What Is Spacetime? How Was It Born?

Abstract

Cosmology has been evolving over an extended period. Even long before the rise of modern science, cosmology was primarily dominated by philosophical concepts concerning the relationship between the Earth and celestial bodies, as well as the nature of the universe itself. Since the formulation of the General Theory of Relativity (GTR), physics has deeply engaged in discussions about the universe’s history, often intertwined with the concepts of space and time. Even within the framework of the standard theory known as Big Bang cosmology, certain philosophical themes, including those related to spacetime, are inherently assumed. In this chapter, I will apply the concept of structural realism about spacetime, a conclusion drawn from contemporary philosophical debates grounded in GTR, to the field of physical cosmology. I will explore the origin of the universe and also delve into the concept of spacetime in the early micro universe, considering the emergence of spacetime.

1. Introduction

1.1 Cosmology and philosophy, cosmology and spacetime

Many people associate the term “cosmology” with physical science, but it has not been too long since this field became integrated into physics. Originally, cosmology was a subject that philosophy grappled with, contemplating the outer space of the Earth and the history of our universe from religious and theological perspectives. Numerous questions about space and time are intertwined with the cosmological viewpoint.

In ancient times, there were cosmological explanations that presupposed the existence of many gods. In Greek mythology, the world began with chaos, from which emerged primordial deities like Gaia, Eros, and Nýxis followed by their descendant deities personified from human and natural concepts [1]. For instance, Kronos presided over time, Poseidon over the sea, and Zeus was the father of the sky. Our universe was believed to be shaped by these various gods.

In addition to these myths, serious metaphysical questions have always been pondered: How does the Earth exist in the universe? Moreover, how is the Earth positioned within the universe? At the very least, the distinction between these two realms has been evident since ancient times—on one hand, the Earth, where human beings and other creatures reside, and on the other hand, the sky, inhabited by various gods. Because the world above was starkly separated from the world below, the notion that the Earth is round rather than flat had a profound impact on people. During this era, many might have believed that the Earth was the center of the universe.

Cosmology continued to evolve worldwide, moving beyond various myths. In fact, the theory of a spherical Earth is said to have been proposed by Pythagoras as early as the sixth century BC to explain the movements of stars [2, 3]. Geocentrism became dominant not only in ancient Greece but also in other regions.

In the fourth century BC, Aristotle presented a worldview where the Earth, terra, was centered, and surrounding it were concentrically arranged aqua, aer, ignis, and celestial spheres [4]. This model depicted the universe as having regular and universal circular motions of celestial bodies made of ether and quintessence above the moon, along with irregular and diverse motions of matter composed of the four elements below the moon. For instance, in this perspective, matter falls downward because terra exerts a pull on it, akin to a description of gravity. Each object, including celestial bodies, moves toward its designated destination based on teleology. Geocentrism held sway until Heliocentrism gained powerful support. Cosmology used to be a domain addressed by philosophers and thinkers, partly to ascertain the Earth’s unique place in the universe.

Generally, astronomy, the oldest science concerning the space beyond Earth, which evolved from ancient times, is widely believed to have truly transitioned into cosmology during the sixteenth century with the work of Nicolaus Copernicus. It took approximately 30 years for Copernicus to formulate the heliocentric theory, which asserts that the Earth is one of many planets orbiting the sun, just like the rest of the celestial bodies in the universe [5]. Copernicus directly observed the movements of stars and mathematically calculated their orbits, a pursuit in the realm of astronomy.

Copernicus’s work served as a catalyst for subsequent developments in physics. His findings later served as inspiration for Tycho Brahe, who accomplished exceptionally accurate calculations of planets, satellites, and fixed stars using a variety of observation instruments before the advent of astronomical telescopes [6]. Thanks to Brahe’s meticulous observations of Mars, Johannes Kepler, one of his students, formulated the renowned Kepler’s First Law [7]. This law stipulates that the orbit of a moving planet is an ellipse, with one of its focal points being the sun. Kepler’s law was probably revolutionary as it contradicted the traditional belief in astronomy, upheld by Christian doctrines, which held that stars moved in perfectly circular orbits in the celestial sphere. Kepler’s significant contribution to astronomy implied that the sun exerts a force that propels planets in their orbits around it. Ultimately, this insight was integrated into modern mechanics through the work of Isaac Newton.

Through this shift in worldview, mathematical physics, which began in the seventeenth century, underwent a significant transformation. Galileo Galilei is often credited with being the first to apply mathematical analysis and descriptions to experimental findings regarding the motion of physical objects. He observed celestial bodies through telescopes, discovering facts about other planets and the sun, such as the satellites around Jupiter, the phases of Venus, and the existence of sunspots [8, 9]. These observations eventually led him to formulate his own heliocentric theory, although it took a considerable amount of time for his ideas to gain wider acceptance [10].

Furthermore, Galilei developed the law of free-fall motion for objects on Earth, introducing a mechanistic perspective on nature that contrasted with the traditional teleological interpretation dating back to Aristotle [11]. René Descartes, a founder of modern philosophy during that era, proposed a mechanistic worldview in which everything in the natural world could be a cause or effect of other things, without invoking final causes (destination) as teleology does [12]. As a result, modern science’s emphasis on empirical evidence had a profound impact on philosophical and metaphysical perspectives.

Newtonian mechanics, a form of natural philosophy, notably through the groundbreaking theory of gravitation, established a connection between terrestrial and celestial phenomena. The law of universal gravitation marked the culmination of this mechanistic viewpoint, and Newton successfully elucidated the behavior of all physical objects with his three laws of motion. Quantities like velocity and acceleration were precisely defined by being linked to the concept of force.

The perspective that both celestial bodies and everyday objects adhere to the same mechanical laws conferred a special authority upon mathematical physics as a means of describing nature and the universe. These advancements in science are attributed to Copernicus, Kepler, Galileo, and Newton, all of whom are considered the leaders of the seventeenth century scientific revolution, fundamentally altering the landscape of the philosophy of science. Consequently, cosmology ultimately transitioned into the realm of empirical science.

However, metaphysical questions that extend beyond empirical facts have always been left out. While mechanical laws certainly facilitated the comprehension and prediction of celestial motion, they did not provide insight into the underlying reasons for the functioning of gravitational forces. To capture a holistic understanding of the universe, progressively more assumptions were required. It was not until two centuries later, with Albert Einstein’s formulation of the General Theory of Relativity (GTR), that the scientific study of cosmology, delving into the history of the universe, truly evolved. During Newton’s era, cosmology remained susceptible to theological or superempirical presuppositions.

Indeed, in Philosophiæ Naturalis Principia Mathematica, a three-volume book authored by Newton in 1687, he posited the existence of an objective absolute spacetime [13]. This concept aimed to describe physical phenomena geometrically, rather than algebraically. In addition to the law of universal gravitation and the three laws of motion, Newton introduced this concept, even though he stated in the book, “Hypotheses non fingo,” meaning “I frame no hypotheses.”

Absolute spacetime encompasses a four-dimensional (4D) framework comprising three-dimensional (3D) infinitely expanding space and one-dimensional (1D) time. Characterized by Euclidean geometry across the universe, this overarching space already uniformly and universally exists independently of any matter it might encompass. Here, matter means physical objects such as balls, planets, and light¹. Within this absolute space, time flows uniformly at a constant rate throughout the universe, regardless of the presence or absence of material changes. For Newton, even in the absence of any matter in the world, an empty spacetime, akin to a vacuum state, could theoretically exist as a container for flowing time. This concept represented an ultimate background that extended beyond empirical facts.

Absolute spacetime also has relevance to one of the most profound cosmological questions: How did our universe come into being? Newton postulated that God (a divine presence) might require this spacetime container to perceive matter, as if space served as a kind of sensory center (sensorium dei). This notion is hinted at in his work Optics, written in 1704 ([14], p. 15). Therefore, one could deduce that, according to this cosmological perspective, absolute spacetime was either a primary or a preexisting entity, and matter appeared within it. It is important to note, however, that Newton himself possibly held the belief that the universe remained unchanged eternally, with no beginning or no end.

In philosophical discourse, the standpoint advocating the existence of spacetime, much like Newton’s concept of absolute spacetime, is classified as substantivalism. This term denotes a form of realism regarding spacetime. Indeed, Newton is considered the progenitor of substantivalists. For proponents of substantivalism, the universe is understood as encompassing the entirety of the spacetime region within which matter resides.

Conversely, relationism argues that spacetime merely comprises a set of relationships among material entities. In contrast to Newton’s stance, Gottfried Wilhelm Leibniz, as the progenitor of relationists, rejected the introduction of assumptions beyond empirical evidence, such as the notion of absolute spacetime [14]. He contended that space can be reduced to relative positional relations among material objects, and that time only arises once changes between these objects occur. According to his viewpoint, the concept of an empty spacetime or vacuum state devoid of matter holds no meaning. The idea of spacetime comes into being secondarily, oriented toward observing matter. This implies a framework of relative spacetime theory, rather than an absolute one. In essence, this constitutes a form of antirealism concerning spacetime.

Leibniz endorsed his theory of relative spacetime by invoking the Principle of Sufficient Reason (PSR), which originates in Section 32 of his work the Monadology, penned in 1714 [15]. He challenged the concept of God Newton appealed to. Surely, if God were omniscient and omnipotent, even a sensory center would be unnecessary. He is often said to hold the belief that everything must possess a reason for its existence or state.

In a scenario involving absolute spacetime, if God were to create a universe, it would be unique and distinct from all other potential universes. For instance, by shifting the entirety of the original universe, I call universe A, 3 feet to the east in absolute space with all relative positional relationships between objects retained, a new universe, let us say universe B, would arise. This prompts a comparison between the original universe (universe A) and the moved universe (universe B). Here arises a question: why did God opt to create universe A instead of universe B, despite his omnipotence allowing the creation of either or any other universe? The PSR contends that there must inherently exist a reason for this choice.

However, there is no explaining why one universe is preferred over all possible alternatives. To address this quandary, Leibniz employed a reductio ad absurdum approach. This reasoning led him to reject the notion of absolute spacetime, arguing that universe A could not truly be distinguished from universe B [14].

Relationism offers a distinct response to the aforementioned question. According to Leibniz’s theory of relative spacetime, two worlds with identical relative positional relations among objects, like universe A and B, are considered indistinguishable. Notably, the distinction between universe A and universe B becomes epistemologically elusive since only the relative relations and their changes can be directly observed. This conclusion is encapsulated in the Principle of the Identity of Indiscernibles (PII), as expressed by Leibniz in 1686 [16]². In essence, within this framework of relationist cosmology, matter is theorized to have originated from a state of nothingness at the beginning of the universe.

In the notion of velocity, relationism focuses solely on the relative aspect. When considering the physical motion of an object, the use of a reference frame, typically represented by a coordinate system with arbitrary three-dimensional coordinates including an origin for any space points at different time intervals, is essential. However, the values assigned to an object’s position, velocity, and acceleration are contingent upon the chosen coordinate system, highlighting a relational perspective. For example, physical conditions, such as an object being at rest (with zero velocity) or in motion (with nonzero velocity), are influenced by the coordinate system. Thus, this relational perspective supports the relationist worldview.

In contrast, proponents of absolute spacetime may argue that these conditions are determined absolutely based on the object’s position within the objective and overarching background. Consequently, while an infinite number of coordinate systems can be established, some among them are fixed in true space, allowing for the description of the true physical motion using a specific coordinate system known as the absolute standard of a reference frame at rest. Substantivalism introduces the notion of “absolute rest,” a concept absent in relationism.

The absolute standard of a reference frame at rest can give rise to many different yet indiscernible states. For substantivalists who assume absolute spacetime, the world C, in which all objects move uniformly in a specific direction at a defined speed as observed from the absolute standard of a reference frame at rest, must be different from the world D, where all objects are at rest according to the coordinate system. This distinction holds true, even if the relative positions between all objects are the same. In both worlds, objects undergo linear uniform motion at the same speed (nonzero in C and zero in D).

The notion of absolute rest describes more than empirical aspects. For relationists who do not acknowledge absolute spacetime, the worlds C and D are identical if the relative positions and velocities of all objects remain the same. It becomes nonsensical to assert that all objects are moving equally or at rest, as in both scenarios, objects possess equal relative zero velocities in relation to one another. In other words, in terms of physical or empirical aspects, worlds C and D share the commonality of objects having no acceleration or being subjected to no applied force. Consequently, the discrepancy between these two worlds, which substantivalism emphasizes, cannot be observed or experimentally tested in an epistemological sense.

However, the debate between substantivalism and relationism is not always confined to metaphysical discussions beyond empirical or physical facts. Moreover, circular motion presents a compelling example of force in action. In response to Leibniz, Samuel Clarke, a representative of Newton’s ideas, introduced the concept of centrifugal force acting on rotating objects in his correspondence with Leibniz in the years 1715 and 1716 [14, 18]. Consider two universes: universe E, where all objects rotate around a fixed point with a constant angular velocity, and universe F, where all objects are at rest, while the relative positions between all objects remain the same in both E and F. While relationists do not require a distinction between E and F, akin to their perspective on cases A and B, or C and D, there exists a noteworthy contrast. In universe E, objects rotating around the point experience a centripetal force or acceleration, resulting in the perception of a centrifugal force like the “Coriolis force.” Conversely, all objects in universe F do not encounter such forces. This is a physical distinction!

Even when relative positions and velocities remain the same, two universes are discernible in terms of whether or not forces are genuinely at work. The centrifugal force is a consequence of rotations within the absolute space or absolute standard of rest system, as Clarke suggested³. The existence of these forces gives rise to two universes that present not only metaphysical differences but also physically distinct attributes.

This explanation of a functional force relies on the existence of absolute spacetime, and the assumption of spacetime became a contentious topic among physicists and philosophers for an extended period. Immanuel Kant, a German philosopher, stated in his renowned work Critique of Pure Reason, written in 1781, that spacetime is not an objective background but rather a subjective framework inherent to human perception [19]. According to him, the Euclidean nature of space is an a priori assumption that serves as the basis for contemplating epistemological phenomena, rather than an empirically derived fact. The notion of absolute spacetime being objectively real yet empirically unverifiable was challenging to embrace, especially during that era.

However, it is also true that unlike equal-speed linear uniform motion, uniform circular motion, along with forces that can actually be experienced, poses significant challenges for relationism, as opposed to Leibniz’s view. Unfortunately, Leibniz had passed away before addressing this counterargument, which led to the conclusion of the correspondence without his response. Nevertheless, his relationalist view continued to influence subsequent thought. Ernst Mach, an Austrian physicist in the nineteenth century, took an extreme stance known as positivism or phenomenalism, asserting that only observed facts or sensory data perceived through human senses should be accepted in scientific theories. Mach’s worldview can be seen as reflecting Leibnizian relationism and offering a response to this issue.

Mach also rejected the concept of rotations with respect to absolute space. Aligned with his extreme commitment to empiricism, in his book titled Mechanik in ihrer Entwicklung; historisch-kritisch dargestellt written in 1883, he aimed to reconstruct Newtonian mechanics solely based on relations directly perceived between macroscopic physical objects, without assuming the existence of invisible concepts like space or time [20]. He argued that motion should be defined solely based on relations to the entire universe (Mach principle⁴), and if circular motion is to have any meaning, objects would rotate relative to some point in the universe—much like the Earth’s revolution around the sun or the entire solar system’s rotation around the center of Milky Way galaxy, rather than rotating in relation to absolute space. In this view, genuine forces such as centripetal force would only function due to these relative circular motions between objects in the broader system. Mach’s relationism stripped physical or empirical phenomena of any sense of absoluteness.

Mach’s influence extended significantly to physics and philosophical worldviews. He is known for his rejection of atomism, asserting that the micro atoms composing macro objects are invisible, and positing these micro atoms necessitates additional superempirical factors. This skepticism toward atomism brought him into opposition with the mechanical worldview originally proposed by Descartes and could have posed challenges for the development of quantum physics. However, it also made valuable contributions to physics. Einstein, influenced by Mach’s relational perspective, laid the foundation for the theory of relativity [21]. Indeed, the Special Theory of Relativity (STR) can be seen as a culmination of rejecting absolute reference frames such as the absolute standard of a reference frame at rest, sparking a new scientific revolution regarding our understanding of spacetime. Absolute spacetime was eventually eliminated from physics nearly two centuries later.

On the other hand, Mach’s positivism facilitated new epistemological approaches, like phänomenologie, for philosophers. It also posed a fundamental question about the purpose of science, which remains a contemporary task in the philosophy of science. Debates concerning the realism of spacetime have significantly impacted various domains of philosophy and physics, whether directly or indirectly.

Even in contemporary cosmology, the origins of the universe remain unresolved. Contemporary cosmology rests on the foundations of GTR and Quantum Gravity Theory (QGT), utilizing relativistic cosmology for the macro universe and quantum cosmology for the early micro universe, respectively. Nevertheless, philosophical interpretations of spacetime go on, even within these theories. The debate between substantivalism and relationism concerning the realism of absolute spacetime, originating from Newton and Leibniz, was initially confined to the framework of Newtonian mechanics until Einstein eliminated absolute spacetime from physical theories through his proposition of the STR. However, even after the concept of absolute spacetime faded, discussions about the realism of spacetime remained, albeit taking on new forms. Therefore, the fundamental cosmological question also remains: How did our spacetime come into being?

I would like to present my own response to this question. In Section 2, I will provide a concise exploration of macro spacetime realism within the context of GTR. I will then extend this viewpoint to relativistic cosmology, assessing the state of outer space throughout the history of our universe and offering an answer to the question. Moving on to Section 3, within QGT, I intend to touch upon the modern concept of spacetime emergence, considering the way this is related to the answer in Section 2. In essence, I aim to analyze contemporary cosmology through the lens of contemporary philosophy of spacetime.

2. Relativistic cosmology and realism of spacetime

Scientific cosmology has rapidly evolved both theoretically and experimentally, seeking to unravel the origins and potential destinies of the universe. Approximately a century has elapsed since Einstein introduced GTR, and it is within this span of time that significant advancements have taken place. These advances have been driven not only by classical GTR but also by quantum physics, starting with quantum mechanics. Below, I will provide a rough overview of the progression in cosmological research ([22], pp. 6-7).

1. Right after Einstein formalized GTR, he devised a static model of the universe in which the universe has a finite size using his own equation known as Einstein equation [23].

2. Einstein introduced what is known as the cosmological constant, which brought about a repulsive force rather than an attractive one, into his equation. This adjustment aimed to avoid the depiction of the universe contracting under the influence of gravity [23].

3. When evidence of the expansion of the universe was indicated by redshift through Edwin Hubble’s observations [24], it led to the belief that the universe had a beginning. Based on the current elemental composition ratios, George Gamow and others proposed the Big Bang theory, suggesting that the early universe was in a state of high density and high temperature [25].

4. Observations of the cosmic microwave background radiation confirmed the predicted elemental ratios proposed by the Big Bang theory, validating the accuracy of the theory [26]⁵.

5. With the development of elementary particle cosmology, the inflation theory became the standard model to address issues within Big Bang cosmology such as the horizon problem and the flatness problem, suggesting that the universe underwent rapid inflationary expansion from an initial microscopic quantum state to its present macroscopic state [27].

Advancements in scientific technology throughout the twentieth century have led to an improvement in observational precision. At present, even for the early universe during periods where direct observation is not possible (due to light being bound to elements and therefore not reaching us), various quantum cosmological theories have been proposed from the perspective of particle physics.

However, in these cases, metaphysics still has room to influence physics. For example, in 1, it is based on the belief in the eternal nature of the universe, rather than empirical reasons, that Einstein maintained a static model of the universe. Einstein aimed to render the universe static and adjusted his equation in 2. Despite observations confirming that the universe was indeed expanding, as evidenced in 3, it is intriguing that his cosmological constant continued to exist in cosmology as vacuum energy or a form of dark energy without being discarded. Recent observations of supernovae reveal not only the universe’s expansion but also its accelerated expansion [28], and so an enigmatic form of energy is responsible for explaining this phenomenon (I will take up this case in 2.1). In fact, the universe derived from 2 was not solely static but also artificially closed and finite, deduced from various other presuppositions. Notably, the finiteness of the universe was incorporated to sustain Mach’s principle, allowing the model to avoid the boundary conditions imposed on points at infinity, which are unobservable. Even within a single cosmological model, certain arbitrary beliefs are included.

While numerous observation outcomes, as 4 and 5 show, narrowed down the options for cosmological models, several contradictory models persist. Following the rejection of the static model, a steady-state cosmology was proposed [29, 30]. In this new theory, the matter density of the universe remains constant as it expands. In a different sense, the universe was kept eternal or stable. This is despite the fact that the Big Bang theory, along with inflation in the early universe, is now standard.

In the next subsection, I will focus on the notion of spacetime within the context of GTR in order to comprehend the core concepts of expansion discussed above. This notion significantly influences the modern cosmological worldview.

2.1 What is spacetime in GTR?

In twentieth-century physics, the concept of spacetime underwent a radical transformation. Unlike the persistence of the eternal universe concept or cosmological constant, the notion of absolute spacetime vanished entirely from physics. This change was a result of another scientific revolution, accompanied by a paradigm shift brought about by STR and GTR from the standpoint of the philosophy of science [31]. Given this transformation, one must reevaluate the nature of spacetime.

Certainly, this transformation did not occur all at once in the theory but unfolded step by step through STR and GTR. Initially, STR dismantled the absolute standard of a reference frame at rest and asserted the equivalence of all reference frames known as inertial frames with only two principles. This equivalence demonstrated that simultaneity and the rate of time’s passage were coordinate-dependent [32]. While this declaration immediately negated the existence of absolute time, many, including Einstein, continued to regard three-dimensional space as Euclidean until the analysis of a noninertial frame obtained by rotating an inertial frame [33, 34]⁶. In fact, it took Einstein approximately 10 years, from the proposal of STR in 1905 to the formulation of GTR in 1916 [36].

General Theory of Relativity (GTR) required an even more abstract formulation using mathematics. With the collaboration of mathematician Marcel Grossman, Einstein conceived the mathematical description of the gravitational field, which could be understood as being synonymous with geometric functions [37]. Consequently, the concept of gravity became integrated into the geometry of spacetime, with the former presupposed by the latter:

There can be no space nor any part of space without gravitational potentials; for these confer upon space its metrical qualities, without which it cannot be imagined at all. ([38], p. 618).

The paradigm of spacetime gradually shifted from Euclidean to non-Euclidean. The depiction of curved spacetime as described by GTR through Riemannian geometry allowed for a more physical conception of spacetime. Spacetime appeared to transcend its role as a passive container, evolving into a dynamic medium that interacted with matter according to GTR. Riemannian geometry, being non-Euclidean in nature, offered mathematical constructs like manifolds and metrics to describe physical spacetime. The relationship between mathematical tools and physical aspects became more intricate than in Newton’s era⁷. It is vital to ascertain what exactly refers to physical spacetime.

In the case of Newtonian mechanics, the special coordinate, namely the absolute standard of a reference frame at rest, can directly correspond to absolute spacetime. That is to say, the coordinate system, functioning as a mathematical tool, can refer to physical spacetime itself. Intuitively, the entity we typically regard as substance remains unchanged across different times, even though its state might vary. For instance, as Mauro Dorato points out, “A traffic light, (…), being a substance, remains the same across a change from yellow to red” ([40], p. 1626). Space consists of infinite spatial points and this view can be applied to these points:

If absolute spacetime is considered a substance, these points labeled (x, y, z) persist through time t, remaining the same across different moments in time ([41], p. 56).

Each space point stands as an independent entity with its own distinct identity⁸. This substantivalist viewpoint aligns with Descartes’ definition of substance, which is taken over from the Aristotelian worldview in modern philosophy:

when we conceive a substance, we understand nothing else than an entity which is in such a way that it needs no other entity in order to be [43].

Consequently, the entirety of absolute spacetime, composed of each individual substance, can also be regarded as a substance. Each spatial coordinate value on the absolute standard of a reference frame at rest refers to an individual space point.

However, when dealing with spacetime in GTR, things are not as straightforward. Unlike the absolute standard of a reference frame at rest, which assumes a flat infinite spacetime, there exists a disparity between the mathematical coordinate system and physical spacetime. While a coordinate system may indeed be essential for describing spacetime, in Riemannian geometry, spacetime encompasses geometric information from global topology to local metrics. No longer is there an objective spatial coordinate value to correspond to a space point, as space and time are subjective due to their coordinate-dependent nature, as stated in STR. At least, the notion that space points persist through time no longer holds true.

It becomes necessary to amalgamate space and time into the holistic four-dimensional spacetime, as portrayed in a Minkowski diagram. However, this diagram serves as a conceptual tool to elucidate the relativity of space and time, rather than a model for referring to concrete physical spacetime. Furthermore, this applies to an ideal flat spacetime, not the more general curved spacetime. A mere coordinate system is insufficient to adequately describe physical spacetime, regardless of whether it is a four-dimensional system with time and three-dimensional Cartesian coordinates or a system with time and three-dimensional polar coordinates.

Spatiotemporal features are articulated as attributes of spacetime points within GTR. The Einstein equation, a cornerstone of GTR, is formulated as follows:

$$R_{\mathit{\mu \nu }}-\frac{1}{2}{g}_{\mathit{\mu \nu }}R=8\pi {\mathit{GT}}_{\mathit{\mu \nu }}.$$E1

Here, G represents the gravitational constant, and the indices $\mathit{\mu \nu }$ correspond to t, x, y, z. On the left-hand side, $R_{\mathit{\mu \nu }}$ stands for the Ricci tensor, R denotes the Ricci scalar, and $g_{\mathit{\mu \nu }}$ represents the metric tensor. All of these elements encapsulate the geometric information of spacetime. On the right-hand side, $T_{\mathit{\mu \nu }}$ signifies the energy-momentum tensor, which characterizes nongravitational fields—specifically, material fields like the electromagnetic field. The metric tensor $g_{\mathit{\mu \nu }}$ is defined throughout the spacetime manifold, yielding the gravitational field. Importantly, it can be interpreted as essential physical attributes of each spacetime point labeled (t, x, y, z), rather than being merely mathematical constructs [40, 44, 45, 46, 47, 48]. Therefore, each point possesses local properties derived from $g_{\mathit{\mu \nu }}$.

Furthermore, these local properties are extrinsic attributes of one point in relation to others, rather than intrinsic characteristics specific to that single point. To be more precise, the spacetime interval ds is expressed as a local relationship between infinitesimal neighboring points in terms of distance, as illustrated by $g_{\mathit{\mu \nu }}$ in the following equation:

$${\mathit{ds}}^2=g_{\mathit{\mu \nu }}{\mathit{dx}}^{\mu }{\mathit{dx}}^{\nu }.$$E2

What I intend to convey is that temporal distinctiveness does not stem from intrinsic identity, but rather from extrinsic identity obtained in relation to other parts. If this temporal distinctiveness is applied to space points as well, each of these points is recognized in relation to others, rather than possessing a distinct identity of its own. While this aligns with substantivalist thought, it is relatively akin to the structural realism I will later introduce. However, for the sake of simplicity, in this chapter, I assume that substances have their own, or intrinsic identities.

“Locality provided by the metric field tensor gives rise to topology,” forming the spacetime manifold that serves as the framework for describing the universe ([49], p. 14)⁹. In classical GTR, spacetime points are identified through the metric, not as coordinate values in a particular coordinate system. This implies that each spacetime point lacks intrinsic physical properties on its own.

Without metrical information $g_{\mathit{\mu \nu }}$, it would be impossible to determine which coordinate values marked as (t, x, y, z) (such as p and q) in a given coordinate system correspond to which physical spacetime points. In fact, mathematical coordinate values change through coordinate transformations, which imply automorphisms on the manifold. However, in line with general covariance, the fundamental principle of GRT, it is ensured that all coordinate systems should be equivalent. In other words, a spacetime point is referred to by various coordinate values, depending on the coordinate system employed. Hence, solely coordinate values lack any intrinsic or inherent physical information. They can only correspond to physical spacetime points through the metric, which provides extrinsic local information. Even Einstein himself took 2 years to fully embrace this novel spacetime perspective¹⁰. The fundamental essence of physical spacetime being rooted in locality would be characterized in contemporary discussions within the context of the philosophy of spacetime.

In addition, if we set STR aside and focus solely on GTR, the independence of spacetime as a distinct entity becomes uncertain. Since $R_{\mathit{\mu \nu }}$ and R are derived from $g_{\mathit{\mu \nu }}$, which describes spatiotemporal properties, it can be construed that the left-hand side of the equation delineates spacetime with its gravitational/metric field, while the right-hand side encompasses matter¹¹. Admittedly, I have previously acknowledged that spacetime appears to function as a dynamic medium interacting with matter, as illustrated by Eq. (1), yet this perception might be somewhat superficial. This is because Einstein’s equation allows for boundary concepts between spacetime and matter, specifically vacuum energy or dark energy mentioned earlier.

The narrative commences with the introduction of the cosmological constant. When Einstein incorporated a cosmological constant in 2, Eq. (1) underwent a slight modification:

$$R_{\mathit{\mu \nu }}-\frac{1}{2}{g}_{\mathit{\mu \nu }}R+\mathrm{\Lambda }{\mathit{g}}_{\mathit{\mu \nu }}.$$E3

Here, Λ represents the cosmological constant, serving as a repulsive force. But how did it come about? A cosmological constant can be perceived as an additional Lagrangian term distinct from ${\mathcal{L}}_m$ for ordinary matter in the entire action, which underpins the derivation of Eqs. (1) and (3):

$$S=\int \left[\frac{1}{16\mathit{\pi G}}\left(R-2\mathrm{\Lambda }\right)+{\mathcal{L}}_m\right]\sqrt{-g}d\mathrm{\Omega }.$$E4

Here, g denotes the determinant of the metric tensor $g_{\mathit{\mu \nu }}$ and $d\mathrm{\Omega }=\mathit{\text{cdtdxxydz}}$. The second term on the right-hand side represents a mathematical degree of freedom, which can be incorporated into the Einstein-Hilbert action (the first term) like a constant of integration. In Eq. (1), one can set Λ = 0. As described in 2, this cosmological constant was initially intended to function as a repulsive force to counteract gravity’s attractive one.

However, the crucial question pertains to the attribution of this effect within the universe—namely, is it attributed to spacetime or to matter? For instance, if one perceives a vacuum state as a system where $T_{\mathit{\mu \nu }}=0$, then Λ could be attributed to an empty spacetime. Hence, $\mathrm{\Lambda }{g}_{\mathit{\mu \nu }}$ might be viewed as an effect of vacuum energy, suggesting that spacetime possesses nongravitational energy as an inherent property. This energy facilitates the universe’s expansion even in the absence of material objects or fields. However, an opposite perspective can also be taken.

As long as one grasps Eq. (4), Λ is not linked to the derivation of R, or the spacetime side. Hence, in Eq. (3), the cosmological constant or vacuum energy can also be attributed to the matter side, as follows:

$$R_{\mathit{\mu \nu }}-\frac{1}{2}{g}_{\mathit{\mu \nu }}R=8\mathit{\pi G}\left(T_{\mathit{\mu \nu }}-\frac{\mathrm{\Lambda }}{8\mathit{\pi G}}{g}_{\mathit{\mu \nu }}\right)=8\mathit{\pi G}{\tilde{T}}_{\mathit{\mu \nu }}.$$E5

Mathematically, this equation can be obtained by simply moving the term $\mathrm{\Lambda }{g}_{\mathit{\mu \nu }}$ from the left-hand side to the right-hand side. However, it can also be inferred that vacuum energy is a form of matter. In this perspective, $-\frac{\mathrm{\Lambda }}{8\mathit{\pi G}}{g}_{\mathit{\mu \nu }}$ corresponds to an unusual medium with negative pressure. This negative pressure is believed to exert a repulsive force counteracting gravity throughout the entire universe, as indicated by the calculation result of Eq. (5). In essence, whichever side vacuum energy is attributed to, both interpretations hold valid physical implications.

Vacuum energy is generalized to invisible dark energy, which is present throughout the universe, spanning all spatiotemporal regions. Λ itself is not always required to be constant, and the cosmological constant faces a significant issue from a theoretical perspective in quantum physics¹². Therefore, vacuum energy has been expanded into the concept of dark energy, which encompasses not only constant but also variable quantities to explain the universe’s acceleration. However, the fundamental question remains: Should the impact of dark energy be interpreted as an attribute of spatiotemporal properties or material properties?

This underdetermination, or the inability to definitively judge one of the two interpretations as correct solely based on observations and data, renders the philosophy of spacetime more contentious than physics¹³. The question of whether this medium is attributed to spacetime or matter might not hold great importance, as its empirical function in explaining cosmological phenomena, such as the universe’s expansion, remains consistent. However, from a philosophical standpoint, the very sameness becomes a problem. When it comes to these unknown yet physically existing entities, the demarcation between spacetime and matter is no longer distinct within the framework of GTR.

Spacetime might be inseparable not only from gravitational fields but also from matter. Einstein probably initially intended to reject spacetime without matter, or physical states with ${\mathit{T}}_{\mathit{\mu \nu }}=0$, in accordance with the relationist spirit. However, he made a concession to such models as De Sitter’s universe, which is one of the solutions of Eq. (3) with ${\mathit{T}}_{\mathit{\mu \nu }}=0$ (but ${\tilde{T}}_{\mathit{\mu \nu }}\ne 0$) proposed by Willem de Sitter as a vacuum solution with only the cosmological constant [59, 60, 61] right after proposing the static model. Einstein suggested that behind the established geometry of spacetime, there should be some material entities [62, 63]. But Einstein presupposed matter distributed in a specific region known as “the equator,” not the aforementioned invisible entities spanning all spatiotemporal regions in accordance with Mach’s principles. However, this equator is not physically accessible, and empirical reasons for the existence of such matter were not established¹⁴. Expressed from a viewpoint of philosophy of science, Einstein’s presupposition was too ad hoc to defend Mach’s principle. Consequently, the concept of empty spacetime became even more ambiguous.

The twentieth-century field theory tells us that both spacetime and matter are described by fields [35]. Quantum theory has the most specific features of fields, but even when limited to classical physics, fields took the place of matter and spacetime. In fact, as I have already shown, in GTR spacetime cannot be described without a gravitational field, and this field leads to a global structure of manifold with topology. Einstein recognized this as well:

If we imagine the gravitational field, i.e. the functions $g_{\mathit{ik}}$ to be removed, there does not remain a space of the type (I) [Minkowski spacetime], but absolutely nothing, and also no “topological space”. For the functions $g_{\mathit{ik}}$ describe not only the field, but at the same time the topological and metrical structural properties of the manifold ([64], p. 155).

Furthermore, matter can also be referred to by material fields, expressed as $T_{\mathit{\mu \nu }}$ or ${\mathcal{L}}_m$ rather than by discrete bodies. In classical physics, there are two representative fields—one is the gravitational/metric field, and the other is the electromagnetic field identified as light. Within classical physics, these fields may have been the most fundamental candidates for describing the physical world, namely the universe, rather than spacetime and matter.

Finally, Einstein understood that spacetime was a structural quality of the field, not subordinate to matter. Given the ambiguity between matter and spacetime, it may have been natural for Einstein to strive to establish a Unified Field Theory (UFT), which attempts to unify these fields into one, in the mid-1920s after relinquishing Mach’s principle ([65]; cf. [35], p. 101)¹⁵. Whether or not this interpretation is accurate depends on the development of physics itself, or whether UFT will eventually be realized in the future. However, even when discussing philosophical realism, the entire field should be regarded as central entities, as expressed below:

There is no such thing as an empty space, i.e. a space without field. Space-time does not claim existence on its own, but only as a structural quality of the field ([64], p. 156).

The priority of matter over spacetime or even the equivalence of both notions may be irrelevant, suggesting that only the fields exist in the world. What we perceive as spacetime and matter may merely be epistemological forms used to represent the underlying structure.

In short, when asked the question “Is spacetime real? Does spacetime exist?”, the answer can be simplified to the existence of a geometric structure in which each point possesses only extrinsic local properties in relation to others, established by the gravitational field. Exemplification or instantiation of this structure is indifferent—whether it is attributed to pure spacetime, matter, dark energy, or any other factor¹⁶. What holds significance is the structure itself, referring to something present in the physical world and described mathematically [40]. This interpretation, structural realism concerning spacetime, distinguishes itself from traditional substantivalism and relationism while it gets the best of both worlds.

2.2 What is expanding?

In simple terms, an expanding universe literally means that the scale of the universe is increasing. The greater the distance between galaxies and clusters of galaxies distributed in the universe becomes, the larger the universe itself becomes. As I mentioned in 3, observations show that the redshift, which is the increase in wavelength of light reaching Earth, indicates how fast a star or galaxy from which the light was emitted is moving away. This information indirectly helps us determine the rate of expansion. Initially, I would like to accept the expansion of the universe as an empirical fact.

Well, now that the spacetime picture in GTR has been established, I will explore the theory of an expanding universe using this structural perspective. In fact, the reason behind the universe’s expansion, even if it holds true, still lacks a satisfactory explanation. One model that attempts to explain the expanding universe is based on the concept of space itself. Let me elaborate on this view, which ultimately leads to the fundamental question: How did spacetime come into existence?

In order to discuss the entire universe, one needs to establish a basic model for the global universe system. In standard cosmology, the strong cosmological principle is initially applied to the system—our universe exhibits isotropy and homogeneity on a global scale¹⁷. Under this cosmological principle, a specific metric tensor $g_{\mathit{\mu \nu }}$ from Eq. (2) can be derived as the Friedmann-Lemaître-Robertson-Walker metric (FLRW metric)¹⁸:

$${\mathit{ds}}^2=c^2{\mathit{dt}}^2-a^2\left(t\right)\left[\frac{{\mathit{dr}}^2}{1-{\mathit{Kr}}^2}+r^2\left(d{\theta }^2+{sin}^2\mathit{\theta d}{\varphi }^2\right)\right].$$E6

This metric is represented in a four-dimensional coordinate notation as (t, r,$\theta ,\varphi$), where the time coordinate is denoted as t, and the three-dimensional space is parameterized using polar coordinates (r,$\theta ,\varphi$). The spatial metric in three dimensions can be expressed using the constant spatial curvature K. If the universe possesses a finite and closed space, the value of K is positive, whereas if the universe has an infinite and open space, the value of K is equal to or less than zero standardly¹⁹. It should be noted that in the spatial coordinate system, known as the co-moving coordinate system, the scale factor a(t) changes as proper time t for observers increases, causing expansion or contraction of the universe. That is, the scale of the spatial coordinate system changes in time for observers at rest with respect to the moving spatial coordinate system. Additionally, it is important to mention that the origin of the polar coordinates is set at the observer’s position (Earth in the Milky Way galaxy).

The FLRW metric is often likened to a balloon model. It is the scale factor a(t) that drives the expansion of the universe. This factor, as its name implies, serves as a kind of ruler for measuring the scale of the spatial coordinate system. When the scale factor increases, the scale of the system also increases, regardless of whether the system or the universe itself is finite or infinite. The scale of the spatial coordinate system corresponds to the scale of the universe. Thus, this coordinate system is referred to as a co-moving coordinate system. Galaxies and clusters of galaxies move in accordance with this spatial coordinate system, their positions on the system remaining unchanged, much like the marks on a balloon moving apart as the balloon expands.

This explanation of the universe’s expansion is straightforward and easy to understand, but it does carry the potential for misunderstanding. The explanation, reliant on a co-moving system, seeks to offer a rationale for the behavior of the scale factor a(t) by solving Eqs. (1) and (5), presupposing the isotropy and homogeneity of the universe based on the cosmological principle²⁰. In the calculation, $T_{\mathit{\mu \nu }}$ is expressed as a combination of various ideal fluids, characterized by zero viscosity and constant densities, including dark energy, as observed from the spatial coordinate system²¹. However, if measured using a spatial unit of the co-moving coordinate system linked to the matter’s scale, these densities and volumes remain constant. As I mentioned in the previous subsection, $T_{\mathit{\mu \nu }}$ cannot describe discrete entities like galaxies. This ideal model is insufficient to describe the actual distribution of matter.

Furthermore, this model does not provide an explanation for the fundamental reason behind the current expansion of the universe. For whether the value of a(t) increases or decreases depends on particular additional assumptions, such as the initial conditions a(0) = 0 and a(t) > 0. In other words, solutions for a(t) as increasing functions, signifying the universe’s expansion, are obtained under the premise that the universe possesses a beginning (t = 0) with a singularity (a(0) = 0). The balloon model utilizing the FLRW metric does not inherently provide the explanation for the cause of this expansion, but just expresses the expansion itself.

The notion that outer space is expanding only presents this model on a surface level. Phenomenally in physics, the alteration in the scale of the universe is merely assumed. This model employs a co-moving coordinate system to depict this alteration and describes the expansion as if the spatial unit were increasing due to the growth of a(t). It is essential to bear in mind that this spatial depiction is confined to the co-moving coordinate system. The positions of galaxies in the universe are not changed by expanding spatial forces or similar factors. Observers arbitrarily fix the spatial coordinate system on their positions. The expansion of the universe is not a consequence of the space itself expanding, but rather it is caused by other factors such as energy, entropy, or other elements that are not accounted for in this ideal symmetric model.

Expanding space is merely a metaphor. The idea that the spatial coordinate system expands in accordance with the increase of the time parameter is a mathematical representation rather than a physical phenomenon. In this context, space does not possess the nature of a substance like absolute spacetime. While General Relativity Theory (GRT) describes the dynamic nature of spacetime, it does not attribute material characteristics to it, as illustrated by the traffic light example mentioned in the previous subsection.

Let us revisit the characteristics of the absolute spacetime that Newton presupposed:

If absolute spacetime is considered a substance, these points labeled (x, y, z) persist through time t, remaining the same across different moments in time ([41], p. 56).

If there is a concept of expanding space in the universe, each spatial point labeled (r,$\theta ,\varphi$), which remains constant over time t, appears to move away from one another, as if each spatial coordinate value corresponds to an individual substance. However, this depiction is primarily a lengthy assumption.

For instance, in GTR, space points are not the same across different times. In the co-moving coordinate system, let us consider a space point labeled p with spatial coordinates $\left(x_p,y_p,z_p\right)$ at time coordinate value $t_1$ in the FLRW metric. This space point is not identical to another point labeled p at the same coordinate system but at a different time coordinate value $t_2$, as space points do not persist through time. Point p at $t_1$ and point p at $t_2$ coincidentally possess the same spatial coordinates within the framework of the co-moving coordinate system (Figure 1). According to the principle of general covariance, even if a different coordinate system were applied to these two points, physical phenomena should be described in the same manner. Hence, there is no need to assert their identity since these two spacetime points labeled $\left(t_1,x_p,y_p,z_p\right)$ and $\left(t_2,x_p,y_p,z_p\right)$ in the co-moving system are assigned distinct spatial coordinates in the new coordinate system.

Figure 1.

The explanation of an expanding universe with a co-moving coordinate system.

Upon investigation, the co-moving coordinate system (t, r,$\theta ,\varphi$) can be converted into the static coordinate system (t,$r^{\prime },\theta ,\varphi$), keeping the scale of the spatial coordinate system unaltered through the coordinate transformation ($r^{\prime }=a\left(t\right)r$):

$${\mathit{ds}}^2=\left(c^2-H^2\left(t\right){r^{\prime }}^2\right){\mathit{dt}}^2+2H\left(t\right)\mathit{dtd}{r}^{\prime }-\frac{d{r^{\prime }}^2}{1-\frac{K{r^{\prime }}^2}{a^2\left(t\right)}}-{r^{\prime }}^2\left(d{\theta }^2+{sin}^2\mathit{\theta d}{\varphi }^2\right).$$E7

Here, H(t) represents the Hubble constant, and it can be expressed using a(t) as $H\left(t\right)=\frac{a\dot{(}t)}{a\left(t\right)}$ (where $a\dot{(}t)$ is the first derivative of a(t) with respect to t).

This coordinate system, with the term dtdr present, is not synchronized with respect to time unless H(t) = 0. Consequently, the progression of time varies at each point in space, and defining simultaneity becomes impossible.

This new static coordinate system (t,$r^{\prime },\theta ,\varphi$) seems quite complex, as time t from the co-moving coordinate system is still in use, but it must provide a framework to describe physical phenomena. General covariance allows this coordinate system to be equivalent to the original one. It is certain that in the static coordinate system, a spatial component ($r^{\prime },\theta ,\varphi$) does not expand. As time t increases, the nonzero spatial curvature $\frac{K}{a^2\left(t\right)}$ can change²², but the scale of the spatial coordinate system remains constant. In this coordinate system, space never expands; matter only moves away. A space point labeled p at time $t_1$ in the co-moving system and another at time $t_2$ in the same system are labeled different spatial coordinate values in the static coordinate system, respectively.

However, the co-moving coordinate system might be more than just a convenient tool for describing the universe’s system. Each spatial coordinate value in the co-moving coordinate system corresponds to a physical position. It directly corresponds to the position of each galaxy. If the universe is isotropic and homogeneous, using this coordinate system is very natural and logical. In this sense, the co-moving coordinate system might be the most suitable model for representing spacetime itself.

The co-moving coordinate system saves the cosmological principle. Hubble also demonstrated through observational data on redshift that the farther galaxies are from Earth, the faster they move away, with velocities proportional to their distance from Earth [24]. Intuitively, these results seem to suggest that all celestial bodies are moving away from the center of the universe, namely our Milky Way galaxy, consid-ered seemingly a special place, which contradicts the cosmological principle. However, if the universe truly expands like a balloon, these data are consistent with the cosmological principle as Figure 1 shows it. Thus, if there is no expanding space behind matter, there is no explanation for why the universe is expanding in this manner. This reduction to the absurd resembles a no-miracle argument, asserting that expanding space is real which appeals to Inference to the Best Explanation [68], even though the distinction between material positions and space points, or spacetime and matter, remains ambiguous. The balloon model not only explains the way phenomena occur but also provides a clear worldview on the expanding space concept.

However, the structural interpretations discussed in the previous subsection refute the notion of expanding space. Certainly, the spatial coordinate values in the co-moving coordinate system might refer to the physical positions of galaxies. The co-moving coordinate system could suggest the existence of expanding space behind matter. But let us remember the perspective of structural realism. Structural realism does not concern itself with what the concrete attribution of the spatiotemporal structure is.

In this context, it is crucial to recall that each spacetime point lacks intrinsic physical properties on its own. They possess only extrinsic or relational local properties provided by the metric. A coordinate value devoid of metric information cannot uniquely refer to a specific spacetime point, even though it refers to a position within the universe.

For a structural realist, both the co-moving coordinate system and the static coordinate system are merely different forms to describe the same spatiotemporal structure. The notion that two spatial points are uniquely identified or persist through time is not upheld by structural realism, which is the reasonable interpretation within classical spacetime, particularly within GTR. The co-moving coordinate system is essentially a convenient mathematical aid, but the entirety of four-dimensional spacetime can be described using other complex coordinate systems, such as the static coordinate system.

Structural realism concerning spacetime informs us that the abstract mathematical representation, like the FLRW metric derived from gravitational fields, undoubtedly refers to something physical in the universe, but it does not assume a specific form with size and shape, akin to a physical substance like the expanding universe. Many mathematical tools might represent this structure, but attributing a single physical entity exclusively to any one of them is not in line with the essence of structural realism, even though the distinction between spacetime and matter might be ambiguously claimed by structural realists. Expanding space is not endorsed by the structural worldview [22, 69].

If space were to expand as a substance, this expansion should have had a starting point. Assuming a singularity like a(0) = 0 when t = 0 signifies that the universe, encompassing all material entities, came into being from a size of zero at t = 0, in accordance with classical perspectives²³. Thus, the beginning of the universe aligns with the beginning of spacetime. In this interpretation, the void from which the universe occurred is not equivalent to space itself, but rather it denotes nothingness or something distinct from space devoid of matter.

However, the structural interpretation that negates the existence of an expanding universe furnishes an alternate picture of the universe’s beginning:

Even if the universe contracted to a point of zero, space did not similarly contract. Conse-quently, at the beginning, a spatiotemporal structure could be postulated to exist around the singularity [69].

In this picture, the vacuum underlying the universe, which indicates the region distributed by matter, might be synonymous with spacetime devoid of matter. This is because spatiotemporal structures can exist independently of matter, akin to Minkowski metric and Schwarzschild metric, which serve as vacuum solutions of Eq. (1). In this sense, spacetime constitutes the universal background of the universe from the perspective of structural realism.

This is precisely the answer I aim to present to the fundamental question. Structural realism concerning spacetime does not accept the notion of the beginning of spacetime in cosmology. It is quite intriguing that this audacious assertion is derived not from QGT addressing the question, but from philosophical concepts rooted in GTR.

3. Are we going to the non-spacetime picture?

Spacetime is often said to emerge from more fundamental entities in Quantum Gravity Theory (QGT). In simpler terms, spatial properties, such as distance and continuity, cease to hold true in microscale regions like the Planck scale. This arises because the manner in which gravitational fields integrate quantum effects within these domains remains unclear. Consequently, the notion that spacetime is not fundamental necessitates a reconsideration of conventional empirical concepts such as observations and causality. These concepts must be interpreted in a manner that accounts for the absence of spacetime.

This task also influences both physics and philosophy. The novel worldview has the potential to induce the next scientific revolution, as physics is on the verge of revisiting the fundamental nature of spacetime since the advent of general relativity [31]. This, in essence, is the primary role of physics, not metaphysics nor philosophy. But this challenge also prompts philosophers to contemplate the existence of empirical coherence [70] and the distinction between mathematical entities and physical entities, or physical salience [71] in QGT.

If Quantum Gravity Theory (QGT) can formulate consistent theories that integrate quantum theory and GTR, or formalize GTR in microscale regions, this impending revolution could present a coherent framework elucidating the emergence of spacetime. This framework would address the question of which aspects of a spatiotemporal structure are taken to these microscale regions.

The term “emergence” is highly ambiguous and controversial. In physics, emergence pertains not only to entities and properties but also to phenomena and theories that are “novel” and “robust” in relation to more fundamental ones [72]. In other words, spacetime is more than just a composition of constituent elements; it encompasses spatiotemporal properties that cannot be explained solely by the functions of these constituents. Moreover, the discourse concerning the emergence of spacetime takes on different aspects within various types of QGTs.

In the previous section, the conclusion was reached that the structural realism of spacetime, as derived from GTR, contradicts the notion of expanding space within relativistic cosmology. This perspective strongly asserts that space is distinct from a substance-like entity which came into being from the singularity. It does not endorse the notion that a spatiotemporal structure could not disappear at the universe’s birth. If this interpretation is valid, what lessons can we draw from Quantum cosmology’s exploration of the early micro universe?

QGT is closely related to Quantum cosmology because, following the standard Big Bang cosmology, the early universe contracted into a small size, irrespective of whether space also underwent contraction. This implies that some of the QGTs describe the early universe near the singularity, searching for the origin of the universe or spacetime. Toward the conclusion of this chapter, I will illustrate how the structural view addresses this non-spacetime theory in Quantum cosmology.

However, it is worth noting that structural realism about spacetime is rooted in GTR, whereas QGT does not necessarily presuppose GTR in macroscopic regions. Indeed, Super String Theory (SST), one of the QGTs, operates with a distinct classical field theory that becomes approximately consistent with GRT under high-energy conditions. This theory establishes a definite spatiotemporal structure as a background for a string or other material entities by disassociating gravity from spacetime, quantifying gravitational fields along with other fields like electromagnetic ones. SST might be the most coherent QGT, yet its concept of spacetime starkly diverges from the structural interpretation that equates spatiotemporal structures with gravitational fields, leaving aside the fact that, in SST, spacetime extends to 11 dimensions rather than the usual 4.

Hence, I find it exceedingly challenging to establish a connection between the notion of “spacetime” in SST and the spacetime discussed in the previous section. However, SST can be viewed as a theory addressing the emergence of spacetime, which is recognized as T-duality. In contrast to ordinary spatial domains, whose sizes are quantifiable, the target space employed as a backdrop in SST operates on a lower level of observability and can possess physically equivalent yet distinct length scales, a perplexing concept [73]. In the microscale regions of this target space, concepts like distance lose their clear definition. This serves as an illustration of more fundamental entities exhibiting non-spatiotemporal attributes.

As other more fundamental entities, which spacetime is emergent from, concepts, such as the spin network or the spin foam in Loop Quantum Gravity (LQG) and the causal set in Causal Set Theory (CST), have arisen. LQG and CST share the common goal of elucidating the constituents of spacetime itself—akin to the notion of a space or spacetime atom. The spin network characterizes nodes connected with edges within an abstract graph, representing discrete volume and surface elements of the space atom. The causal set, on the other hand, depicts spacetime points arranged within a causal relation that conveys a temporal dimension in a coherent manner, as if all points collectively compose a unified space. These depictions seemingly suggest that spacetime possesses substance-like attributes, reminiscent of the old substantivalism, which appears to be at odds with the structural perspective. It is worth noting that both LQG and CST reflect theoretical structures within the framework of GTR²⁴.

Especially, LQG directly quantizes metric, but there is no isomorphic correspondence between structure of spin network and the spatiotemporal one because through quantization processes, volume or surface area, from which continuity of space is emergent, is defined and spatiotemporal properties such as locality are lost [75]²⁵. Hence, there is a mathematical discontinuity between spin network and spacetime, which is the cause of emergence.

However, a spatiotemporal structure does not change into spin network through quantization. Indeed, LQG begins with the Einstein-Hilbert action calculated from $g_{\mu \nu }$ in Eq. (4), a spatiotemporal structure. To enable canonical quantization, the action needs to be transformed into the Palatini-Holst action [77], which is classically equivalent to it. When defining the configuration variable $A_a^i$ called the Ashtekar variables, its canonical conjugate momentum is expressed using the scalar density of the triad field${\tilde{E}}_a^i$. With this Astekar representation, the new action is given as below:

$$\begin{array}{l}S=\frac{1}{8\mathit{\pi G\beta }}\int d^{4}x\{{\tilde{E}}_i^a\left({\mathcal{L}}_t{A}_a^i\right)+N{\mathrm{ϵ}}_{\mathit{ijk}}{\tilde{E}}_i^a{\tilde{E}}_j^b{F}_{\mathit{ab}}^k+N^a{\tilde{E}}_i^b{F}_{\mathit{ab}}^i+{\lambda }^i\left(D_a{\tilde{E}}_i^a\right)\}.\end{array}$$E8

Here, $\beta$ is the Barbero-Immirzi parameter, N, $N^a$, and ${\lambda }^j$ are undetermined constants of the Lagrangian, ${\mathcal{L}}_t$ represents the Lie derivative, ε_ijk is the Levi-Civita epsilon (antisymmetric tensor according to Einstein’s summation convention), $F_{\mathit{ab}}^i$ is the differential form of $A_a^i$ given analogously by Young-Mills theory for the electromagnetic field, and $D_a$ represents the covariant derivative²⁶. This is simply a change of variables, and Eq. (8) still describes a classical structure. Furthermore, even when canonical variables $\left(A_a^i,{\tilde{E}}_i^a\right)$ are quantized, there exists an isomorphic correspondence between structures, quantum numbers, and classical numbers. Quantized spacetime is mathematically connected to classical spacetime through the structural interpretation.

The structure of a spin network can be derived from the calculation of quantized spacetime/gravitational field, not from classical spacetime. Through the quantization of Eq. (8), a wave function $\mathrm{\Phi }$ with a variable of $A_a^i$ is associated in relation to other known quantizations of different fields. However, to establish conditions without spacetime geometry as a background, in which physical quantities can be defined, a more abstract and complex state space is required. It is the spin network which is described in this unique space, and the discrete physical quantities are variables related to quantized original spacetime, namely the volume and surface areas of units of space.

Therefore, the spin network could be a space/spacetime atom, but not space/spacetime itself. It is because space atoms are described in another space detached from a metrical system of Eq. (8) in which spacetime itself is described. This abstract system does not constitute a spatiotemporal region. Two different foundations have different concepts. Consequently, spatiotemporal properties emerge independently of what a spin network describes, to some extent [75].

Ultimately, could QGT potentially describe the origin of spacetime itself in the future? If we simplify the idea, if an expanding space contracts into a singularity along with matter²⁷, QGT might describe a space atom of extremely small size. However, this concept of the size of a space atom is developed as a spin network, within a different abstract system. As long as there remains a conceptual discreteness between the spin network and spacetime, even space atoms resembling a substance could remain consistent with the structural realism about spacetime. This is because the latter is independent of the properties of the former.

Is this abstract foundational space a physical region? In physics, there exist various types of space beyond the four-dimensional spacetime, such as high-dimensional space, topological space, and phase space. Let us call these abstract space non-spacetime now. The concept of state/Hilbert space was a conceptual or mathematical one used to describe the behavior of a system. However, QGT blurs the boundaries of this abstract role even further. If spin networks and causal sets are indeed real as space atoms, where would they exist? Does this question make sense without a spatiotemporal structure? Can we entertain the idea of a universe or physical regions existing out of spacetime?

If we consider physical regions to encompass the entire universe, does the concept of emergent spacetime allow for the inclusion of non-space regions as part of the physical ones, rather than mere mathematical models or tools? This notion has already been introduced in modern physics: for example, when quantizing the electromagnetic field, photons oscillate not in physical space but within an abstract space formed by the physical quantities of the field itself [80]. However, virtual particles like photons correspond to fields that exist within spacetime, and the empirical aspects or phenomena associated with these particles occur within a spacetime region.

When it comes to space atoms, the possibility of non-spacetime regions comes into play. Causality might serve as a bridge connecting the spatiotemporal universe to a non-spatiotemporal one, potentially through the context of the big bang. To consider the behaviors of spin networks and causal sets as genuine phenomena, it might be necessary to expand the scope of physics even further!

Acknowledgments

This work was supported by Research Grants of Suntory Foundation in Japan. Thank you very much.