Greg Egan's Orthogonal trilogy takes place in a universe governed by different physical laws, where time and space are fundamentally equivalent. Its spacetime geometric structure differs from our universe by a single sign: the metric signature in our universe is \((-,+,+,+)\), whereas in the Orthogonal universe it is \((+,+,+,+)\). Therefore, its spacetime is a 4-dimensional Riemannian manifold, also known as a “Riemannian universe”. Based on this first principle, the author constructs its unique physics, chemistry, ecology, and society. There are over 80,000 words of rigorous mathematical derivations on its companion website, scaled at the level of graduate theoretical physics.
This article will discuss its background settings in simpler terms at the level of university physics. Readers uninterested in derivations can focus solely on the bolded text and sections without formulas. Readers concerned about “conceptual” spoilers are advised to finish at least Book I and the first 6 chapters of Book II before reading (unless encountering significant difficulties understanding the novels), or to read only the final Commentary section. This article adopts geometrized units where the spacetime conversion factor (ie. the speed of light in our universe, or the speed of blue light in the Orthogonal universe) and the gravitational constant are set to 1 (I.p79). Three diagrams in Book I are crucial for understanding the entire trilogy: the worldline of the Peerless on p157, the luxagen potential field on p192, and the origin of the arrow of time on p338. When cross-referencing the novels, paginations of Gollancz paperback are used, where the three “Afterword” are at I.p361, II.p389 ,and III.p359; “Book I, page 79” is abbreviated as “I.p79”, “Book III, Chapter 33” as “III-33”, and so forth.
The essence of special relativity is that physical quantities such as time and space, or energy and momentum, are not absolute. They pair up to form 4-dimensional vectors (4-vectors), and these vectors, or spacetime coordinate systems, rotate depending on the velocity (i.e., the Lorentz transformation). The projection of a vector onto the coordinate axes gives the observed values of the physical quantities, and paired quantities can transform into one another at different velocities. However, the inner product of two vectors is a scalar that remains invariant under rotation, known as a Lorentz invariant—the most common example being the inner product of a vector with itself, the square root of whose absolute value yields the vector's length.
For instance, a time interval and spatial intervals form a 4-vector \((\Delta t, \Delta x, \Delta y, \Delta z)\), and its inner product with itself is the spacetime interval, which remains invariant under different velocities or frames of reference. In our universe, this inner product is \(-\Delta t^2 + \Delta x^2 + \Delta y^2 + \Delta z^2\), whereas in the Orthogonal universe, it is \(\Delta t^2 + \Delta x^2 + \Delta y^2 + \Delta z^2\) — the difference in the metric signature between the two universes corresponds exactly to the signs of the terms in the inner product. In our universe, the spacetime interval can be positive or negative, termed spacelike and timelike, respectively. A Lorentz transformation, which preserves the inner product, cannot change one type of interval into another. The Orthogonal universe, however, has no distinction between spacelike and timelike, nor does it possess a light cone where the spacetime interval is zero.
Generally, 4-vectors with equal self-inner products form a two-sheeted hyperboloid in our universe (for timelike vectors). A special direction called “time” exists, and Lorentz transformations cannot map the lower sheet to the upper sheet, meaning there is an absolute distinction between past and future. In the Orthogonal universe, they form a hypersphere; there is no special direction, and any arbitrary direction can act as “time”.
Let us now consider only 2D spacetime. For an object with velocity \(v\), the distance it travels per unit time is \(v\), and its worldline (referred to as “history” in the novels) is a straight line with a slope of \(1/v\). Its length in our universe is \(\sqrt{1 - v^2}\) (for \(v < 1\)), whereas in the Orthogonal universe it is \(\sqrt{1 + v^2}\). However, in its own frame of reference, it remains stationary, and its worldline is parallel to the time axis, yet its length remains the same—this length is the proper time experienced by the object itself, which is \(\sqrt{1 \mp v^2}\) for our universe and the Orthogonal universe, respectively. Therefore, in the Orthogonal universe, a moving object will experience more time, exactly opposite to our universe. Note that the formula for our universe implies a speed upper limit of 1, while the Orthogonal universe has no such constraint; velocity can reach infinity, allowing the vector to rotate by 90 degrees and causing time and space to swap. Infinite velocity may appear special, but it simply means the direction of the worldline is perpendicular to the time axis. Since time and space are essentially the same in the Orthogonal universe, this direction is no different from any other. Similarly, the worldline can rotate to a negative slope, moving backward in time.
More generally, the Lorentz transformation in our universe is a rotation in hyperbolic space (ie. Lorentz boost), whereas in the Orthogonal universe, it is a rotation in Euclidean space. The tangent of the rotation angle \(\theta\) is precisely the velocity:
It is easy to verify that in the low-speed limit where \(v \to 0\), this reduces to the Galilean transformation. Thus, the Orthogonal universe follows Newton's laws of motion in non-relativistic scenarios. Furthermore, if we apply the substitution \(x \to ix\) (similar to a Wick rotation; quantities derived from length like \(v, p\) are substituted accordingly), it becomes the Lorentz transformation of our universe (which is essentially a rotation matrix in hyperbolic space). As seen from the spacetime interval expression, equations in our universe can generally be converted to their Orthogonal counterparts via this substitution. Conversely, one can first perform relativistic derivations in the Orthogonal universe—which is intuitively much simpler geometrically—and then switch back to the conclusions for our universe via a pseudo-Wick rotation. This could be a “hack” for learning and understanding the theory of relativity!
Note that as long as \(v \Delta x / \Delta t < -1\) (i.e., the angle between the two vectors on the spacetime diagram is greater than 90 degrees), the signs of \(\Delta t'\) and \(\Delta t\) will be opposite. Therefore, the Lorentz transformation in the Orthogonal universe does not preserve chronological order, allowing for the violation of causality.
All 4-vectors obey the same Lorentz transformation, so the vertical and horizontal axes of the spacetime diagram can also be replaced with energy and momentum, or temporal frequency and spatial frequency. Half of the diagrams in the novels are spacetime diagrams of this kind, though many don't explicitly draw the axes. When 4-vectors in our universe rotate on the diagram, their lengths appear to change; this is because the spacetime diagram projects the non-Euclidean spacetime of our universe onto a flat plane. However, in the Orthogonal universe, spacetime is a 4-dimensional Euclidean space. Simply rotating a vector on the spacetime diagram completes the reference frame transformation, as well as the mutual conversion between physical quantities like time and space, or energy and momentum. Hence, the novels refer to special relativity as “rotational physics”.
Velocity addition is simply the composition of two rotations:
(Applying the substitution \(v \to iv\) yields the relativistic velocity addition formula for our universe—which is fundamentally the angle addition formula in hyperbolic space.)
For an object undergoing variable motion, its worldline is a curve. The tangent represents its proper time axis, the inverse of the tangent's slope corresponds to its velocity, and the length of the curve corresponds to the proper time experienced by the object. Now suppose the worldline is a circle of radius \(\rho\). From geometric relationships, the acceleration \(a = \frac{\mathrm{d}v}{\mathrm{d}t} = \frac{\mathrm{d}(\tan\theta)}{(\rho \mathrm{d}\theta)\cos\theta} = \frac{1}{\rho \cos^3\theta}\). In the instantaneous inertial frame where the object is momentarily at rest, \(\theta = 0\), so \(a = 1/\rho\): this is the proper acceleration experienced by the object itself. Thus, an object whose worldline is a circle will experience a constant equivalent gravity, the magnitude of which is the inverse of the circle's radius. (Transforming this conclusion to our universe: a curve on a spacetime diagram that maintains a constant distance from a fixed point is a hyperbola.)
In the worldline of the Peerless (I.p157), the voyage is divided into five phases:
In the Orthogonal universe's version of the twin paradox, the elapsed time for a straight worldline is shorter than for a curved one; the mountain can experience an arbitrarily long time, during which time on the home world does not pass at all.
The energy-momentum vector \(\boldsymbol{P} = (E, p)\) is a 4-vector (i.e., 4-momentum). Therefore, its length is a Lorentz invariant, remaining constant regardless of velocity or reference frame transformations. From the stationary case where \(p = 0\), we know this length is the mass (traditional textbooks and pop science often call it rest mass, but the novels and this text follow modern conventions, discarding the concept of relativistic mass). Thus, we obtain the relationship between energy, mass, and momentum:
This means the mass shell is a circle, rather than a hyperbola as in our universe. In the low-speed limit where \(p \to 0\), \(E = m - \frac{p^2}{2m}\), containing a rest energy term and a kinetic energy term with a sign opposite to classical mechanics results. (The novels sometimes call relativistic energy true energy to distinguish it from traditional energy with the opposite sign.) The greater the velocity, the smaller the energy. This implies that all matter has an inherent tendency toward instability; it can increase its kinetic or thermal energy simply by emitting light (I-6, 19). Mass sets the upper limit for energy and momentum; even accelerating to infinite velocity only requires kinetic energy equivalent to the mass. For a massless object, both energy and momentum must be 0, so all particles including photons must possess non-zero mass. (In our universe, the minus sign in the above equation becomes a plus sign, allowing energy and momentum to approach infinity, and permitting massless particles.)
This relationship between energy and velocity seems bizarre, but it becomes natural when you consider that “time” is essentially a fourth spatial dimension, and “energy” is essentially the component of momentum along that dimension. The magnitude of velocity represents the ratio of the spatial component to the temporal component as an object moves through 4D spacetime; therefore, naturally, the greater the velocity, the smaller the energy (because our universe is a non-Euclidean spacetime, the conclusion for this final step is exactly the opposite).
On a spacetime diagram, the conservation of energy and momentum is represented by the total 4-momentum of the initial and final states being equal—identical in both magnitude and direction. Now consider a rocket with an exhaust velocity \(v_e\). As it accelerates and loses mass, the 4-momentum vector representing the rocket rotates toward the spatial axis, and its length shortens; simultaneously, the arrow representing the exhaust rotates alongside it, maintaining a constant angle (see diagram in III-2)—because the rocket consistently ejects exhaust at a relative velocity \(v_e\). In a minuscule timeframe, if the rocket's mass decreases from \(m\) to \(m - \mathrm{d}m\), the sum of the arrow representing the rocket's final state and the exhaust arrow equals the rocket's initial state arrow, meaning the three form a triangle. (Note here that the length of the exhaust arrow is not \(\mathrm{d}m\), because mass is not conserved in chemical reactions—see I-6.) From geometric relationships, \(\mathrm{d}m \cdot v_e = \mathrm{d}m \tan\alpha = m \mathrm{d}\theta = m \mathrm{d}(\tan^{-1}v)\). Integrating this gives the relativistic rocket equation:
(Using a pseudo-Wick rotation to change the trigonometric function to a hyperbolic function yields the corresponding equation for our universe: \(\ln(m_i / m_f) = \frac{\tanh^{-1}v}{v_e}\).) When \(v\) reaches infinity, the ratio of initial to final mass is \(e^{\pi / 2v_e}\). I.p165 states that burning 2/3 of the mass can accelerate the ship to infinity; this requires an exhaust velocity of 1.4, which corresponds to ultraviolet light (see next section). Having gas particles exceed the speed of light seems unrealistic, but for a system with a positive temperature, the average particle speed would naturally be greater than 1 (see Macroscopic Phenomena section).
Note that in the special case where \(\mathrm{d}m = 0\), the initial state, final state, and exhaust can still form a triangle. This is the “eternal flame”, which can continue to emit light and accelerate without consuming fuel.
So far we only discussed the transformation of scalars and vectors under 4-dimensional rotations. Spinors follow a different rule (see Quantum Mechanics section). For a physical law to be self-consistent, its equations must satisfy Lorentz covariance; that is, all physical quantities in the equation must transform cooperatively under 4D rotation according to their respective transformation rules, while the mathematical form of the equation remains unchanged.
The temporal frequency and spatial frequency of light form a 4-vector \((\nu, \kappa)\), so its length is a constant, which is the maximum value of \(\nu\) when \(\kappa=0\):
Therefore, light has an upper limit for frequency and a lower limit for wavelength. (I-5, Appendix II). From this dispersion relation, the group velocity can be calculated as \(c_g = -d\nu / d\kappa = \kappa / \nu = -1 / c_p\). It is opposite in direction to the phase velocity, and their magnitudes are reciprocals of each other; the two are perpendicular on a spacetime diagram. (II-6; a simulator can be found here)
These relationships can also be derived geometrically. In the diagram below, assuming the spacetime interval between wavefronts is fixed at 1, the pulse direction remains perpendicular to them, and different speeds of light correspond to different tilt directions of the wavefronts on the spacetime diagram. The spatial and temporal intervals of the wavefronts are the wavelength and period, which are AB and AC in the diagram; their ratio is the phase velocity, and according to similar triangles, its length equals BE. The spatial and temporal intervals of the pulse are AD and DE in the diagram; their ratio is the group velocity, and according to similar triangles, its length equals CE. By the geometric mean theorem, BE \(\cdot\) CE = 1.
The speed of light mentioned in the novels defaults to the group velocity. Thus we have \(\nu^2 + \kappa^2 = \nu^2(1 + c^2) = \nu_{\max}^2\), so the speed of light must vary with frequency. The speed of light can be any value from 0 to infinity, with the frequency reaching its maximum when the speed is 0. The author sets the visible light speed range from 0.53 (called red light) to 1.33 (called purple light). Light with a speed of 1 is called blue light; conversely, the speed of blue light \(c_b\) is the constant we set to 1 in the geometrized units of the Orthogonal universe (I.p96). To transform the formulas in this article into familiar forms and apply them to specific numerical calculations, simply insert the appropriate powers of \(c_b\) to make the equation dimensionally sound—for example, “the length of a 4-momentum is its mass” would become \(E^2 + p^2 c_b^2 = m^2 c_b^4\). Thus, in standard units, to maintain the Lorentz invariance of a 4-vector, a proportionality factor \(c_b\) is needed between its temporal and spatial components, hence the name “spacetime conversion factor”.
In quantum mechanics, a photon's 4-momentum as a particle and its 4-frequency as a wave are essentially the same thing, following the same Lorentz transformation, differing only by a coefficient: \((E, p) = h(\nu, \kappa)\), \(m_\gamma = h\nu_{\max} = \hbar\omega_{\max}\) (II-6, 18). This coefficient corresponds to Planck's constant, called Patrizia's constant in the novels. This is the physical meaning of the upper limit of light frequency—it corresponds to the photon mass \(m_\gamma\).
For light sources and receivers at different velocities, both parties perceive different speeds of light in their respective frames of reference, resulting in different colors, which is equivalent to the Doppler effect. The “light cone” rotates along with the reference frame (using the intuitive definition of a light cone: the collection of light seen and emitted by an observer; in our universe, its essence is a spacetime interval of zero, thus fundamentally different from the Orthogonal universe). Furthermore, from the perspective of both reference frames, the emission time should be earlier than the reception time; otherwise, causality is violated, and the light cannot be seen. Hurtler is an exception (see Starry Sky section), and there are more exceptions in the latter two books.
The inner product of the 4-wavevector (equal to \(2\pi\) times the 4-frequency) and the 4-position vector is also a Lorentz invariant, which is the phase \(\phi = 2\pi(\nu t + \kappa x) = \omega t + kx\). It also differs from our universe by a sign, so wavefronts of equal phase in the novels are all drawn as straight lines with negative slopes on the spacetime diagram.
The length of light's 4-wavevector is a constant, so the sum of the squares of the frequencies in the four dimensions is constant. From this, the light equation (called the “Nereo equation” in the novels) can be guessed as (I-6, 11):
where \(\boldsymbol{A}\) and \(\boldsymbol{J}\) are the 4-potential and 4-current density. It resembles Maxwell's equations but has two differences: the spatial terms have opposite signs, and there is an extra \(\omega_{\max}^2\) term. The former comes from the metric signature of the Orthogonal universe, and the latter comes from the non-zero photon mass—equivalent to the Proca equation. It is an elliptic partial differential equation, requiring boundary conditions in the time dimension rather than initial conditions to solve, creating prediction difficulties (I-8).
In the source-free case, the equation simplifies to \(\partial_t^2 \varphi + \partial_x^2 \varphi + \partial_y^2 \varphi + \partial_z^2 \varphi + \omega_{\max}^2 \varphi = 0\). Obviously, its solution is \(\varphi_0 e^{i(\boldsymbol{k} \cdot \boldsymbol{r} \pm \omega t)}\), where \(|\boldsymbol{k}|^2 + \omega^2 = \omega_{\max}^2\), and the positive and negative signs correspond to light moving forward and backward in time, respectively (retarded and advanced waves). Note that cases where \(|\boldsymbol{k}|\) or \(\omega\) is greater than \(\omega_{\max}\) still satisfy the equation; in this case, the frequency in some dimension must be imaginary, causing an exponential explosion of the wave. To avoid this, the universe must be finite in both time and space, and its topology must be a compact manifold like a 4-sphere (\(S^4\)) or a 4-torus (\(T^4\)) (I-6, 8, Afterword).
Without loss of generality, let's assume the universe is a 4-dimensional hypercube of side length \(L\) and satisfies periodic boundary conditions, making it equivalent to a 4-torus. The periodic boundary conditions force the frequency and wavevector to take discrete values: \(\omega = \frac{2\pi n_t}{L}, k_x = \frac{2\pi n_x}{L}, k_y = \frac{2\pi n_y}{L}, k_z = \frac{2\pi n_z}{L}\). Substituting into the equation yields \(n_t^2 + n_x^2 + n_y^2 + n_z^2 = (L\nu_{\max})^2 = (L/\lambda_{\min})^2\). For the equation to have a solution, \(L/\lambda_{\min}\) (i.e., the ratio of the universe's size to the lower limit of light's wavelength) must first be the square root of some positive integer. In number theory, it can be proven that a positive integer can always be decomposed into the sum of squares of four integers, and the number of decomposition methods is usually around the same order of magnitude as the number itself. Thus, the existence and abundance of solutions are guaranteed; the values of frequency and wavevector are almost continuous, but the total number is finite. For finite universes of other shapes, the ratio of the universe's size to the lower limit of light's wavelength similarly needs to satisfy a certain relationship to ensure the light equation has free wave solutions (I-8, III-14).
In the presence of a source, there are two types of charges with opposite signs, referred to in the novels as “source intensity”. Particles carrying a unit charge are called luxagens. The potential field around a stationary luxagen is corrugated, being a Coulomb field superimposed with an oscillation:
It can be seen that the force between like-sign luxagens is attractive at close range, and as distance increases, it oscillates back and forth between attraction and repulsion. The oscillation period is roughly \(\lambda_{\min})\); the corrugation of the potential field originates from the non-zero photon mass, and in fact, its expression can be directly obtained from the Yukawa potential via a pseudo-Wick rotation. A moving luxagen will radiate light with a frequency equal to its motion frequency. (I-11, 19, see also chapter Riemannian Electromagnetism on the companion website.)
The corrugated luxagen potential field makes constructive interference difficult, making it hard to form macroscopic static electricity and currents, hence there is no electrical or electronic equipment. Similarly, there are no metals; materials are various kinds of “stone”. Static attraction phenomena can only be observed on extremely small particles in a zero-gravity environment (I-16). It also implies that luxagens of the same sign and their electromagnetic interactions alone could potentially form atomic, molecular, and lattice structures, without the need to introduce other elementary particles or nuclear forces. Therefore, only two elementary particles appear in the novels: photons and luxagens.
On a spacetime diagram, the range visible to the human eye is the area enclosed by the past light cones of red light and purple light. For a celestial body with a velocity less than the speed of red light, only the segment of its worldline from purple to red lies within the visible light cone region. The projection of its two intersection points with the light cone region onto the spatial plane is the width of the star trail, and dividing this by the distance to the vertex of the cone gives the angular size. In the non-relativistic case where the transverse velocity \(v\) is much less than the speed of light, the angular size of the star trail equals the transverse distance the star moves during the time difference between the arrival of red and purple light, divided by the distance to the star, which is \(v/c_{\text{red}} - v/c_{\text{purple}} \approx 1.13 v/c_b\). It is easy to verify that the angular size of the star trail obtained geometrically from the spacetime diagram matches this result in the low-speed limit. If the star's speed is one-thousandth of the speed of blue light (a ratio comparable to our Sun's speed orbiting the galactic center), the angular size of the star trail is 4 arcminutes, making its color distinguishable to the naked eye.
The angular size of a star trail depends only on the star's velocity and is independent of distance. Therefore, for distant stars, the color of the star trail can be resolved, but the relative motion of the star cannot be observed; for very slow and very close celestial bodies (like the Object Tamara spots in II-2), the color of the star trail cannot be resolved, but its motion can be observed. In Book II, the angular size of the Object's star trail is less than the telescope's resolution, say 0.1 arcseconds (about twice the Hubble telescope's resolution), giving an upper limit for its transverse velocity of about 100 strides/pause. Since it moves 2 arc-pauses per day, the upper limit of its distance is about 8 gross severances. This is how Tamara made her estimation.
On the other hand, if not limited to visible light observation, the star trail would extend. From the formula above, if the observation spectrum is expanded to the ultraviolet limit where velocity is infinite, the angular size of the star trail becomes \(v/c_{\text{red}}\), only extending by half. However, the infrared limit end of the star trail would extend to infinity. Thus, using infrared imaging can resolve the star trails of ultra-low-speed celestial bodies, thereby determining their velocity and consequently their distance (II-5).
By measuring the angular position of any color in the star trail, its light speed can be obtained, while the corresponding wavelength for that color can be measured by a grating (called “light comb” in the book). This is the principle of the experimental scheme proposed by Yalda in I-3. Replacing the grating with a prism can accomplish the same experiment, but since the dispersion of the prism material is unknown, the deflection angle cannot be converted to a wavelength, so the prism must first be calibrated with a grating.
For celestial bodies moving faster than purple light, their worldline generally has four or two intersection points with the visible light cone region. The star trail is divided into two roughly symmetrical halves; from the outside to the center, the color shifts from purple toward red. In the former case, the two halves of the star trail are separated, with two red inner endpoints. In the latter case, the two halves merge into a single star trail, connecting to the other half before the center turns red. This is exactly what a Hurtler looks like—as the visible light cone region moves forward along the time axis, its intersection with a fixed horizontal worldline starts as a line gradually lengthening, with purple at both ends. As the line lengthens, the center gradually turns from purple to red, eventually splitting into two lines (I-7). However, from the perspective of the celestial body itself, half of the star trail is emitting light into the past, and the human eye cannot see such time-reversed light, so only half of the star trail is visible. When the two halves are not separated, the star trail will be purple at one end and cut off at a certain color at the other; these cutoff points caused by the arrow of time are coplanar in space, forming a colorful “transition circle” or “new horizon” in the celestial sphere (I-17). The two halves of a Hurtler are both visible because it interacts with space dust in the solar system that has an opposite time arrow.
For the case where the celestial body's speed is between red and purple light, see the analysis in I-15. Also see the video on the companion website.
During the voyage of the Peerless, at position 1, home cluster is within the visible light cone region, while the orthogonal cluster is not. At position 2, after reaching the light speed range, the orthogonal cluster enters the light cone region (the probe in I-12 operates on the same principle). At position 3, the velocity reaches infinity; at this time, only the forward-facing half of the celestial sphere (\(z' > 0\)) can see the home cluster, forming a “transition circle” at the intersection of the two hemispheres; the orthogonal cluster is only visible in the other half of the sky (\(z' < 0\)). Position 4 is “Ancestor Day”, where the starry sky matches what is seen from the home world. At position 5, similarly, only the forward-facing half of the celestial sphere (\(z'' < 0\)) contains the home cluster, but the side where the home world is located cannot be seen; the orthogonal cluster is invisible due to the opposite arrow of time, making the other half of the sky a “dark hemisphere”.
In statistical mechanics, temperature is defined by entropy: \(T = 1 / \frac{\partial S}{\partial E}\). In the Orthogonal universe, for a system with fixed particle number, when entropy changes as a function of energy, the state with maximum entropy is when worldlines show no preference for a specific direction, so “time” and “space” proportions are equal, and the average velocity is along the blue light direction \(v=1\). At this point, \(\frac{\partial S}{\partial E} = 0\), and the temperature is infinite. \(v < 1\) and \(v > 1\) correspond to negative and positive temperatures, respectively. A positive temperature (\(\frac{\partial S}{\partial E} > 0\)) system gains entropy when acquiring energy, while a negative temperature (\(\frac{\partial S}{\partial E} < 0\)) system gains entropy when losing energy. Therefore, under an entropy-increasing arrow of time, positive temperature systems tend to increase in energy, negative temperature systems tend to decrease in energy (increasing kinetic energy), and matter tends to emit light and heat. Energy will flow from negative temperature objects to positive temperature objects. (I-9, see also chapter Riemannian Thermodynamics on the companion website.) Its reverse process, where matter cools by absorbing light, will not occur naturally, as it is an entropy-decreasing process (I-6).
The average speed of particles in a positive temperature system is greater than the speed of blue light, easily exceeding planetary escape velocities. Therefore, the cores of stars are not gaseous, but solid (III-4). The absence of a speed limit also implies the non-existence of black holes.
The luxagen potential field acts as Coulomb attraction at very close distances, and also has minimum points at distances that are integer multiples of the lower limit of light's wavelength (I.p192 diagram). These two types of minimums form the basis of atoms and crystals, respectively: when other same-sign luxagens sit in the closeup deep abyss, they form atoms; when they sit in the farther valleys, they form lattices (I-16). However, luxagens radiate light when moving, which logically should make them accelerate faster and faster, releasing more and more light and heat, eventually tearing apart the atom or lattice. The problem Yalda raised in I-19 corresponds to the stability problem of the Rutherford atomic model—except in our universe, electrons slow down after radiating light and eventually fall into the nucleus. Both require quantum mechanics to resolve. However, in the Orthogonal universe, achieving matter stability is much more difficult—light energy and thermal energy have opposite signs, and energy conservation allows matter to spontaneously emit light and heat, which is a positive feedback process. Therefore, rocks, biological bodies, and chemical reactions are all highly prone to explosion, and plant photosynthesis is a controlled manifestation of this process.
A spherical shell composed of luxagens can almost perfectly cancel the external field when its radius is appropriate, thereby forming gas molecules (I-13, see also chapter Riemannian Electromagnetism on the companion website). Therefore, all gases are inert and do not participate in chemical reactions; combustion and animals do not need “oxygen”.
The requirement for matter stability prevents small molecules from existing stably (see Quantum Mechanics section for details), so there is no water; liquids are liquid crystals or “resins”, and highly fluid liquids would quickly emit light and explode (I-1, III-3).
Plant photosynthesis involves emitting light rather than absorbing it; soil and food are primarily used to provide free energy (low entropy). Because there is no electric current, animals can only use optical fibers as nerves (called “pathway” in Book II). Since there is no respiration or blood, humans can survive in a vacuum without equipment, needing only to solve heat dissipation issues.
Because emitting light and heat is a positive feedback process, cooling is crucial for biological bodies; they might even explode if bodily functions are impaired. There are no birds in this world, animals have no fur, and people do not wear clothes, all due to the cooling problem.
The reproductive method of the people in this alien world is for the mother to fission into four offspring (so octofurcate is an insult, though during famines they can only split into two), two females and two males, forming two pairs of cos. Each pair of co's then reproduces the next generation. The infertile males are only half the size of the females; they serve as triggers in the reproductive process and are responsible for raising the offspring (thus their genders do not exactly correspond to ours; “males” are in some sense closer to worker bees, though Book II implies the reproductive process might also involve sexual reproduction elements). The names of a pair of co's are also paired; the author uses Italian names with obvious masculine/feminine distinctions for translation. A female dying in childbirth is called the way of women, while a male dying naturally of old age is called the way of men. Naturally human can live up to their thirties, but traditional females mostly choose to reproduce between 12 and 16 years old (hence 1 generation), or they might delay it until after 24 (2 generations).
Yalda raised the problem of matter stability in I-19: according to classical theory, a luxagen oscillating in a potential well will radiate light at its orbital frequency, thereby gradually gaining more and more kinetic and thermal energy, so a solid cannot remain stable. To address this, quantum mechanics must be introduced, treating the luxagen as a wave, and linking the particle's energy to the wave's frequency via Patrizia's constant (equivalent to Planck's constant). Thus, a luxagen confined in a potential well is a standing wave; its states and corresponding energies can only take discrete values, and it only radiates light when transitioning between states, with the energy equal to the energy difference between states.
In our universe, the ground state is the state closest to the bottom of the potential well. A particle in the ground state cannot lower itself to a lower energy state by radiating light, so electrons do not fall into the nucleus, and matter remains stable. But in the Orthogonal universe, the signs of potential and kinetic energy are opposite to that of light energy; a particle can climb all the way up the potential valley by radiating light, so how can there be a stable state? The answer lies in the fact that photons in the Orthogonal universe have an energy upper limit (i.e., photon mass). Therefore, as long as the gap between two energy levels is much greater than this upper limit, multiple photons must be radiated simultaneously to climb to the upper level. At this point, the lower level can become a stable state. The requirement for matter stability is the existence of an energy gap several times the photon mass, making quantum transition a multi-photon process. This is a significant difference from our universe. Conversely, plants or liberators can lower the energy gap to around the photon mass, causing the matter to lose its stability.
Now let's guess the specific form of the wave equation. Patrizia's constant links the 4-momentum of the luxagen to the 4-wavevector of the luxagen wave, so \((E, p) = (\hbar\omega, \hbar k)\), and the corresponding operator is \((i\hbar\partial_t, i\hbar\partial_x, i\hbar\partial_y, i\hbar\partial_z) = (i\hbar\partial_t, i\hbar\nabla)\). In the low-speed approximation, energy, momentum, and potential energy satisfy \(E = m - |p|^2/2m - V' = -|p|^2/2m - V\) (the last step redefines the zero point of potential energy). Replacing energy and momentum with operators in the above equation yields \(i\hbar\partial_t \Psi = \frac{\hbar^2}{2m} \nabla^2 \Psi - V\Psi\), which is equivalent to the Schrödinger equation. Since the Orthogonal universe behaves identically to our universe in the non-relativistic limit, except that the sign of classical energy is reversed, as long as energy is defined oppositely to relativistic “true energy”, the Schrödinger equations for both are identical, and the results are the same. (Therefore, for most contents in II-15, 18, 21, 28, 37, you only need to reverse the energy sign to match our universe, which will not be elaborated here.) The potential field of a single luxagen can be approximated as a Coulomb potential at very close distances, and as a harmonic potential near the minimum points at integer multiples of light's minimum wavelength. Atoms and crystals correspond to these two cases respectively.
Like in our universe, luxagens are fermions, obeying the Pauli exclusion principle (called rule of one or Assunto principle in the novels); each quantum state can be occupied by at most one luxagen. This degeneracy pressure ensures that solids do not collapse into a single point.
Consider a \(Z\)-luxagen atom, approximating its potential field with a Coulomb potential and ignoring shielding effects. Its energy levels are the quasi-hydrogen atom levels \(-\frac{Z^2}{n^2} E_0\), and the degeneracy of each level is \(2n^2\). Here, \(E_0 = \frac{m_l \alpha^2}{4}\), determined by the luxagen mass \(m_l\) and the fine-structure constant \(\alpha\). When all energy levels are filled, \(Z = 2(1 + 4 + \cdots + n_{\max}^2) \approx \frac{2n_{\max}^3}{3}\) (assuming \(Z\) is large). The gap between the highest level and the next one is \(-\frac{Z^2}{(n_{\max}+1)^2} E_0 + \frac{Z^2}{n_{\max}^2} E_0 \approx \frac{4Z}{3} E_0\). The atomic stability condition requires this gap to be several times the photon mass. According to the novels' setting, this multiplier is not less than 4; meanwhile, the luxagen mass is \(3/10\) of the photon mass (this ratio was chosen to keep the number of luxagens required for a stable solid as small as possible), and the fine-structure constant is \(1/137\) like our universe. Thus, the stability condition is expressed as \(\frac{4Z}{3} \frac{m_l \alpha^2}{4} > 4m_\gamma\), rearranging gives \(Z > \frac{12}{\alpha^2} \frac{m_\gamma}{m_l} \approx 8 \times 10^5\).
Consider a 3D quantum harmonic oscillator; its energy levels are \((n_x + n_y + n_z + \frac{3}{2})\hbar\omega\), with a spacing of \(\hbar\omega\). For different combinations of \(n_x, n_y, n_z\), the quantum states are different, but the energy can be the same, meaning degeneracy exists (II.p78). For real crystals, the potential energy deviates from the harmonic potential, and a degenerate level expands into an energy band composed of many closely spaced levels, but the gap between adjacent bands remains roughly \(\hbar\omega\)—where \(\omega\) is determined by the lattice size. The final conclusion shows that the solid stability condition \(\hbar\omega > 4m_\gamma\) can be guaranteed when there are no fewer than 100 luxagens per lattice point, and the crystal radius is greater than 150 times the minimum wavelength of light. (See chapter Riemannian Quantum Mechanics on the companion website for details.) This means that a stably existing solid must contain at least 100 million luxagens. It can be seen that stable small atoms or small molecules do not exist. Therefore, small molecule liquids like water do not exist, and atomic spectra will be extremely complex.
Suppose the system is initially in a bound eigenstate \(|i\rangle\), and transitions to a free state \(|f\rangle\) under the perturbation of a light field \(V e^{i\omega t}\). For a single-photon process, Fermi's Golden Rule gives the transition probability per unit time as \(w = \frac{2\pi}{\hbar} |\langle f | V | i \rangle|^2 \delta(E_f - E_i - \hbar\omega)\). It can be seen that the transition probability is proportional to the light intensity, and the probability of not transitioning is \(e^{-wt}\). In Carla's tarnish experiment, one transition simultaneously emits four or five photons, making it a multi-photon process that requires higher-order perturbation treatment. The final conclusion is: for an \(n\)-photon process, the transition probability is proportional to the \(n\)-th power of the light intensity, and the number of untransitioned particles decays exponentially over time (II-18).
For a particle with spin \(s\), its possible spin projections are \(-s, -s+1, \cdots, s-1, s\), a total of \(2s+1\) types, which is its number of polarizations. Photon has a spin of 1 and three polarizations, so each polarizing filter attenuates 1/3 of natural light (I.p48, 83). These three polarizations are mutually orthogonal in space, and are orthogonal to the photon's worldline direction in 4D spacetime, so it is not a transverse wave in 3D space, but a transverse wave in a 4D sense. (In our universe, the photon mass is 0, and the longitudinal polarization state is suppressed, so light is a transverse wave with only two polarizations.) Luxagen has a spin of 1/2 and two polarizations: spin up and spin down (II-33).
The novels assume particles satisfy the spin-statistics theorem: the total wave function for multiple identical particles must be either symmetric (in which case particles have integer spin, called bosons) or antisymmetric (in which case particles have half-integer spin, called fermions) with respect to the exchange of any two particles (II-37). Luxagens are fermions, so they must either have an antisymmetric position wave function and a symmetric spin wave function, or a symmetric position wave function and an antisymmetric spin wave function. In the former case, luxagens tend to stay away from each other, increasing the inter-luxagen potential energy, thereby lowering the system's energy. Therefore, a state with uniform spin direction is a stable state. (II-39; this “exchange interaction” also exists in our universe, but the last two steps are reversed: when electrons stay away from each other, potential energy decreases, and energy decreases accordingly.)
The norm of a quaternion \(v = a + bi + cj + dk\) is \(|v| = \sqrt{a^2 + b^2 + c^2 + d^2}\). Therefore, 4-vectors in the Orthogonal universe can be represented by quaternions. For example, the norm of 4-momentum is \(|P| = \sqrt{E^2 + p_x^2 + p_y^2 + p_z^2} = m\).
In the Orthogonal universe, Lorentz transformations are rotations in 4D space, forming the SO(4) group. It can be represented by a rotation matrix (see Special Relativity section above), or by quaternions, using a pair of unit quaternions to left-multiply and right-divide the quaternion representing the 4-vector: \(v \to q_L v q_R^{-1}\) (Book II, Appendix III). The spin of a luxagen is a rotation in a 2D complex space, forming the SU(2) group. It can be represented by spinors, or by quaternions, using a unit quaternion to left-multiply the quaternion representing the left-handed or right-handed spinor: \(\psi_L \to q_L \psi_L\) or \(\psi_R \to q_R \psi_R\). This means that a rotation in 4D space can be decomposed into two rotations in 2D complex spaces. Note that when both \(q_L\) and \(q_R\) take negative signs, \(q_L v q_R^{-1}\) remains unchanged, which means that turning half a circle in the 2D complex space corresponds to turning a full circle in 4D space, i.e., the luxagen spin is 1/2. In group theory language, the direct product of two SU(2) groups forms a double cover of the SO(4) group.
Notice that \(\psi_L \psi_R^{-1}\) transforms as \(q_L \psi_L (q_R \psi_R)^{-1} = q_L (\psi_L \psi_R^{-1}) q_R^{-1}\), so it is also a 4-vector. The angle between two 4-vectors is fixed; assuming it is parallel to the 4-momentum, they only differ by a coefficient: \(P = m\psi_L \psi_R^{-1}\). Taking the norm verifies that the coefficient is indeed mass. Replacing momentum with operators and moving \(\psi_R\) to the left yields the Dirac equation in Weyl spinor representation. (II-33, see also chapter Riemannian Quantum Mechanics on the companion website.)
Quantum field theory in the Orthogonal universe is fundamentally the same as in ours. From its perspective, only fields are physical entities, particles are just excited states of fields, thus providing a natural explanation for wave-particle duality and particle indistinguishability (II-21, 37). Field theory must be compatible with relativity, so fields must possess Lorentz covariance. They can be classified according to the transformation rules they obey into scalar fields, vector fields, spinor fields, etc., with corresponding representative particles being spin-0 particles (none appear in the novels, but there is a hint on II.p251), spin-1 photons, and spin-1/2 luxagens, obeying the Klein-Gordon equation, Nereo equation, and Dirac equation, respectively.
The ground state of a quantum harmonic oscillator has a zero-point energy of \(\frac{1}{2}\hbar\omega\). The electromagnetic field (vector field) and luxagen field (spinor field) can be second-quantized into similar systems, and the sum of the zero-point energies of all modes constitutes the “vacuum energy”, which corresponds to the cosmological constant (see General Relativity section). Every mode of a free electromagnetic field contributes \(\frac{1}{2}\hbar\omega_i\) to the vacuum energy, while every mode of a free luxagen field contributes \(-\frac{1}{2}\hbar\omega_i\) (II-21, III-14, 33). In our universe, vacuum energy is an infinite value requiring renormalization; but in the Orthogonal universe, finite size leads to a finite number of modes, so the vacuum energy is also finite, with its specific sign depending on the ratio of the number of modes of the two types. The above discusses a “free vacuum” without interactions; a more precise calculation must include interactions, using Feynman diagrams (called “diagram calculus” in the novels) to calculate the contribution of virtual particle pair creation and annihilation to vacuum energy order by order (III-18, 20).
Consider the two diagrams in II-25; they are essentially Feynman diagrams. The first diagram describes positive and negative luxagens annihilating into a pair of photons, and the second describes the collision of a photon and a luxagen. However, the only difference between the two diagrams is a 90-degree rotation of the spacetime diagram, so they can essentially be viewed as the same process. Note that the arrow direction of the negative luxagen in the first diagram is opposite to the time direction, which means antiparticles can be viewed as particles traveling backward in time. Therefore, after the orthogonal cluster circles the universe in the time dimension, it becomes antimatter (I.p338). Photons are their own antiparticles, so they do not become antimatter and possess no arrows in the diagrams.
Time and space in the Orthogonal universe are fundamentally identical, so “the distinction between past, present, and future is only a stubbornly persistent illusion”. Spacetime can be viewed as a static 4D structure, like a tapestry, where time does not “flow”; however, any direction can be defined as “time”, cutting out a series of 3D slices orthogonal to it, and linking these slices together constitutes “time evolution”. Thus, we obtain time and space through a “3+1 decomposition” in a “block universe” (I-11, III-7; a demo can be found here).
Since spacetime is fundamentally the same, no underlying physical laws distinguish between time and space; any direction in 4D spacetime can be defined as “time”. However, in a certain locality of the universe, it is possible for the worldlines of all particles to follow roughly the same direction, making it a locally special direction. This special direction can further distinguish between forward and backward: worldlines gradually become chaotic and scatter in the forward direction (entropy increases), while they gradually converge and organize in the backward direction (entropy decreases). This is how the arrow of time (arrow of entropy) emerges. This “unnatural” ordered arrangement and one-way convergence of worldlines originates from a much more ordered initial state, namely a entropy minimum (equivalent to the Big Bang in our universe). Whether this seemingly “unnatural” entropy minimum is a “brute fact” or has a “natural” origin remains an unsolved mystery in our universe, and the main plot of Book III is to unravel this mystery in the Orthogonal universe. People believe time has a direction and entropy-increasing processes are irreversible because everything around them shares the same local entropy arrow, which dictates the direction of macroscopic physical processes and causality in human understanding—between two adjacent states, we habitually view the low-entropy state as the cause and the high-entropy state as the effect, making predictions accordingly. On the other side of the entropy minimum, the direction of the entropy arrow reverses, swapping “past” and “future”; yet from the perspective of an entity obeying that arrow, entropy is still increasing. Since the Orthogonal universe is finite, the arrows from these two sections will eventually meet and form a closed loop. Under conflicting entropy arrows, a macroscopic process might appear as an entropy increase from one side's perspective and an entropy decrease from the other's. It is also possible that from neither perspective is it an entropy increase—because worldlines no longer have a clear convergence direction—making definitive predictions impossible, only that situations with more microstates are more likely to occur. (I.p86, p139, III-20~23)
Besides the entropy arrow (thermodynamic arrow) emerging from the statistical behavior of particle clusters, there is also an independent luxagen arrow (Nereo arrow) for single charged particles. The luxagen arrow does not change direction along the entire worldline, distinguishing between matter and antimatter. When the Peerless reaches infinite velocity, on its outward journey, its luxagen arrow is opposite to the orthogonal cluster, while their entropy arrows are the same. Therefore, the matter in the orthogonal cluster is antimatter and visible to Peerless (I.p338). On the return journey, both arrows reverse. Thus, travelers can safely visit the orthogonal worlds without annihilating, but they cannot see the light emitted by the orthogonal stars (violating causality) unless using a “time reversal camera”. And on the time-reversed orthogonal world, they experience many entropy-decreasing phenomena that appear causally inverted.
In the Orthogonal universe, worldlines can be closed, allowing for time travel. To avoid paradoxes, events must remain self-consistent. (This corresponds to closed timelike curves and the Novikov self-consistency principle in our universe.) Mathematically, this means the initial value problem becomes a boundary value problem, needing to satisfy both past and future boundary conditions simultaneously. This global constraint requires particles in every corner of the universe to “conspire”, leading to many counter-intuitive phenomena (II-26, III-5, 20~23). Furthermore, for given boundary conditions, there are infinitely many self-consistent solutions, leading to unpredictability. The novels resolve this through the “censorship hypothesis”, suggesting that different self-consistent solutions have different probabilities of occurring, and the spontaneous creation of complexity is far less likely (III-5, 8).
The Einstein field equations in the Orthogonal universe have the same form as in ours (the only difference is the metric signature):
where \(R_{\mu\nu}\), \(R\), \(g_{\mu\nu}\), and \(T_{\mu\nu}\) are the Ricci curvature tensor, scalar curvature, metric tensor, and energy-momentum stress tensor, respectively. It also yields Newton's law of universal gravitation in the classical limit (called “Vittorio theory” in the novels).
The gravitational deflection of light can be derived from the field equations. Suppose a particle at an infinite distance has a velocity \(v\) and grazes a star of mass \(M\) with an impact parameter \(b\) (called “offset” in the diagram in III.p181), causing a small-angle deflection. Classical theory gives the deflection angle as \(\frac{2M}{bv^2}\): the particle deflects toward the star, and the higher the velocity, the smaller the deflection angle. General relativity in our universe gives the deflection angle as \(\frac{2M}{bv^2}(1+v^2)\): when \(v\) is the speed of light, this is twice the classical theoretical value. Using a pseudo-Wick rotation, the deflection angle in the Orthogonal universe is \(\frac{2M}{bv^2}(1-v^2)\): when \(v < 1\) (reddish light), the deflection direction aligns with classical theory but with smaller magnitude; when \(v = 1\) (blue light), there is no deflection; when \(v > 1\) (purplish light), the deflection direction is opposite to classical theory (III-15, 20). In our universe, the deflection angle given by general relativity is only double the classical theoretical value, making it hard to distinguish in low-precision experiments, which is why the original Eddington experiment has always been controversial. In the Orthogonal universe, it is very easy to determine experimentally which theory is correct.
In cosmology, galaxies are treated as particles, composing an ideal fluid. In a comoving frame of reference, \(g_{\mu\nu} = \text{diag}(1,1,1,1)\) and \(T_{\mu\nu} = \text{diag}(\rho, p, p, p)\), where \(\rho\) and \(p\) are energy density and pressure, respectively. Taking the trace of the field equations yields: \(R - \frac{4R}{2} = -R = 8\pi(\rho + 3p)\). Substituting \(R\) back into the field equations gives \(R_{00} = 8\pi\rho + \frac{1}{2}R = 4\pi(\rho - 3p)\). In the low-entropy limit, all worldlines point along the time axis, so \(p = 0\); in the high-entropy limit, the four dimensions are indistinguishable, so \(p = \rho\). Therefore, the sign of \(R\) is opposite to \(\rho\): positive energy density causes negative scalar curvature, and negative energy density causes positive scalar curvature. When \(\rho > 0\), \(R_{00}\) is positive at low entropy, causing worldlines to converge in the time direction; at high entropy, it is negative, causing worldlines to diverge (III-4, 6).
For ordinary matter, typically \(\rho > 0\), but vacuum energy also contributes to energy density (equivalent to dark energy or adding a cosmological constant term to the field equations). Photons have positive zero-point energy, and luxagens have negative zero-point energy; the vacuum energy is the sum of all possible modes of both. In a 4-sphere (\(S^4\)) universe, the vacuum energy is positive, leading to negative scalar curvature. However, in a 4-torus (\(T^4\)) universe, the number of luxagen modes would increase by a factor of 16, making the vacuum energy negative, thereby making the universe's total energy density negative, which in turn leads to positive scalar curvature. On the other hand, a theorem in Riemannian geometry states that \(T^4\) does not permit positive scalar curvature, and while \(S^4\) permits negative scalar curvature, its sectional curvature cannot be non-positive everywhere. Therefore, the universe must be a 4-sphere with non-uniform curvature. This guarantees a entropy minimum (III-14, 33).
In our history, the refinement of electromagnetic laws gave birth to Maxwell's equations, thereby laying the foundation for special relativity; stoichiometry and Brownian motion provided scientific evidence for atomic theory; the deflection of cathode rays in electromagnetic fields led to the discovery of the electron; atomic spectra and the ultraviolet catastrophe became the catalysts for the establishment of quantum mechanics. In the Orthogonal universe, none of these pathways are feasible: because macroscopic electromagnetic fields do not exist naturally, electromagnetic laws and electron deflection cannot be discovered; because there are no small atoms and small molecules, Dalton's atomic theory cannot be born, nor do simple Rydberg formula relationships exist in spectra; because there is no water to create a “weightless” environment, Brownian motion of micro-particles can only be observed in space; because frequency has an upper limit, the ultraviolet catastrophe does not exist. Therefore, the novels envision unique paths of scientific development: vacuum dispersion of light → rotational physics (special relativity) → Nereo equation (Maxwell's equations) → luxagens (electrons), tarnish effect (photoelectric effect), Compton scattering → photons and wave-particle duality → quantum mechanics and Schrödinger equation → lasers → optical solids (artificial atoms) → spectral research → Zeeman effect → spin and Pauli exclusion principle → Dirac equation. Once these two cornerstones are established, the development of quantum field theory and general relativity is not much different from our universe.
The inhabitants of the alien world use a base-12 counting system and units (Appendix I of each volume). If we assume their stride and pause correspond roughly to our meter and second, their body sizes are similar to ours, while their reaction times and lifespans are only a fraction of ours. Under this assumption, the planet's radius, rotation period, orbital radius, orbital period, and gravity are 1/2, 1/4, 1/3, 1/3, and 2 times that of Earth, respectively. The speed of blue light is 3/4 of our speed of light, and the visible light wavelength range is on the same order of magnitude as ours. \(\nu_{\max}\) and \(\lambda_{\min}\) are approximately 400 THz and 500 nm, respectively. If we further assume that Patrizia's constant in the Orthogonal universe equals our Planck's constant, then the masses of photons and luxagens are on the order of \(10^{-36}\) kg, which is 5 orders of magnitude lower than our electron mass. The solid lattice spacing is about \(\lambda_{\min}/2\), which is 3 orders of magnitude larger than ours.
This section primarily discusses trivias in the novels. Many are highly specialized and profound; only brief hints can be provided here. Interested readers should have no difficulty finding detailed explanations with the help of AI.
1. p12 [worms...swept across the sky in a slow whirl but coming no closer to their destinations] This line shows how “star trails” in the Orthogonal universe differ from ours: the directions of the lines are not tangent to the circles in time-lapse photography formed by planet rotation.
3. p50 [changing the shape of that mirror] This concept is similar to the adaptive optics systems of astronomical telescopes, both using deformable mirrors to precisely correct the wavefront of light.
3. p52 [some portion of the trail’s spread of velocities should have been favored] Light in the Orthogonal universe has both transverse and longitudinal wave components. After a pair of crossed polarizing filters removes the transverse waves, the remaining “unpolarized light” is the longitudinal wave (but later, the longitudinal wave is treated as the third polarization). Thus, akin to the aether theory in our history, early wave theories considered light similar to mechanical waves, propagating through a solid medium. Following this analogy, the speeds of the longitudinal and transverse waves of light should differ, so the variation of light speed with color could be explained by different proportions of longitudinal and transverse waves; however, experiments refuted this. This implies that light is not a mechanical wave. See also p92.
3. p53 [Gemma and Gemmo] This is equivalent to Rømer first measuring the speed of light through the eclipses of Jupiter's moons in the 17th century.
5. p89 [light's velocity] Note that velocity of light here is group velocity, which is the reciprocal of the phase velocity obtained by dividing the wavelength by the period.
5. p93 [any kink of physics...would have to work just as well if you picked it up and rotated it in four dimensions] That is, physical laws must satisfy Lorentz covariance, remaining invariant under 4D rotation. The reasoning process in this chapter can be summarized as follows: from astronomical observations, the dispersion relation of light is obtained, finding that wavelength and light speed satisfy the equation \(\lambda^2 = a_1/c_g^2 + a_2\); choosing natural units so both coefficients equal 1, thus \(\lambda^2 = 1/c_g^2 + 1\); substituting \(\lambda = 1/\kappa\) and \(c_g = -d\nu/d\kappa\), solving yields \(\nu^2 + \kappa^2 = 1\) (in the novel, this formula is derived via geometric methods, detailed in Appendix II); geometrically, this indicates that light can be viewed as a set of wavefronts with a fixed spacing of 1. The “color” of light changes as the wavefronts rotate in the spacetime coordinate system, and the projection of the wavefront spacing onto the coordinate axes is the temporal frequency and spatial frequency; this shows that time and space are not absolute and can transform into each other in the form of rotations.
7. p106 [three and a half pauses...A gross and a half severances] Based on the speeds of red and purple light, the Hurtler's distance can be calculated to be about 237 severances, only 1% of the distance to the sun.
8. p113 [errors in your prediction] This discusses the well-posedness of partial differential equations, i.e., under what conditions the solution to the equation is existed, unique, and stable. The wave equation of a string is of hyperbolic type and is well-posed for initial conditions. The light equation in the Orthogonal universe is of elliptic type and is well-posed for boundary conditions; if only initial conditions are provided, tiny errors in the conditions will be exponentially amplified into massive errors in the solution.
8. p123 [the sum of squares] According to Lagrange's four-square theorem and Jacobi's four-square theorem, a positive integer \(n\) can definitely be decomposed into the sum of the squares of four integers, and the number of decomposition methods is 8 times the sum of all factors of \(n\) that are not divisible by 4.
8. p128 [largest proper factor] This number only has two large prime factors, equivalent to RSA encryption.
9. p144 [At about two-thirds of the maximum total energy] Macroscopic Phenomena section has already pointed out that entropy reaches its maximum when the particle velocity is 1. At this point, the momentum is \(\sqrt{2}/2\) times the mass (i.e., “about two-thirds of maximum total energy”).
9. p133 [Once it's creating light, it's lost to the ordinary world] A negative temperature system will continuously transfer energy to photons while simultaneously gaining an equal amount of kinetic energy. This positive feedback process of emitting light and heat will continue until the system's particles exceed the speed of blue light and become a positive temperature.
10. p144 [The time that passes on the rocket] The worldline of constant proper acceleration is a circle, with the radius equal to the reciprocal of the acceleration. In the novel, this radius is one year, from which it can be deduced that its gravitational acceleration is 20 strides per square pause.
12. p205 [crossed the void faster than anything but a Hurtler] This statement is actually inaccurate; it only has 4/5 of the speed of blue light.
12. p208 [probe] The probe's worldline is similar to that of the Peerless, with the difference that: because it only accelerates to 4/5 of the speed of blue light, the 1/4 circumference segments now become arcs with a central angle of \(\tan^{-1}(4/5)\); the two horizontal straight lines for the outward and return journeys now become two short diagonal lines tangent to the arcs (the diagram on this page draws the diagonal line phase of the outward journey). The total duration for the ground frame is \(4 \times 4 / \sqrt{4^2 + 5^2} \approx 2.5\) years, plus a few bells experienced during the diagonal line phases.
12. p218 [six pauses...ten dozen strides per pause faster] This also shows that the gravitational acceleration in this world is 20 strides per square pause.
15. p265 [an evenly ploughed field] See chapter Dual Pythagorean Theorem on the companion website; this is the basis for the derivation in Chapter 5 and Appendix II.
15. p268 [Definition of‘now’] This discusses whether the ground frame or the mountain frame should be used as the standard for simultaneity. If the former, the current duration in the ground frame is \(\sin 0.5 \approx 0.48\) years, thus 10 days short of half a year.
16. p289 [mottled appearance of sand under polarized light] This is actually using a polarizing microscope to distinguish single crystals from polycrystals.
16. p298 [question of stability] The tennis racket theorem states that a rigid body's rotation is only stable around its principal axes of maximum and minimum moments of inertia. If dissipation is included, only the axis with the maximum moment of inertia is a stable axis of rotation.
17. p302 [inverted bowl] The field of view of the flying mountain now corresponds to rotating the pyramid in the diagram on p273 by 90 degrees. Thus, the incident light from ordinary stars represented by the half of the pyramid exposed above ground is coming from the future direction and invisible. Therefore, from the mountain, star trails of ordinary stars can only be seen in the half celestial sphere in the forward direction. The intersection line between the pyramid and the ground corresponds to the “new horizon” seen from the mountain.
19. p337 [mechanical stability] See p227; mechanical stability requires the spherical shell to be at a potential energy minimum. The novels do not provide an answer to this problem; the author's explanation is that the external field of real gas molecules is not perfectly canceled, but is negligibly small.
19. p342 [retrace its path exactly] Bertrand's theorem points out that only the gravitational potential and the harmonic oscillator potential can form closed orbits. Therefore, as long as the potential energy is not of these two forms, the orbit will contain low-frequency components.
1. p8 [transition circle] i.e., the “new horizon” mentioned in I-17. See Starry Sky section.
3. p17 [Cornelio line] See the final diagram in I-9.
3. p25 [If not the wavelengths, what about the frequencies?] Note that the wavelength and frequency of light in the Orthogonal universe are not in a simple inverse proportion. Ivo's inability to immediately convert wavelength into frequency in Chapter 25 is for the same reason.
5. p35 [extend a visible color trail by a factor of a dozen...it would have to be moving at little more than a jogging pace] From the estimation in Starry Sky section, if it still cannot be resolved after the angular size of the star trail is expanded 12 times, the upper limit of the Object's relative speed is about 8 strides/pause.
5. p35 [we’ll slip out of orthogonality with the home world] Equivalent to tilting on the spacetime diagram by an angle whose tangent equals the Object's relative velocity divided by the speed of blue light (which is comparable to the angular size of the Object's star trail). From the previous text, it can be inferred that the upper limit of this tilt angle is about 0.1 arcseconds, so 2 million years on the mountain equaling 1 year on the home world, which is actually not a big problem. The main issue is that adjusting the mountain's speed is inconvenient.
6. p40 [ motion of the pulse is not the motion of the wavefronts] See the relationship between group velocity and phase velocity derived in Light section.
8. p55 [a three-year holiday] The turnaround phase lasts 3.14 years, during which the direction of gravity turns 90 degrees.
9. p60 [three arc-bells...the view perpendicular to the beam] 3 arc-bells means 90 degrees.
9. p63 [the valleys shallower...making the luxagens roll back and forth more slowly] Consider a luxagen oscillating in a potential valley \(V = Af(r)\). In the equation of motion, performing the time scale transformation \(t \to t/\sqrt{A}\) can eliminate \(A\). Therefore, the motion period is inversely proportional to \(\sqrt{A}\). If scattered light comes from such a luxagen, halving the incident light intensity logically should cause its frequency to change significantly.
10. p69 [brain stops repeating itself] This answers the final question in I-9: after the brain initiates fission, it is broken down and reabsorbed, but the fission process continues.
12. p82 [the greatest angle of deflection always turns out the same] This is consistent with the conclusions of classical and relativistic elastic scattering problems in our universe: the sine of the maximum deflection angle equals the mass ratio of the two particles (see Landau's Mechanics §17, Classical Theory of Fields §2-5). Substituting the data from the book yields a maximum deflection angle of \(\sin^{-1} 0.3 \approx 17.5^\circ \approx 7\) arc-chimes.
15. p103 [assumed that the luxagen would be moving slowly] This indicates the non-relativistic case, resulting in the Schrödinger equation.
15 The discussion in this chapter can be summarized as follows: applying the wave-particle duality of light to luxagens; using standing waves to explain quantization phenomena; constructing wave equations; solving the harmonic oscillator problem; explaining matter stability; superposition and time evolution of wave functions (to be elaborated in Chapter 18).
18. p125 [I make them as simple as possible...But no simpler.] This is a famous quote attributed to Einstein. Later in Chapter 28, Faraday's "what use is a newborn baby" is also adapted.
18. p125 [a luxagen with access to just two energy levels] This describes Rabi oscillations in a two-level system: under the drive of incident light, the system's transition between energy levels manifests as periodic oscillation.
18. p126 [just measured in different units] Hence the natural unit system where \(c_b=1, \hbar=1\).
18. p127 [free waves] If the final state of the transition is not a discrete energy level like a two-level system, but a continuous spectrum like free waves, the time dependence of the transition behavior is no longer Rabi oscillation, but exponential decay described by Fermi's Golden Rule: because there are infinitely many final states, once a transition to a final state occurs, it is very difficult to return to the initial state.
18. p128 [a rule of thumb] i.e., the statistical interpretation of the wave function. The "interactions" mentioned later serve the functions of decoherence and measurement.
18. p131 [the argument everyone had once thought settled in the days of Giorgio and Yalda was refusing to lie quietly in its grave] The wave-particle duality problem will be ultimately resolved in the quantum field theory developed in Chapter 21.
21. p143 [power series calculations] i.e., using higher-order perturbation to calculate transition probabilities.
21. p147 [subtract that value] Implies renormalization in quantum field theory by subtracting vacuum energy.
21. p148 [apply Patrizia’s principle for a second time] Implies second quantization. Assunto completed the second quantization of bosons; Onesto hopes to do the same for fermions.
22. p153 [echoes] The curved floor and ceiling of the workshop form two whispering galleries. Compare to its description in III-15.
22. p157 [the rendezvous point didn’t lie in that plane] The initial orbit of the Gnat is perpendicular to the Peerless's rotation axis, and the transition circle (horizon) is perpendicular to the Peerless's travelling direction, so the two are in the same plane.
23. p160 [geometric frequency shift] Equivalent to the Doppler effect; see Light section.
23. p164 [If you swap its positive luxagens for negative ones...grains of the two minerals might stick together half a wavelength closer than usual] See the diagrams in I-19. Potential energy reaches a minimum when the distance between a pair of like-sign luxagens is approximately \(\lambda_{\min}\), and when the distance between a pair of opposite-sign luxagens is approximately \(\lambda_{\min}/2\).
23. p166 [The sky from the Peerless need never be flat again...benefits of parallax] Because the parallax on the mountain is too small, only angles can be measured, making it difficult to measure distance (see Chapter 2), so the sky can be said to be two-dimensional.
23. p167 [a family of lines that would complete the elegant geometrical construction] In 3D space, two points moving at constant speeds form a pair of affine point rows, and the lines connecting corresponding points envelope a hyperbolic paraboloid.
25. p189 [tilted its energy-momentum vector steeply enough to make it lie at three tenths its original height] i.e., the diagram on the next page. However, the diagram exaggerates the tilt angle of the luxagen vector: for a luxagen with a velocity much less than the speed of light, the vector can be considered vertical. Therefore, the height of the photon vector generated by annihilation equals the length of the luxagen vector, which is 3/10 of the length of the photon vector.
28. p215 [It's that difficult?] The Orthogonal universe lacks simple monatomic systems like hydrogen atoms and noble gases. The constituent units of gases and solids all contain millions of luxagens, so the spectra are extremely complex. At the end of Chapter 31, Carla envisions artificially constructing the “simplest solid system” to facilitate spectroscopy.
29. p219 [spread horizontally] Genetic genes use light as a carrier, spreading like memes, so horizontal gene transfer is common, even across species.
31. p230 [feed the luxagens in at one end] The problem Romolo posed corresponds to initial state preparation in quantum state manipulation. Patrizia's solution is consistent with Carlo's concept of the light recorder: converting temporal signals into spatial signals.
32. p236 [Breeding is an exchange of information.] This hints at their method of inheritance. In daily life foreign genes are acquired through horizontal gene transfer; during mating, males may introduce some genes into females via light signals. This form of reproduction can be seen as a combination of asexual and sexual reproduction.
33. p251 [imagine a case where they’re parallel instead] The cases where the field vector is perpendicular and parallel to the energy-momentum vector correspond to the 4D transverse wave solution and the longitudinal wave (scalar wave) solution of the Nereo equation, respectively. The corresponding particles are the photon and the spin-0 particle.
33. p252 [vector multiplication and division] This skips over a piece of science history mentioned in Appendix III and the Afterword: because 4-vectors and Lorentz transformations in the Orthogonal universe can be conveniently represented by quaternions, quaternion multiplication and division were used for 4-vector calculations shortly after the birth of rotational physics.
33 The discussion in this chapter can be summarized as follows: experiments show luxagens have two polarizations, requiring a physical quantity that satisfies covariance (i.e., the equation remains unchanged under 4D rotation, performed in this chapter by left-multiplying and/or right-dividing by unit quaternions) to describe them; this physical quantity cannot be a vector or a scalar, because they yield three polarizations or one; after decomposing a 4D rotation into rotations in two 2D complex planes, “leftor/rightor” can be constructed, which leads to a “new wave equation” satisfying covariance, namely the Dirac equation; finally, proving that leftor/rightor physically corresponds to the intrinsic angular momentum of the luxagen, i.e., spin. For details see here.
35. p275 [photon was effectively colliding with a significant portion of the mirrorstone] This discussion corresponds to the Pound-Rebka experiment, which used the Mössbauer effect to measure gravitational redshift.
35. p280 [transmuting all our spare calmstone into sunstone] I-11 mentions ancient alchemists craving for eternal flame. “Transmuting” is clearly borrowed from their terminology, just as our "nuclear transmutation" borrows from alchemical terms.
39. p330 [encrypted digest] Equivalent to a hash code and digital signature.
44. p374 [testing the luxagen field theory] In our universe, quantum field theory is also the most successful theory in predictive accuracy, with agreement with experiments reaching up to 12 digits.
2. p7 [it’s not clear how we can begin to decelerate in the approach to the home world] The first half of the final 1/4 circle in the diagram requires the rocket exhaust to point toward the past to produce the necessary acceleration. Therefore, the exhaust arrows for this part are not drawn; instead, it is labeled "How?". Readers might recall that Benedetta's probe in I-12 successfully returned, but its flight path did not include a complete circle, so this problem did not exist for it.
2. p8 [the fastest radiation we can actually detect] After the mountain decelerates to a certain extent, the orthogonal star becomes invisible. See Starry Sky section.
3. p24 [flight plan] The diagram shows the fastest flight path for rendezvous and docking with a uniformly accelerating target. For details, see chapter Soft Interception on the companion website.
6. p64 [A torus was Yalda’s preferred model] In I-8, the spherical universe model was rejected by Yalda because of its excessively strong "see a world in a grain of sand" prediction capability, but it is picked up again in this book. The author's explanation is: although in principle, for a spherical universe, measuring wave values in an arbitrarily small 3-sphere can extrapolate to the entire universe, this requires extremely high measurement precision and is therefore not truly practical.
6. p64 [a four-sphere with negative curvature...impossible geometrically] What Agata proved (referred to as the “theorems on sectional curvature” in Chapter 8) is equivalent to the Cartan-Hadamard theorem, which implies that the sectional curvature of \(S^4\) cannot be non-positive everywhere.
6. p68 [message system] Egan envisioned an almost identical system in his short story “The Hundred Light-Year Diary”.
8. p79 [cosmic censor] This concept is borrowed from the Cosmic Censorship Hypothesis proposed by Penrose.
14. p117 [assume the right kind of relationship between the mass of a photon and the dimensions of the cosmos] This refers to the condition for the Nereo equation to have free wave solutions. See Electromagnetism section.
14. p117 [trying to resurrect it] Alludes to Einstein's views on the cosmological constant.
17. p143 [the blazing rim of the home cluster star trails appeared horizontal as the Surveyor ascended toward the dark hemisphere] Currently the mountain's gravity comes from centrifugal force, so while on the mountain, “downward” is perpendicular to the mountain's axis, and the transition circle is in the vertical direction. When the Surveyor accelerates toward the dark hemisphere, the direction of gravity turns 90 degrees, making the transition circle horizontal.
18. p145 [a few stints since they’d passed the one-quarter mark in the duration of their outward journey] A quarter of the outward journey is 1.5 years. Reaching orthogonality takes 1.57 years, a difference of 0.07 years \(\approx 3\) stints.
18. p146 [About a stint before mid-turnaround, our line of simultaneity would have had just the right slope] From the diagram, the moment of reunion occurs in the mountain's present, meaning the mountain's line of simultaneity passes through the point of reunion. The slope of the first line of simultaneity is 2 years / 6 generations \(\approx 0.028\). On the semicircle of the turnaround phase, the corresponding arc length is 1 year \(\times 0.028 \approx 1.2\) stints.
18. p149 [true vacuum] This paragraph describes an “adiabatic switch”, starting from the vacuum state (ground state) of a non-interacting system and introducing interactions through an infinitely slow process to evolve into an interacting vacuum.
18. p150 [Sanctified by the ancestors’ gaze or not] Currently home cluster cannot be seen from the Peerless, but can be seen from the Surveyor. See Starry Sky section.
19. p155 [four-dimensional polyhedron] The author's official website has a 3D version of this program.
20. p170 [laws of conservation] Virtual particles in Feynman diagrams can violate conservation laws.
20. p176 [that principle] Namely, Feynman's explanation of Fermat's principle using path integrals.
20. p181 [pity the poor cosmos] This adapts an anecdote about Einstein: when asked what he would do if the Eddington experiment disproved general relativity, he said, “Then I would feel sorry for the dear Lord. The theory is correct anyway.”
21. p191 [The closer we can stay to thermal equilibrium, the more predictable things should be.] The equilibrium state has the maximum number of microstates and the highest probability of occurrence.
22. p209 [measure the detailed distribution of thermal vibrations in the soil they’d brought with them] This discusses "Loschmidt's demon." The microscopic positions and velocities of all particles in the soil determine its future evolution, which macroscopically manifests as the arrow of time, i.e., entropy increases in the future direction. If a demon reverses the velocities of all particles, its arrow of time would reverse.
25. p215 [The outlets will have to be on the base of the mountain] The message system must be aimed at the dark hemisphere, which is opposite to the travelling direction.
26. p259 [there couldn’t be a deeper level] Referring to Bell's Theorem regarding hidden variable theories.
28. p288 [a stable ball of luxagens] See I-13.
28. p289 [signal was scrambled beyond recovery] This is similar to atmospheric phase noise, where variations in atmospheric refractive index cause signal phase jitter.
28. p289 [rise toward the axis and diffuse] This is how enriched uranium centrifuges work.
28. p289 [each light wave that passed over it distorted its shape sufficiently to spoil the usual cancellation between the luxagens, and the secondary wave generated by that process combined with the first to slow it down] This describes the microscopic mechanism of the refractive index: electromagnetic waves polarize the medium, driving charged particles to vibrate, and the resulting secondary waves interfere with the original incident wave, ultimately forming a slower wave.
33. p338 [more complicated experiments] Alludes to neutron interferometry experiments.
33. p340 [additional structure...sixteen possibilities] Namely, spin structure. \(S^n\) has only 1 spin structure, while \(T^n\) has \(2^n\).
33. p340 [you can’t have a torus that’s positively curved everywhere] This is a theorem proven by Richard Schoen and Shing-Tung Yau.