfrac{GM}{c^2}

This bound can be generalized to the case \Lambda\neq 0 as follows

R>\dfrac{2GM}{c^2}\left(1-\Lambda R^2-\dfrac{1}{9}\left(1-\dfrac{c^2\Lambda R^3}{GM}\right)^2\right)^{-1}

Cosmic repulsion with \Lambda>0 will blow anything apart unless distance is below the cosmic R_\Lambda. Now, HUP provides

M>\dfrac{\hbar}{c}R_\Lambda=M_\Lambda=\dfrac{\sqrt{3}}{3}M_{dS}

Cosmic repulsion with \Lambda>0 plus the BH horizon condition R>R_S provides an upper bound

M<\dfrac{c^2}{G}R_\Lambda=\dfrac{\hbar c^2}{G\hbar}R_\Lambda=\dfrac{\hbar c}{G}\dfrac{1}{M_\Lambda}=\dfrac{M_p^2}{M_\Lambda}

Therefore, the existence of a cosmological constant implies mass itself is confined to the interval

M_\Lambda\leq M\leq \dfrac{M_p^2}{M_\Lambda}

due to the HUP and the BH horizon, i.e., due to Quantum Mechanics and General Relativity. In fact, we can be more precise using the so called Kottler-Schwarzschild-de Sitter black hole solution (sometimes referred as Schwarzschild-de Sitter solution only)

ds^2=-c^2\Delta (r)dt^2+\dfrac{dr^2}{\Delta (r)}+r^2(d\theta^2+\sin^2\theta d\phi^2)

with

\Delta (r)=1-\dfrac{2GM}{c^2 r}-\dfrac{1}{3}\Lambda r^2

The static condition for the metric implies that there are 2 real roots for \Delta (r), say r=r_B and r=r_C. If there are 2 roots, we have

3M\sqrt{\Lambda}<\dfrac{c^2}{G}

and this implies a maximal force bound, since simple algebra form it yields

\dfrac{1}{3}Mc^2\sqrt{\Lambda}<\dfrac{c^4}{9G}=F_{m,static}=F_{ms}

A critical case happens when the 2 roots are the same number, i.e., when r_B=r_C, something that is usually referred as Nariai case

r_B=r_C=\dfrac{1}{\sqrt{\Lambda}}

and it results in the solution known as dS_2\times S^2, i.e., the product of the 2-dimensional dS space with the 2-sphere. Otherwise, when we have two different positive and real roots, we have

R_s\leq r_B\leq L_{dS}

or equivalently

\dfrac{2GM}{c^2}\leq r_B\leq \dfrac{1}{\sqrt{\Lambda}}

and

\dfrac{1}{\sqrt{\Lambda}}\leq r_C\leq \sqrt{\dfrac{3}{\Lambda}}

or equivalently

L_{dS}\leq r_C\leq L_\Lambda

In terms of force, that length bounds imply the relationships we were seeking

(6)        \boxed{F_\Lambda(r=r_B)\leq F_\Lambda (r=r_C=r_{dS})\leq F_{\Lambda}(r=r_D=L_\Lambda)\leq F_{ms}}

or

(7)        \boxed{\dfrac{2GM^2}{3}\Lambda\leq \dfrac{Mc^2}{3}\sqrt{\Lambda}\leq \dfrac{\sqrt{3}Mc^2}{3}\sqrt{\Lambda}\leq \dfrac{c^4}{9G}}

Remark: The addition of a cosmological constant does NOT alter the existence of a maximal force, it only changes the dimensionless coefficient. As you can easily check

F_{ms}=\dfrac{c^4}{9G}=\dfrac{4}{9}\left(\dfrac{c^4}{4G}\right)=\left(\dfrac{2}{3}\right)^2F_m

This result can be easily extended to extra dimensions, and you can also work out the influence on it of higher dimensional extended objects (p-branes)…But that, it is another story.

Let me add the references I liked the most from the paper (in addition to Schiller’s e-book, Motion Mountain):

1) Mod. Phys, Lett. A 15, 2153 (2000).  Mak M. K., Dobson Jr. P. N., Harko T.

2) Nariai H., 1951. Sci. Rep. Tohoku Univ. 35, 62. (I love Japanese people and papers like this from his national science).

3) Buchdahl H.A., 1959. Phys. Rev. 116, 1027.

4) Cvetic M., Gibbons G.W., Pope C.N.(2011). Class. Quant. Grav. 28, 195001.

Additional references can be found in the paper if you read it.

In summary:

1st. There seems to be a mysterious maximal force (maximal Q-magnitude) operating in Nature. If it is classical or quantum in origin, I believe, it is an open question. I am more inclined to think the foundation is quantum than classical, but due to the reboot of emergent approach even in Quantum Mechanics, we should be careful about what we are trying to understand. Emergent gravity and Quantum Mechanics are studied in recent times.

2nd. The inclusion of a cosmological constant does not alter the existence of the maximal force (power). It only changes a dimensionless factor and the physical magnitude being maximized. This result holds in any spacetime dimensions.

3rd. The existence of the cosmological constant (de Sitter radius) is connected to a new set of “fundamental” units: the dS/cosmological units. In fact, it is also tied to a MINIMAL force (quantum?), not only to the existence of a maximal force. This result is well known to geeks out there studying de Sitter (extended) relativity and its generalizations. In fact, we do know that Lie algebra stability principles imply that the maximal extension of the Poincare group is the dS (AdS) group under very general conditions. This topic of extended kinematical groups is known since the 20th century (specially, due to the pioneer works by Bacry and Levy-Leblond) and I will discuss some day here.

See you in my next blog post!

May the Maximal Force (Q) be with you! 

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