Quantum Gravity in a Nutshell1

DEDICATION

To my wife Wanyana Ritah,

My sons Odhran & Leander ,

and lastly to Carlo Rovelli

PREFACE

There is a need for a book on a Quantum Theory of Gravity that is not directed at specialists but, rather, sets out the concepts underlying this subject for a broader scientific audience and conveys joy in their beauty. The author has written with this goal in mind, and has succeeded admirably. This wonderful and exciting book is optimal for physics graduate students and researchers. The physical explanations are exceedingly well written and integrated with formulas. Quantum Gravity is the next big thing and this book will help the reader understand and use the theory.

Author’s Note

Our search for ultimate understanding—the Quantum Theory of Gravity—has long been the quest of such great scientists as Aristotle, Newton, Einstein, Hawking and many others, and is expected to transform science, providing clarity and understanding that is unknown today, ideally via one single overlooked principle in nature. So far, this quest has produced theories such as Special Relativity, General Relativity and Quantum Mechanics, and such recent proposals as Dark Matter and Dark Energy in cosmology. Yet these all suffer serious internal problems and compatibility issues with each other, introducing even more questions, mysteries and paradoxes—and often even violations of our laws of physics upon closer examination. As a result, the Quantum Theory of Gravity continues to elude us, leaving a fractured and divided scientific community with no clear direction forward. This has also resulted in the mathematisation of physics which has resulted in the reduction of the cosmos to a mathematical entity, which has not only confused physicists but accounts for their worst and most distracting assertions. This book makes a first case for the latter, with clear discussions exposing the flaws in the above concepts and more, while stepping back to take a good look at the scientific legacy we have inherited.

We are probably asking the wrong questions at the moment, nevertheless it is impossible to resist the temptation to try. After all, the other fundamental forces – except gravity – fit very neatly with quantum mechanics.

Balungi Francis 2018

1. Solving Quantum Gravity

To the intra-atomic movement of electrons, atoms would have to radiate not only electromagnetic but also gravitational energy if only in tiny amounts. As this is hardly true in nature, it appears that quantum theory would have to modify not only Maxwellian electrodynamics, but also the new theory of gravitation.

Albert Einstein

The development of a quantum theory of gravity began in 1899 with Max Planck’s formulation of Planck scales of mass, time, and length. During this period, the theories of quantum mechanics, quantum field theory and general relativity had not yet been developed. This means that Planck himself had no idea about what he had just developed-behind the Black board. Planck was not aware of quantum gravity and what it would mean for physicists. But he had just coined in formula one of the starting point for the holy grail of physics.

After P.Bridgman’s disapproval of Planck’s units in 1922, Albert Einstein having published the General Relativity theory, a few months after its publication he noted that to the intra-atomic movement of electrons, atoms would have to radiate not only electromagnetic but also gravitational energy if only in tiny amounts, as this is hardly true in nature, it appears that quantum theory would have to modify not only Maxwellian electrodynamics, but also the new theory of gravitation. This showed Einstein’s interest in the unification of Planck’s quantum theory with his newly developed theory of Gravitation.

Then in 1933 came Bronstein’s cGh-plan as we know it today. In his plan he argued a need for Quantum Gravity. In his own words he stated: After the relativistic quantum theory is created, the task will be to develop the next part of our scheme that is, to unify quantum theory (h), special relativity (c) and the theory of gravitation (G) into a single theory. Thus the theory of quantum gravity is expected to be able to provide a satisfactory description of the microstructure of space time at the so called Planck scales, at which all fundamental constants of the ingredient theories, c (speed of light), h ( Planck constant) and G ( Newton’s constant), come together to form units of mass, length and time.

Therefore the need for the theory of quantum gravity is crucial in understanding nature, from the smallest to the biggest particle ever known in the universe. For example, we can describe the behavior of flowing water with the long- known classical theory of hydrodynamics, but if we advance to smaller and smaller scales and eventually come across individual atoms, it no longer applies. Then we need quantum physics just as a liquid consists of atoms Daniel Oriti in this case imagines space to be made up of tiny cells or atoms of space and a new theory of quantum gravity is required to describe them fully.

The aim of this book is to develop a theory capable of explaining the quantum behavior of the gravitational fields and thereafter explain the problems involving a combination of very high energy and very small dimensions of space such as, the behavior of Black holes and the study of the properties of the early universe.

For us to solve quantum gravity (QG), we need to address, understand and resolve in detail the problems brought about by the failure of the general theory of relativity (GR). Below I outline briefly where GR breaks down and later I resolve each of these problems with applications.

(1) General relativity fails to explain details near or beyond space-time singularities. That is, for high or infinite densities where matter is enclosed in a very small volume of space.  Abhay Ashtekar says that; when you reach the singularity in general relativity, physics just stops, the equations break down

(2)General relativity fails to account for dark matter.

(3) General relativity also fails to be quantized.

Singularity Resolution in Quantum Gravity

The demand for consistency between a quantum description of matter and a geometric description of spacetime, as well as the appearance of singularities (where curvature length scales become microscopic), indicate the need for a full theory of quantum gravity. For example; for a full description of the interior of black holes, and of the very early universe, a theory is required in which gravity and the associated geometry of space-time are described in the language of quantum physics. Despite major efforts, no complete and consistent theory of quantum gravity is currently known, even though a number of promising candidates exist.

The first step towards the development of a quantum theory of gravity lies in studying the kind of physics behind black holes which are born when normal stars die or which were formed in regions of high energy density in the early universe. Black holes on the other hand, are completely collapsed stars that is, stars that could not find any means to hold back the inward pull of gravity and therefore collapse to a singularity.

This section is aimed at answering questions like; i) Do objects continually collapse to a singularity or there is a limiting distance below which the very notions of space and length cease to exist?

Theorem:- A star more than three times the size of our Sun collapses in this way; the gravitational forces of the entire mass of a star overcomes the electromagnetic forces of individual atoms and so collapse inwards. If a star is massive enough it will continue to collapse creating a Black hole, where the whopping of space time is so great that nothing can escape not even light, it gets smaller and smaller. The star in fact gets denser as atoms even subatomic particles literally get crashed into smaller and smaller space, and its ending point is of course a space time singularity.

––––––––

The appearance of singularities in any physical theory is an indication that either something is wrong or we need to reformulate the theory itself. Singularities are like dividing something by zero. One such theory plagued by singularities is the General theory of relativity (GR) and the problems in GR arise from trying to deal with a universe that is zero in size (infinite densities). However, quantum mechanics suggests that there may be no such thing in nature as a point in space-time, implying that space-time is always smeared out, occupying some minimum region. The minimum smeared-out volume of space-time is a profound property in any quantized theory of gravity and such an outcome lies in a widespread expectation that singularities will be resolved in a quantum theory of gravity. This implies that the study of singularities acts as a testing ground for quantum gravity.

Loop quantum gravity (LQG) suggests that singularities may not exist. LQG states that due to quantum gravity effects, there must be a minimum distance beyond which the force of gravity no longer continues to increase as the distance between the masses become shorter or alternatively that interpenetrating particle waves mask gravitational effects that would be felt at a distance. It must also be true that under the assumption of a corrected dynamical equation of LQ cosmology and brane world model, for the gravitational collapse of a perfect fluid sphere in the commoving frame, the sphere does not collapse to a singularity but instead pulsates between a maximum and minimum size, avoiding the singularity.

Additionally, the information loss paradox is also a hot topic of theoretical modeling right now because it suggests that either our theory of quantum physics or our model of black holes is flawed or at least incomplete. and perhaps most importantly, it is also recognized with some prescience that resolving the information paradox will hold the key to a holistic description of quantum gravity, and therefore be a major advance towards a unified field theory of physics.

The paradox, as formulated, arises from considerations of the ultimate fate of the information that falls into a black hole: does it disappear as it falls into the black hole singularity? As well, what happens to the information of a black hole when it evaporates to nothing due to Hawking radiation? If a black hole loses all of its energy, then all of the information about all of the particles that fell in it would be lost as well. Of course the disappearance of information would be a violation of conservation laws of energy, which states that no energy or information can be destroyed.

Planck stars

To resolve the black hole singularities and the information paradox. We consider the possibility that the energy of a collapsing star and any additional energy falling into the hole could condense into a highly compressed core with density of the order of the Planck density. If this is the case, the gravitational collapse of a star does not lead to a singularity but to one additional phase in the life of a star: a quantum gravitational phase where the  gravitational attraction is balanced by a quantum pressure.

Since the energy density or pressure is expressed as force per unit surface area of a star we have,

Therefore nature appears to enter the quantum gravity regime when the energy density of matter reaches the Planck scale. The point is that this may happen well before relevant lengths become planckian. For instance, a collapsing spatially compact universe bounces back into an expanding one. The bounce is due to a quantum-gravitational repulsion which originates from the modified Heisenberg uncertainty, and is akin to the force that keeps an electron from falling into the nucleus. The above given statement is based on the following facts:

The resolution of classical singularities under the assumption of a maximal acceleration has been studied using canonical methods for Rindler, Schwarzschild, Reissner-Nordstrom, Kerr-Newman and Friedman-Lemaitre metrics.

To reconcile quanum mechanics with general relativity, we develop a quantum geometry in relativistic phase space (Rindler space) in which the maximal (proper) acceleration of a particle is modified to read,

Where, c is the constant speed of light, r is the linear dimension of a particle , α is the coupling constant (or size of the extra dimensions), n is a positive number (or the extra dimension number and is the flux in the extra dimension

This acceleration is based on an assumption, that particles are extended objects, never to be identified with mathematical points in ordinary space. This acceleration is important because it cures strong singularities that plague general relativity. This acceleration is also a straight forward consequence of our modified uncertainty relation given as,

,

Where r represents the size of a star, in this case-horizon radius, p is the momentum of a particle approaching or falling into the hole of a star, α is the coupling constant and n is positive. From the above given uncertainty principle, we derive the planck length. such that when the momentum , the gravitational coupling constant for gravitational interactions is and finally n=1/2. We get the planck length as the minimum length of space-time as

––––––––

Therefore from the uncertainity principle, the repulsion force is given by,

Therefore bounce does not happen when the universe is of planckian size, as was previously expected; it happens when the matter energy density reaches the Planck density in this way,

Let the surface area of a star be,  then the matter energy density will be given as,

For a Schwarzschild black hole with radius  and . We have a maximum energy density value wnen n=1 given as,

At this energy density, a Planck star is formed. The key feature of this theoretical object is that this repulsion arises from the energy density, not the Planck length, and starts taking effect far earlier than might be expected. This repulsive 'force' is strong enough to stop the collapse of the star well before a singularity is formed, and indeed, well before the Planck scale for distance. Since a Planck star is calculated to be considerably larger than the Planck scale for distance, this means there is adequate room for all the information captured inside of a black hole to be encoded in the star, thus avoiding information loss.

The analogy between quantum gravitational effects on

Cosmological and black-hole singularities has been exploited to study if and how quantum gravity could also resolve the r = 0 singularity at the center of a collapsed star, and there are good indications that it does. For example, from the modified uncertainty principle, when the momentum of a particle or matter falling into a black hole is Planckian  where is the Planck mass, we have,

Where is the Planck length. Taking  we have the size of a star as,

Where m is the mass of the star and n is positive. For instance, if n = 1/6, a stellar-mass black hole would collapse to a Planck star with a size of the order of centimeters. This is very small compared to the original star in fact, smaller than the atomic scale  but it is still more than 30 orders of magnitude larger than the Planck length. This is the scale on which we are focusing here. The main hypothesis here is that a star so compressed would not satisfy the classical Einstein equations anymore, even if huge compared to the Planck scale. Because its energy density is already planckian.

Singularity Resolution under the Assumption of Maximal Acceleration and Minimal length for both the Schwarzschild and Reissner- Nordstrom Black Hole

Under the assumption of  ( where is the coupling constant), in the Caianeillo maximum acceleration model ( ) , we derive the minimum radius to which a gravitating body can collapse in the commoving frame for both the Schwarzschild and Reissner-Nordstrom Black hole.

In the context of a geometrical unification of quantum mechanics and general relativity in phase space, Caianiello was the first person to propose the existence of a maximal proper acceleration for massive particles. Caianiello was able to derive the value for the maximum acceleration of a particle of rest mass m from the time-energy uncertainty relation. Caianiello model was based on two assumptions; and  for (3).

Applications of Caianiello’s model include cosmology, the dynamics of accelerated strings, neutrino oscillations and the determination of a lower neutrino mass bound. There is also evidence for maximal acceleration and singularity resolution in covariant loop quantum gravity found by Rovelli and Vidotto.

In this book we propose an adhoc assumption of  where is the coupling constant. This differs from Caianiello's model assumption of . Therefore the maximum acceleration(3) will be given by,

(4)

Where, r is the smallest possible distance between any two masses. In this book r takes values for the Schwarzschild and Reissner-Nordstrom radius.

Equation (4) given above reduces to the value that was earlier derived by Caianiello under two conditions;

(i) When   and for a Schwarzschild Black hole of mass M. Where is the gravitational coupling constant .

(ii) When and for a Reissner-Nordstrom Black hole. Where   is the electromagnetic coupling constant .

Maximal Acceleration in Quantum Gravity

Considering the event horizon of a Reissner-Nordstrom black hole of radius and gravitational coupling . Then substituting in (4), the growing acceleration approaching a classical singularity in the Reissner-Nordstrom metric is bounded by the existence of a maximal acceleration of;

(5)

Where e is charge on an electron, is the permittivity of free space and ћ is the reduced Planck constant.

Considering the event horizon of a Schwarzschild black hole of radius and gravitational coupling . Then substituting in (4), the growing acceleration approaching a classical singularity in the Schwarzschild metric is bounded by the existence of a maximal acceleration of;

(6)

Minimal Radius in Quantum Gravity

Because of the equivalence principle in the case of gravitational interaction, we propose to show here that the existence of a minimal length for both a Reissner and Schwarzschild Black hole is a straight forward consequence of our maximal acceleration value (4). In Newtonian law (center of mass system)

Where, R is the radius of a Black hole ( In this case the minimum radius