LOG#189. Fundamental challenges.

The marriage between gravity and quantum mechanics is “complicated”. The best physicists and brightest minds have tried, but only with partial success. String/superstring theory, now M-theory, is a curious story. The another story is canonical quantum gravity, or loop quantum gravity in its more modern formulation. Let me go back-wards in time.

During the 20th century, we have created the two greatest theories and frameworks of the Human Science. They have names you know very well. Quantum Mechanics (that crazy theory) and relativity (oh, Einstein’s baby! Isn’t it?).

Quantum Mechanics is the microscopic theory of matter (molecules, atoms, nuclei, elementary particles) and energy. Despite what you have heard, there is no problem with making Quantum Mechanics a relativistic theory. It is called Quantum Field Theory (QFT). The Standard Model is a QFT covering every subatomic (till know) force (electroweak and strong), but it does NOT cover gravity.

Relativity, and here I mean GENERAL relativity, is the simplest theory that is consistent locally with both, special relativity and the equivalence principle. The equivalence principle, as Sheldon tries to explain to Penny in certain The Big Bang theory episode, states that there is only a notion of mass. Gravitational masses and inertial masses are the same. Well, indeed, there are 3 notions of mass. 2 gravitational, and one inertial. Anyway, it shows that, to an incredible precision, these masses are the same, with an accuracy of 1 part in $10^{12-14}$ . So, for all practical purposes, and for the common experiments or experiences we have, they are equal. Let me remember you the 3 masses:

Active gravitational mass. $\int g\cdot dS=-4\pi G_N M$ .
Passive gravitational mass. $P=Mg$ .
Inertial mass. $F=Ma$ .

I have assumed, but the result is completely general, that $D=4$ . This theory of general relativity is only built on the general covariance principle and the equivalence principle (some sort of Mach principle contextualized in the 20th theory…). General relativity provides Newton theory as an approximation, but it also includes new effects and an infinite set (in principle) of perturbative corrections. It also explains gravity at large or very large scales (well, indeed, excepting the dark matter or dark energy, that are just plugged by hand into the equations, with no explanation, ad hoc). General relativity explains and describes the expansion of the Universe, predicts the existence of black holes, and gravitational waves (note that there is NO gravitational waves in newtonian gravity!).

Hard marriages use to accomplish big challenges. The biggest of the challenges is that the theory predicts itself its own fall-down. The fall-down events are indeed the structure and dynamics of black holes. It makes us to ask: what is gravity? What is gravitational quantum mechanics? What is, at last, quantum mechanics?

Gravity, the first fundamental force we discovered. It was Newton genius (and likely also the hidden role of Hooke, the most hated person by Newton, defenestrated) who discovered gravity, but some people discuss from time to time if it was discovered by Hooke…Anyway, returning to our theme, Newton (and/or Hooke) managed to unify terretrial mechanics with celestial mechanics. And it yields Universal Gravitation. In fact, there were suspicions by Galileo (the first modern scientist), supported by the Kepler and Copernicus data and observations, that the geocentric view of the Universe was unsustainable. Likely, even some Greek people like Hypparcus also knew it a part of all this stuff.

Gravity is an engine! The simplest way to realize it by Galileo: take any inclined plane. An inclined plane is a machine that transforms height $H$ into energy. In fact $E\propto v^2\propto H$ . You can do it yourself. Pick any round object and an inclined plane. Leave the object at certain $H$ and observe what happens. That was done by Galileo hundreds of times! The coolest thing about this observation is that the fuel it uses is…Gravity. Technicaly, as you learn from High School:

$E_m(H)=E_m(0)$

$\dfrac{1}{2}mv_0^2+mgH=\dfrac{1}{2}Mv^2+mgH_0$

Then, you get $v=\sqrt{2gH}$ , and $\dfrac{1}{2}mv^2=mgH$ . Have you ever imagined that inclined planes were engines fueled by gravity? It is something beautiful. These arguments produce a conceptual revolution in philosophy. And a conceptual revolution in physics, due to the introduction of the notion of energy by Leibniz (another Newton’s foe, but this time, protected such as he managed to teach you differentials and calculus from a more intuitive way than that fluxion calculus by Newton). Descartes, trying to make energy more geometrical, introduced the notion of linear momentum ( $p=mv$ ). In fact, Leibniz’s monads have a secret link with his findings in physics and mathematics (physmatics, yeah). Monads are equivalent to energy. Monads are the dynamical atoms of the Universe in those times. As a consequence of this thing, the same any inclined explains, it allows us to get a clock. As the $YM^2=Gravity$ says, double the bet. Take 2 inclined planes. You can make any object to “oscillate” between 2 arbitrary inclined planes, if you do it properly. Furthermore, it turns out that the own solar system, as the ancient ones taught us (but using religion or myths, not math, not physics or physmatics), any pair of celestial bodies can be used as CLOCKS. Using the universal law of gravity, and the notion of centripetal force, you obtain

$m\dfrac{v^2}{R}=G_N\dfrac{Mm}{R^2}$

And from this equation, you get the Kepler 3rd law, relating period (time, clock, tick tack, tick tack,…) with distance between two astronomical objects:

$T^2=\dfrac{4\pi^2}{GM_t}R^3$

I am assuming the space-time is 4D, so the space is 3d. Using the same trick, but with atoms instead of gravitating objects, you see that the same is also true in atomic physics. Any pair of bodies also act as clocks in motion. Since Aristotle times, celestial bodies have served us (well, indeed from much more ancient times), eternal clocks. However, as we do know not, they are not eternal, only they life much longer than a human life. Even more, the whole Universe is a machine made by gravity, taking the machine as an equation

$F_N=G_N\dfrac{M_1M_2}{R_{12}^2}$

From this viewpoint, the force $F_N$ is associated to motion. The gravitational constant $G_N$ is the universal conversion factor, the energy or fuel is mass (energy in relativity) $M_1M_2$ , and the height is just $R_{12}$ , similarly to the inclined plane. Every celestial body falls down. Even the Doctor falls down, despite he regenerates. The fascinating thing with $G_N$ is that it is not a pure number, it has dimensions. It is not like the Reynolds number in fluid theory, or $\pi$ in mathematics. It is similar to other constant, $G_F$ , the Fermi constant for weak interactions. Fermi constant and the weak interaction explains radioactive decay. Due to weak interactions, some particles are unstable and not eternal, e.g., the neutron. Some grand unification theories (GUTs) predice that even protons do decay (but they hay a very long lifetime). Mathematically speaking

$G_N=L^3T^{-2}M^{-1}$

and $G_N$ has units of $m^3s^{-2}kg^{-1}=Nm^2/kg^2$ . In general relativity, we have something just a bit better. As we merge space and time into space-time, if we do not distinguish length and time, we must introduce a new scale or conversion factor. It is the speed of light $c=299792458m/s$ . In fact, $c=LT^{-1}$ . Physical dimensions are categories (category theory fans reading me just know?) that matter into physics. They quantities. Extra example: the fine structure constant is a pure number, created in this way

$\alpha_{em}=\dfrac{K_Ce^2}{\hbar c}\approx \dfrac{1}{137}$

With units such as $c=\hbar=K_C=1$ you obtain $\alpha \cdot 137=e^2$ .

What is a black hole (BH)? Well, the details are NOT fully now. But there is some ways to understand a BH withe the fashion I have explained here. A black hole is something that traps light. A black hole does not let light to escape. BH are interesting because light is interesting. So, well, what is space-time? What is light? Light are waves or quanta of electromagnetic fields. You are reading me through the screen of your computer or e-device. The device is at certain distance $d$ from your eyes. You are seeing me in the past (OK, only after some minutes since I published and a few fractions of a second. If your eyes are about a meter from the screen, the light uses about 3 nanoseconds to arrive to your eyes and some negligible time to be process in your brain. See any portrait of Nature, a nice girl/boy means that the light travels from them towards us. But, since the speed of light IS finite, it requires some time $t_L=d/c$ . That is common knowledge, and it is used by astronomers, astrophysicists, and collider physicists from all over the world. See the stars (the sun or any other) means to see into our past, or the past of the Universe. You have a time machine every night. For visionaries, a question: would you look forward and away because you could travel with $v>c$ violating special relativity in 3+1 space-time? Hint: multi-temporal relativity and other forgotten relativities (I promise, I will post about them in the future) can go beyond the speed of light, with a price.

Telescopes are real time-machines. The night-sky is a wonderful cosmic TV! It is a screen. Not just like a cinema or theater, but also a nice screen. Space-time is a network. It is just like a continuum limit of a lattice made up with light rays. To understand the space-time IS to interchange concepts like length $L$ or time $T$ in a dual way, using the speed of light. In the same way, it shows than mass and energy are the same thing as well, only a conversion factor is necessary. And as you know, $E=Mc^2$ , don’t you? Moreover, as any physical velocity of the universe, excepting massless force carriers, is less than $c$ , force carriers travel to the speed of light if they are massless and gravity alters not only matter, but also to light rays since they carry energy $E=\hbar \omega$ . Now, we have two constants $G_N, c$ to play with. So,

$\dfrac{L^3}{T^2}=L\dfrac{L^2}{T^2}=L c^2$

Therefore, using the definition of $G_N$ and its dimensions, we can form a gravitational length associated to any relativistic object. Up to a pure number, it reads

$L_G=\dfrac{GM}{c^2}$

Essentially, experts might say, it is half of the Schwarzschild radius. Note that if the speed of light were infinite, the gravitational length would be ZERO. In reality, the Schwarzschild radius is about

$R_S=2.95\left(\dfrac{M}{M_\odot}\right)km$

and the gravitational length defined above is

$L_G=1.48\left(\dfrac{M}{M_\odot}\right)km$

This characteristic length does exist for ANY object in the Universe. Even for you. Neglect any other interaction, and plug a mass there. For instant, for the sun is 1.5 km, for Earth is about 4 mm (the Schwarzschild radii of these objects are 3 km and 8 mm, aproximately). In real life you get a short “life”. Gravitational lifetime is much shorter. The hidden reason you don’t notice your gravitational length is that real life scales are much bigger than gravity scales. However, if you were ablo to turn-off any interaction excepting gravity, your length would be $L_G$ . Did Marvel know this? Did Ant-Man know? We have arrived to a surprising new result. Black holes are what is left if you turn-off everything excepting gravity. BH have only mass (or some charges as I told you in the previous post). Of course, this is the no-hair theorem, that could be wrong, or softly wrong, with the recent new researches.

Stage two is Quantum Mechanics. Classical Mechanics is a sort of code or tool that allow us the identify between what is possible it happens, and what it do happens. During the 18th and 19th century, analytical mechanics introduces a new object into physics. It is called the action. Action is usually denoted by $S$ (please do not confuse it the entropy $S$ , another $S$ ). Action allows derive the Newton laws from other principle, much more general. It is called the minimal action principle. It is an analogue to the maximal entropy (Max-Ent) principle in thermodynamics. Then, what is action? $S$ is not for Superman, hehehe. Geeknerd quote: “It is not an S. In (our) my planet it means hope (quantization!)”. Dimensions of action are:

$S=MT\cdot c^2=ML^2T^{-1}$

Conceptually, action is a simple thing. Any action is the product of the energy you use in some process by the time the process lasts. Using space-time dimensions, action is:

$S=MTc^2=(ML)c^2$

that is, action IS linear momentum times spatial size or a type of angular momentum times a phase (adimensional phases are). The quantum revolution is born. There exists a MINIMAL VALUE for the action. That is named quantum of action. That is $\hbar$ or $h$ . Classical physics is just the theory in which the minimal action is ZERO, or just, you take the limit in which the quantum of action is zero. Classically, there is nothing that avoids you a finite variation of energy in a instant ( $\Delta t=0$ ). Quantum Mechanics is different. The new axiom or postulate (physical or mathematic) is that you get a minimal quantum of action $h$ or $\hbar=h/2\pi\approx 1.1\cdot 10^{-34}J\cdot s$ . This happened in 1900. Max Planck had to do it in order to explain the blackbody radiation. Thus, energy quantization is not truly fundamental. Action QUANTIZATION is MORE FUNDAMENTAL. To any minimal action, you get a new quantum length scale. It is the Compton or Quantum wavelength

$\lambda_C=\lambda_Q=\dfrac{\hbar}{Mc}$

Old atomists believed that atoms were static, and that mass was a sum of invariable atoms. The new atomism, granted by Quantum Mechanics (and its statistical interpretation), states that atoms are dynamical. This was highlighted by L. Boltzmann, but he was bullied with terrible results for his own life.

That action is a sum of quanta of action is something you could remember from school, if you learned about the Bohr-Sommerfeld correction to the Bohr atom. That quanta of action are what matters, drives you into a deep question about the localization of energy. After all, quantum length is $L_Q\propto 1/M$ , and it is independent of $G_N$ . Einstein, in 1905, envisioned how to apply the localization of photons with light quanta to explain the otherwise paradoxical photoelectric effect. Photoelectric effect is quantum energy being localized. Now, we have three scales for any object. Its own normal size $L$ , the quantum size $L_Q$ , and its gravitational size $L_G$ . $L>>L_G>>L_Q$ in general. For instance, any electron has $L_Q>>L_G>>L$ . Therefore, the electron IS quantum. Real life for humans is generally classical $L$ . How many atoms of action you have for these objects? Imagine some object, like a house. It has a mass M. Houses has a size $L$ . The number of quanta of action of houses you get is, for normal houses:

$N=\dfrac{ML}{\hbar}c$

If the house shows to be gravitational, the number of gravitational houses (boxes) are

$N_G=\dfrac{ML_G}{\hbar} c$

and if the house shows to be quantum

$N_Q=\dfrac{ML_Q}{\hbar}c=1$

The ideal house, or box, with zero size does not exist in our Universe. If you do the same for the electron, the platonic electron has normal size non-null (you have ensembles of electrons in atoms!), gravitational electrons do not exist (to our knowledge, $L_G<1$ ), and for quanta of electrons you easily get $N_Q=1$ . Of course, the deep question is why elephants, cars, houses, humans or entities are NOT electrons. Quantum fluctuations are minimal changes of action, i.e., $N\rightarrow N+1$ or $N\rightarrow N-1$ , but you can get others exciting more quanta. The macroscopic world, for elephants, cars, houses, humans, or entities is a world where $N\sim N+1$ . When $N\neq N+1$ you get the QM weirdo. Quantum particles, like the electron. Thus, what is Quantum Gravity? There is no a unique answer. Experts differ about it. According to these lines, since gravity introduce $L_G$ and atoms of action $N_G$ depending on mass in a universal way, quantum gravity must be a theory with quanta of action for certain quanta of action, these quanta of action are UNKNOWN yet, but they are provided in an invariant way by the equations

$N_G=\dfrac{ML_G}{\hbar}c=\dfrac{ G_NM^2}{\hbar c}=\dfrac{G_NE^2}{\hbar c^5}=\left(\dfrac{E}{E_P}\right)^2$

where the Planck energy is about $10^{19}GeV=10^{16}TeV$ . Thus, quantum gravity introduces a fundamental new number into physics. The minimal number of quantu of action for gravity $N_G$ . It is not $1$ but $N_G(M)$ . It depends on $G_N, \hbar, c$ . When gravity is turned on, we have to push up the energy. For classical physics, this new quanta of action are zero, for quantum physics is about $1$ , but for quantum gravity you have to lift it up. To get this easily, you can see yourself, than when you put $L_Q=L_G$ you get the Planck mass, or equivalently, you get the Planck length. If you do it, you will convince yourself of all this thing. Additionally, if you introduce the vacuum energy density as being not-null, you introduce a new quantity, the cosmological constant. It introduces you new length scales (how many scales are there after all?). The idea is that even the vacuum has a “minimal mass” or energy. One option is to use the scale:

$M_W=\dfrac{\hbar}{c}\sqrt{\dfrac{\Lambda}{3}}=M_PR_P/R_W$

Note that $G_N$ is not there, so it is not gravitational at all! Also, you can introduce the scale

$R_\Lambda=\sqrt{\dfrac{3}{\Lambda}}=R_U\approx 10^{26}m$

Compare it to the Planck length, $10^{-35}m$ . The classical radius of the electron is about $10^{-15}m$ , similar to the nuclear size. From $M_W$ you can get a scale $R_W$ , and take, e.g., $R=(R_P^2R_W)^{1/3}\sim 1 fm$ , where $R_P=L_P$ is the Planck length, and you get just like the nuclear size again. The remaining scale with gravity is

$M_W^{'}=\dfrac{c^2}{G_N}\sqrt{\dfrac{3}{\Lambda}}$

The radii or length scales:

$R_W=\sqrt{3/\Lambda}$

$R_W^{'}=\dfrac{\hbar c}{G_N}\sqrt{\dfrac{\Lambda}{3}}=L_P^2/L_\Lambda$

$R_P=L_P=\sqrt{G\hbar/c^3}$

represents different systems. The quantity $g=c^3/G\hbar\Lambda\sim 10^{-120}-10^{-122}$ is really tiny. Note that $M_WM_W^{'}=M_P^2$ , $R_WR_W^{'}=R_P^2$ and that there are also a duality of $M_W$ alone with respect to $M_P$ . How many mass scales are there, then? How many quanta of action? The cosmological quanta of action is very mysterious. You could find

$n_\Lambda=\dfrac{M_\Lambda}{M_W}=N^{1/4}\sim 10^{30}$

$N_\Lambda=\dfrac{M_W^{'}}{M_\Lambda}=N^{3/4}\sim 10^{90}$

Remark: $R_W^{'}<<L_P$ . Thus, for elementary particles in quantum mechanics one should have $M_\Lambda<M<M_P$ , for QM holes (?) one could guess

$M_P<M<M_\Lambda^{'}$ one could expect classical black holes. Let me put the five scales of mass into numbers:

1st. $M_W=\dfrac{\hbar}{c}\sqrt{\dfrac{\Lambda}{3}}\approx 10^{-67}g\approx 6\cdot 10^{-35} eV/c^2$

2nd. $M_\Lambda=\sqrt{M_PM_W}\approx 10^{-35} g\approx 6 meV/c^2$ .

3rd. $M_P=\sqrt{\hbar c/G_N}\approx 10^{-5}g=6\cdot 10^{27} eV/c^2=6\cdot 10^{6} ZeV/c^2$ .

4th. $M_\Lambda^{'}=M_P^2/M_\Lambda\approx 10^{25}g\approx 6\cdot 10^{57} eV/c^2$ .

5th. $M_W^{'}=M_P^2/M_W\approx 10^{56}g\approx 6\cdot 10^{88} eV/c^2$ .

There are two additional mass scales, you can make:

6th. $M_T=\left(\dfrac{\hbar^2\sqrt{\Lambda}}{G_N}\right)^{1/3}\approx 10^{-28}kg\approx 60 MeV/c^2$ . Note that there is no speed of light here!

7th. $M_T^{'}=c\left(\dfrac{\hbar}{G_N^2\sqrt{\Lambda}}\right)^{1/3}\approx 10^{12}kg\approx 6\cdot 10^{47} eV/c^2$ .

As final ending, let me point out something I missed in previous posts. The Beck deduction of $\Lambda$ with quantum information theory uses the formula

$\Lambda=\dfrac{m_e^6G_N^2}{\alpha^6 \hbar^4}$

It has been deduced as well by Harko, using diffent arguments, to provide

$\Lambda=\dfrac{L_p^4}{r_e^6}=\dfrac{\hbar^2 G_N^2m_e^2c^6}{e^{12}}$

and it gives, as I mentioned, the observed value $\Lambda\sim 10^{-56}cm^{-2}$ . Of course, if this number and formula is correct or just a numerological coincidence, must be elucidated yet.

In summary, quantum mechanics and general relativity ARE important because:

Quantum mechanics introduces quanta of action. It is not only that energy is quantized. Everything is, at the end, quantized. Quantum mechanics is the theory to consider if you want to describe something light and small.
General relativity introduces space-time and $G_N, c^2$ on equal footing. It is the theory you need in order to describe something BIG and MASSIVE.
Quantum gravity is the theory you need to know how to describe particles that are both, small and very massive. No one had complete success. Maybe, quantum mechanics and general relativity as they are formulated have to be modified. Are QM and GR effective theories at the end? Is space-time (or the quantum of action) emergent?

Some (right or not, who knows?) hints:

The measurement problem.
QFT is not (to my knowledge) affected by maximal acceleration/force, $A=Mc^3/\hbar=Ec/\hbar$ . Maximal or critical acceleration plus newtonian gravity gives you Planck length (check it yourself!).
Quantum rest is impossible due to the HUP or GUP.
Quantum fluctuations of vacuum implies a fluctuating non-null vacuum energy density. QFT gives a wrong result…By 122 orders of magnitude. SUSY does not solves it all…
Electrons can not be extremal black holes, or can they? The Schwarzschild radius of any electron is about $10^{-57}m$ . The RN radius for an electron $R_Q^2=K_CG_Ne^2/c^4$ is about $R_Q\sim 10^{-36}m<L_P$ .
Einstein idea of elementary particles as singularities can not be correct. Can they? The issue of geodesic motion in general relativity of such a particle is crucial.
Primordial black holes (PMB), i.e., black holes created in the early Universe not as result of the remaining hypermassive stars but as result of the fluctuations in density due to inflation or post-inflation could be the dark matter we are searching for. The PMB window is essential in the range of mass $10^{14}-10^{23}kg$ .

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The Spectrum Of Riemannium

The Art Of Science and Physmatics. This is the geeknerd way!

LOG#189. Fundamental challenges.

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