Humans have puzzled over the possibility of an objective moral theory since the dawn of philosophical inquiry. Many argue that morality is socially constructed rather than based in physical laws. Ethical relativists like Foucault (Wilkin, 1999) and sophists like Protagoras (Guthrie, 1969) have proposed that fundamental ethical truths do not exist. Along a similar line of reasoning, Richard Joyce contends that biases imposed by human evolutionary psychology invalidate the concept of fundamental morality since all ethical ideas depend on our evolutionary history (Joyce, 2001). But certain nondualist approaches to cognitive philosophy may provide a new perspective on this notion, a perspective in which ethical truths are both fundamental and relative. I propose that, if consciousness and emotion derive directly from physical processes, then objective morality exists, but continuously varies over space and time.

     For my theory of objective morality, I assume the panpsychist idea that conscious experience is a property of matter, analogous to mass or charge (Seager, 1995). According to this view, specific physical patterns may give rise to specific emotional states in a one-to-one correspondence, making qualia omnipresent. It should be noted that if panpsychism is true, the stochastic flickers of qualia associated with most matter would likely feel primitive enough that they would be unrecognizable by human standards. Next, I apply an authenticity-driven form of existentialism to this panpsychist perspective, allowing physically-encoded emotions to take on intrinsic meaning. In this writing, I exclude “meaninglessness” from existentialism and instead focus on the aspects of existentialism which involve people creating meaning for themselves (Holt, 2012). The distribution of emotional states across the universe may serve as a starting point for a physical description of ethics. By combining panpsychism with physicalism and existentialism, morality can be linked to physics in an objective fashion.

     Despite its past association with metaphysics, panpsychism is rekindling among contemporary thinkers as an explanation for consciousness (Strawson, 2006). Integrated information theory or IIT (Oizumi, Albantakis, & Tononi, 2014) is a mathematical formulation which seeks to quantify consciousness using the information arising from dynamical systems. Galen Strawson has argued that IIT implies panpsychism since all physical structures contain some amount of information (Strawson, 2006). Furthermore, Adam Barrett has proposed modifications to IIT which may help account for fundamental physical interpretations of the universe like quantum field theory (Barrett, 2014). Panpsychic descriptions of reality are reentering philosophical and scientific discourse as new data are acquired and new theoretical interpretations develop.

     But many still find the concept that inanimate objects may possess primitive qualia to be absurd. To counter this presumption, consider a fragment of quartz resting on a ridge. As the sun rises, photons excite the atoms on the crystal’s surface, causing thermal oscillations to propagate into the quartz. This thermal diffusion is modulated by crystallographic defects, causing a heterogeneous distribution of heat inside the rock. As dusk falls, the quartz fragment begins to cool, emitting heat at varying rates across the surface. The particular rates are influenced directly by this quartz specimen’s pattern of internal defects. Next, consider a mouse, also located on the ridge. As the sun rises, photons excite the retinaldehyde molecules in the mouse’s eyes, triggering signal transduction via electrochemical systems. This signal moves into the mouse’s brain, where it propagates through a series of neural pathways, causing a heterogeneous distribution of neural activity. Soon, the signal’s interaction with preexisting brain structures is translated into a motor action; the mouse blinks and looks away from the bright illumination. The particular motor response is modulated by the structural organization of this mouse’s brain at the given time. The quartz and the mouse both receive sensory inputs, process them according to internal properties, and then give motor outputs. Although the rock’s “brain” is much more disorganized and chaotic than the mouse’s brain, it operates by the same basic principles and could plausibly experience a primitive form of consciousness. As such, the possibility of panpsychism cannot be readily dismissed as absurd or metaphysical.

     Another prominent objection to panpyschism arises from brain processes which occur in a subconscious fashion. For instance, activity in the primary visual cortex (V1) does not correlate with conscious visual experience except for a few special cases (Crick & Koch, 1995) (Boyer, Harrison, & Ro, 2005) (Boehler, Schoenfeld, Heinze, & Hopf, 2008). However, the presence of subconscious neural events does not necessarily indicate that the said events are subconscious from the viewpoint of their associated anatomies. Instead, anatomical structures like V1 may experience their own independent qualia. The full informational content of their perceptions may not be transmitted or translated into the brain areas like the prefrontal cortex (PFC) which can be identified with a patient’s sense of self (Mitchell, Banaji, & Macrae, 2005). Of course, some data does transfer into higher brain regions to facilitate processes like vision, but the information undergoes an extensive series of transformations before arriving at the PFC and other regions associated with conscious processing. For this reason, the “unconscious anatomies” objection is insufficient to invalidate panpsychism.

     While existentialism is often linked to both authenticity in developing a personal outlook on life and to a sense of nihilistic meaningless, I use a modified interpretation of existentialism within this essay. I discard the nihilistic component of existentialism and enhance its emotional authenticity to operate in a manner analogous to Cartesian view around the ontology of thought, cogito ergo sum (Murdoch, Cottingham, Descartes, & Stoothoff, 1985). My interpretation of existentialism revolves around idea that emotions possess intrinsic validity in the same way Descartes argued for the intrinsic validity of consciousness. By applying this form of existentialism to physicalist panpsychism, an objective theory of morality gains feasibility.

     My proposed ethical theory does not mean that morality is uniform throughout the cosmos. Equipping the distribution of emotional states or “affective manifold” with existentialism allows for objective morality to continuously vary as informatic patterns change over space and time. Manifolds are mathematical objects which can be informally described as deformed versions of spaces such as the real line, plane, or three-dimensional Euclidean space. Although the affective manifold would not generally exhibit universal emotional uniformity, particular conformations of the manifold may still represent ultimate maxima or minima over all reality. An ultimate maximum may resemble the Buddhist concept of Nirvana, while an ultimate minimum may resemble a hell far worse than any imagined by the world’s religions. These extrema describe states of absolute moral “good” and absolute moral “bad” in a fashion which is fundamentally linked to physics. While this theory connects morality to fundamental processes in the universe, the said morality is also dependent on spatiotemporal coordinates. Ethical truths vary over the affective manifold except when the manifold holds a constant value across the entire cosmos as in the Nirvana and hell cases. In this way, morality may simultaneously take on objective and relative characteristics. Natural law exists, but a given natural law rarely takes on the same form across any two distinct subsets of spacetime.

     As an example, consider a subset of spacetime which includes only a human named Zebadiah during a specified eight-second time interval. Zebadiah feels unjustly underpaid at his job. During the eight-second period, he notices a fifty-dollar bill sitting on the desk of his boss, Enrique. He is alone and could easily snatch the money without consequences. Zebadiah pockets the fifty-dollar bill and does not experience guilt. He feels that this action is morally correct. Inside the spatial bounds of Zebadiah’s brain and the temporal bounds of the eight-second interval, this action is indeed morally correct on a fundamental level. But after the eight seconds have passed, Zebadiah recalls that Enrique’s wife is ill and cannot work, so Enrique is struggling financially. Zebadiah feels a rush of guilt and places the money back on Enrique’s desk. In this new time period, the Zebadiah-specific natural law has changed. Unknown to Zebadiah, Enrique was watching the entire time via a newly-installed security camera. During both time intervals, Enrique feels that Zebadiah is only taking a moral action if he does not steal the money. From an existential viewpoint, all the described perspectives are valid. Panpsychism connects this existentialism to physical processes, supporting moral objectivity on given subsets of the affective manifold.

     Morality is objective, but not absolute except for the specific situations in which the affective manifold is configured in a universally desirable or undesirable manner. Using a physics-based form of panpsychism along with an interpretation of existentialism which grants intrinsic validity to a subject’s emotions, the affective manifold emerges as a possible source of objective ethics. In most distributions of emotional states over the cosmos, ethical truths undergo continuous variation as spatial and temporal coordinates change. In this way, morality can be quite different for distinct persons and even for individual persons at different times. However, it remains an absolute ethical truth that reconfiguring the affective manifold such that all matter in the universe is optimized for generating a supremely positive emotional state represents a morally ideal action. Likewise, reconfiguring the affective manifold to generate a supremely negative emotional state represents an absolute moral wrong. My objective theory of morality may provide a new perspective on the study of ethics and so help to more effectively derive insights around contemporary ethical problems.

References

Barrett, A. (2014). An integration of integrated information theory with fundamental physics. Frontiers in Psychology. Retrieved from https://www.frontiersin.org/article/10.3389/fpsyg.2014.00063

Boehler, C. N., Schoenfeld, M. A., Heinze, H.-J., & Hopf, J.-M. (2008). Rapid recurrent processing gates awareness in primary visual cortex. Proceedings of the National Academy of Sciences, 105(25), 8742 LP-8747. Retrieved from http://www.pnas.org/content/105/25/8742.abstract

Boyer, J. L., Harrison, S., & Ro, T. (2005). Unconscious processing of orientation and color without primary visual cortex. Proceedings of the National Academy of Sciences of the United States of America, 102(46), 16875 LP-16879. Retrieved from http://www.pnas.org/content/102/46/16875.abstract

Crick, F., & Koch, C. (1995). Are we aware of neural activity in primary visual cortex? Nature, 375(6527), 121–123.

Guthrie, W. K. C. (1969). The Sophists. Cambridge University Press.

Holt, K. (2012). Authentic Journalism? A Critical Discussion about Existential Authenticity in Journalism Ethics. Journal of Mass Media Ethics, 27(1), 2–14. http://doi.org/10.1080/08900523.2012.636244

Joyce, R. (2001). The Myth of Morality. Cambridge University Press.

Mitchell, J. P., Banaji, M. R., & Macrae, C. N. (2005). The Link between Social Cognition and Self-referential Thought in the Medial Prefrontal Cortex. Journal of Cognitive Neuroscience, 17(8), 1306–1315. http://doi.org/10.1162/0898929055002418

Murdoch, D., Cottingham, J., Descartes, R., & Stoothoff, R. (Eds.). (1985). Principles of Philosophy. In The Philosophical Writings of Descartes (Vol. 1, pp. 177–292). Cambridge: Cambridge University Press. http://doi.org/DOI: 10.1017/CBO9780511805042.007

Oizumi, M., Albantakis, L., & Tononi, G. (2014). From the Phenomenology to the Mechanisms of Consciousness: Integrated Information Theory 3.0. PLOS Computational Biology, 10(5), e1003588. Retrieved from https://doi.org/10.1371/journal.pcbi.1003588

Seager, W. E. (1995). Consciousness, information, and panpsychism. Journal of Consciousness Studies, 2(3).

Strawson, G. (2006). Realistic monism: why physicalism entails panpsychism.

Wilkin, P. (1999). Chomsky and Foucault on Human Nature and Politics: An Essential Difference? Social Theory and Practice, 25(2), 177–210. Retrieved from http://www.jstor.org/stable/23559137

     Neuroecologies, cybernetic systems which combine biological and technological components, may streamline human interactions with their environment and counteract issues like pollution and climate change. Today, humans occupy too many ecological niches for long-term sustainability. If we continue to outcompete other species, we will observe deadly systemic bifurcations that lead to desertification, coastal submergence, and even more widespread extinctions. However, the interdependence of existing socioeconomic structures makes them difficult to uproot in order to avert catastrophe. To overcome this challenge, I propose that humans must integrate into their environments without sacrificing existing structures, instead metamorphosing those structures into more sustainable forms through improved technologies. In particular, the worlds of electronics, nanoscience, and synthetic biology must merge into the meadows, forests, and oceans. As written by Richard Brautigan in his 1967 poem, All Watched Over by Machines of Loving Grace, the “mammals and computers [will] live together in mutually programming harmony.”

     Traditionally, environmentalist viewpoints have divorced technology and nature, framing human influence as corrupting and impure. However, this represents an incorrect and ultimately counterproductive approach. Humans and their creations are no more separate from nature than termites and their mounds, beavers and their dams, or bacteria and their metabolic byproducts. While human impact on the environment has occurred at a rapid pace, this does not mean that humans are inherently less ethical than other organisms. Framing the technological humans as evil is a mistake since this ideology “drives underground” many possible solutions to environmental challenges.

     Many naturalists have supported anti-interference arguments with the fact that some technology has contributed to environmental damages. However, framing any particular technology as representative of all technology is misguided in an analogous way to cultural prejudice. Few would advocate judging the entire population of China based on an interaction with a single Chinese person. Likewise, it should be clear that the technologies which have contributed to climate change should not represent the philosophical concept of technology as a whole.

     Unlike most other organisms, humans possess a potent ability to simulate possible future scenarios. This ability arises from internal biological cognition, social exchange of knowledge, and technological augmentations. On its own, the human brain can generate plans for behavior many years into the future. In societies, networks of human brains act as a form of distributed superintelligence, agglomerating and refining individual predictions (Lenartowicz, 2017). The outsourcing of cognitive tasks to computers, pencils, and other technologies provides an immense expansion to the human noosphere. This combined global cortex has the power to analyze, predict, and act to tackle today’s greatest challenges.

     Biological and technological systems often share characteristics. From a network perspective, both usually possess a small-world structure (Sporns, 2009) (Newman, 2003). Small-world networks (supplementary Fig. S1C-D) have small mean path lengths and high clustering. For reference, mean path length is the average number of jumps taken to traverse between any pair of nodes in a network and clustering measures the connectivity among a given node’s nearest-neighbors (supplementary Fig. S1A-B). In addition, both biological and technological networks tend to exhibit resilience against random removal of nodes and vulnerability to targeted removal of highly interconnected nodes (Newman, 2003). Such network properties show remarkable universality across technology and biology.

     From a control theory perspective, biological and technological systems share modules and protocols (Csete, 2002). Modules are subsystems that utilize interfaces to other modules and work with some degree of independence. Biological modules can be seen in macromolecular complexes, membrane-bound compartments, cells, oscillating neural circuits, organs, as well as in ecosocial structures like packs of wolves and coral reefs. Technological modules arise in circuit components like flip-flops, in microprocessors, in network-connected computational devices, stations on assembly lines, and in components of infrastructure like hospitals and schools.

     Protocols are rules which manage the flow of ordered operations within systems (Csete, 2002). Biological protocols arise from information encoded in nucleic acids, epigenetic regulatory networks, neural ensembles, and ecological competition structures. Within such biological and technological protocols, feedback loops are universal. Feedback facilitates stable oscillations, system-scale or subsystem-scale adaptation to external stimuli, and discourse among modules.

     Since biology and technology operate by common principles, the engineering techniques applied in technological systems can be adapted to engineering biological systems and engineering interfaces between systems. Of course, doing this will also require recognizing differences and working to optimize the characteristics of all the systems involved in order to more seamlessly merge our technology and environment.

     For illustrative purposes, I will briefly outline a potential urban implementation of neuroecologies, a fictional city called Las Futuras. The skyscrapers of Las Futuras are wrapped in a silky, biocompatible polymer (Fig. 1B). Embedded in the polymeric matrix are nanomachines which recycle chemical waste products into nutrients, antioxidants, water, and molecules that chelate toxins and facilitate their traffic to more intensive treatment nodules. Cybernetic fireflies and bioluminescent trees illuminate the metropolis at night. There are no roads, the ground is overgrown with sensory flowers that monitor conditions and transmit data through conductive root networks, glistening metabolic blobs which assemble swarms of nanorobotic gardeners on command (Fig. 1C), and gossamer membranes for collecting solar energy.

     Enhanced herbivores and insects wander freely, instinctively consuming the fruitlike globules growing out of the buildings. There is no predation as all predators have either been transformed into herbivores or their populations have been phased out by reproductive methods. Because of this and the fact that the animals may live for hundreds or even thousands of years, their fertility rates are modulated by AI savants who continuously oversee the organismic network and prevent catastrophic bifurcations from triggering issues like overpopulation and starvation.

     The reason that few humanoid creatures are visible is that they exist elsewhere. Inside the skyscrapers, dense cores of computronium (Fig. 1A) simulate vast and beautiful worlds, kaleidoscopic utopias in which humanity’s billions and their AGI children may reside without overconsumption of resources. These people do have the option of entering android bodies and exploring realspace, but this exploration usually involves visiting locations other than Earth. As such, the neuroecologies remain sparsely populated by androids and their resource demands are minimal.

Figure 1 (A) Top view of the speculative urban neuroecology dubbed Las Futuras. Some skyscrapers (hexagonal structures) contain computronium for simulating uploaded minds and their virtual worlds, others manufacture materials and machines as needed by the system, still others recycle waste products. (B) Mesoscale view of a region within Las Futuras. Engineered trees exhibit bioluminescence. Nanomachines which recycle atmospheric pollutants are found in the walls of the skyscrapers. (C) Ground-level view of a region within Las Futuras. Fruitlike globules provide nutrients for cybernetic wildlife, including modified microorganisms and nanorobots. Solar membranes gather energy for local use, though more extensive solar farms exist outside the metropolitan area. Wriggling wormlike creatures tend to various macroscale tasks. Sensory flowers (not shown) monitor ambient conditions and route information into the conductive root network. This extensive system of roots provides a method of rapidly transporting data into the noosphere. Most organisms and devices are compatible with the roots and can easily link their nervous systems to the network.

     Although this kind of world might seem distant, there is a real possibility that it will be achievable before the end of the 21^st century. As a result of exponential trends in computer science and bioinformatics, many futurists argue that we are approaching a technological singularity. If this event occurs, it will be marked by an intelligence explosion which radically transforms life on Earth and beyond. The technological singularity may come from artificial superintelligence, self-aware computer networks, biological enhancement, or from the merging of humans and machines (Vinge, 1993). National Medal of Technology recipient Ray Kurzweil proposes that the technological singularity may occur as early as 2045, providing cognitive abilities up to a billion-fold more powerful than the entirety of today’s human race combined (122). This prediction may initially sound outrageous, but Kurzweil has shown a remarkable ability to accurately envision future events as evidenced by his numerous successful predictions; for instance, former world chess champion Garry Kasparov losing his games against IBM’s Deep Blue chess computer in 1997. If the singularity comes to pass, it will provide the intellectual and robotic resources necessary to merge technology and the environment.

     For some, my vision of the future might initially sound unsettling as it is a very different world than the world to which we are accustomed. But throughout evolutionary history, the ecological scene has continuously metamorphosed. In the Precambrian, aquatic metazoans (multicellular organisms) were only just beginning to emerge and the land was populated only by cyanobacterial mats (Horodyski and Knauth, 1994). Today, a rich and beautiful biosphere thrives across the air, land, and sea. The prospect of transformation need not be frightening. Technology is a part of nature, facilitating the metamorphosis. We have the ability to carry out this transformation, guiding unstable human and nonhuman ecologies into unified, sustainable neuroecologies.

References

Brautigan. “All Watched Over by Machines of Loving Grace.” All Watched Over by Machines of Loving Grace, Communication Company, 1967.

Bullmore, Ed, and Olaf Sporns. “Complex brain networks: graph theoretical analysis of structural and functional systems.” Nature Reviews Neuroscience, vol. 10, no. 3, Apr. 2009, 186–198., doi:10.1038/nrn2575.

Csete, M. E. “Reverse Engineering of Biological Complexity.” Science, vol. 295, no. 5560, Jan. 2002, pp. 1664–1669., doi:10.1126/science.1069981.

Horodyski, R. J., and L. P. Knauth. “Life on Land in the Precambrian.” Science, vol. 263, no. 5146, 1994, pp. 494–498., doi:10.1126/science.263.5146.494.

Kurzweil, R. (2005). The Singularity is Near: When Humans Transcend Biology. New York: Viking.

Lenartowicz, Marta. “Creatures of the semiosphere: A problematic third party in the ‘humans plus technology’ cognitive architecture of the future global superintelligence.” Technological Forecasting and Social Change, vol. 114, 2017, pp. 35–42., doi:10.1016/j.techfore.2016.07.006.

Newman, Mark. “The Structure and Function of Complex Networks.” SIAM Review, vol. 45, no. 2, Mar. 2003, pp. 167–256., doi:10.1137/S003614450342480.

Vinge, Vernor. “The Coming Technological Singularity: How to Survive in the Post-human Era.” Interdisciplinary Science and Engineering in the Era of Cyberspace, Dec. 1993.

Supplementary Figure S1 (A) The clustering coefficient for a node is the number of links among its nearest neighbors divided by the maximum number of links among those neighbors. (B) The path length between two nodes is the minimum number of edges which must be traversed in order to reach one node from the other. (C) Small-world networks are extremely common in biological and technological systems. Here, a randomly generated small-world network is depicted. (D) Small-worldness is proportional to the mean clustering coefficient over the mean path length. Note that the N^-1 terms cancel since they are the same in the numerator and the denominator for a given network with N nodes.

Nanowire arrays restore vision in blind mice

• Nanowire arrays were developed and implanted in the retinas of blind mice to replace photoreceptor cells (A).
• The nanowires consisted of titanium dioxide wires decorated with gold nanoparticles.
  • TiO₂ nanowires are known to generate photocurrents upon exposure to UV light.
  • Decorating the wires with gold nanoparticles enhances this effect and allows for blue and green visible light to also trigger photocurrents.
  • The photocurrents for blue and green light were much lower than the photocurrents for near-UV light, but still high enough to be useful.
• Patch-clamp recording was used to measure the responses of retinal ganglion cells (RGCs) in controls and in mice equipped with nanowires to near-UV, blue, and green light.
  • The RGCs in wild-type mice were responsive to near-UV, blue, and green light.
  • RGCs with nanowires were responsive to near-UV light (B).
  • RGCs with nanowires were responsive to blue and green light (C), though the latencies were longer compared to near-UV case, likely because of their smaller photocurrents.
• The responses of nanowire-equipped RGCs to varying light spot diameters were also tested (D).
  • For near-UV light, RGCs could be reliably activated with spots of about 50-100 µm (and larger spots).
  • For blue and green light, RGCs could be reliably activated with spots of about 200-300 µm (and larger spots).
• Pupillary reflex improvements in the nanowire-carrying mice supported the conclusion that the nanowire implants had partially restored vision.

Reference: Tang, J., Qin, N., Chong, Y., Diao, Y., Y., Wang, Z., … Zheng, G. (2018). Nanowire arrays restore vision in blind mice. Nature Communications, 9(1). doi:10.1038/s41467-018-03212-0

Surface chemistry governs cellular tropism of nanoparticles in the brain

• The interactions of polylactic acid (PLA) nanoparticles with the brain microenvironment were studied. It should be noted that this paper also investigated PLA nanoparticle interactions with glioblastoma tumors, but that topic will not be discussed in this summary.
• PLA nanoparticles were infused into rat brains. Four types of PLA nanoparticle formulation were tested, each bearing a different functional group.
  • PLA nanoparticles coated with polyethylene glycol (PEG), hyperbranched glycerol (HPG), HPG-CHO (aldehydes instead of vicinal diols), and unmodified PLA nanoparticles were examined.
  • PLA-PEG and PLA-HPG nanoparticles have decreased immunogenicity. PLA-HPG-CHO nanoparticles are less immunogenic as well as exhibiting bioadhesivity.
  • The nanoparticles were engineered to be brain-penetrating. They possessed neutral or negative surface charge, diameters of under 100 nm, hydrodynamic diameters (a measure which incorporates the movement of polymer chains in the nanoparticles) of under 150 nm, and did not aggregate in cerebrospinal fluid. In addition, they were fluorescently labeled with the chemical dye DiA.
  • All nanoparticle formulations diffused over volumes of about 40 mm³. The diffusion volumes were also fairly homogenous, though some minor variation occurred (A).
• Interactions with neurons, astrocytes, and microglia were tested. Nanoparticles were infused into the brain and allowed to diffuse. Relative abundances of nanoparticles in the three cell types were analyzed using fluorescence-activated cell sorting to measure mean florescence intensity (MFI) within individual cells (B).
  • PLA-PEG and PLA-HGP nanoparticle uptake was distributed fairly evenly among the cell types. They also exhibited lower total uptake than PLA-HPG-CHO nanoparticles.
  • In addition to showing higher total uptake, PLA-HPG-CHO nanoparticles exhibited preferential uptake by microglial cells and decreased uptake by neurons.
• Confocal fluorescence microscopy was used to image tissue samples with infused nanoparticles.
  • Neuron morphology was not significantly affected by any of the nanoparticle types.
  • PLA, PLA-PEG, and PLA-HPG-CHO (but not PLA-HPG) nanoparticles caused astrocytes to display upregulated GFAP protein, indicating some level of immunogenicity (C).
  • PLA and PLA-HPG-CHO nanoparticles caused amoeboid morphology in microglia, indicating that they had entered an immunologically active state. PLA-PEG and PLA-HGP nanoparticles allowed microglia to stay in a ramified (branched) state, indicating lack of immunological response (D).
  • In the displayed confocal microscopy images (taken after 4 hours of diffusion), nanoparticles are stained red, nuclei blue, and GFAP white.

Reference: Song, E., Gaudin, A., King, A. R., Seo, Y., Suh, H., Deng, Y., … Saltzman, W. M. (2017). Surface chemistry governs cellular tropism of nanoparticles in the brain. Nature Communications, 8, 15322. doi:10.1038/ncomms15322

Andromeda Spaceways Magazine #70 can be accessed at the link below. Visit pages 89-90 to read my poem, The Sonata Machine.

ASM70

 

Manifolds

• Although differential geometry usually involves smooth manifolds, topological manifolds provide the foundation for understanding smooth manifolds.
• Topological manifolds are a type of topological space which must satisfy the conditions of Hausdorffness, second-countability, and paracompactness (see Notes on Topology).
• In addition, topological manifolds must be locally homeomorphic to Euclidean space, ℝⁿ.
• In order to be locally homeomorphic to Euclidean space, the formalisms below must hold.

• This means that, for every point x in a locally Euclidean topological space X, there exists an open set U such that x belongs to U and there exists a homeomorphism h from ℝⁿ to U (and its inverse). These homeomorphisms are called charts. The combination of charts which covers X is called an atlas.
• For an atlas, two charts can overlap on a manifold. The intersection of these two charts is an open set which maps to the same region of Euclidean space. Transition maps are composition functions f(g^-1) = f∘g^-1 and g(f^-1) = g∘f^-1 which map the open sets in ℝ²to the manifold (by the inverse function) and then map back to Euclidean space (by the function which describes the other homeomorphism). This is also called a coordinate transformation.

• If a manifold’s transition maps are all at least once differentiable (C¹), then the manifold is called a differentiable manifold. If the transition maps are all infinitely differentiable (C^∞), then the manifold is called a smooth manifold.

Dot products on smooth manifolds

• The metric tensor allows generalization of the dot product for vectors on manifolds and is given by the equation below. Here, u and v are input vectors. The superscript and subscript indicate that uⁱ is represented as a column vector and v_j is represented as a row vector (covector to uⁱ) inside the summation. The matrix g_ij is a coefficient matrix which modifies the initial coordinate system for a given manifold. The summation is over matrix indices i and j where i=j.

• To see how this equation works, consider using metric tensor to describe the dot product of two vectors in ℝ². In this case, g_ij is simply the 2×2 identity matrix.

• For manifolds in general, the coefficient matrix g_ij is usually not the identity matrix. Instead, the entries of g_ij are given by the equation below involving the Jacobian matrix J.

• To use this equation, new coordinates must be defined in terms of Euclidean coordinates. In order to understand this, consider the example of polar coordinates. The polar coordinate formulas are operated on by the Jacobian. When inputted into the formula for the coefficient matrix, they simplify to the result below.

• Any set of alternative coordinates written in terms of Euclidean coordinates can be used to generate a g_ij matrix by this process.
• An interesting application of the metric tensor is to compute the generalized dot product of two vectors on a surface (2-manifold). The vectors “start out” in ℝ², but then are mapped onto the surface, which is embedded in ℝ³. To accomplish this, the ℝ² coordinates are represented in ℝ³ by using zero for the z-components. Note that the plane is assumed to be perpendicular to the surface in this scenario.

• Given a pair of vectors u and v tangent to a point on a parametric surface (2-manifold), the lengths of the vectors and the angle between them can be computed using the metric tensor. The lengths are given in the top equations and the angle is given in the bottom equations (below).

Arc lengths on smooth manifolds

• The metric tensor can be used to determine the distance between the points γ(t₁) and γ(t₂) on a manifold. The vector-valued function γ(t) defines a parametric curve on the manifold. Here, g_ij is generated using the Jacobian of the parametric functions in γ(t). The components γⁱ and γ^j are equivalent when i=j since they represent the same components. The absolute value is included to keep the term under the square root positive.

• Consider the example of a parametric curve in ℝ² which is mapped onto a 2-manifold embedded in ℝ³. Here, the arc length of such a curve is computed.

Integration on smooth manifolds

• Integration can be performed on surfaces (2-manifolds) using surface integrals. Furthermore, this method can be generalized to volumes and hypervolumes on n-dimensional smooth manifolds.
• The surface integral of a region on a 2-manifold embedded in ℝ³ can be computed by the equation below. The double-struck bars indicate magnitude. Here, the surface is given as a function z(x,y).

• It should be noted that r may represent any parameterized parametrized surface (not just a surface given as z(x,y)). The more general case of a surface embedded in ℝ³ is given below.

• Using some algebraic manipulations, the surface integral can be rewritten as the equation below. The matrix g represents the metric tensor (generated by applying the Jacobian to the parameterized function r).

• In this form, the surface integral can easily be generalized to higher dimensions so as to integrate volumes and hypervolumes on smooth manifolds. Below, a volume integral on a smooth 3-manifold is given.

• For integrating a hypervolume on an n-dimensional smooth manifold, a generalized version of the surface and volume integrals can be used.