The fundamental “players” in the cosmological arena are microscopic particles and the interactions by which they exchange well-defined quantum numbers. Many of the critical properties of micro-physics can be determined by their behaviors in nearly flat space-time, as described by Minkowski. There are several requirements that a successful model of fundamental processes should fulfill. Among these characteristics are:

1. • Describes quantum phenomenology: Quantum mechanics successfully describes the subtle behaviors of matter and energies undergoing microscopic exchanges. Quantum behaviors inherently have aspects beyond measurement.

2. • Conservation properties: Most particles have internal quantum numbers, like charge, lepton number, baryon number, etc., that are carried undiminished throughout complicated interactions with other particles. In addition, they carry properties like mass and spin, whose kinematic transformations under space-time transformations are well-defined, and which satisfy composite conservation laws (energy, momentum, angular momentum) for sufficiently isolated homogeneous and isotropic systems.

3. • Unitarity: Despite being unable to follow quantum coherent particle properties while that coherence is maintained, the evolution of those properties is described in a manner that ensures conservation of probability. This means that these properties do not just “pop” into or out of existence between the detections and interrelations of the particles.

4. • Cluster decomposability and classical correspondence: Classical physics is quite successful in describing much of common phenomenology. Classical models exploit those characteristics of a system that can be isolated, studied, and parameterized independent of observation.

5. […]

The incorporation of quantum mechanics into gravitational dynamics introduces perplexing issues into modern physics. In contrast to other interactions like electromagnetism, the classical trajectory of a gravitating system is independent of the mass coupling to the gravitational field. As previously discussed, this allows the gravitation of arbitrary test particles to be described in terms of local geometry only, the basis of general relativity. Thus, the geometrodynamics of classical general relativity are most directly expressed using localized geodesics. However, quantum dynamics incorporate measurement constraints that disallow complete localization of physical systems. A coherent quantum system is not represented by a path or a classical trajectory; rather, it self-interferes throughout regions. This complicates the use of classical formulations in describing inherently quantum processes.

In addition, the equations of general relativity are complex and non-linear in the interrelations between sources and geometry, which makes solutions of even classical systems complicated. The key to describing complex systems is to determine the most useful set of parameters and coordinates that give concise predictive explanations of those systems. This chapter will develop tools for examining quantum behaviors in gravitating systems.

Quantum coherence and gravity

The behaviors of quantum objects in Minkowski space-time are well understood, despite a lack of consensus on the various interpretations (Copenhagen, many worlds, etc.) of the underlying fundamentals of the quantum world, or concerns of the completeness of quantum theory. There have also been tests of systems modeled by equations that involve both Newton's gravitational constant GN and Planck's constant ħ, as described in Section 2.3.1.

One of the principles of modern physics that is most adhered to is the expectation that models constructed to describe the phenomena of the physical universe should not depend upon any absolute frame of reference. The discovery of the CMB radiation perhaps demonstrates a counter example to this supposition, due to the preferred frame at rest relative to the energy content of the universe during its initial phase of expansion, as will be discussed in the next chapter. However, for most phenomena, the co-variance of the laws modeling those phenomena is consistent with the expectation of independence of the fundamental physics from the particular frame of reference utilized by the observer. This principle is embodied in the concept of complementarity [23] in the description of black holes. In its most direct expression, complementarity simply states that no observer should ever witness a violation of a law of nature. In particular, one expects that for a freely falling observer, there should be no local effects of gravitation as espoused by the principles of equivalence and relativity.

In this treatment, a horizon will always be a light-like surface that globally separates causally disconnected regions of space-time. Since light itself is characterized by both classical and quantum properties, geometries with horizons offer insights into the subtle relationships between general relativity and quantum physics. Generally, horizons can be only globally (not locally) defined, which means that local experiments performed by freely falling, inertial observers cannot detect the presence of such horizons.

This book is the result of decades of efforts by the author to understand physics at its most basic foundations, and to construct models that can address some of the unanswered questions about gravity, quantum mechanics, and the spectrum of fundamental particles. It is felt that foundational explorations should examine the conceptual and philosophical basis of a discipline. As such, less emphasis was placed upon the mathematics of complex calculations, while more emphasis was placed upon the consistencies and critiques of basic premises, in the preparation of this book. As a scientist, there is often a tendency to be drawn towards mathematical and formal pursuits, simply because of the beauty and elegance of mathematics. Such approaches sometimes bypass difficult conceptual approaches and thought experiments, or develop speculative formulations just because of their elegance. This book attempts to establish a balance towards the exploration of basic concepts and puzzles.

The prior book on black holes by Lenny Susskind and myself serves as an introduction to quantum physics and relativity for static geometries, as well as information in horizon physics. However, some of my more recent explorations indicate that qualitative modifications in descriptions of the physics occur once dynamics has been incorporated. This then calls into question any intuitions from static geometries that have been used to imply that those very geometries cannot be static. This book incorporates dynamic, spatially coherent geometries, as well as expanding upon the well-established foundations of the prior work.

Groups and special relativity

Fundamentals of group theory

Properties of groups

A group is a set of elements that have the following properties:

1. • Contains the identity transformation 1;

2. • If E and E′ are elements in the group, then there exists a group operation (generically called “multiplication”) that always produces an element of the group, E″ = E′ · E (closure);

3. • For every transformation element E, there exists in the group an inverse element E-1, where E-1 • E = 1;

4. • The group operation is associative, E′ · (E′ · E) = (E″ · E′) · E.

A particular type of group satisfies an additional property. For an abelian group, the group operation yields the same answer regardless of the order, E′ · E = E · E′. Often, a subset of the elements within a group satisfies all four group properties. This subset is referred to as a subgroup.

Generally, two different groups are isomorphic if there is a one-to-one relationships between the elements of the groups with regards to group operations. More generally, if a set of elements in one group are in direct relationship with one element in another, the groups are homomorphic.

A particular class of groups is quite useful for describing transformations in quantum physics. Invertible N × N matrices (i.e., matrices with non-vanishing determinants) form the set of linear groups.

Foundations of special relativity

The special theory of relativity has had a profound impact upon notions of time and space within the scientific and philosophic communities. This well-established model of local coordinate transformations in the universe is built upon two fundamental postulates:

1. • The principle of relativity: the laws of physics apply in all inertial reference systems;

2. • The universality of the speed of light: the speed of light in a vacuum is the same for all inertial observers, regardless of the motion of the source or observer.

The principle of relativity is not unique to the special theory of relativity; indeed it is assumed within Galilean relativity. However, if the equations of electrodynamics described by Maxwell's equations describe laws of nature, then the second postulate immediately follows from the first, since Maxwell's equations predict a universal speed of propagation of electromagnetic waves in a vacuum. The consequences of these postulates will be developed briefly.

Lorentz transformations

One of the most direct routes towards developing the transformations satisfying the postulates of special relativity involves examining the distance traveled by a propagating light pulse: (Δx)2 + (Δy)2 + (Δz)2 = (Δct)2.

From special to general relativity

In general relativity, the “force of gravity”, which directly couples to a gravitating body through its mass, is replaced by relationships between geometric coordinates. This can be done because the inertial mass that relates the acceleration of a body from rest (a purely geometrical aspect) to the force through Newton's second law F = ma, is the same as the gravitational mass that couples the gravitational acceleration g to that body Fgravitation = mg. Newton tested this equivalence using various pendulums, and Eotvos [75] in 1889 verified the equivalence of inertial and gravitational mass to better than one part in 109. Gravity attracts different masses in a way that results in the differing masses having the same accelerations. This tenet embodies the equivalence principle, which will be discussed next. Since bodies of vastly differing constitutions and masses gravitate equivalently, one can then construct the trajectories of general gravitating masses in terms of geometric geodesies (special curves in the space-time), independent of the mass, charge, or internal structure of the gravitating body.

The principle of equivalence

The principle of equivalence forms the conceptual foundation of early formulations of the theory of general relativity. For present purposes, the principle of equivalence will be stated as follows: At every space-time point in an arbitrary gravitational field, it is possible to choose a locally inertial coordinate system such that (within a sufficiently small region of that point) the laws of nature take the same form as in an unaccelerated Minkowski coordinate system in the absence of gravity. Such an assertion inherently relates the inertial mass to the gravitational mass.