Through the Wormhole on Spacetime Surface | Part 1: Geometry to Rule Them All

Geometry has played a significant role in our understanding of space and form throughout history. Euclid established the principles of geometry, while Descartes introduced Cartesian coordinates, merging geometry with algebra. Clifford algebra and Riemann's non-Euclidean geometry further expanded geometric language. The theory of relativity began with Galileo's observations on relative motion and was later developed by Lorentz, Poincaré, and Einstein. Einstein's theory of special relativity introduced the concept of spacetime and the mass-energy equivalence. General relativity explained gravity as the curvature of spacetime caused by mass and energy. Differential geometry and tensor calculus were essential mathematical frameworks in understanding general relativity. Einstein integrated Minkowski's work to describe a four-dimensional curved universe. Part 1. Geometry to rule them all. Geometry, commonly seen as the mathematical language of space and form, has deepened our understanding of immediate surroundings and the expansive cosmos. This mathematical domain employs tools like rulers, compasses, coordinates, dimensional objects, shapes, and transformations to elucidate spatial relationships. The roots of geometry extend to ancient civilizations where rudimentary land measurement techniques, astronomical calculations, and navigation were used. The systematic and organized principles of geometry, however, were established by the ancient Greek scholar Euclid in his Elements series around 300 BC. In this work, Euclid offered rigorous definitions, postulates, and axioms that laid the logical groundwork for geometry. He derived a set of theorems and corollaries from these foundations that covered plane geometry, solid geometry, number theory, and proportion. Euclid's work has mostly stood the test of time, influencing mathematical thought for over two millennia. Fast forwarding 2,000 years from Euclid's time, geometry underwent a transformation with the introduction of Cartesian coordinates, a system devised by Rene Descartes and later adopted by Sir Isaac Newton and Gottfried Wilhelm Leibniz in their theory of infinitesimal calculus in the 17th and 18th century. This innovation heralded a new era in natural philosophy by fusing geometry with algebra, enriching our understanding of the physical world. The Cartesian system allowed for translating geometric problems into mathematical equations, simplifying symbolic manipulation and solution discovery. In the mid-19th century, English mathematician and philosopher William Clifford took the symbiosis between geometry and algebra a step further. Clifford enhanced geometric language by expressing it in algebraic forms through what became known as Clifford algebra. Unlike Cartesian algebra, Clifford's framework unified various geometric entities within a single structure, including scalars, vectors and bivectors, the latter being a type of geometric object that captures area, much like vectors capture length. The geometric product, a key operation in Clifford algebra, merged these entities, thereby streamlining the mathematics behind rotations and reflections in three dimensions. Clifford's contributions found applications in higher dimensional spaces, relativity and quantum mechanics, notably in the study of spinors, mathematical entities that describe particle spins and the analysis of fermionic fields, which characterize a subset of subatomic particles in string models. In the mid-19th century, German mathematician Bernhard Riemann expanded the scope of geometry beyond the Euclidean plane, exploring curved surfaces and higher dimensions. His foundational work in non-Euclidean geometry, where the parallel line axiom and the rule of triangle angles summing to 180 degrees no longer held, became instrumental in developing relativity theories. Relativity. The discussion on relative motion began as early as the 17th century, most notably with the work of Galileo Galilei, an Italian astronomer, philosopher and physicist. Galileo's thought experiment laid out the basic principles that would later form the foundation for the theory of relativity. His Salviati thought experiment, presented in the book Dialogue Concerning the Two Chief World Systems in 1632, shows that the motion of various objects inside a stationary or uniformly moving ship appears the same to an external observer. For example, flying butterflies, jumping men, dripping water or swimming fish in a bowl experience no change due to the uniform motion of the ship, provided the observed phenomena are below deck and thus isolated from external forces like wind and air resistance. In simple terms, two systems moving relative to each other without acceleration are equivalent. This idea evolved into Galileo's relativity principle, focusing on relative motion characteristics. It postulates that the fundamental laws of physics are constant for observers in inertial frames, whether in motion or at rest. Two centuries later, in a broader sense, these principles found application in theories involving electromagnetic forces and the ether. The understanding of relativity advanced further with the work of Hendrik Lorentz, a Dutch physicist and mathematician whose contributions spanned the late 19th and early 20th century. Lorentz focused on transformation equations, now known as Lorentz transformations. These equations describe how two observers' measurements of time and space are related, assuming they are moving at a constant velocity relative to each other. Through these equations, Lorentz alluded to the interconnection between time, space and motion. Around the same period, French mathematician and theoretical physicist Henri Poincaré also explored relativity. In his 1905 paper The Principles of Mathematical Physics, Poincaré examined the principle of relativity, its interpretations and its challenges. Engaged in discussions on the fundamental principles of physics, especially in the context of electromagnetism and motion, Poincaré contributed to the conversation on the relativity of simultaneity and the use of Lorentz transformations. His work set the stage for further developments by Einstein. The theory of relative motion was already evolving in the scientific community when the German physicist Albert Einstein displayed a keen ability to identify and solve open problems in physics. In a productive culmination, he published four groundbreaking papers in 1905, one of which was the theory of special relativity, released on September 26. Einstein's theory of special relativity diverged from previous theories on relative motion by introducing two clear postulates. A. The laws of physics are invariant for all inertial observers and B. The speed of light in a vacuum is constant for all observers, regardless of the motion of the light source or observer. This challenged the prevailing ether hypothesis and the notion of absolute simultaneity. Einstein's special relativity fused space and time into spacetime, showing that measurements of time and space depend on the relative motion between the observer and the event. This led to the physical interpretation of phenomena like time dilation and length contraction, which Lorentz had already speculated upon in the context of ether theories. Additionally, special relativity introduced the mass-energy equivalence principle, encapsulated by the equation E equals mc squared, highlighting the convertibility of mass and energy. While other prominent scientists of the era could have formulated special relativity, the theory of general relativity presented a unique challenge that perhaps only Einstein could have met. This perspective often omits the contributions of German physicist Max Abraham, Finnish physicist Gunnar Nordstrom, German mathematician David Hilbert, and Swiss mathematician Marcel Grossmann, all of whom were also working on a new theory of gravitation and Einstein was in communication with them. The development of general relativity was not a solitary endeavor. It had its proponents. Even if many physicists were initially skeptical and did not see any future in Einstein's work with gravity. Nordstrom's work, perhaps the most notable among the independent theories, became an unreferenced footnote in history due to its unsuccessful predictions concerning the behavior of light rays under gravitation and its lack of covariance compared to Einstein's theory. Nordstrom's theory was based on Minkowski's formulation of spacetime and scalar fields, whereas Einstein already employed tensors which were more capable of describing gravity. Hilbert's work, in turn, was developed concurrently with Einstein's theory of gravity at its later formulation and was fundamentally inspired by Einstein's core ideas on gravity. In 1915, Einstein presented his most refined thoughts on the forces of gravity within the framework of general relativity. He suggested that gravity could be understood as a manifestation of spacetime curvature caused by mass and energy, which blurred the traditional concept of force. Einstein needed to venture into new mathematical territories to substantiate his physical insights regarding the equivalence principle. The earlier realization was that all objects fall at the same rate in a vacuum, irrespective of their mass. For example, in a vacuum, a feather and a bowling ball dropped from a tower would reach the ground simultaneously, absent any resistance. Einstein's new principle maintained that the laws of physics must be uniform across all coordinate systems, not just in a freely falling, non-rotating laboratory, but in a broader array of physical situations. Inertial mass is identical to gravitational mass. General relativity requires understanding two mathematical frameworks. A. differential geometry and B. tensor calculus. Differential geometry studies curves and surfaces in spaces of two or more dimensions, focusing on their properties under smooth deformations. Tensor calculus involves mathematical objects that generalize vectors and can describe physical quantities in a way that is independent of any particular coordinate system. These frameworks offer the tools needed to describe the interactions between mass energy and the fabric of space-time from any observer's perspective. Initially hesitant, Einstein eventually integrated the work of his mathematics teacher, Minkowski, into his theories of relativity. Minkowski formulated a unified system for space and time to describe hyperbolic space-time in Einstein's theories of relativity. These areas of mathematics, previously largely separate from physics, provided Einstein with the tools to articulate his concept of a four dimensional curved universe through field equations relating mass energy to geometry. The geometrization concept is intriguing. It posits that objects with mass, charge and momentum can be understood as localized energy causing curvatures in a four-dimensional coordinate system. If everything traditionally considered physical things, fields and particles can be expressed as geometric entities, does this suggest that the nature of reality is fundamentally mathematical? And do our methods of perception and measurement merely give the illusion of rigid bodies with centers of mass energy? These profound ontological questions remain a subject of ongoing research. A multidisciplinary approach involving philosophical rigor, empirical research and mathematical modeling is essential for addressing such questions. Newton unified the motion of terrestrial objects like falling apples with the orbital movements of celestial bodies under a single set of equations. This gave a concrete interpretation to the ancient hermetic axiom, as above so below. Einstein, however, extended this unification by merging space and time, entities long considered separate by natural philosophers. While Newton might have had inklings of local causality, he left its application to gravity unresolved. Einstein picked up this challenge after his successful work with light quanta. His completed framework replaced Newton's instantaneous action at a distance model of gravity with one rooted in local causality. In Einstein's relativity theories, objects trace geodesics, the most straightforward paths through curved spacetime. This makes absolute time and simultaneity irrelevant due to the constant speed of light, which serves as the basis for signal causation. The principle of local causality is maintained. Initial influences affect only immediate neighboring objects, with subsequent effects propagating up to 300,000 kilometers per second, affecting both gravitational and electromagnetic phenomena, including visible light. The shift from Newtonian to Einsteinian mechanics is not without its complexities. General relativity is intricate, layered with mathematical formulations, physical interpretations, and ontological questions, some of which we will explore later. Quantum mechanics. While general relativity governs large-scale phenomena, quantum mechanics rules the microscopic realm. The birth of quantum mechanics dates back to the early 20th century when classical physics failed to explain certain behaviors at atomic and subatomic levels. Max Planck's quantization of energy in 1900, along with Einstein's explanation of the photoelectric effect in 1905, marked a significant departure from classical physics. Further developments in quantum mechanics were fueled by contributions from physicists and mathematicians like Niels Bohr, Werner Heisenberg, Erwin Schrödinger, Max Born, Wolfgang Pauli, and Paul Dirac. Geometry also plays a role in quantum mechanics, as evidenced by the formulation of the Dirac equation, which integrates both quantum mechanics and special relativity to describe the behavior of fermions with spin half, and the Yang-Mills equation, which extends the concept of gauge invariance and geometry into the realm of quantum field theory, providing a framework for understanding the strong and weak nuclear forces. Quantum field theory, the foundation of the standard model of particle physics, employs group theory and differential geometry to describe elementary particle interactions. The structure of quantum states and the rules governing their evolution owe much to geometric principles encapsulated within an abstract infinite dimensional space named after Hilbert, who shifted his focus to quantum mechanics after working on relativity first. This shift of interest was widespread in the physics community, so much so that relativity theories became less attractive than quantum mechanics for decades. Even Einstein received his Nobel Prize in 1921 for explaining the photoelectric effect which relates to quantum mechanics, not relativity. In the Hilbert space, each quantum state is represented as a vector. Physical quantities relevant to these states, like energy or momentum, are expressed as operators. This abstract mathematical framework gives rise to quantum principles such as superposition. This principle states that a quantum system can exist in multiple states simultaneously, each with its associated probability until an observation measurement occurs. Superposition is a concept shared with classical wave mechanics, but treated uniquely in quantum mechanics. Entanglement. A phenomenon where particles in a quantum system become correlated so that one particle state instantaneously correlates with another state, no matter the distance separating them. This does not imply faster than light communication or influence, as the correlation is observed rather than used for signal propagation. The outcome of one measurement is instantaneously known once the other is measured, but this does not transmit any usable information faster than light. Probability. Quantum mechanics uses probability amplitudes, which are complex numbers associated with the likelihood of finding a system in a particular state. The square of the amplitude's magnitude gives the actual probability of an event occurring, a fundamental departure from classical probabilities. While classical probabilities sum directly, quantum probabilities, amplitudes, must be added and then squared, reflecting the wave-like interference patterns unique to quantum systems. Decoherence. This concept describes the transition of a quantum system to a classical state as it interacts with its environment. This causes the system to lose its quantum superposition and behave more predictably. Decoherence helps to explain why quantum effects are not generally observed in macroscopic objects. Tunneling. A quantum effect where particles have a probability of passing through potential barriers, even when they lack the energy to do so in classical physics. This phenomenon is essential in various quantum technologies, such as scanning tunneling microscopes or quantum computing. It helps to explain several natural processes, such as nuclear fusion in stars or biological enzyme action in cells. Quantum jump. This refers to the sudden probabilistic transition from one quantum state to another, often resulting in the emission or absorption of energy in discrete quanta. These jumps are characterized by their whole number discretization, as the energy levels of quantum systems are quantized. In atomic systems, for instance, electrons jump between fixed orbits or energy levels, and the energy difference between these levels is emitted or absorbed as a photon with a frequency directly proportional to the energy difference, reflecting the discrete nature of quantum states. General relativity and quantum mechanics represent contrasting paradigms. General relativity operates under smooth continuous mechanics, while quantum mechanics involves the discretization of quantum numbers, not space-time inherently, quantum jumps and entanglement. This disparity presents a challenge for those attempting to merge these theories. Either one must be modified, or a new framework that accommodates both general relativity and quantum mechanics must be developed without compromising their current accuracy, applicability and sophistication levels. In the standard model, each type of elementary particle, be it a boson, quark or lepton, corresponds to a unique field that permeates space-time. These fields can be thought of as the fabric of the universe, with each particle type representing a specific vibrational pattern within this fabric. Certain symmetries, known as gauge symmetries, govern the interactions among elementary particles. These symmetries dictate the laws of interaction and maintain the consistency of the physical laws that apply to the particles. Gauge symmetries are expressed through mathematical constructs called Lie groups, named after Norwegian mathematician Sophus Lie, who lived in the 17th century. Lie groups capture continuous symmetries and provide a geometric framework for understanding the structure and behavior of gauge fields, mediating particle interactions. The concept of gauge fields is formalized using the mathematical notion of fiber bundles, a key element in differential topology. In a fiber bundle, each point of a base space is connected to a unique geometric structure called a fiber. In particle physics, the base space represents space-time, while the fibers correspond to the various possible states or configurations of a gauge field at each geometric point. This is the basis for describing particles as point-like in the standard model. Both matter particles called fermions and force carrier particles called bosons emerge from a synthesis of empirical data categorized by the features of the particles that have been collected in experiments over the last century and theoretical foundations. Fermions, comprising quarks and leptons, are distinguished by their half-integer spins and compliance with the Pauli exclusion principle. In contrast, bosons stand out for their integer spins and significantly for their derivation from symmetry group representations. String models, often called string theory, aim to unify quantum mechanics and general relativity and extend this geometric perspective. The basic entities are one-dimensional strings that vibrate in multi-dimensional space-time rather than point-like particles. Like musical notes produced by a vibrating string, the various vibrational patterns of these strings are thought to correspond to different types of particles. The mathematics used to describe these vibrations and their interactions is steeped in geometry. It involves group theory, manifold topologies and Riemann surfaces as strings moving through space, creating world sheets. So, particles in the string model have a more profound theoretical origin than in the standard model. The standard, bosonic string and superstring models emerged in the 1970s when Einstein and contemporaries like Nordstrom, Hilbert, Theodor Kaluza, Max Born, Leopold Infeld, Hermann Weyl, Gustav Mee and Dirac sought to unify electromagnetism and relativity. Some even early atomic models. They lacked vital information unavailable until after the 1960s. Only electrons, protons and photons and a bit later, neutrons and positrons were known at that time. Discoveries made post-1960, such as identifying quarks and gluons, the partial understanding of strong interaction colour forces, the Higgs mechanism and the confirmation of the universe's expansion were gaps in previous knowledge and stymied unification efforts. New insights provided by the understanding of strong and weak nuclear forces, the principles of spontaneous symmetry breaking and the non-local behaviour of entangled particles have opened perplexing new avenues for the unification of general relativity and quantum mechanics. These complexities have left a highly intricate puzzle for future researchers to solve. Topology. Topology, sometimes called rubber sheet geometry, is a branch of mathematics that originated in the 18th century. Its pioneering contributions came from Swiss mathematician Leonhard Euler, German mathematician August Ferdinand Möbius and Riemann. Unlike classical geometry, which focuses on distances and angles, topology is concerned with properties of space that remain unchanged under continuous transformations. A key concept in topology is the idea of open sets, collections of points defined by the condition that any point in the set has a neighbourhood entirely within that set. Neighbourhoods are sets around a point extending to a certain limit, embodying the idea of closeness without requiring a metric for measuring distance. In topological terms, continuity means that when points from one set map onto another, points that approximate in the original set remain proximate in the target set. This preserves the essential relationships between topological spaces during transformations. A set of points becomes a topological space when one specifies which subsets are open. These open sets satisfy certain conditions to formalise our intuitive understanding of geometric concepts, like convergence and continuity. A classical question in topology concerns the number of holes in a straw. Typically, a straw is modelled as a cylinder. In topology, it is considered equivalent to a torus, a doughnut shape or a coffee cup with one handle because one can be deformed into the other without tearing or gluing. Both have a single hole and are topologically equivalent. Defining open sets becomes a practical step when examining the straw's surface as a topological space. In this context, an open set is a collection of points on the straw's surface, where each point is surrounded by a small region lying entirely within the set, away from the boundary. This concept of open sets is crucial as it helps understand how the surface behaves under continuous transformations. For instance, stretching or bending the straw is a continuous transformation as it does not add or remove holes. Such transformations retain the topological equivalence between the original and transformed object, emphasising the value of open sets in studying continuous transformations without resorting to non-continuous operations like tearing or gluing. Theoretical physics took a nuanced turn in the 1960s when American scientist John Archibald Wheeler introduced the term geometrodynamics. Wheeler proposed the concept of quantum or spacetime foam, suggesting that spacetime might have a fluctuating, intricate structure at extremely tiny scales, constantly undergoing topological changes. The relationship between topology and geometry has significant implications in theoretical physics. This narrative will extend as we delve into the theory of topological geometrodynamics and its unique approach to geometrisation. As the conversation progresses through the subsequent interview, we will touch upon an even more ambitious unification effort to merge classical and quantum physics, number theory, and potentially even cognition.