Chronology protection conjecture

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Chronology_protection_conjecture

The chronology protection conjecture is a hypothesis first proposed by Stephen Hawking that laws of physics beyond those of standard general relativity prevent time travel on all but microscopic scales—even when the latter theory states that it should be possible (such as in scenarios where faster than light travel is allowed). The permissibility of time travel is represented mathematically by the existence of closed timelike curves in some solutions to the field equations of general relativity. The chronology protection conjecture should be distinguished from chronological censorship under which every closed timelike curve passes through an event horizon, which might prevent an observer from detecting the causal violation (also known as chronology violation).

Etymology

In a 1992 paper, Hawking uses the metaphorical device of a "Chronology Protection Agency" as a personification of the aspects of physics that make time travel impossible at macroscopic scales, thus apparently preventing temporal paradoxes. He says:

It seems that there is a Chronology Protection Agency which prevents the appearance of closed timelike curves and so makes the universe safe for historians.

The idea of the Chronology Protection Agency appears to be drawn playfully from the Time Patrol or Time Police concept, which has been used in many works of science fiction such as Poul Anderson's series of Time Patrol stories or Isaac Asimov's novel The End of Eternity, or in the television series Doctor Who. "The Chronology Protection Case" by Paul Levinson, published after Hawking's paper, posits a universe that goes so far as to murder any scientists who are close to inventing any means of time travel. Larry Niven, in his short story ‘Rotating Cylinders and the possibility of Global Causality Violation’ expands this concept so that the universe causes environmental catastrophe, or global civil war, or the local sun going nova, to any civilisation which shows any sign of successful construction.

General relativity and quantum corrections

Many attempts to generate scenarios for closed timelike curves have been suggested, and the theory of general relativity does allow them in certain circumstances. Some theoretical solutions in general relativity that contain closed timelike curves would require an infinite universe with certain features that our universe does not appear to have, such as the universal rotation of the Gödel metric or the rotating cylinder of infinite length known as a Tipler cylinder. However, some solutions allow for the creation of closed timelike curves in a bounded region of spacetime, with the Cauchy horizon being the boundary between the region of spacetime where closed timelike curves can exist and the rest of spacetime where they cannot. One of the first such bounded time travel solutions found was constructed from a traversable wormhole, based on the idea of taking one of the two "mouths" of the wormhole on a round-trip journey at relativistic speed to create a time difference between it and the other mouth (see the discussion at Wormhole#Time travel).

General relativity does not include quantum effects on its own, and a full integration of general relativity and quantum mechanics would require a theory of quantum gravity, but there is an approximate method for modeling quantum fields in the curved spacetime of general relativity, known as semiclassical gravity. Initial attempts to apply semiclassical gravity to the traversable wormhole time machine indicated that at exactly the moment that wormhole would first allow for closed timelike curves, quantum vacuum fluctuations build up and drive the energy density to infinity in the region of the wormholes. This occurs when the two wormhole mouths, call them A and B, have been moved in such a way that it becomes possible for a particle or wave moving at the speed of light to enter mouth B at some time T₂ and exit through mouth A at an earlier time T₁, then travel back towards mouth B through ordinary space, and arrive at mouth B at the same time T₂ that it entered B on the previous loop; in this way the same particle or wave can make a potentially infinite number of loops through the same regions of spacetime, piling up on itself. Calculations showed that this effect would not occur for an ordinary beam of radiation, because it would be "defocused" by the wormhole so that most of a beam emerging from mouth A would spread out and miss mouth B.^[7] But when the calculation was done for vacuum fluctuations, it was found that they would spontaneously refocus on the trip between the mouths, indicating that the pileup effect might become large enough to destroy the wormhole in this case.

Uncertainty about this conclusion remained, because the semiclassical calculations indicated that the pileup would only drive the energy density to infinity for an infinitesimal moment of time, after which the energy density would die down. But semiclassical gravity is considered unreliable for large energy densities or short time periods that reach the Planck scale; at these scales, a complete theory of quantum gravity is needed for accurate predictions. So, it remains uncertain whether quantum-gravitational effects might prevent the energy density from growing large enough to destroy the wormhole. Stephen Hawking conjectured that not only would the pileup of vacuum fluctuations still succeed in destroying the wormhole in quantum gravity, but also that the laws of physics would ultimately prevent any type of time machine from forming; this is the chronology protection conjecture.

Subsequent works in semiclassical gravity provided examples of spacetimes with closed timelike curves where the energy density due to vacuum fluctuations does not approach infinity in the region of spacetime outside the Cauchy horizon. However, in 1997 a general proof was found demonstrating that according to semiclassical gravity, the energy of the quantum field (more precisely, the expectation value of the quantum stress-energy tensor) must always be either infinite or undefined on the horizon itself. Both cases indicate that semiclassical methods become unreliable at the horizon and quantum gravity effects would be important there, consistent with the possibility that such effects would always intervene to prevent time machines from forming.

A definite theoretical decision on the status of the chronology protection conjecture would require a full theory of quantum gravity as opposed to semiclassical methods. There are also some arguments from string theory that seem to support chronology protection, but string theory is not yet a complete theory of quantum gravity. Experimental observation of closed timelike curves would of course demonstrate this conjecture to be false, but short of that, if physicists had a theory of quantum gravity whose predictions had been well-confirmed in other areas, this would give them a significant degree of confidence in the theory's predictions about the possibility or impossibility of time travel.

Other proposals that allow for backwards time travel but prevent time paradoxes, such as the Novikov self-consistency principle, which would ensure the timeline stays consistent, or the idea that a time traveler is taken to a parallel universe while their original timeline remains intact, do not qualify as "chronology protection".

Quantum mechanics of time travel

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Quantum_mechanics_of_time_travel

The theoretical study of time travel generally follows the laws of general relativity. Quantum mechanics requires physicists to solve equations describing how probabilities behave along closed timelike curves (CTCs), theoretical loops in spacetime that might make it possible to travel through time.

In the 1980s, Igor Novikov proposed the self-consistency principle. According to this principle, any changes made by a time traveler in the past must not create paradoxes. If a time traveler tries to change the past, the laws of physics will ensure that history remains consistent. This means that the outcomes of events will always align with the traveler’s actions in a way that prevents any contradictions.

However, Novikov's self-consistency principle may be incompatible when considered alongside certain interpretations of quantum mechanics, particularly two fundamental principles of quantum mechanics, unitarity and linearity. Unitarity ensures that the total probability of all possible outcomes in a quantum system always sums to 1, preserving the predictability of quantum events. Linearity ensures that quantum evolution preserves superpositions, allowing quantum systems to exist in multiple states simultaneously.

There are two main approaches to explaining quantum time travel while incorporating Novikov's self-consistency. The first approach uses density matrices to describe the probabilities of different outcomes in quantum systems, providing a statistical framework that can accommodate the constraints of CTCs. The second approach involves state vectors, which describe the quantum state of a system. This approach sometimes leads to concepts that deviate from the conventional understanding of quantum mechanics.

Deutsch's prescription for closed timelike curves (CTCs)

In 1991, David Deutsch proposed a method to explain how quantum systems interact with closed timelike curves (CTCs) using time evolution equations. This method aims to address paradoxes like the grandfather paradox, which suggest that a time traveler who stops their own birth would create a contradiction. One interpretation of Deutsch's approach is that it implies the time traveler might end up in a parallel universe rather than their own, although the formalism itself does not explicitly require the existence of parallel universes.

Method overview

To analyze the system, Deutsch divided it into two parts: a subsystem outside the CTC and the CTC itself. To describe the combined evolution of both parts over time, he used a unitary operator (U). This approach relies on a specific mathematical framework to describe quantum systems. The overall state is represented by combining the density matrices (ρ) for both the subsystem and the CTC using a tensor product (⊗). Notably, Deutsch assumed no initial correlation between these two parts. While this assumption breaks time symmetry (meaning the laws of physics wouldn't behave the same forwards and backwards in time), Deutsch justifies it using arguments from measurement theory and the second law of thermodynamics.

Deutsch's proposal uses the following key equation to describe the fixed-point density matrix (ρCTC) for the CTC:

${\displaystyle {\rho }_{\text{CTC}}={\text{Tr}}_A\left[U\left({\rho }_A\otimes {\rho }_{\text{CTC}}\right)U^{†}\right]}$.

The unitary evolution involving both the CTC and the external subsystem determines the density matrix of the CTC as a fixed point, as represented by this equation. The trace operation (${\displaystyle {Tr}_A}$) indicates that we are considering the partial trace over the subsystem outside the CTC, focusing on the state of the CTC itself.

Ensuring Self-Consistency

Deutsch's proposal ensures that the CTC always returns to a self-consistent state after a loop. This means that the overall state of the CTC remains consistent. However, this raises concerns. If a system retains memories after traveling through the CTC, it could create complex scenarios where it appears to have experienced different possible pasts.

Furthermore, Deutsch's method might not work with common probability calculations in quantum mechanics, like path integrals, unless we take into account the chance that the system goes through different paths that all lead to the same outcome. There can also be multiple solutions (fixed points) for the system's state after the loop, introducing a form of randomness (nondeterminism). Deutsch suggested using the solution with the highest entropy, which aligns with the natural tendency of systems to evolve towards higher entropy states.

To calculate the final state outside the CTC, a specific mathematical operation (trace) considers only the external system's state after the combined evolution of both the external system and the CTC. The tensor product (⊗) of the density matrices for both systems describes this combined evolution. Then, a unitary time evolution operator (U) is applied to the whole system.

Implications and criticisms

Deutsch's approach has intriguing implications for paradoxes like the grandfather paradox. Consider a scenario in which everything, except a single quantum bit (qubit), travels through a time machine and flips its value according to a specific operator:

${\displaystyle U=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)}$.

Deutsch argues that the solution maximizing von Neumann entropy (a measure of how scrambled or mixed the information in the qubit is) is the most relevant. In this case, the qubit becomes a mix of starting at 0 and ending at 1, or vice versa. Deutsch's interpretation, which can align with the many-worlds view of quantum mechanics, avoids paradoxes because the qubit travels to a different parallel universe after interacting with the CTC.

Researchers have explored the potential of Deutsch's ideas. Deutsch's CTC time travel, if possible, might allow computers near a time machine to solve problems far beyond classical computers, but the feasibility of CTCs and time travel remains a topic of debate and further research is needed.

Despite its theoretical nature, Deutsch's proposal has faced significant criticism. For instance, Tolksdorf and Verch demonstrated that quantum systems without CTCs can still achieve Deutsch's criterion with high accuracy. This finding casts doubt on the uniqueness of Deutsch's criterion for quantum simulations of CTCs as theorized in general relativity. Their research showed that classical systems governed by statistical mechanics could also meet these criteria, implying that the peculiarities attributed to quantum mechanics might not be essential for simulating CTCs. Based on these results, it appears that Deutsch's criterion is not specific to quantum mechanics and may not be a good way to figure out the possibilities of real-time travel or how quantum mechanics might make it possible. Consequently, Tolksdorf and Verch argue that their findings doubt the validity of Deutsch's explanation of his time travel scenario using the many-worlds interpretation.

Lloyd's prescription: Post-selection and time travel with CTCs

Seth Lloyd proposed an alternative approach to time travel with closed timelike curves (CTCs), based on "post-selection" and path integrals. Path integrals are a powerful tool in quantum mechanics that involve summing probabilities over all possible ways a system could evolve, even if those paths don't strictly follow a single timeline. Unlike classical approaches, path integrals allow for consistent histories even with CTCs. Lloyd argues that focusing on the state of the system outside the CTC is more relevant.

He proposes an equation that explains the transformation of the density matrix, which represents the system's state outside the CTC, following a time loop:

${\displaystyle {\rho }_f=\frac{C{\rho }_i{C}^{†}}{\text{Tr}\left[C{\rho }_i{C}^{†}\right]}}$, where ${\displaystyle C={\text{Tr}}_{\text{CTC}}\left[U\right]}$.

In this equation:

• ${\displaystyle {\rho }_f}$ is the density matrix of the system after interacting with the CTC.
• ${\displaystyle {\rho }_i}$ is the initial density matrix of the system before the time loop.
• ${\displaystyle C}$ is a transformation operator derived from the trace operation over the CTC, applied to the unitary evolution operator ${\displaystyle U}$.

The transformation relies on the trace, a specific mathematical operation within the CTC that reduces a complex matrix to a single number. If this trace term is zero (${\displaystyle \text{Tr}\left[C{\rho }_i{C}^{†}\right]=0}$), the equation has no solution, indicating an inconsistency like the grandfather paradox. Conversely, a non-zero trace leads to a unique solution for the external system's state.

Thus, Lloyd's approach ensures self-consistency and avoids paradoxes by allowing only histories consistent with the system's initial and final states. This aligns with the concept of post-selection, where only certain outcomes are considered based on predetermined criteria, effectively filtering out paradoxical scenarios.

Entropy and computation

Michael Devin (2001) proposed a model that incorporates closed timelike curves (CTCs) into thermodynamics, suggesting it as a potential way to address the grandfather paradox. This model introduces a "noise" factor to account for imperfections in time travel, proposing a framework that could avoid paradoxes.

Devin's model posits that each cycle of time travel involving a quantum bit (qubit) carries a usable form of energy, termed "negentropy" (negative entropy, representing a decrease in disorder). The model suggests that the amount of negentropy is proportional to the noise level introduced during time travel. This implies that a time machine could potentially extract work from a thermal bath in proportion to the negentropy generated.

Moreover, Devin's model indicates that a time machine could significantly reduce the computational effort required to solve complex problems, such as cracking codes through trial and error. CTCs could allow for a more efficient computation process because the system can effectively "reuse" information from different timelines, leading to faster problem-solving capabilities.

However, the model also predicts that as the noise level approaches zero, the usable energy and computational power will become infinitely large. This implies that conventional computational complexity classes, which categorize problems based on their difficulty for classical computers, might not apply to time machines with very low noise levels. Devin's model is entirely theoretical and speculative and has not been confirmed by experimental evidence.

Polyhedral skeletal electron pair theory

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Polyhedral_skeletal_electron_pair_theory

In chemistry the polyhedral skeletal electron pair theory (PSEPT) provides electron counting rules useful for predicting the structures of clusters such as borane and carborane clusters. The electron counting rules were originally formulated by Kenneth Wade, and were further developed by others including Michael Mingos; they are sometimes known as Wade's rules or the Wade–Mingos rules. The rules are based on a molecular orbital treatment of the bonding. These rules have been extended and unified in the form of the Jemmis mno rules.

Predicting structures of cluster compounds

The structure of the butterfly cluster compound [Re₄(CO)₁₂]²⁻ conforms to the predictions of PSEPT.

Different rules (4n, 5n, or 6n) are invoked depending on the number of electrons per vertex.

The 4n rules are reasonably accurate in predicting the structures of clusters having about 4 electrons per vertex, as is the case for many boranes and carboranes. For such clusters, the structures are based on deltahedra, which are polyhedra in which every face is triangular. The 4n clusters are classified as closo-, nido-, arachno- or hypho-, based on whether they represent a complete (closo-) deltahedron, or a deltahedron that is missing one (nido-), two (arachno-) or three (hypho-) vertices.

However, hypho clusters are relatively uncommon due to the fact that the electron count is high enough to start to fill antibonding orbitals and destabilize the 4n structure. If the electron count is close to 5 electrons per vertex, the structure often changes to one governed by the 5n rules, which are based on 3-connected polyhedra.

As the electron count increases further, the structures of clusters with 5n electron counts become unstable, so the 6n rules can be implemented. The 6n clusters have structures that are based on rings.

A molecular orbital treatment can be used to rationalize the bonding of cluster compounds of the 4n, 5n, and 6n types.

4n rules

Ball-and-stick models showing the structures of the boron skeletons of borane clusters.

The following polyhedra are closo polyhedra, and are the basis for the 4n rules; each of these have triangular faces. The number of vertices in the cluster determines what polyhedron the structure is based on.

Number of vertices	Polyhedron
4	Tetrahedron
5	Trigonal bipyramid
6	Octahedron
7	Pentagonal bipyramid
8	D2d (trigonal) dodecahedron (snub disphenoid)
9	Tricapped trigonal prism
10	Bicapped square antiprismatic molecular geometry
11	Edge-contracted icosahedron (octadecahedron)
12	Icosahedron (bicapped pentagonal antiprism)

Using the electron count, the predicted structure can be found. n is the number of vertices in the cluster. The 4n rules are enumerated in the following table.

Electron count	Name	Predicted structure
4n − 2	Bicapped closo	n − 2 vertex closo polyhedron with 2 capped (augmented) faces
4n	Capped closo	n − 1 vertex closo polyhedron with 1 face capped
4n + 2	closo	closo polyhedron with n vertices
4n + 4	nido	n + 1 vertex closo polyhedron with 1 missing vertex
4n + 6	arachno	n + 2 vertex closo polyhedron with 2 missing vertices
4n + 8	hypho	n + 3 vertex closo polyhedron with 3 missing vertices
4n + 10	klado	n + 4 vertex closo polyhedron with 4 missing vertices

Pb²⁻
₁₀

When counting electrons for each cluster, the number of valence electrons is enumerated. For each transition metal present, 10 electrons are subtracted from the total electron count. For example, in Rh₆(CO)₁₆ the total number of electrons would be 6 × 9 + 16 × 2 − 6 × 10 = 86 – 60 = 26. Therefore, the cluster is a closo polyhedron because n = 6, with 4n + 2 = 26.

S²⁺
₄

Other rules may be considered when predicting the structure of clusters:

1. For clusters consisting mostly of transition metals, any main group elements present are often best counted as ligands or interstitial atoms, rather than vertices.
2. Larger and more electropositive atoms tend to occupy vertices of high connectivity and smaller more electronegative atoms tend to occupy vertices of low connectivity.
3. In the special case of boron hydride clusters, each boron atom connected to 3 or more vertices has one terminal hydride, while a boron atom connected to two other vertices has two terminal hydrogen atoms. If more hydrogen atoms are present, they are placed in open face positions to even out the coordination number of the vertices.
4. For the special case of transition metal clusters, ligands are added to the metal centers to give the metals reasonable coordination numbers, and if any hydrogen atoms are present they are placed in bridging positions to even out the coordination numbers of the vertices.

In general, closo structures with n vertices are n-vertex polyhedra.

To predict the structure of a nido cluster, the closo cluster with n + 1 vertices is used as a starting point; if the cluster is composed of small atoms a high connectivity vertex is removed, while if the cluster is composed of large atoms a low connectivity vertex is removed.

To predict the structure of an arachno cluster, the closo polyhedron with n + 2 vertices is used as the starting point, and the n + 1 vertex nido complex is generated by following the rule above; a second vertex adjacent to the first is removed if the cluster is composed of mostly small atoms, a second vertex not adjacent to the first is removed if the cluster is composed mostly of large atoms.

Os₆(CO)₁₈, carbonyls omitted

Example: Pb²⁻
₁₀

Electron count: 10 × Pb + 2 (for the negative charge) = 10 × 4 + 2 = 42 electrons.

Since n = 10, 4n + 2 = 42, so the cluster is a closo bicapped square antiprism.

Example: S²⁺
₄

Electron count: 4 × S – 2 (for the positive charge) = 4 × 6 – 2 = 22 electrons.

Since n = 4, 4n + 6 = 22, so the cluster is arachno.

Starting from an octahedron, a vertex of high connectivity is removed, and then a non-adjacent vertex is removed.

Example: Os₆(CO)₁₈

Electron count: 6 × Os + 18 × CO – 60 (for 6 osmium atoms) = 6 × 8 + 18 × 2 – 60 = 24

Since n = 6, 4n = 24, so the cluster is capped closo.

Starting from a trigonal bipyramid, a face is capped. The carbonyls have been omitted for clarity.

B
₅H⁴⁻
₅, hydrogen atoms omitted

Example: B
₅H⁴⁻
₅

Electron count: 5 × B + 5 × H + 4 (for the negative charge) = 5 × 3 + 5 × 1 + 4 = 24

Since n = 5, 4n + 4 = 24, so the cluster is nido.

Starting from an octahedron, one of the vertices is removed.

The rules are useful in also predicting the structure of carboranes. Example: C₂B₇H₁₃

Electron count = 2 × C + 7 × B + 13 × H = 2 × 4 + 7 × 3 + 13 × 1 = 42

Since n in this case is 9, 4n + 6 = 42, the cluster is arachno.

The bookkeeping for deltahedral clusters is sometimes carried out by counting skeletal electrons instead of the total number of electrons. The skeletal orbital (electron pair) and skeletal electron counts for the four types of deltahedral clusters are:

• n-vertex closo: n + 1 skeletal orbitals, 2n + 2 skeletal electrons
• n-vertex nido: n + 2 skeletal orbitals, 2n + 4 skeletal electrons
• n-vertex arachno: n + 3 skeletal orbitals, 2n + 6 skeletal electrons
• n-vertex hypho: n + 4 skeletal orbitals, 2n + 8 skeletal electrons

The skeletal electron counts are determined by summing the total of the following number of electrons:

• 2 from each BH unit
• 3 from each CH unit
• 1 from each additional hydrogen atom (over and above the ones on the BH and CH units)
• the anionic charge electrons

5n rules

As discussed previously, the 4n rule mainly deals with clusters with electron counts of 4n + k, in which approximately 4 electrons are on each vertex. As more electrons are added per vertex, the number of the electrons per vertex approaches 5. Rather than adopting structures based on deltahedra, the 5n-type clusters have structures based on a different series of polyhedra known as the 3-connected polyhedra, in which each vertex is connected to 3 other vertices. The 3-connected polyhedra are the duals of the deltahedra. The common types of 3-connected polyhedra are listed below.

5n cluster: P₄

5n + 3 cluster: P₄S₃

5n + 6 cluster: P₄O₆

Number of vertices	Type of 3-connected polyhedron
4	Tetrahedron
6	Trigonal prism
8	Cube
10	Pentagonal prism
12	D2d pseudo-octahedron (dual of snub disphenoid)
14	Dual of triaugmented triangular prism (K5 associahedron)
16	Square truncated trapezohedron
18	Dual of edge-contracted icosahedron
20	Dodecahedron

The 5n rules are as follows.

Total electron count	Predicted structure
5n	n-vertex 3-connected polyhedron
5n + 1	n – 1 vertex 3-connected polyhedron with one vertex inserted into an edge
5n + 2	n – 2 vertex 3-connected polyhedron with two vertices inserted into edges
5n + k	n − k vertex 3-connected polyhedron with k vertices inserted into edges

Example: P₄

Electron count: 4 × P = 4 × 5 = 20

It is a 5n structure with n = 4, so it is tetrahedral

Example: P₄S₃

Electron count 4 × P + 3 × S = 4 × 5 + 3 × 6 = 38

It is a 5n + 3 structure with n = 7. Three vertices are inserted into edges

Example: P₄O₆

Electron count 4 × P + 6 × O = 4 × 5 + 6 × 6 = 56

It is a 5n + 6 structure with n = 10. Six vertices are inserted into edges

6n rules

As more electrons are added to a 5n cluster, the number of electrons per vertex approaches 6. Instead of adopting structures based on 4n or 5n rules, the clusters tend to have structures governed by the 6n rules, which are based on rings. The rules for the 6n structures are as follows.

S₈ crown

Total electron count	Predicted structure
6n – k	n-membered ring with k⁄2 transannular bonds
6n – 4	n-membered ring with 2 transannular bonds
6n – 2	n-membered ring with 1 transannular bond
6n	n-membered ring
6n + 2	n-membered chain (n-membered ring with 1 broken bond)

Example: S₈

Electron count = 8 × S = 8 × 6 = 48 electrons.

Since n = 8, 6n = 48, so the cluster is an 8-membered ring.

6n + 2 cluster: hexane

Hexane (C₆H₁₄)

Electron count = 6 × C + 14 × H = 6 × 4 + 14 × 1 = 38

Since n = 6, 6n = 36 and 6n + 2 = 38, so the cluster is a 6-membered chain.

Isolobal vertex units

Provided a vertex unit is isolobal with BH then it can, in principle at least, be substituted for a BH unit, even though BH and CH are not isoelectronic. The CH⁺ unit is isolobal, hence the rules are applicable to carboranes. This can be explained due to a frontier orbital treatment. Additionally there are isolobal transition-metal units. For example, Fe(CO)₃ provides 2 electrons. The derivation of this is briefly as follows:

• Fe has 8 valence electrons.
• Each carbonyl group is a net 2 electron donor after the internal σ- and π-bonding are taken into account making 14 electrons.
• 3 pairs are considered to be involved in Fe–CO σ-bonding and 3 pairs are involved in π-backbonding from Fe to CO reducing the 14 to 2.

Bonding in cluster compounds

closo-B
₆H²⁻
₆

MO diagram of B
₆H²⁻
₆ showing the orbitals responsible for forming the cluster. Pictorial representations of the orbitals are shown; the MO sets of T and E symmetry will each have two or one additional pictorial representation, respectively, that are not shown here.

The boron atoms lie on each vertex of the octahedron and are sp hybridized. One sp-hybrid radiates away from the structure forming the bond with the hydrogen atom. The other sp-hybrid radiates into the center of the structure forming a large bonding molecular orbital at the center of the cluster. The remaining two unhybridized orbitals lie along the tangent of the sphere like structure creating more bonding and antibonding orbitals between the boron vertices. The orbital diagram breaks down as follows:

The 18 framework molecular orbitals, (MOs), derived from the 18 boron atomic orbitals are:

• 1 bonding MO at the center of the cluster and 5 antibonding MOs from the 6 sp-radial hybrid orbitals
• 6 bonding MOs and 6 antibonding MOs from the 12 tangential p-orbitals.

The total skeletal bonding orbitals is therefore 7, i.e. n + 1.

Transition metal clusters

Transition metal clusters use the d orbitals for bonding. Thus, they have up to nine bonding orbitals, instead of only the four present in boron and main group clusters. PSEPT also applies to metallaboranes

Clusters with interstitial atoms

Owing their large radii, transition metals generally form clusters that are larger than main group elements. One consequence of their increased size, these clusters often contain atoms at their centers. A prominent example is [Fe₆C(CO)₁₆]^2-. In such cases, the rules of electron counting assume that the interstitial atom contributes all valence electrons to cluster bonding. In this way, [Fe₆C(CO)₁₆]^2- is equivalent to [Fe₆(CO)₁₆]^6- or [Fe₆(CO)₁₈]^2-.

Grok

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Grok

Grok (/ˈɡrɒk/) is a neologism coined by American writer Robert A. Heinlein for his 1961 science fiction novel Stranger in a Strange Land. While the Oxford English Dictionary summarizes the meaning of grok as "to understand intuitively or by empathy, to establish rapport with" and "to empathize or communicate sympathetically (with); also, to experience enjoyment", Heinlein's concept is far more nuanced, with critic Istvan Csicsery-Ronay Jr. observing that "the book's major theme can be seen as an extended definition of the term." The concept of grok garnered significant critical scrutiny in the years after the book's initial publication. The term and aspects of the underlying concept have become part of communities such as computer science.

Descriptions in Stranger in a Strange Land

Critic David E. Wright Sr. points out that in the 1991 "uncut" edition of Stranger, the word grok "was used first without any explicit definition on page 22" and continued to be used without being explicitly defined until page 253 (emphasis in original). He notes that this first intensional definition is simply "to drink", but that this is only a metaphor "much as English 'I see' often means the same as 'I understand'". Critics have bridged this absence of explicit definition by citing passages from Stranger that illustrate the term. A selection of these passages follows:

Grok means "to understand", of course, but Dr. Mahmoud, who might be termed the leading Terran expert on Martians, explains that it also means, "to drink" and "a hundred other English words, words which we think of as antithetical concepts. 'Grok' means all of these. It means 'fear', it means 'love', it means 'hate' – proper hate, for by the Martian 'map' you cannot hate anything unless you grok it, understand it so thoroughly that you merge with it and it merges with you – then you can hate it. By hating yourself. But this implies that you love it, too, and cherish it and would not have it otherwise. Then you can hate – and (I think) Martian hate is an emotion so black that the nearest human equivalent could only be called mild distaste.

Grok means "identically equal". The human cliché "This hurts me worse than it does you" has a distinctly Martian flavor. The Martian seems to know instinctively what we learned painfully from modern physics, that observer acts with observed through the process of observation. Grok means to understand so thoroughly that the observer becomes a part of the observed – to merge, blend, intermarry, lose identity in group experience. It means almost everything that we mean by religion, philosophy, and science and it means as little to us as color does to a blind man.

The Martian Race had encountered the people of the fifth planet, grokked them completely, and had taken action; asteroid ruins were all that remained, save that the Martians continued to praise and cherish the people they had destroyed.

All that groks is God.

Etymology

Robert A. Heinlein originally coined the term grok in his 1961 novel Stranger in a Strange Land as a Martian word that could not be defined in Earthling terms, but can be associated with various literal meanings such as "water", "to drink", "to relate", "life", or "to live", and had a much more profound figurative meaning that is hard for terrestrial culture to understand because of its assumption of a singular reality.

According to the book, drinking water is a central focus on Mars, where it is scarce. Martians use the merging of their bodies with water as a simple example or symbol of how two entities can combine to create a new reality greater than the sum of its parts. The water becomes part of the drinker, and the drinker part of the water. Both grok each other. Things that once had separate realities become entangled in the same experiences, goals, history, and purpose. Within the book, the statement of divine immanence verbalized among the main characters, "thou art God", is logically derived from the concept inherent in the term grok.

Heinlein describes Martian words as "guttural" and "jarring". Martian speech is described as sounding "like a bullfrog fighting a cat". Accordingly, grok is generally pronounced as a guttural gr terminated by a sharp k with very little or no vowel sound (a narrow IPA transcription might be [ɡɹ̩kʰ]). William Tenn suggests Heinlein in creating the word might have been influenced by Tenn's very similar concept of griggo, earlier introduced in Tenn's story Venus and the Seven Sexes (published in 1949). In his later afterword to the story, Tenn says Heinlein considered such influence "very possible".

Adoption and modern usage

In computer programmer culture

Uses of the word in the decades after the 1960s are more concentrated in computer culture, such as an InfoWorld columnist in 1984 imagining a computer saying, "There isn't any software! Only different internal states of hardware. It's all hardware! It's a shame programmers don't grok that better."

The Jargon File, which describes itself as "The Hacker's Dictionary" and has been published under that name three times, puts grok in a programming context:

When you claim to "grok" some knowledge or technique, you are asserting that you have not merely learned it in a detached instrumental way but that it has become part of you, part of your identity. For example, to say that you "know" Lisp is simply to assert that you can code in it if necessary – but to say you "grok" Lisp is to claim that you have deeply entered the world-view and spirit of the language, with the implication that it has transformed your view of programming. Contrast zen, which is a similar supernatural understanding experienced as a single brief flash.

The entry existed in the very earliest forms of the Jargon File in the early 1980s. A typical tech usage from the Linux Bible, 2005 characterizes the Unix software development philosophy as "one that can make your life a lot simpler once you grok the idea".

The book Perl Best Practices defines grok as understanding a portion of computer code in a profound way. It goes on to suggest that to re-grok code is to reload the intricacies of that portion of code into one's memory after some time has passed and all the details of it are no longer remembered. In that sense, to grok means to load everything into memory for immediate use. It is analogous to the way a processor caches memory for short term use, but the only implication by this reference was that it was something a human (or perhaps a Martian) would do.

The main web page for cURL, an open source tool and programming library, describes the function of cURL as "cURL groks URLs".

The book Cyberia covers its use in this subculture extensively:

This is all latter day usage, the original derivation was from an early text processing utility from so long ago that no one remembers but, grok was the output when it understood the file. K&R would remember.

The keystroke logging software used by the NSA for its remote intelligence gathering operations is named GROK.

One of the most powerful parsing filters used in Elasticsearch software's logstash component is named grok.

A reference book by Carey Bunks on the use of the GNU Image Manipulation Program is titled Grokking the GIMP.

In counterculture

See also: Counterculture of the 1960s

• Tom Wolfe, in his book The Electric Kool-Aid Acid Test (1968), describes a character's thoughts during an acid trip: "He looks down, two bare legs, a torso rising up at him and like he is just noticing them for the first time ... he has never seen any of this flesh before, this stranger. He groks over that ..."

• In his counterculture Volkswagen repair manual, How to Keep Your Volkswagen Alive: A Manual of Step-by-Step Procedures for the Compleat Idiot (1969), dropout aerospace engineer John Muir instructs prospective used VW buyers to "grok the car" before buying.

• The word was used numerous times by Robert Anton Wilson in his works The Illuminatus! Trilogy and Schrödinger's Cat Trilogy. For instance, in The Eye in the Pyramid, volume one of Illuminatus:

I caught the references to Aristotle, the old man of the tribe with his unfortunate epistemological paresis, and also to that feisty little lady I always imagine is really the lost Anastasia, but I still didn’t grok. “What do you mean?” I asked (...)

• And in The Trick Top Hat, volume two of Schrödinger's Cat:

Williams went on. "You've got to think of time ripples, as well as space ripples, to grok the quantum world. ..."

Temporal paradox

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Temporal_paradox

A temporal paradox, time paradox, or time travel paradox, is a paradox, an apparent contradiction, or logical contradiction associated with the idea of time travel or other foreknowledge of the future. While the notion of time travel to the future complies with the current understanding of physics via relativistic time dilation, temporal paradoxes arise from circumstances involving hypothetical time travel to the past – and are often used to demonstrate its impossibility.

Types

"Causal loop" redirects here. For the cause and effect diagram, see causal loop diagram. For the plot device, see time loop.

Temporal paradoxes fall into three broad groups: bootstrap paradoxes, consistency paradoxes, and Newcomb's paradox. Bootstrap paradoxes violate causality by allowing future events to influence the past and cause themselves, or "bootstrapping", which derives from the idiom "pull oneself up by one's bootstraps." Consistency paradoxes, on the other hand, are those where future events influence the past to cause an apparent contradiction, exemplified by the grandfather paradox, where a person travels to the past to prevent the conception of one of their ancestors, thus eliminating all the ancestor's descendants. Newcomb's paradox stems from the apparent contradictions that stem from the assumptions of both free will and foreknowledge of future events. All of these are sometimes referred to individually as "causal loops." The term "time loop" is sometimes referred to as a causal loop, but although they appear similar, causal loops are unchanging and self-originating, whereas time loops are constantly resetting.

Bootstrap paradox

A bootstrap paradox, also known as an information loop, an information paradox, an ontological paradox, or a "predestination paradox" is a paradox of time travel that occurs when any event, such as an action, information, an object, or a person, ultimately causes itself, as a consequence of either retrocausality or time travel.

Backward time travel would allow information, people, or objects whose histories seem to "come from nowhere". Such causally looped events then exist in spacetime, but their origin cannot be determined. The notion of objects or information that are "self-existing" in this way is often viewed as paradoxical. Everett gives the movie Somewhere in Time as an example involving an object with no origin: an old woman gives a watch to a playwright who later travels back in time and meets the same woman when she was young, and gives her the same watch that she will later give to him. An example of information which "came from nowhere" is in the movie Star Trek IV: The Voyage Home, in which a 23rd-century engineer travels back in time, and gives the formula for transparent aluminum to the 20th-century engineer who supposedly invented it.

Predestination paradox

Smeenk uses the term "predestination paradox" to refer specifically to situations in which a time traveler goes back in time to try to prevent some event in the past.

Grandfather paradox

The consistency paradox or grandfather paradox occurs when the past is changed in any way, thus creating a contradiction. A common example given is traveling to the past and intervening with the conception of one's ancestors (such as causing the death of the parent beforehand), thus affecting the conception of oneself. If the time traveler were not born, then it would not be possible for the traveler to undertake such an act in the first place. Therefore, the ancestor lives to offspring the time traveler's next-generation ancestor, and eventually the time traveler. There is thus no predicted outcome to this. Consistency paradoxes occur whenever changing the past is possible. A possible resolution is that a time traveller can do anything that did happen, but cannot do anything that did not happen. Doing something that did not happen results in a contradiction. This is referred to as the Novikov self-consistency principle.

Variants

The grandfather paradox encompasses any change to the past, and it is presented in many variations, including killing one's past self. Both the "retro-suicide paradox" and the "grandfather paradox" appeared in letters written into Amazing Stories in the 1920s. Another variant of the grandfather paradox is the "Hitler paradox" or "Hitler's murder paradox", in which the protagonist travels back in time to murder Adolf Hitler before he can instigate World War II and the Holocaust. Rather than necessarily physically preventing time travel, the action removes any reason for the travel, along with any knowledge that the reason ever existed.

Physicist John Garrison et al. give a variation of the paradox of an electronic circuit that sends a signal through a time machine to shut itself off, and receives the signal before it sends it.

Newcomb's paradox

Main article: Newcomb's paradox

Newcomb's paradox is a thought experiment showing an apparent contradiction between the expected utility principle and the strategic dominance principle. The thought experiment is often extended to explore causality and free will by allowing for "perfect predictors": if perfect predictors of the future exist, for example if time travel exists as a mechanism for making perfect predictions then perfect predictions appear to contradict free will because decisions apparently made with free will are already known to the perfect predictor. Predestination does not necessarily involve a supernatural power, and could be the result of other "infallible foreknowledge" mechanisms. Problems arising from infallibility and influencing the future are explored in Newcomb's paradox.

Proposed resolutions

Logical impossibility

Even without knowing whether time travel to the past is physically possible, it is possible to show using modal logic that changing the past results in a logical contradiction. If it is necessarily true that the past happened in a certain way, then it is false and impossible for the past to have occurred in any other way. A time traveler would not be able to change the past from the way it is, but would only act in a way that is already consistent with what necessarily happened.

Consideration of the grandfather paradox has led some to the idea that time travel is by its very nature paradoxical and therefore logically impossible. For example, the philosopher Bradley Dowden made this sort of argument in the textbook Logical Reasoning, arguing that the possibility of creating a contradiction rules out time travel to the past entirely. However, some philosophers and scientists believe that time travel into the past need not be logically impossible provided that there is no possibility of changing the past, as suggested, for example, by the Novikov self-consistency principle. Dowden revised his view after being convinced of this in an exchange with the philosopher Norman Swartz.

Illusory time

Consideration of the possibility of backward time travel in a hypothetical universe described by a Gödel metric led famed logician Kurt Gödel to assert that time might itself be a sort of illusion. He suggests something along the lines of the block time view, in which time is just another dimension like space, with all events at all times being fixed within this four-dimensional "block".

Physical impossibility

Sergey Krasnikov writes that these bootstrap paradoxes – information or an object looping through time – are the same; the primary apparent paradox is a physical system evolving into a state in a way that is not governed by its laws. He does not find these paradoxical and attributes problems regarding the validity of time travel to other factors in the interpretation of general relativity.

Self-sufficient loops

A 1992 paper by physicists Andrei Lossev and Igor Novikov labeled such items without origin as Jinn, with the singular term Jinnee. This terminology was inspired by the Jinn of the Quran, which are described as leaving no trace when they disappear. Lossev and Novikov allowed the term "Jinn" to cover both objects and information with the reflexive origin; they called the former "Jinn of the first kind", and the latter "Jinn of the second kind". They point out that an object making circular passage through time must be identical whenever it is brought back to the past, otherwise it would create an inconsistency; the second law of thermodynamics seems to require that the object tends to a lower energy state throughout its history, and such objects that are identical in repeating points in their history seem to contradict this, but Lossev and Novikov argued that since the second law only requires entropy to increase in closed systems, a Jinnee could interact with its environment in such a way as to regain "lost" entropy. They emphasize that there is no "strict difference" between Jinn of the first and second kind. Krasnikov equivocates between "Jinn", "self-sufficient loops", and "self-existing objects", calling them "lions" or "looping or intruding objects", and asserts that they are no less physical than conventional objects, "which, after all, also could appear only from either infinity or a singularity."

Novikov self-consistency principle

Main article: Novikov self-consistency principle

The self-consistency principle developed by Igor Dmitriyevich Novikov expresses one view as to how backward time travel would be possible without the generation of paradoxes. According to this hypothesis, even though general relativity permits some exact solutions that allow for time travel that contain closed timelike curves that lead back to the same point in spacetime, physics in or near closed timelike curves (time machines) can only be consistent with the universal laws of physics, and thus only self-consistent events can occur. Anything a time traveler does in the past must have been part of history all along, and the time traveler can never do anything to prevent the trip back in time from happening, since this would represent an inconsistency. The authors concluded that time travel need not lead to unresolvable paradoxes, regardless of what type of object was sent to the past.

Top: original billiard ball trajectory. Middle: the billiard ball emerges from the future, and delivers its past self a strike that averts the past ball from entering the time machine. Bottom: The billiard ball never enters the time machine, giving rise to the paradox, putting into question how its older self could ever emerge from the time machine and divert its course.

Physicist Joseph Polchinski considered a potentially paradoxical situation involving a billiard ball that is fired into a wormhole at just the right angle such that it will be sent back in time and collides with its earlier self, knocking it off course, which would stop it from entering the wormhole in the first place. Kip Thorne referred to this problem as "Polchinski's paradox". Thorne and two of his students at Caltech, Fernando Echeverria and Gunnar Klinkhammer, went on to find a solution that avoided any inconsistencies, and found that there was more than one self-consistent solution, with slightly different angles for the glancing blow in each case. Later analysis by Thorne and Robert Forward showed that for certain initial trajectories of the billiard ball, there could be an infinite number of self-consistent solutions. It is plausible that there exist self-consistent extensions for every possible initial trajectory, although this has not been proven. The lack of constraints on initial conditions only applies to spacetime outside of the chronology-violating region of spacetime; the constraints on the chronology-violating region might prove to be paradoxical, but this is not yet known.

Novikov's views are not widely accepted. Visser views causal loops and Novikov's self-consistency principle as an ad hoc solution, and supposes that there are far more damaging implications of time travel. Krasnikov similarly finds no inherent fault in causal loops but finds other problems with time travel in general relativity. Another conjecture, the cosmic censorship hypothesis, suggests that every closed timelike curve passes through an event horizon, which prevents such causal loops from being observed.

Parallel universes

The interacting-multiple-universes approach is a variation of the many-worlds interpretation of quantum mechanics that involves time travelers arriving in a different universe than the one from which they came; it has been argued that, since travelers arrive in a different universe's history and not their history, this is not "genuine" time travel. Stephen Hawking has argued for the chronology protection conjecture, that even if the MWI is correct, we should expect each time traveler to experience a single self-consistent history so that time travelers remain within their world rather than traveling to a different one.

David Deutsch has proposed that quantum computation with a negative delay—backward time travel—produces only self-consistent solutions, and the chronology-violating region imposes constraints that are not apparent through classical reasoning. However Deutsch's self-consistency condition has been demonstrated as capable of being fulfilled to arbitrary precision by any system subject to the laws of classical statistical mechanics, even if it is not built up by quantum systems. Allen Everett has also argued that even if Deutsch's approach is correct, it would imply that any macroscopic object composed of multiple particles would be split apart when traveling back in time, with different particles emerging in different worlds.