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119-786: While other theories assert that consciousness emerges as the complexity of the computations performed by cerebral neurons increases, Orch OR posits that consciousness is based on non-computable quantum processing performed by qubits formed collectively on cellular microtubules, a process significantly amplified in the neurons. The qubits are based on oscillating dipoles forming superposed resonance rings in helical pathways throughout lattices of microtubules. The oscillations are either electric, due to charge separation from London forces , or magnetic, due to electron spin —and possibly also due to nuclear spins (that can remain isolated for longer periods) that occur in gigahertz , megahertz and kilohertz frequency ranges. Orchestration refers to

238-543: A pair of black holes merging . The simplest type of such a wave can be visualized by its action on a ring of freely floating particles. A sine wave propagating through such a ring towards the reader distorts the ring in a characteristic, rhythmic fashion (animated image to the right). Since Einstein's equations are non-linear , arbitrarily strong gravitational waves do not obey linear superposition , making their description difficult. However, linear approximations of gravitational waves are sufficiently accurate to describe

357-674: A "non-computable" influence embedded in the Planck scale of spacetime geometry. Penrose claimed that such information is Platonic , representing pure mathematical truths, which relates to Penrose's ideas concerning the three worlds: the physical, the mental, and the Platonic mathematical world. In Shadows of the Mind (1994), Penrose briefly indicates that this Platonic world could also include aesthetic and ethical values, but he does not commit to this further hypothesis. The Penrose–Lucas argument

476-570: A body in accordance with Newton's second law of motion , which states that the net force acting on a body is equal to that body's (inertial) mass multiplied by its acceleration . The preferred inertial motions are related to the geometry of space and time: in the standard reference frames of classical mechanics, objects in free motion move along straight lines at constant speed. In modern parlance, their paths are geodesics , straight world lines in curved spacetime . Conversely, one might expect that inertial motions, once identified by observing

595-489: A candidate for the physical basis of the non-computable process that he hypothesized to exist in the brain. However, he disliked the random nature of environmentally induced collapse, as randomness was not a promising basis for mathematical understanding. Penrose proposed that isolated systems may still undergo a new form of wave function collapse, which he called objective reduction (OR). Penrose sought to reconcile general relativity and quantum theory using his own ideas about

714-448: A computational system is a complex object which consists of three parts. First, a mathematical dynamical system D S {\displaystyle DS} with discrete time and discrete state space; second, a computational setup H = ( F , B F ) {\displaystyle H=\left(F,B_{F}\right)} , which is made up of a theoretical part F {\displaystyle F} , and

833-560: A computer, or by considering small perturbations of exact solutions. In the field of numerical relativity , powerful computers are employed to simulate the geometry of spacetime and to solve Einstein's equations for interesting situations such as two colliding black holes. In principle, such methods may be applied to any system, given sufficient computer resources, and may address fundamental questions such as naked singularities . Approximate solutions may also be found by perturbation theories such as linearized gravity and its generalization,

952-508: A curiosity among physical theories. It was clearly superior to Newtonian gravity , being consistent with special relativity and accounting for several effects unexplained by the Newtonian theory. Einstein showed in 1915 how his theory explained the anomalous perihelion advance of the planet Mercury without any arbitrary parameters (" fudge factors "), and in 1919 an expedition led by Eddington confirmed general relativity's prediction for

1071-530: A curved generalization of Minkowski space. The metric tensor that defines the geometry—in particular, how lengths and angles are measured—is not the Minkowski metric of special relativity, it is a generalization known as a semi- or pseudo-Riemannian metric. Furthermore, each Riemannian metric is naturally associated with one particular kind of connection, the Levi-Civita connection , and this is, in fact,

1190-539: A curved geometry of spacetime in general relativity; there is no gravitational force deflecting objects from their natural, straight paths. Instead, gravity corresponds to changes in the properties of space and time, which in turn changes the straightest-possible paths that objects will naturally follow. The curvature is, in turn, caused by the energy–momentum of matter. Paraphrasing the relativist John Archibald Wheeler , spacetime tells matter how to move; matter tells spacetime how to curve. While general relativity replaces

1309-597: A gravitational field (cf. below ). The actual measurements show that free-falling frames are the ones in which light propagates as it does in special relativity. The generalization of this statement, namely that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames, is known as the Einstein equivalence principle , a crucial guiding principle for generalizing special-relativistic physics to include gravity. The same experimental data shows that time as measured by clocks in

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1428-471: A gravitational field— proper time , to give the technical term—does not follow the rules of special relativity. In the language of spacetime geometry, it is not measured by the Minkowski metric . As in the Newtonian case, this is suggestive of a more general geometry. At small scales, all reference frames that are in free fall are equivalent, and approximately Minkowskian. Consequently, we are now dealing with

1547-567: A large area of the brain. He further postulated that the action of this large-scale quantum activity is the source of 40 Hz gamma waves , building upon the much less controversial theory that gap junctions are related to the gamma oscillation. In a study Hameroff was part of, Jack Tuszyński of the University of Alberta demonstrated that anesthetics hasten the duration of a process called delayed luminescence, in which microtubules and tubulins re-emit trapped light. Tuszyński suspects that

1666-450: A massive central body M is given by A conservative total force can then be obtained as its negative gradient where L is the angular momentum . The first term represents the force of Newtonian gravity , which is described by the inverse-square law. The second term represents the centrifugal force in the circular motion. The third term represents the relativistic effect. There are alternatives to general relativity built upon

1785-779: A number of exact solutions are known, although only a few have direct physical applications. The best-known exact solutions, and also those most interesting from a physics point of view, are the Schwarzschild solution , the Reissner–Nordström solution and the Kerr metric , each corresponding to a certain type of black hole in an otherwise empty universe, and the Friedmann–Lemaître–Robertson–Walker and de Sitter universes , each describing an expanding cosmos. Exact solutions of great theoretical interest include

1904-488: A path to achieving general anesthesia". A highly disputed theory put forth in the mid-1990s by Hameroff and Penrose posits that consciousness is based on quantum vibrations in tubulin/microtubules inside brain neurons. Computer modeling of tubulin's atomic structure found that anesthetic gas molecules bind adjacent to amino acid aromatic rings of non-polar π-electrons and that collective quantum dipole oscillations among all π-electron resonance rings in each tubulin showed

2023-595: A predecessor to Orch OR in The Emperor's New Mind , coming to the problem from a mathematical viewpoint and in particular Gödel's theorem, but lacked a detailed proposal for how quantum processes could be implemented in the brain. Stuart Hameroff separately worked in cancer research and anesthesia , which gave him an interest in brain processes. Hameroff read Penrose's book and suggested to him that microtubules within neurons were suitable candidate sites for quantum processing, and ultimately for consciousness. Throughout

2142-453: A problem, however, as there is a lack of a self-consistent theory of quantum gravity . It is not yet known how gravity can be unified with the three non-gravitational forces: strong , weak and electromagnetic . Einstein's theory has astrophysical implications, including the prediction of black holes —regions of space in which space and time are distorted in such a way that nothing, not even light , can escape from them. Black holes are

2261-459: A prohibitive energy requirement. In 2022, a group of Italian physicists conducted several experiments that failed to provide evidence in support of a gravity-related quantum collapse model of consciousness, weakening the possibility of a quantum explanation for consciousness. Biology-based criticisms have been offered, including a lack of explanation for the probabilistic release of neurotransmitter from presynaptic axon terminals and an error in

2380-455: A prolonged excitation to diffuse through microtubules farther than expected, which did not occur when repeated under anesthesia. However, diffusion results have to be interpreted carefully, since even classical diffusion can be very complex due to the wide range of length scales in the fluid filled extracellular space. At high concentrations (~5 MAC ) the anesthetic gas halothane causes reversible depolymerization of microtubules. This cannot be

2499-618: A purely physical process occurring inside a closed physical system called a computer . Turing's 1937 proof, On Computable Numbers, with an Application to the Entscheidungsproblem , demonstrated that there is a formal equivalence between computable statements and particular physical systems, commonly called computers . Examples of such physical systems are: Turing machines , human mathematicians following strict rules, digital computers , mechanical computers , analog computers and others. An alternative account of computation

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2618-400: A real part B F {\displaystyle B_{F}} ; third, an interpretation I D S , H {\displaystyle I_{DS,H}} , which links the dynamical system D S {\displaystyle DS} with the setup H {\displaystyle H} . General relativity General relativity , also known as

2737-715: A relativistic theory of gravity. After numerous detours and false starts, his work culminated in the presentation to the Prussian Academy of Science in November 1915 of what are now known as the Einstein field equations, which form the core of Einstein's general theory of relativity. These equations specify how the geometry of space and time is influenced by whatever matter and radiation are present. A version of non-Euclidean geometry , called Riemannian geometry , enabled Einstein to develop general relativity by providing

2856-437: A rule. "Medium-independence" requires that the property can be instantiated by multiple realizers and multiple mechanisms, and that the inputs and outputs of the mechanism also be multiply realizable . In short, medium-independence allows for the use of physical variables with properties other than voltage (as in typical digital computers); this is imperative in considering other types of computation, such as that which occurs in

2975-625: A spectrum with a common mode peak at 613 T Hz . Simulated presence of 8 different anesthetic gases abolished the 613 THz peak, whereas the presence of 2 different nonanesthetic gases did not affect the 613 THz peak, from which it was speculated that this 613 THz peak in microtubules could be related to consciousness and anesthetic action. Another study that Hameroff was a part of claims to show "anesthetic molecules can impair π-resonance energy transfer and exciton hopping in 'quantum channels' of tryptophan rings in tubulin, and thus account for selective action of anesthetics on consciousness and memory". In

3094-502: A study published in August 2024, an undergraduate group led by a Wellesley College professor found that rats given epothilone B , a drug that binds to microtubules, took over a minute longer to fall unconscious when exposed to an anesthetic gas. Orch OR has been criticized both by physicists and neuroscientists who consider it to be a poor model of brain physiology . Orch OR has also been criticized for lacking explanatory power ;

3213-466: A stunning signature of superradiance in a micron-scale biological system." Marlan Scully , a physicist well-known for his work in the field of theoretical quantum optics, said "We will certainly be examining closely the implications for quantum effects in living systems for years to come." The study states that "by analyzing the coupling with the electromagnetic field of mega-networks of tryptophans present in these biologically relevant architectures, we find

3332-490: A university matriculation examination, and, despite the shortness of the book, a fair amount of patience and force of will on the part of the reader. The author has spared himself no pains in his endeavour to present the main ideas in the simplest and most intelligible form, and on the whole, in the sequence and connection in which they actually originated." General relativity can be understood by examining its similarities with and departures from classical physics. The first step

3451-465: A warm and noisy environment, in which quantum effects typically are not expected to take place. The results of the study were theoretically predicted and then experimentally confirmed by the researchers. Majed Chergui , who led the experimental team, stated that "It's a beautiful result. It took very precise and careful application of standard protein spectroscopy methods, but guided by the theoretical predictions of our collaborators, we were able to confirm

3570-539: A wave train traveling through empty space or Gowdy universes , varieties of an expanding cosmos filled with gravitational waves. But for gravitational waves produced in astrophysically relevant situations, such as the merger of two black holes, numerical methods are presently the only way to construct appropriate models. General relativity differs from classical mechanics in a number of predictions concerning orbiting bodies. It predicts an overall rotation ( precession ) of planetary orbits, as well as orbital decay caused by

3689-526: Is Minkowskian , and the laws of physics exhibit local Lorentz invariance . The core concept of general-relativistic model-building is that of a solution of Einstein's equations . Given both Einstein's equations and suitable equations for the properties of matter, such a solution consists of a specific semi- Riemannian manifold (usually defined by giving the metric in specific coordinates), and specific matter fields defined on that manifold. Matter and geometry must satisfy Einstein's equations, so in particular,

Orchestrated objective reduction - Misplaced Pages Continue

3808-423: Is a scalar parameter of motion (e.g. the proper time ), and Γ μ α β {\displaystyle \Gamma ^{\mu }{}_{\alpha \beta }} are Christoffel symbols (sometimes called the affine connection coefficients or Levi-Civita connection coefficients) which is symmetric in the two lower indices. Greek indices may take the values: 0, 1, 2, 3 and

3927-445: Is a universality of free fall (also known as the weak equivalence principle , or the universal equality of inertial and passive-gravitational mass): the trajectory of a test body in free fall depends only on its position and initial speed, but not on any of its material properties. A simplified version of this is embodied in Einstein's elevator experiment , illustrated in the figure on the right: for an observer in an enclosed room, it

4046-402: Is based on the propagation of light, and thus on electromagnetism, which could have a different set of preferred frames . But using different assumptions about the special-relativistic frames (such as their being earth-fixed, or in free fall), one can derive different predictions for the gravitational redshift, that is, the way in which the frequency of light shifts as the light propagates through

4165-407: Is close enough for the tubulin π electrons to become quantum entangled . During entanglement, particle states become inseparably correlated. Hameroff originally suggested in the fringe Journal of Cosmology that the tubulin-subunit electrons would form a Bose–Einstein condensate . He then proposed a Frohlich condensate , a hypothetical coherent oscillation of dipolar molecules. However, this too

4284-499: Is curved. The resulting Newton–Cartan theory is a geometric formulation of Newtonian gravity using only covariant concepts, i.e. a description which is valid in any desired coordinate system. In this geometric description, tidal effects —the relative acceleration of bodies in free fall—are related to the derivative of the connection, showing how the modified geometry is caused by the presence of mass. As intriguing as geometric Newtonian gravity may be, its basis, classical mechanics,

4403-405: Is defined in the absence of gravity. For practical applications, it is a suitable model whenever gravity can be neglected. Bringing gravity into play, and assuming the universality of free fall motion, an analogous reasoning as in the previous section applies: there are no global inertial frames . Instead there are approximate inertial frames moving alongside freely falling particles. Translated into

4522-419: Is found throughout the works of Hilary Putnam and others. Peter Godfrey-Smith has dubbed this the "simple mapping account." Gualtiero Piccinini's summary of this account states that a physical system can be said to perform a specific computation when there is a mapping between the state of that system and the computation such that the "microphysical states [of the system] mirror the state transitions between

4641-445: Is impossible to decide, by mapping the trajectory of bodies such as a dropped ball, whether the room is stationary in a gravitational field and the ball accelerating, or in free space aboard a rocket that is accelerating at a rate equal to that of the gravitational field versus the ball which upon release has nil acceleration. Given the universality of free fall, there is no observable distinction between inertial motion and motion under

4760-560: Is known as gravitational time dilation. Gravitational redshift has been measured in the laboratory and using astronomical observations. Gravitational time dilation in the Earth's gravitational field has been measured numerous times using atomic clocks , while ongoing validation is provided as a side effect of the operation of the Global Positioning System (GPS). Tests in stronger gravitational fields are provided by

4879-404: Is mass. In special relativity, mass turns out to be part of a more general quantity called the energy–momentum tensor , which includes both energy and momentum densities as well as stress : pressure and shear. Using the equivalence principle, this tensor is readily generalized to curved spacetime. Drawing further upon the analogy with geometric Newtonian gravity, it is natural to assume that

Orchestrated objective reduction - Misplaced Pages Continue

4998-456: Is merely a limiting case of (special) relativistic mechanics. In the language of symmetry : where gravity can be neglected, physics is Lorentz invariant as in special relativity rather than Galilei invariant as in classical mechanics. (The defining symmetry of special relativity is the Poincaré group , which includes translations, rotations, boosts and reflections.) The differences between

5117-430: Is now associated with electrically charged black holes . In 1917, Einstein applied his theory to the universe as a whole, initiating the field of relativistic cosmology. In line with contemporary thinking, he assumed a static universe, adding a new parameter to his original field equations—the cosmological constant —to match that observational presumption. By 1929, however, the work of Hubble and others had shown that

5236-492: Is termed computable , while the statement or calculation itself is referred to as a computation . Turing's definition apportioned "well-definedness" to a very large class of mathematical statements, including all well-formed algebraic statements , and all statements written in modern computer programming languages. Despite the widespread uptake of this definition, there are some mathematical concepts that have no well-defined characterisation under this definition. This includes

5355-495: Is the Shapiro Time Delay, the phenomenon that light signals take longer to move through a gravitational field than they would in the absence of that field. There have been numerous successful tests of this prediction. In the parameterized post-Newtonian formalism (PPN), measurements of both the deflection of light and the gravitational time delay determine a parameter called γ, which encodes the influence of gravity on

5474-409: Is the parametrized post-Newtonian (PPN) formalism, which allows quantitative comparisons between the predictions of general relativity and alternative theories. General relativity has a number of physical consequences. Some follow directly from the theory's axioms, whereas others have become clear only in the course of many years of research that followed Einstein's initial publication. Assuming that

5593-471: Is the realization that classical mechanics and Newton's law of gravity admit a geometric description. The combination of this description with the laws of special relativity results in a heuristic derivation of general relativity. At the base of classical mechanics is the notion that a body 's motion can be described as a combination of free (or inertial ) motion, and deviations from this free motion. Such deviations are caused by external forces acting on

5712-602: The curvature of spacetime is directly related to the energy and momentum of whatever present matter and radiation . The relation is specified by the Einstein field equations , a system of second-order partial differential equations . Newton's law of universal gravitation , which describes classical gravity, can be seen as a prediction of general relativity for the almost flat spacetime geometry around stationary mass distributions. Some predictions of general relativity, however, are beyond Newton's law of universal gravitation in classical physics . These predictions concern

5831-432: The Einstein notation , meaning that repeated indices are summed (i.e. from zero to three). The Christoffel symbols are functions of the four spacetime coordinates, and so are independent of the velocity or acceleration or other characteristics of a test particle whose motion is described by the geodesic equation. In general relativity, the effective gravitational potential energy of an object of mass m revolving around

5950-609: The Gödel universe (which opens up the intriguing possibility of time travel in curved spacetimes), the Taub–NUT solution (a model universe that is homogeneous , but anisotropic ), and anti-de Sitter space (which has recently come to prominence in the context of what is called the Maldacena conjecture ). Given the difficulty of finding exact solutions, Einstein's field equations are also solved frequently by numerical integration on

6069-498: The Penrose–Lucas argument leaves the question of the physical basis of non- computable behaviour open. Most physical laws are computable, and thus algorithmic. However, Penrose determined that wave function collapse was a prime candidate for a non-computable process. In quantum mechanics , particles are treated differently from the objects of classical mechanics . Particles are described by wave functions that evolve according to

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6188-566: The Schrödinger equation . Non-stationary wave functions are linear combinations of the eigenstates of the system, a phenomenon described by the superposition principle . When a quantum system interacts with a classical system—i.e. when an observable is measured—the system appears to collapse to a random eigenstate of that observable from a classical vantage point. If collapse is truly random, then no process or algorithm can deterministically predict its outcome. This provided Penrose with

6307-539: The brain or in a quantum computer . A rule, in this sense, provides a mapping among inputs, outputs, and internal states of the physical computing system. In the theory of computation , a diversity of mathematical models of computation has been developed. Typical mathematical models of computers are the following: Giunti calls the models studied by computation theory computational systems, and he argues that all of them are mathematical dynamical systems with discrete time and discrete state space. He maintains that

6426-451: The execution of computer algorithms . Mechanical or electronic devices (or, historically , people) that perform computations are known as computers . Computer science is an academic field that involves the study of computation. The notion that mathematical statements should be 'well-defined' had been argued by mathematicians since at least the 1600s , but agreement on a suitable definition proved elusive. A candidate definition

6545-682: The field equation for gravity relates this tensor and the Ricci tensor , which describes a particular class of tidal effects: the change in volume for a small cloud of test particles that are initially at rest, and then fall freely. In special relativity, conservation of energy –momentum corresponds to the statement that the energy–momentum tensor is divergence -free. This formula, too, is readily generalized to curved spacetime by replacing partial derivatives with their curved- manifold counterparts, covariant derivatives studied in differential geometry. With this additional condition—the covariant divergence of

6664-472: The general theory of relativity , and as Einstein's theory of gravity , is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics . General relativity generalizes special relativity and refines Newton's law of universal gravitation , providing a unified description of gravity as a geometric property of space and time , or four-dimensional spacetime . In particular,

6783-473: The post-Newtonian expansion , both of which were developed by Einstein. The latter provides a systematic approach to solving for the geometry of a spacetime that contains a distribution of matter that moves slowly compared with the speed of light. The expansion involves a series of terms; the first terms represent Newtonian gravity, whereas the later terms represent ever smaller corrections to Newton's theory due to general relativity. An extension of this expansion

6902-454: The scalar gravitational potential of classical physics by a symmetric rank -two tensor , the latter reduces to the former in certain limiting cases . For weak gravitational fields and slow speed relative to the speed of light, the theory's predictions converge on those of Newton's law of universal gravitation. As it is constructed using tensors, general relativity exhibits general covariance : its laws—and further laws formulated within

7021-429: The summation convention is used for repeated indices α {\displaystyle \alpha } and β {\displaystyle \beta } . The quantity on the left-hand-side of this equation is the acceleration of a particle, and so this equation is analogous to Newton's laws of motion which likewise provide formulae for the acceleration of a particle. This equation of motion employs

7140-571: The 1990s, the two collaborated on the Orch OR theory, which Penrose published in Shadows of the Mind (1994). Hameroff's contribution to the theory derived from his study of the neural cytoskeleton , and particularly on microtubules. As neuroscience has progressed, the role of the cytoskeleton and microtubules has assumed greater importance. In addition to providing structural support, microtubule functions include axoplasmic transport and control of

7259-413: The actual motions of bodies and making allowances for the external forces (such as electromagnetism or friction ), can be used to define the geometry of space, as well as a time coordinate . However, there is an ambiguity once gravity comes into play. According to Newton's law of gravity, and independently verified by experiments such as that of Eötvös and its successors (see Eötvös experiment ), there

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7378-401: The base of cosmological models of an expanding universe . Widely acknowledged as a theory of extraordinary beauty , general relativity has often been described as the most beautiful of all existing physical theories. Henri Poincaré 's 1905 theory of the dynamics of the electron was a relativistic theory which he applied to all forces, including gravity. While others thought that gravity

7497-688: The brain would undergo effective wave function collapse due to environmental interaction long before it could influence neural processes (the "warm, wet and noisy" argument, as it later came to be known). He determined the decoherence timescale of microtubule entanglement at brain temperatures to be on the order of femtoseconds , far too brief for neural processing. Christof Koch and Klaus Hepp also agreed that quantum coherence does not play, or does not need to play any major role in neurophysiology . Koch and Hepp concluded that "The empirical demonstration of slowly decoherent and controllable quantum bits in neurons connected by electrical or chemical synapses, or

7616-429: The calculated number of the tubulin dimers per cortical neuron. In 2014, Penrose and Hameroff published responses to some criticisms and revisions to many of the theory's peripheral assumptions, while retaining the core hypothesis. Computation A computation is any type of arithmetic or non-arithmetic calculation that is well-defined. Common examples of computation are mathematical equation solving and

7735-504: The cell's movement, growth and shape. Orch OR combines the Penrose–Lucas argument with Hameroff's hypothesis on quantum processing in microtubules. It proposes that when condensates in the brain undergo an objective wave function reduction, their collapse connects noncomputational decision-making to experiences embedded in spacetime's fundamental geometry. The theory further proposes that the microtubules both influence and are influenced by

7854-403: The computational states." Philosophers such as Jerry Fodor have suggested various accounts of computation with the restriction that semantic content be a necessary condition for computation (that is, what differentiates an arbitrary physical system from a computing system is that the operands of the computation represent something). This notion attempts to prevent the logical abstraction of

7973-766: The configuration of the microtubule lattice might be suitable for quantum error correction , a means of resisting quantum decoherence. In 2009, Reimers et al. and McKemmish et al., published critical assessments. Earlier versions of the theory had required tubulin-electrons to form either Bose–Einsteins or Frohlich condensates, and the Reimers group noted the lack of empirical evidence that such could occur. Additionally they calculated that microtubules could only support weak 8 MHz coherence. McKemmish et al. argued that aromatic molecules cannot switch states because they are delocalised; and that changes in tubulin protein-conformation driven by GTP conversion would result in

8092-406: The connection that satisfies the equivalence principle and makes space locally Minkowskian (that is, in suitable locally inertial coordinates , the metric is Minkowskian, and its first partial derivatives and the connection coefficients vanish). Having formulated the relativistic, geometric version of the effects of gravity, the question of gravity's source remains. In Newtonian gravity, the source

8211-551: The conventional activity at the synapses between neurons. Hameroff proposed that microtubules were suitable candidates for quantum processing. Microtubules are made up of tubulin protein subunits. The tubulin protein dimers of the microtubules have hydrophobic pockets that may contain delocalized π electrons . Tubulin has other, smaller non-polar regions, for example 8 tryptophans per tubulin, which contain π electron-rich indole rings distributed throughout tubulin with separations of roughly 2 nm. Hameroff claims that this

8330-560: The deflection of starlight by the Sun during the total solar eclipse of 29 May 1919 , instantly making Einstein famous. Yet the theory remained outside the mainstream of theoretical physics and astrophysics until developments between approximately 1960 and 1975, now known as the golden age of general relativity . Physicists began to understand the concept of a black hole, and to identify quasars as one of these objects' astrophysical manifestations. Ever more precise solar system tests confirmed

8449-430: The difference of the spacetime curvature of these states in the universe's fine-scale structure. Orchestrated objective reduction has been criticized from its inception by mathematicians, philosophers, and scientists. The criticism concentrated on three issues: Penrose's interpretation of Gödel's theorem ; Penrose's abductive reasoning linking non-computability to quantum events; and the brain's unsuitability to host

8568-420: The discovery of an efficient quantum algorithm for computations performed by the brain, would do much to bring these speculations from the 'far-out' to the mere 'very unlikely'." In response to Tegmark's claims, Hagan, Tuszynski and Hameroff claimed that Tegmark did not address the Orch OR model, but instead a model of his own construction. This involved superpositions of quanta separated by 24 nm rather than

8687-474: The emergence of collective quantum optical effects, namely, superradiant and subradiant eigenmodes. ... our work demonstrates that collective and cooperative UV excitations in mega-networks of tryptophans support robust quantum states in protein aggregates, with observed consequences even under thermal equilibrium conditions." In an experiment, Gregory D. Scholes and Aarat Kalra of Princeton University used lasers to excite molecules within tubulins, causing

8806-452: The emission of gravitational waves and effects related to the relativity of direction. In general relativity, the apsides of any orbit (the point of the orbiting body's closest approach to the system's center of mass ) will precess ; the orbit is not an ellipse , but akin to an ellipse that rotates on its focus, resulting in a rose curve -like shape (see image). Einstein first derived this result by using an approximate metric representing

8925-500: The end-state for massive stars . Microquasars and active galactic nuclei are believed to be stellar black holes and supermassive black holes . It also predicts gravitational lensing , where the bending of light results in multiple images of the same distant astronomical phenomenon. Other predictions include the existence of gravitational waves , which have been observed directly by the physics collaboration LIGO and other observatories. In addition, general relativity has provided

9044-555: The energy–momentum tensor, and hence of whatever is on the other side of the equation, is zero—the simplest nontrivial set of equations are what are called Einstein's (field) equations: G μ ν ≡ R μ ν − 1 2 R g μ ν = κ T μ ν {\displaystyle G_{\mu \nu }\equiv R_{\mu \nu }-{\textstyle 1 \over 2}R\,g_{\mu \nu }=\kappa T_{\mu \nu }\,} On

9163-446: The equivalence principle holds, gravity influences the passage of time. Light sent down into a gravity well is blueshifted , whereas light sent in the opposite direction (i.e., climbing out of the gravity well) is redshifted ; collectively, these two effects are known as the gravitational frequency shift. More generally, processes close to a massive body run more slowly when compared with processes taking place farther away; this effect

9282-456: The exceedingly weak waves that are expected to arrive here on Earth from far-off cosmic events, which typically result in relative distances increasing and decreasing by 10 − 21 {\displaystyle 10^{-21}} or less. Data analysis methods routinely make use of the fact that these linearized waves can be Fourier decomposed . Some exact solutions describe gravitational waves without any approximation, e.g.,

9401-408: The exterior Schwarzschild solution or, for more than a single mass, the post-Newtonian expansion), several effects of gravity on light propagation emerge. Although the bending of light can also be derived by extending the universality of free fall to light, the angle of deflection resulting from such calculations is only half the value given by general relativity. Closely related to light deflection

9520-433: The first non-trivial exact solution to the Einstein field equations, the Schwarzschild metric . This solution laid the groundwork for the description of the final stages of gravitational collapse, and the objects known today as black holes. In the same year, the first steps towards generalizing Schwarzschild's solution to electrically charged objects were taken, eventually resulting in the Reissner–Nordström solution , which

9639-412: The general relativistic framework—take on the same form in all coordinate systems . Furthermore, the theory does not contain any invariant geometric background structures, i.e. it is background independent . It thus satisfies a more stringent general principle of relativity , namely that the laws of physics are the same for all observers. Locally , as expressed in the equivalence principle, spacetime

9758-484: The geometry of space. Predicted in 1916 by Albert Einstein, there are gravitational waves: ripples in the metric of spacetime that propagate at the speed of light. These are one of several analogies between weak-field gravity and electromagnetism in that, they are analogous to electromagnetic waves . On 11 February 2016, the Advanced LIGO team announced that they had directly detected gravitational waves from

9877-402: The halting problem and the busy beaver game . It remains an open question as to whether there exists a more powerful definition of 'well-defined' that is able to capture both computable and 'non-computable' statements. Some examples of mathematical statements that are computable include: Some examples of mathematical statements that are not computable include: Computation can be seen as

9996-469: The history of the universe and have provided the modern framework for cosmology , thus leading to the discovery of the Big Bang and cosmic microwave background radiation. Despite the introduction of a number of alternative theories , general relativity continues to be the simplest theory consistent with experimental data . Reconciliation of general relativity with the laws of quantum physics remains

10115-430: The hypothetical process by which connective proteins, such as microtubule-associated proteins (MAPs), influence or orchestrate qubit state reduction by modifying the spacetime-separation of their superimposed states. The latter is based on Penrose's objective-collapse theory for interpreting quantum mechanics, which postulates the existence of an objective threshold governing the collapse of quantum states , related to

10234-441: The image), and a set of events for which such an influence is impossible (such as event C in the image). These sets are observer -independent. In conjunction with the world-lines of freely falling particles, the light-cones can be used to reconstruct the spacetime's semi-Riemannian metric, at least up to a positive scalar factor. In mathematical terms, this defines a conformal structure or conformal geometry. Special relativity

10353-446: The influence of the gravitational force. This suggests the definition of a new class of inertial motion, namely that of objects in free fall under the influence of gravity. This new class of preferred motions, too, defines a geometry of space and time—in mathematical terms, it is the geodesic motion associated with a specific connection which depends on the gradient of the gravitational potential . Space, in this construction, still has

10472-417: The key mathematical framework on which he fit his physical ideas of gravity. This idea was pointed out by mathematician Marcel Grossmann and published by Grossmann and Einstein in 1913. The Einstein field equations are nonlinear and considered difficult to solve. Einstein used approximation methods in working out initial predictions of the theory. But in 1916, the astrophysicist Karl Schwarzschild found

10591-410: The language of spacetime: the straight time-like lines that define a gravity-free inertial frame are deformed to lines that are curved relative to each other, suggesting that the inclusion of gravity necessitates a change in spacetime geometry. A priori, it is not clear whether the new local frames in free fall coincide with the reference frames in which the laws of special relativity hold—that theory

10710-457: The left-hand side is the Einstein tensor , G μ ν {\displaystyle G_{\mu \nu }} , which is symmetric and a specific divergence-free combination of the Ricci tensor R μ ν {\displaystyle R_{\mu \nu }} and the metric. In particular, is the curvature scalar. The Ricci tensor itself is related to

10829-477: The light of stars or distant quasars being deflected as it passes the Sun . This and related predictions follow from the fact that light follows what is called a light-like or null geodesic —a generalization of the straight lines along which light travels in classical physics. Such geodesics are the generalization of the invariance of lightspeed in special relativity. As one examines suitable model spacetimes (either

10948-411: The mapping account of pancomputationalism , the idea that everything can be said to be computing everything. Gualtiero Piccinini proposes an account of computation based on mechanical philosophy . It states that physical computing systems are types of mechanisms that, by design, perform physical computation, or the manipulation (by a functional mechanism) of a "medium-independent" vehicle according to

11067-455: The matter's energy–momentum tensor must be divergence-free. The matter must, of course, also satisfy whatever additional equations were imposed on its properties. In short, such a solution is a model universe that satisfies the laws of general relativity, and possibly additional laws governing whatever matter might be present. Einstein's equations are nonlinear partial differential equations and, as such, difficult to solve exactly. Nevertheless,

11186-446: The mechanism of anesthetic action, however, because human anesthesia is performed at 1 MAC . (Neither Penrose or Hameroff claim that depolymerization is the mechanism of action for Orch OR.) At ~1 MAC halothane, reported minor changes in tubulin protein expression (~1.3-fold) in primary cortical neurons after exposure to halothane and isoflurane are not evidence that tubulin directly interacts with general anesthetics, but rather shows that

11305-442: The more general Riemann curvature tensor as On the right-hand side, κ {\displaystyle \kappa } is a constant and T μ ν {\displaystyle T_{\mu \nu }} is the energy–momentum tensor. All tensors are written in abstract index notation . Matching the theory's prediction to observational results for planetary orbits or, equivalently, assuring that

11424-552: The much smaller separations stipulated for Orch OR. As a result, Hameroff's group claimed a decoherence time seven orders of magnitude greater than Tegmark's, although still far below 25 ms. Hameroff's group also suggested that the Debye layer of counterions could screen thermal fluctuations, and that the surrounding actin gel might enhance the ordering of water, further screening noise. They also suggested that incoherent metabolic energy could further order water, and finally that

11543-432: The observation of binary pulsars . All results are in agreement with general relativity. However, at the current level of accuracy, these observations cannot distinguish between general relativity and other theories in which the equivalence principle is valid. General relativity predicts that the path of light will follow the curvature of spacetime as it passes near a star. This effect was initially confirmed by observing

11662-459: The ordinary Euclidean geometry . However, space time as a whole is more complicated. As can be shown using simple thought experiments following the free-fall trajectories of different test particles, the result of transporting spacetime vectors that can denote a particle's velocity (time-like vectors) will vary with the particle's trajectory; mathematically speaking, the Newtonian connection is not integrable . From this, one can deduce that spacetime

11781-502: The passage of time, the geometry of space, the motion of bodies in free fall , and the propagation of light, and include gravitational time dilation , gravitational lensing , the gravitational redshift of light, the Shapiro time delay and singularities / black holes . So far, all tests of general relativity have been shown to be in agreement with the theory. The time-dependent solutions of general relativity enable us to talk about

11900-765: The phenomenon has a quantum origin, with superradiance being investigated as one possibility (in a 2024 study, superradiance was confirmed to occur in networks of tryptophans , which are found in microtubules). Tuszyński told New Scientist that "We're not at the level of interpreting this physiologically, saying 'Yeah, this is where consciousness begins,' but it may." The 2024 study, called Ultraviolet Superradiance from Mega-Networks of Tryptophan in Biological Architectures and published in The Journal of Physical Chemistry , confirmed superradiance in networks of tryptophans. Large networks of tryptophans are

12019-444: The philosopher Patricia Churchland wrote, "Pixie dust in the synapses is about as explanatorily powerful as quantum coherence in the microtubules." David Chalmers argues against quantum consciousness. He instead discusses how quantum mechanics may relate to dualistic consciousness . Chalmers is skeptical that any new physics can resolve the hard problem of consciousness . He argues that quantum theories of consciousness suffer from

12138-508: The possible states. The rough threshold for OR is given by Penrose's indeterminacy principle: Thus, the greater the mass–energy of the object, the faster it will undergo OR and vice versa. Mesoscopic objects could collapse on a timescale relevant to neural processing. An essential feature of Penrose's theory is that the choice of states when objective reduction occurs is selected neither randomly (as are choices following wave function collapse) nor algorithmically. Rather, states are selected by

12257-565: The possible structure of spacetime . He suggested that at the Planck scale curved spacetime is not continuous, but discrete. He further postulated that each separated quantum superposition has its own piece of spacetime curvature , a blister in spacetime. Penrose suggests that gravity exerts a force on these spacetime blisters, which become unstable above the Planck scale of 10 − 35 m {\displaystyle 10^{-35}{\text{m}}} and collapse to just one of

12376-511: The preface to Relativity: The Special and the General Theory , Einstein said "The present book is intended, as far as possible, to give an exact insight into the theory of Relativity to those readers who, from a general scientific and philosophical point of view, are interested in the theory, but who are not conversant with the mathematical apparatus of theoretical physics. The work presumes a standard of education corresponding to that of

12495-430: The principle of equivalence and his sense that a proper description of gravity should be geometrical at its basis, so that there was an "element of revelation" in the manner in which Einstein arrived at his theory. Other elements of beauty associated with the general theory of relativity are its simplicity and symmetry, the manner in which it incorporates invariance and unification, and its perfect logical consistency. In

12614-510: The proteins controlling tubulin production are possible anesthetic targets. Further proteomic study reports 0.5 mM [C]halothane binding to tubulin monomers alongside three dozens of other proteins. In addition, modulation of microtubule stability has been reported during anthracene general anesthesia of tadpoles. The study called Direct Modulation of Microtubule Stability Contributes to Anthracene General Anesthesia claims to provide "strong evidence that destabilization of neuronal microtubules provides

12733-609: The quantum phenomena required by the theory, since it is considered too "warm, wet and noisy" to avoid decoherence . In 1931, mathematician and logician Kurt Gödel proved that any effectively generated theory capable of proving basic arithmetic cannot be both consistent and complete . In other words, a mathematically sound theory lacks the means to prove itself. In his first book concerning consciousness, The Emperor's New Mind (1989), Roger Penrose argued that equivalent statements to "Gödel-type propositions" had recently been put forward. Partially in response to Gödel's argument,

12852-446: The same premises, which include additional rules and/or constraints, leading to different field equations. Examples are Whitehead's theory , Brans–Dicke theory , teleparallelism , f ( R ) gravity and Einstein–Cartan theory . The derivation outlined in the previous section contains all the information needed to define general relativity, describe its key properties, and address a question of crucial importance in physics, namely how

12971-471: The same weakness as more conventional theories. Just as he argues that there is no particular reason why particular macroscopic physical features in the brain should give rise to consciousness, he also thinks that there is no particular reason why a particular quantum feature, such as the EM field in the brain, should give rise to consciousness either. In 2000 Max Tegmark claimed that any quantum coherent system in

13090-472: The speed of light in vacuum. When there is no matter present, so that the energy–momentum tensor vanishes, the results are the vacuum Einstein equations, In general relativity, the world line of a particle free from all external, non-gravitational force is a particular type of geodesic in curved spacetime. In other words, a freely moving or falling particle always moves along a geodesic. The geodesic equation is: where s {\displaystyle s}

13209-518: The theory can be used for model-building. General relativity is a metric theory of gravitation. At its core are Einstein's equations , which describe the relation between the geometry of a four-dimensional pseudo-Riemannian manifold representing spacetime, and the energy–momentum contained in that spacetime. Phenomena that in classical mechanics are ascribed to the action of the force of gravity (such as free-fall , orbital motion, and spacecraft trajectories ), correspond to inertial motion within

13328-644: The theory's predictive power, and relativistic cosmology also became amenable to direct observational tests. General relativity has acquired a reputation as a theory of extraordinary beauty. Subrahmanyan Chandrasekhar has noted that at multiple levels, general relativity exhibits what Francis Bacon has termed a "strangeness in the proportion" ( i.e . elements that excite wonderment and surprise). It juxtaposes fundamental concepts (space and time versus matter and motion) which had previously been considered as entirely independent. Chandrasekhar also noted that Einstein's only guides in his search for an exact theory were

13447-487: The two become significant when dealing with speeds approaching the speed of light , and with high-energy phenomena. With Lorentz symmetry, additional structures come into play. They are defined by the set of light cones (see image). The light-cones define a causal structure: for each event A , there is a set of events that can, in principle, either influence or be influenced by A via signals or interactions that do not need to travel faster than light (such as event B in

13566-489: The universe is expanding. This is readily described by the expanding cosmological solutions found by Friedmann in 1922, which do not require a cosmological constant. Lemaître used these solutions to formulate the earliest version of the Big Bang models, in which the universe has evolved from an extremely hot and dense earlier state. Einstein later declared the cosmological constant the biggest blunder of his life. During that period, general relativity remained something of

13685-494: The weak-gravity, low-speed limit is Newtonian mechanics, the proportionality constant κ {\displaystyle \kappa } is found to be κ = 8 π G c 4 {\textstyle \kappa ={\frac {8\pi G}{c^{4}}}} , where G {\displaystyle G} is the Newtonian constant of gravitation and c {\displaystyle c}

13804-629: Was criticized by mathematicians, computer scientists, and philosophers, and the consensus among experts in these fields is that the argument fails, with different authors attacking different aspects of the argument. Minsky argued that because humans can believe false ideas to be true, human mathematical understanding need not be consistent and consciousness may easily have a deterministic basis. Feferman argued that mathematicians do not progress by mechanistic search through proofs, but by trial-and-error reasoning, insight and inspiration, and that machines do not share this approach with humans. Penrose outlined

13923-474: Was instantaneous or of electromagnetic origin, he suggested that relativity was "something due to our methods of measurement". In his theory, he showed that gravitational waves propagate at the speed of light. Soon afterwards, Einstein started thinking about how to incorporate gravity into his relativistic framework. In 1907, beginning with a simple thought experiment involving an observer in free fall (FFO), he embarked on what would be an eight-year search for

14042-582: Was proposed independently by several mathematicians in the 1930s. The best-known variant was formalised by the mathematician Alan Turing , who defined a well-defined statement or calculation as any statement that could be expressed in terms of the initialisation parameters of a Turing machine . Other (mathematically equivalent) definitions include Alonzo Church 's lambda-definability , Herbrand - Gödel - Kleene 's general recursiveness and Emil Post 's 1-definability . Today, any formal statement or calculation that exhibits this quality of well-definedness

14161-488: Was rejected by Reimers's group. Hameroff and Penrose contested the conclusion, considering that Reimers's microtubule model was oversimplified. Hameroff then proposed that condensates in microtubules in one neuron can link with microtubule condensates in other neurons and glial cells via the gap junctions of electrical synapses . Hameroff proposed that the gap between the cells is sufficiently small that quantum objects can tunnel across it, allowing them to extend across

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