The time separation function (or Lorentzian distance function) is a fundamental tool used in Lore... more The time separation function (or Lorentzian distance function) is a fundamental tool used in Lorentzian geometry. For smooth spacetimes it is known to be lower semicontinuous, and in fact, continuous for globally hyperbolic spacetimes. Moreover, an axiom for Lorentzian length spaces -a synthetic approach to Lorentzian geometryis the existence of a lower semicontinuous time separation function. Nevertheless, the usual time separation function is not necessarily lower semicontinuous for C 0 spacetimes due to bubbling phenomena. In this paper, we introduce a class of curves called "nearly timelike" and show that the time separation function for C 0 spacetimes is lower semicontinuous when defined with respect to nearly timelike curves. Moreover, this time separation function agrees with the usual one when the metric is smooth. Lastly, sufficient conditions are found guaranteeing the existence of nearly timelike maximizers between two points in a C 0 spacetime.
The discrete spectrum of the Dirac operator for a point electron in the maximal analytically exte... more The discrete spectrum of the Dirac operator for a point electron in the maximal analytically extended Kerr-Newman spacetime is determined in the zero-G limit (zGKN), under some restrictions on the electrical coupling constant and on the radius of the ring-singularity of the zGKN spacetime. The spectrum is characterized by a triplet of integers, associated with winding numbers of orbits of dynamical systems on cylinders. A dictionary is established that relates the spectrum with the known hydrogenic Dirac spectrum. Numerical illustrations are presented. Open problems are listed.
In this paper we review and extend some results in the literature pertaining to spacetime topolog... more In this paper we review and extend some results in the literature pertaining to spacetime topology while naturally utilizing properties of the codimension 2 null cut locus. Our results fall into two classes, depending on whether or not one assumes the presence of horizons. Included among the spacetimes we consider are those that apply to the asymptotically (locally) AdS setting.
Inflationary spacetimes have been argued to be past geodesically incomplete in many situations. H... more Inflationary spacetimes have been argued to be past geodesically incomplete in many situations. However, whether the geodesic incompleteness implies the existence of an initial spacetime curvature singularity or whether the spacetime may be extended (potentially into another phase of the universe) is generally unknown. Both questions have important physical implications. In this paper, we take a closer look at the geometrical structure of inflationary spacetimes and investigate these very questions. We first classify which past inflationary histories have a scalar curvature singularity and which might be extendible and/or non-singular in homogeneous and isotropic cosmology with flat spatial sections. Then, we derive rigorous extendibility criteria of various regularity classes for quasi-de Sitter spacetimes that evolve from infinite proper time in the past. Finally, we show that beyond homogeneity and isotropy, special continuous extensions respecting the Einstein field equations with a perfect fluid must have the equation of state of a de Sitter universe asymptotically. An interpretation of our results is that past-eternal inflationary scenarios are most likely physically singular, except in situations with very special initial conditions.
When developing a quantum theory for a physical system, one determines the system’s symmetry grou... more When developing a quantum theory for a physical system, one determines the system’s symmetry group and its irreducible unitary representations. For Minkowski space, the symmetry group is the Poincaré group, R ⋊ O(1, 3), and the irreducible unitary representations are interpreted as elementary particles which determine the particle’s mass and spin. We determine the symmetry group for Milne-like spacetimes, a class of cosmological spacetimes, to be R×O(1, 3) and classify their irreducible unitary representations. Again they represent particles with mass and spin. Unlike the classification for the Poincaré group, we do not obtain any faster-than-light particles. The factor R corresponds to cosmic time translations. These generate a mass Casimir operator which yields a Lorentz invariant Dirac equation on Milne-like spacetimes. In fact it’s just the original Dirac equation multiplied by a conformal factor Ω. Therefore many of the invariants and symmetries still hold. We offer a new inter...
In this paper, we review and extend some results in the literature pertaining to spacetime topolo... more In this paper, we review and extend some results in the literature pertaining to spacetime topology while naturally utilizing properties of the codimension 2 null cut locus. Our results fall into two classes, depending on whether or not one assumes the presence of horizons. Included among the spacetimes we consider are those that apply to the asymptotically (locally) anti-de sitter (AdS) setting.
Milne-like spacetimes are a class of hyperbolic FLRW spacetimes which admit continuous spacetime ... more Milne-like spacetimes are a class of hyperbolic FLRW spacetimes which admit continuous spacetime extensions through the big bang, τ = 0. The existence of the extension follows from writing the metric in conformal Minkowskian coordinates and assuming that the scale factor satisfies a(τ) = τ + o(τ 1+ε) as τ → 0 for some ε > 0. This asymptotic assumption implies a(τ) = τ + o(τ). In this paper, we show that a(τ) = τ + o(τ) is not sufficient to achieve an extension through τ = 0, but it is necessary provided its derivative converges as τ → 0. We also show that the ε in a(τ) = τ + o(τ 1+ε) is not necessary to achieve an extension through τ = 0.
In this short paper, we review the Dirac equation on the zero-gravity Kerr-Newman spacetime. Our ... more In this short paper, we review the Dirac equation on the zero-gravity Kerr-Newman spacetime. Our main objective is to provide a correspondence between the classification of the bound states for the zGKN spectrum and the usual hydrogenic states 1s 1/2 , 2s 1/2 , etc. of the Hydrogen atom.
with an error on p. 2 in the Introduction. The sentence "We will refer to this double-sheeted ele... more with an error on p. 2 in the Introduction. The sentence "We will refer to this double-sheeted electromagnetic spacetime as the Kerr-Newman spacetime is determined in the zero-G limit (zGKN) spacetime." has been corrected to "We will refer to this double-sheeted electromagnetic spacetime as the zGKN spacetime." In addition, in the title of Sec. V D, the mathematical parameter a appeared in the wrong font. This has been corrected. All online versions of this article were corrected on
The article Remarks on the cosmological constant appearing as an initial condition for Milne-like... more The article Remarks on the cosmological constant appearing as an initial condition for Milne-like spacetimes written by Eric Ling, was originally published electronically on the publisher's internet portal (currently SpringerLink) on 18
The discrete spectrum of the Dirac operator for a point electron in the maximal analytically exte... more The discrete spectrum of the Dirac operator for a point electron in the maximal analytically extended Kerr–Newman spacetime is determined in the zero-G limit (zGKN), under some restrictions on the electrical coupling constant and on the radius of the ring-singularity of the zGKN spacetime. The spectrum is characterized by a triplet of integers, associated with winding numbers of orbits of dynamical systems on cylinders. A dictionary is established that relates the spectrum with the known hydrogenic Dirac spectrum. Numerical illustrations are presented. Open problems are listed.
Milne-like spacetimes are a class of hyperbolic FLRW spacetimes which admit continuous spacetime ... more Milne-like spacetimes are a class of hyperbolic FLRW spacetimes which admit continuous spacetime extensions through the big bang, τ = 0. The existence of the extension follows from writing the metric in conformal Minkowskian coordinates and assuming that the scale factor satisfies a(τ) = τ + o(τ 1+ε) as τ → 0 for some ε > 0. This asymptotic assumption implies a(τ) = τ + o(τ). In this paper, we show that a(τ) = τ + o(τ) is not sufficient to achieve an extension through τ = 0, but it is necessary provided its derivative converges as τ → 0. We also show that the ε in a(τ) = τ + o(τ 1+ε) is not necessary to achieve an extension through τ = 0.
Milne-like spacetimes are a class of $$k = -1$$ k = - 1 FLRW spacetimes which admit continuous sp... more Milne-like spacetimes are a class of $$k = -1$$ k = - 1 FLRW spacetimes which admit continuous spacetime extensions through the big bang. In a previous paper Ling (Found. of Phys. 50:385–428, 2020), it was shown that the cosmological constant appears as an initial condition for Milne-like spacetimes. In this paper, we generalize this statement to spacetimes which share similar geometrical properties with Milne-like spacetimes but without the strong spatially isotropic assumption associated with them. We show how our results yield a “quasi de Sitter” expansion for the early universe which could have applications to inflationary scenarios.
Currently available topological censorship theorems are meant for gravitationally isolated black ... more Currently available topological censorship theorems are meant for gravitationally isolated black hole spacetimes with cosmological constant Λ = 0 or Λ < 0. Here, we prove a topological censorship theorem that is compatible with Λ > 0 and which can be applied to whole universes containing possibly multiple collections of black holes. The main assumption in the theorem is that distinct black hole collections eventually become isolated from one another at late times, and the conclusion is that the regions near the various black hole collections have trivial fundamental group, in spite of there possibly being nontrivial topology in the universe.
We present a physics model for a time-symmetric Milne-like universe. The model is based on the q-... more We present a physics model for a time-symmetric Milne-like universe. The model is based on the q-theory approach to the cosmological constant problem, supplemented by an assumed vacuummatter energy exchange possibly due to quantum-dissipative effects. Without fine-tuning of the initial vacuum energy density, we obtain a harmless big bang singularity (with finite values of the Ricci and Kretschmann curvature scalars) and attractor behavior towards Minkowski spacetime. The time-symmetric spacetime manifold of our model may provide the proper setting for a possible CPT-symmetric universe.
As is well known, constant mean curvature (CMC) spacelike hypersurfaces play an important role in... more As is well known, constant mean curvature (CMC) spacelike hypersurfaces play an important role in solving the Einstein equations, both in solving the contraints and the evolution equations. In this paper we review the CMC existence result obtained by the authors in [10] and consider some new existence results motivated by a conjecture of Dilts and Holst [8]. We also address some issues concerning the conformal structure of cosmological spacetimes.
Milne-like spacetimes are a class of FLRW models which admit C 0 spacetime extensions through the... more Milne-like spacetimes are a class of FLRW models which admit C 0 spacetime extensions through the big bang. The boundary of a Milne-like spacetime can be identified with a null cone in the extension. We find that the comoving observers all emanate from a single point in the extension. This suggests that something physical may have happened there. Next we ask what role these spacetimes have in modern cosmology. We find that Milne-like spacetimes are inconsistent with classical cosmology theory in which a radiation-dominated era follows immediately after the big bang. However, Milne-like spacetimes are consistent with inflationary theory. In fact the very nature of Milne-like spacetimes solves the horizon problem. We note that this has already been observed for the Milne universe in [2] and [25]. Next we show, assuming a finite initial condition on the energy density, that Milne-like spacetimes naturally exhibit a slowroll inflationary era. This makes Milne-like spacetimes good models for inflationary theory. We also show that Milne-like spacetimes still admit C 0 extensions under well-behaved perturbations.
When developing a quantum theory for a physical system, one determines the system's symmetry grou... more When developing a quantum theory for a physical system, one determines the system's symmetry group and its irreducible unitary representations. For Minkowski space, the symmetry group is the Poincaré group, R 4 ⋊ O(1, 3), and the irreducible unitary representations are interpreted as elementary particles which determine the particle's mass and spin. We determine the symmetry group for Milne-like spacetimes, a class of cosmological spacetimes, to be R × O(1, 3) and classify their irreducible unitary representations. Again they represent particles with mass and spin. Unlike the classification for the Poincaré group, we do not obtain any faster-than-light particles. The factor R corresponds to cosmic time translations. These generate a mass Casimir operator which yields a Lorentz invariant Dirac equation on Milne-like spacetimes. In fact it's just the original Dirac equation multiplied by a conformal factor Ω. Therefore many of the invariants and symmetries still hold. We offer a new interpretation of the negative energy states and propose a possible solution to the matter-antimatter asymmetry problem in our universe.
Author(s): Ling, Eric | Advisor(s): Dai, Xianzhe | Abstract: The big bang theory is a model of th... more Author(s): Ling, Eric | Advisor(s): Dai, Xianzhe | Abstract: The big bang theory is a model of the universe which makes the striking prediction that the universe began a finite amount of time in the past at the so called "Big Bang singularity." We explore the physical and mathematical justification of this surprising result. After laying down the framework of the universe as a spacetime manifold, we combine physical observations with global symmetrical assumptions to deduce the FRW cosmological models which predict a big bang singularity. Next we prove a couple theorems due to Stephen Hawking which show that the big bang singularity exists even if one removes the global symmetrical assumptions. Lastly, we investigate the conditions one needs to impose on a spacetime if one wishes to avoid a singularity. The ideas and concepts used here to study spacetimes are similar to those used to study Riemannian manifolds, therefore we compare and contrast the two geometries throughout.
arXiv: General Relativity and Quantum Cosmology, 2018
When developing a quantum theory for a physical system, one determines the system's symmetry ... more When developing a quantum theory for a physical system, one determines the system's symmetry group and its irreducible unitary representations. For Minkowski space, the symmetry group is the Poincar\'e group, $\mathbb{R}^4 \rtimes \text{O}(1,3)$, and the irreducible unitary representations are elementary particles which determine the particle's mass and spin. We determine the symmetry group for Milne-like spacetimes, a class of cosmological spacetimes, to be $\mathbb{R} \times \text{O}(1,3)$, and classify their irreducible unitary representations. Again they represent particles with mass and spin. This yields a Lorentz invariant Dirac equation on Milne-like spacetimes which generalizes the original Dirac equation. In fact it's just the original Dirac equation multiplied by a conformal factor $\Omega$. Many of the same invariants and symmetries still hold for the generalized Dirac equation. We offer a new interpretation of the negative energy states which may have app...
The time separation function (or Lorentzian distance function) is a fundamental tool used in Lore... more The time separation function (or Lorentzian distance function) is a fundamental tool used in Lorentzian geometry. For smooth spacetimes it is known to be lower semicontinuous, and in fact, continuous for globally hyperbolic spacetimes. Moreover, an axiom for Lorentzian length spaces -a synthetic approach to Lorentzian geometryis the existence of a lower semicontinuous time separation function. Nevertheless, the usual time separation function is not necessarily lower semicontinuous for C 0 spacetimes due to bubbling phenomena. In this paper, we introduce a class of curves called "nearly timelike" and show that the time separation function for C 0 spacetimes is lower semicontinuous when defined with respect to nearly timelike curves. Moreover, this time separation function agrees with the usual one when the metric is smooth. Lastly, sufficient conditions are found guaranteeing the existence of nearly timelike maximizers between two points in a C 0 spacetime.
The discrete spectrum of the Dirac operator for a point electron in the maximal analytically exte... more The discrete spectrum of the Dirac operator for a point electron in the maximal analytically extended Kerr-Newman spacetime is determined in the zero-G limit (zGKN), under some restrictions on the electrical coupling constant and on the radius of the ring-singularity of the zGKN spacetime. The spectrum is characterized by a triplet of integers, associated with winding numbers of orbits of dynamical systems on cylinders. A dictionary is established that relates the spectrum with the known hydrogenic Dirac spectrum. Numerical illustrations are presented. Open problems are listed.
In this paper we review and extend some results in the literature pertaining to spacetime topolog... more In this paper we review and extend some results in the literature pertaining to spacetime topology while naturally utilizing properties of the codimension 2 null cut locus. Our results fall into two classes, depending on whether or not one assumes the presence of horizons. Included among the spacetimes we consider are those that apply to the asymptotically (locally) AdS setting.
Inflationary spacetimes have been argued to be past geodesically incomplete in many situations. H... more Inflationary spacetimes have been argued to be past geodesically incomplete in many situations. However, whether the geodesic incompleteness implies the existence of an initial spacetime curvature singularity or whether the spacetime may be extended (potentially into another phase of the universe) is generally unknown. Both questions have important physical implications. In this paper, we take a closer look at the geometrical structure of inflationary spacetimes and investigate these very questions. We first classify which past inflationary histories have a scalar curvature singularity and which might be extendible and/or non-singular in homogeneous and isotropic cosmology with flat spatial sections. Then, we derive rigorous extendibility criteria of various regularity classes for quasi-de Sitter spacetimes that evolve from infinite proper time in the past. Finally, we show that beyond homogeneity and isotropy, special continuous extensions respecting the Einstein field equations with a perfect fluid must have the equation of state of a de Sitter universe asymptotically. An interpretation of our results is that past-eternal inflationary scenarios are most likely physically singular, except in situations with very special initial conditions.
When developing a quantum theory for a physical system, one determines the system’s symmetry grou... more When developing a quantum theory for a physical system, one determines the system’s symmetry group and its irreducible unitary representations. For Minkowski space, the symmetry group is the Poincaré group, R ⋊ O(1, 3), and the irreducible unitary representations are interpreted as elementary particles which determine the particle’s mass and spin. We determine the symmetry group for Milne-like spacetimes, a class of cosmological spacetimes, to be R×O(1, 3) and classify their irreducible unitary representations. Again they represent particles with mass and spin. Unlike the classification for the Poincaré group, we do not obtain any faster-than-light particles. The factor R corresponds to cosmic time translations. These generate a mass Casimir operator which yields a Lorentz invariant Dirac equation on Milne-like spacetimes. In fact it’s just the original Dirac equation multiplied by a conformal factor Ω. Therefore many of the invariants and symmetries still hold. We offer a new inter...
In this paper, we review and extend some results in the literature pertaining to spacetime topolo... more In this paper, we review and extend some results in the literature pertaining to spacetime topology while naturally utilizing properties of the codimension 2 null cut locus. Our results fall into two classes, depending on whether or not one assumes the presence of horizons. Included among the spacetimes we consider are those that apply to the asymptotically (locally) anti-de sitter (AdS) setting.
Milne-like spacetimes are a class of hyperbolic FLRW spacetimes which admit continuous spacetime ... more Milne-like spacetimes are a class of hyperbolic FLRW spacetimes which admit continuous spacetime extensions through the big bang, τ = 0. The existence of the extension follows from writing the metric in conformal Minkowskian coordinates and assuming that the scale factor satisfies a(τ) = τ + o(τ 1+ε) as τ → 0 for some ε > 0. This asymptotic assumption implies a(τ) = τ + o(τ). In this paper, we show that a(τ) = τ + o(τ) is not sufficient to achieve an extension through τ = 0, but it is necessary provided its derivative converges as τ → 0. We also show that the ε in a(τ) = τ + o(τ 1+ε) is not necessary to achieve an extension through τ = 0.
In this short paper, we review the Dirac equation on the zero-gravity Kerr-Newman spacetime. Our ... more In this short paper, we review the Dirac equation on the zero-gravity Kerr-Newman spacetime. Our main objective is to provide a correspondence between the classification of the bound states for the zGKN spectrum and the usual hydrogenic states 1s 1/2 , 2s 1/2 , etc. of the Hydrogen atom.
with an error on p. 2 in the Introduction. The sentence "We will refer to this double-sheeted ele... more with an error on p. 2 in the Introduction. The sentence "We will refer to this double-sheeted electromagnetic spacetime as the Kerr-Newman spacetime is determined in the zero-G limit (zGKN) spacetime." has been corrected to "We will refer to this double-sheeted electromagnetic spacetime as the zGKN spacetime." In addition, in the title of Sec. V D, the mathematical parameter a appeared in the wrong font. This has been corrected. All online versions of this article were corrected on
The article Remarks on the cosmological constant appearing as an initial condition for Milne-like... more The article Remarks on the cosmological constant appearing as an initial condition for Milne-like spacetimes written by Eric Ling, was originally published electronically on the publisher's internet portal (currently SpringerLink) on 18
The discrete spectrum of the Dirac operator for a point electron in the maximal analytically exte... more The discrete spectrum of the Dirac operator for a point electron in the maximal analytically extended Kerr–Newman spacetime is determined in the zero-G limit (zGKN), under some restrictions on the electrical coupling constant and on the radius of the ring-singularity of the zGKN spacetime. The spectrum is characterized by a triplet of integers, associated with winding numbers of orbits of dynamical systems on cylinders. A dictionary is established that relates the spectrum with the known hydrogenic Dirac spectrum. Numerical illustrations are presented. Open problems are listed.
Milne-like spacetimes are a class of hyperbolic FLRW spacetimes which admit continuous spacetime ... more Milne-like spacetimes are a class of hyperbolic FLRW spacetimes which admit continuous spacetime extensions through the big bang, τ = 0. The existence of the extension follows from writing the metric in conformal Minkowskian coordinates and assuming that the scale factor satisfies a(τ) = τ + o(τ 1+ε) as τ → 0 for some ε > 0. This asymptotic assumption implies a(τ) = τ + o(τ). In this paper, we show that a(τ) = τ + o(τ) is not sufficient to achieve an extension through τ = 0, but it is necessary provided its derivative converges as τ → 0. We also show that the ε in a(τ) = τ + o(τ 1+ε) is not necessary to achieve an extension through τ = 0.
Milne-like spacetimes are a class of $$k = -1$$ k = - 1 FLRW spacetimes which admit continuous sp... more Milne-like spacetimes are a class of $$k = -1$$ k = - 1 FLRW spacetimes which admit continuous spacetime extensions through the big bang. In a previous paper Ling (Found. of Phys. 50:385–428, 2020), it was shown that the cosmological constant appears as an initial condition for Milne-like spacetimes. In this paper, we generalize this statement to spacetimes which share similar geometrical properties with Milne-like spacetimes but without the strong spatially isotropic assumption associated with them. We show how our results yield a “quasi de Sitter” expansion for the early universe which could have applications to inflationary scenarios.
Currently available topological censorship theorems are meant for gravitationally isolated black ... more Currently available topological censorship theorems are meant for gravitationally isolated black hole spacetimes with cosmological constant Λ = 0 or Λ < 0. Here, we prove a topological censorship theorem that is compatible with Λ > 0 and which can be applied to whole universes containing possibly multiple collections of black holes. The main assumption in the theorem is that distinct black hole collections eventually become isolated from one another at late times, and the conclusion is that the regions near the various black hole collections have trivial fundamental group, in spite of there possibly being nontrivial topology in the universe.
We present a physics model for a time-symmetric Milne-like universe. The model is based on the q-... more We present a physics model for a time-symmetric Milne-like universe. The model is based on the q-theory approach to the cosmological constant problem, supplemented by an assumed vacuummatter energy exchange possibly due to quantum-dissipative effects. Without fine-tuning of the initial vacuum energy density, we obtain a harmless big bang singularity (with finite values of the Ricci and Kretschmann curvature scalars) and attractor behavior towards Minkowski spacetime. The time-symmetric spacetime manifold of our model may provide the proper setting for a possible CPT-symmetric universe.
As is well known, constant mean curvature (CMC) spacelike hypersurfaces play an important role in... more As is well known, constant mean curvature (CMC) spacelike hypersurfaces play an important role in solving the Einstein equations, both in solving the contraints and the evolution equations. In this paper we review the CMC existence result obtained by the authors in [10] and consider some new existence results motivated by a conjecture of Dilts and Holst [8]. We also address some issues concerning the conformal structure of cosmological spacetimes.
Milne-like spacetimes are a class of FLRW models which admit C 0 spacetime extensions through the... more Milne-like spacetimes are a class of FLRW models which admit C 0 spacetime extensions through the big bang. The boundary of a Milne-like spacetime can be identified with a null cone in the extension. We find that the comoving observers all emanate from a single point in the extension. This suggests that something physical may have happened there. Next we ask what role these spacetimes have in modern cosmology. We find that Milne-like spacetimes are inconsistent with classical cosmology theory in which a radiation-dominated era follows immediately after the big bang. However, Milne-like spacetimes are consistent with inflationary theory. In fact the very nature of Milne-like spacetimes solves the horizon problem. We note that this has already been observed for the Milne universe in [2] and [25]. Next we show, assuming a finite initial condition on the energy density, that Milne-like spacetimes naturally exhibit a slowroll inflationary era. This makes Milne-like spacetimes good models for inflationary theory. We also show that Milne-like spacetimes still admit C 0 extensions under well-behaved perturbations.
When developing a quantum theory for a physical system, one determines the system's symmetry grou... more When developing a quantum theory for a physical system, one determines the system's symmetry group and its irreducible unitary representations. For Minkowski space, the symmetry group is the Poincaré group, R 4 ⋊ O(1, 3), and the irreducible unitary representations are interpreted as elementary particles which determine the particle's mass and spin. We determine the symmetry group for Milne-like spacetimes, a class of cosmological spacetimes, to be R × O(1, 3) and classify their irreducible unitary representations. Again they represent particles with mass and spin. Unlike the classification for the Poincaré group, we do not obtain any faster-than-light particles. The factor R corresponds to cosmic time translations. These generate a mass Casimir operator which yields a Lorentz invariant Dirac equation on Milne-like spacetimes. In fact it's just the original Dirac equation multiplied by a conformal factor Ω. Therefore many of the invariants and symmetries still hold. We offer a new interpretation of the negative energy states and propose a possible solution to the matter-antimatter asymmetry problem in our universe.
Author(s): Ling, Eric | Advisor(s): Dai, Xianzhe | Abstract: The big bang theory is a model of th... more Author(s): Ling, Eric | Advisor(s): Dai, Xianzhe | Abstract: The big bang theory is a model of the universe which makes the striking prediction that the universe began a finite amount of time in the past at the so called "Big Bang singularity." We explore the physical and mathematical justification of this surprising result. After laying down the framework of the universe as a spacetime manifold, we combine physical observations with global symmetrical assumptions to deduce the FRW cosmological models which predict a big bang singularity. Next we prove a couple theorems due to Stephen Hawking which show that the big bang singularity exists even if one removes the global symmetrical assumptions. Lastly, we investigate the conditions one needs to impose on a spacetime if one wishes to avoid a singularity. The ideas and concepts used here to study spacetimes are similar to those used to study Riemannian manifolds, therefore we compare and contrast the two geometries throughout.
arXiv: General Relativity and Quantum Cosmology, 2018
When developing a quantum theory for a physical system, one determines the system's symmetry ... more When developing a quantum theory for a physical system, one determines the system's symmetry group and its irreducible unitary representations. For Minkowski space, the symmetry group is the Poincar\'e group, $\mathbb{R}^4 \rtimes \text{O}(1,3)$, and the irreducible unitary representations are elementary particles which determine the particle's mass and spin. We determine the symmetry group for Milne-like spacetimes, a class of cosmological spacetimes, to be $\mathbb{R} \times \text{O}(1,3)$, and classify their irreducible unitary representations. Again they represent particles with mass and spin. This yields a Lorentz invariant Dirac equation on Milne-like spacetimes which generalizes the original Dirac equation. In fact it's just the original Dirac equation multiplied by a conformal factor $\Omega$. Many of the same invariants and symmetries still hold for the generalized Dirac equation. We offer a new interpretation of the negative energy states which may have app...
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