Orthogonal is a science fiction trilogy by Australian author Greg Egan taking place in a universe where, rather than three dimensions of space and one of time, there are four fundamentally identical dimensions.[1] While the characters in the novels always perceive three of the dimensions as space and one as time, this classification depends entirely on their state of motion, and the dimension that one observer considers to be time can be seen as a purely spatial dimension by another observer.
The plot involves the inhabitants of a planet that comes under threat from a barrage of high-velocity meteors known as 'hurtlers', who launch a generation ship that exploits the distinctive relativistic effects present in this universe which allow far more time to elapse on the ship than passes on the home world, in order for the ship's inhabitants to have enough time to develop the technology needed to protect the planet. The three novels deal with a succession of increasingly advanced scientific discoveries, as well as a number of radical social changes in the culture of the generation ship's passengers.
Technically, the space-time of the universe portrayed in the novels has a positive-definite Riemannian metric, rather than a pseudo-Riemannian metric, which is the kind that describes our own universe.
Greg Egan is an Australian science fiction writer and mathematician, best known for his works of hard science fiction. Egan has won multiple awards including the John W. Campbell Memorial Award, the Hugo Award, and the Locus Award.
Paul J. McAuley is a British botanist and science fiction author. A biologist by training, McAuley writes mostly hard science fiction. His novels dealing with themes such as biotechnology, alternative history/alternative reality, and space travel.
Vector calculus, or vector analysis, is a type of advanced mathematics that has practical applications in physics and engineering. It is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space The term vector calculus is sometimes used as a synonym for the broader subject of multivariable calculus, which spans vector calculus as well as partial differentiation and multiple integration. Vector calculus plays an important role in differential geometry and in the study of partial differential equations. It is used extensively in physics and engineering, especially in the description of electromagnetic fields, gravitational fields, and fluid flow.
The world line of an object is the path that an object traces in 4-dimensional spacetime. It is an important concept of modern physics, and particularly theoretical physics.
The Eternal Champion is a fictional character created by British author Michael Moorcock and is a recurrent feature in many of his speculative fiction works.
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as smooth manifolds with a Riemannian metric. This gives, in particular, local notions of angle, length of curves, surface area and volume. From those, some other global quantities can be derived by integrating local contributions.
In mathematical physics, Minkowski space combines inertial space and time manifolds with a non-inertial reference frame of space and time into a four-dimensional model relating a position to the field.
In the mathematical field of Riemannian geometry, the scalar curvature is a measure of the curvature of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the geometry of the metric near that point. It is defined by a complicated explicit formula in terms of partial derivatives of the metric components, although it is also characterized by the volume of infinitesimally small geodesic balls. In the context of the differential geometry of surfaces, the scalar curvature is twice the Gaussian curvature, and completely characterizes the curvature of a surface. In higher dimensions, however, the scalar curvature only represents one particular part of the Riemann curvature tensor.
In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the requirement of positive-definiteness is relaxed.
In mathematics, a homogeneous space is, very informally, a space that looks the same everywhere, as you move through it, with movement given by the action of a group. Homogeneous spaces occur in the theories of Lie groups, algebraic groups and topological groups. More precisely, a homogeneous space for a group G is a non-empty manifold or topological space X on which G acts transitively. The elements of G are called the symmetries of X. A special case of this is when the group G in question is the automorphism group of the space X – here "automorphism group" can mean isometry group, diffeomorphism group, or homeomorphism group. In this case, X is homogeneous if intuitively X looks locally the same at each point, either in the sense of isometry, diffeomorphism, or homeomorphism (topology). Some authors insist that the action of G be faithful, although the present article does not. Thus there is a group action of G on X which can be thought of as preserving some "geometric structure" on X, and making X into a single G-orbit.
In mathematics, the Chern theorem states that the Euler–Poincaré characteristic of a closed even-dimensional Riemannian manifold is equal to the integral of a certain polynomial of its curvature form.
In mathematics, the signature(v, p, r) of a metric tensor g (or equivalently, a real quadratic form thought of as a real symmetric bilinear form on a finite-dimensional vector space) is the number (counted with multiplicity) of positive, negative and zero eigenvalues of the real symmetric matrix gab of the metric tensor with respect to a basis. In relativistic physics, the v represents the time or virtual dimension, and the p for the space and physical dimension. Alternatively, it can be defined as the dimensions of a maximal positive and null subspace. By Sylvester's law of inertia these numbers do not depend on the choice of basis and thus can be used to classify the metric. The signature is often denoted by a pair of integers (v, p) implying r = 0, or as an explicit list of signs of eigenvalues such as (+, −, −, −) or (−, +, +, +) for the signatures (1, 3, 0) and (3, 1, 0), respectively.
In differential geometry, the Weyl curvature tensor, named after Hermann Weyl, is a measure of the curvature of spacetime or, more generally, a pseudo-Riemannian manifold. Like the Riemann curvature tensor, the Weyl tensor expresses the tidal force that a body feels when moving along a geodesic. The Weyl tensor differs from the Riemann curvature tensor in that it does not convey information on how the volume of the body changes, but rather only how the shape of the body is distorted by the tidal force. The Ricci curvature, or trace component of the Riemann tensor contains precisely the information about how volumes change in the presence of tidal forces, so the Weyl tensor is the traceless component of the Riemann tensor. This tensor has the same symmetries as the Riemann tensor, but satisfies the extra condition that it is trace-free: metric contraction on any pair of indices yields zero. It is obtained from the Riemann tensor by subtracting a tensor that is a linear expression in the Ricci tensor.
In mathematics, the isometry group of a metric space is the set of all bijective isometries from the metric space onto itself, with the function composition as group operation. Its identity element is the identity function. The elements of the isometry group are sometimes called motions of the space.
General relativity is a theory of gravitation developed by Albert Einstein between 1907 and 1915. The theory of general relativity says that the observed gravitational effect between masses results from their warping of spacetime.
The mathematics of general relativity is complex. In Newton's theories of motion, an object's length and the rate at which time passes remain constant while the object accelerates, meaning that many problems in Newtonian mechanics may be solved by algebra alone. In relativity, however, an object's length and the rate at which time passes both change appreciably as the object's speed approaches the speed of light, meaning that more variables and more complicated mathematics are required to calculate the object's motion. As a result, relativity requires the use of concepts such as vectors, tensors, pseudotensors and curvilinear coordinates.
The Clockwork Rocket is a hard science-fiction novel by Australian author Greg Egan and the first part of the Orthogonal trilogy. The novel was published by Night Shade Books on 1 July 2011 with a cover art by Cody Tilson and by Gollancz on 15 September 2011 with a cover art by Greg Egan. The novel describes an alien civilization being threatened by the appearance of hurtling meteors entering their planetary system with an unprecedented speed and the implementation of an unusual plan: All the technology needed for an effective defense shall be developed on board of a generation ship launched into the void while only a few years pass back on the home world in the meantime due to time dilation. This is possible due to different laws for space and time in this universe, in which they have the same signature instead of different ones, or which is alternatively described by a Riemannian instead of a Lorentzian manifold. The consequences on some of the physical concepts needed in the novel including time dilation and radiation, are described by Greg Egan with diagrams in the novel and also his website. The story is continued in The Eternal Flame and The Arrows of Time.
Death's End is a science fiction novel by the Chinese writer Liu Cixin. It is the third novel in the trilogy titled Remembrance of Earth's Past, following the Hugo Award-winning novel The Three-Body Problem and its sequel, The Dark Forest. The original Chinese version was published in 2010. Ken Liu translated the English edition in 2016. It was a finalist for the 2017 Hugo Award for Best Novel and winner of the 2017 Locus Award for Best Science Fiction Novel.
The Eternal Flame is a hard science-fiction novel by Australian author Greg Egan and the second part of the Orthogonal trilogy. The novel was published by Night Shade Books on 26 August 2012 with a cover art by Cody Tilson and by Gollancz on 8 August 2013 with a cover art by Greg Egan. The novel describes the journey of the generation ship Peerless, which has departed in The Clockwork Rocket, and the development of new technology as well as changes of the society on board. An essential task is the construction of an engine not needing any fuel to generate thrust, but instead perfectly balancing out the radiation it emits with the energy this generates. To make such a process work, the universe of the novel is based on a Riemannian instead of a Lorentzian manifold, changing the rules of physics. The details are described by Greg Egan on his website. The story is continued in The Arrows of Time.
Dichronauts is hard science-fiction novel by Australian author Greg Egan. The novel was published by Night Shade Books on 11 July 2017. The novel describes a universe with two time dimensions, one of which correcponds to the time perception of the characters while the other influences their space perception, for example by rotations in this directions to be impossible. Hence a symbiosis of two life forms is necessary, so that they can even see in all directions. Furthermore, many fundamental laws of physics are altered crucially: Objects can roll uphill or not fall over any more when oriented suitably. There is negative kinetic energy and a fourth state of matter. Planets are no longer spherical, but hyperbolic and therefore have three separate surfaces. The details are described by Greg Egan on his website.
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