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Linearized general relativity and the Lanczos potential
Published in arXiv e-prints, 1999
Abstract:“Recently, there has been a revival of interest in the Lanczos potential of the Weyl conformal tensor. Previous work by Novello and Neto has been done with the linearized Lanczos potential as a model of a spin-2 field, which depends on a massless limit of the field. In this paper, we look at an action based on a massless potential, and show that it is classically equivalent to the linearized regime of general relativity, without reference to a massless limit.”
Recommended citation: Daniel Cartin (1999). "Linearized general relativity and the Lanczos potential." arXiv e-prints, gr-qc/9910082.
Self-regulated model of galactic spiral structure formation
Published in Physical Review E, 2001
Abstract:“The presence of spiral structure in isolated galaxies is a problem that has only been partially explained by theoretical models. Because the rate and pattern of star formation in the disk must depend only on mechanisms internal to the disk, we may think of the spiral galaxy as a self-regulated system far from equilibrium. This paper uses this idea to look at a reaction-diffusion model for the formation of spiral structures in certain types of galaxies. In numerical runs of the model, spiral structure forms and persists over several revolutions of the disk, but eventually dies out.”
Recommended citation: Daniel Cartin and Gaurav Khanna (2001). "Self-regulated model of galactic spiral structure formation." Physical Review E 65(1), 016120.
The Lanczos potential as a spin-2 field
Published in arXiv e-prints, 2003
Abstract:“The Lanczos potential $L_{abc}$ acts as a tensor potential for the spin-2 field strength $W_{abcd}$ in an role similar to that of the vector potential $A_a$ for the Maxwell tensor $F_{ab}$. After some general considerations inspired by the example of electromagnetism, we consider the linear spin-2 theory and a Born-Infeld type action in terms of $L_{abc}$.”
Recommended citation: Daniel Cartin (2003). "The Lanczos potential as a spin-2 field." arXiv e-prints, hep-th/0311185.
Generating function techniques for loop quantum cosmology
Published in Classical and Quantum Gravity, 2004
Abstract:“Loop quantum cosmology leads to a difference equation for the wavefunction of a universe, which, in general, has solutions changing rapidly even when the volume changes only slightly. For a semiclassical regime, such small-scale oscillations must be suppressed by choosing the parameters of the solution appropriately. For anisotropic models, this is not possible to do numerically by trial and error; instead, it is shown here, for the Bianchi I LRS model, how this can be done analytically, by using generating function techniques. Those techniques can also be applied to more complicated models, and the results gained allow conclusions about initial value problems for other systems.”
Recommended citation: Daniel Cartin, Gaurav Khanna, and Martin Bojowald (2004). "Generating function techniques for loop quantum cosmology." Classical and Quantum Gravity 21(18), 4495-4509.
Absence of Preclassical Solutions in Bianchi I Loop Quantum Cosmology
Published in Physical Review Letters, 2005
Abstract:“Loop quantum cosmology, the symmetry reduction of quantum geometry for the study of various cosmological situations, leads to a difference equation for its quantum evolution equation. To ensure that solutions of this equation act in the expected classical manner far from singularities, additional restrictions are imposed on the solution. In this Letter, we consider the Bianchi I model, both the vacuum case and the addition of a cosmological constant, and show using generating function techniques that only the zero solution satisfies these constraints. This implies either that there are technical difficulties with the current method of quantizing the evolution equation, or else loop quantum gravity imposes strong restrictions on the physically allowed solutions.”
Recommended citation: Daniel Cartin and Gaurav Khanna (2005). "Absence of Preclassical Solutions in Bianchi I Loop Quantum Cosmology." Physical Review Letters 94(11), 111302.
Separable sequences in Bianchi I loop quantum cosmology
Published in Physical Review D, 2005
Abstract:“In this paper, we discuss the properties of one-parameter sequences that arise when solving the Hamiltonian constraint in Bianchi I loop quantum cosmology using a separation of variables method. In particular, we focus on finding an expression for the sequence for all real values of the parameter, and discuss the preclassicality of this function. We find that the behavior of these preclassical sequences imply time asymmetry on either side of the classical singularity in Bianchi I cosmology.”
Recommended citation: Daniel Cartin and Gaurav Khanna (2005). "Separable sequences in Bianchi I loop quantum cosmology." Physical Review D 72(8), 084008.
Wave functions for the Schwarzschild black hole interior
Published in Physical Review D, 2006
Abstract:“Using the Hamiltonian constraint derived by Ashtekar and Bojowald, we look for preclassical wave functions in the Schwarzschild interior. In particular, when solving this difference equation by separation of variables, an inequality is obtained relating the Immirzi parameter $\gamma$ to the quantum ambiguity $\delta$ appearing in the model. This bound is violated when we use a natural value for $\delta$ based on loop quantum gravity together with a recent proposal for $\gamma$. We also present numerical solutions of the constraint.”
Recommended citation: Daniel Cartin and Gaurav Khanna (2006). "Wave functions for the Schwarzschild black hole interior." Physical Review D 73(10), 104009.
Lattice refining loop quantum cosmology, anisotropic models, and stability
Published in Physical Review D, 2007
Abstract:“A general class of loop quantizations for anisotropic models is introduced and discussed, which enhances loop quantum cosmology by relevant features seen in inhomogeneous situations. The main new effect is an underlying lattice which is being refined during dynamical changes of the volume. In general, this leads to a new feature of dynamical difference equations which may not have constant step-size, posing new mathematical problems. It is discussed how such models can be evaluated and what lattice refinements imply for semiclassical behavior. Two detailed examples illustrate that stability conditions can put strong constraints on suitable refinement models, even in the absence of a fundamental Hamiltonian which defines changes of the underlying lattice. Thus, a large class of consistency tests of loop quantum gravity becomes available. In this context, it will also be seen that quantum corrections due to inverse powers of metric components in a constraint are much larger than they appeared recently in more special treatments of isotropic, free scalar models where they were artificially suppressed.”
Recommended citation: Martin Bojowald, Daniel Cartin, and Gaurav Khanna (2007). "Lattice refining loop quantum cosmology, anisotropic models, and stability." Physical Review D 76(6), 064018.
On the Maximum Sufficient Range of Interstellar Vessels
Published in Journal of the British Interplanetary Society, 2010
Abstract:“This paper considers the likely maximum range of space vessels providing the basis of a mature interstellar transportation network. Using the principle of sufficiency, it is argued that this range will be less than three parsecs for the average interstellar vessel. This maximum range provides access from the Solar System to a large majority of nearby stellar systems, with total travel distances within the network not excessively greater than actual physical distance.”
Recommended citation: D. Cartin (2010). "On the Maximum Sufficient Range of Interstellar Vessels." Journal of the British Interplanetary Society 63, 218-221.
On the maximum sufficient range of interstellar vessels
Published in arXiv e-prints, 2011
Abstract:“This paper considers the likely maximum range of space vessels providing the basis of a mature interstellar transportation network. Using the principle of sufficiency, it is argued that this range will be less than three parsecs for the average interstellar vessel. This maximum range provides access from the Solar System to a large majority of nearby stellar systems, with total travel distances within the network not excessively greater than actual physical distance.”
Recommended citation: Daniel Cartin (2011). "On the maximum sufficient range of interstellar vessels." arXiv e-prints, arXiv:1104.4012.
Numerical Techniques in Loop Quantum Cosmology
Published in SIGMA, 2012
Abstract:“In this article, we review the use of numerical techniques to obtain solutions for the quantum Hamiltonian constraint in loop quantum cosmology (LQC). First, we summarize the basic features of LQC, and describe features of the constraint equations to solve - generically, these are difference (rather than differential) equations. Important issues such as differing quantization methods, stability of the solutions, the semi-classical limit, and the relevance of lattice refinement in the difference equations are discussed. Finally, the cosmological models already considered in the literature are listed, along with typical features in these models and open issues.”
Recommended citation: David Brizuela, Daniel Cartin, and Gaurav Khanna (2012). "Numerical Techniques in Loop Quantum Cosmology." SIGMA 8, 001.
Conserved quantities in isotropic loop quantum cosmology
Published in Europhysics Letters, 2012
Abstract:“We develop an action principle for those models arising from isotropic loop quantum cosmology, and show that there is a natural conserved quantity $Q$ for the discrete difference equation arising from the Hamiltonian constraint. This quantity $Q$ relates the semi-classical limit of the wave function at large values of the spatial volume, but opposite triad orientations. Moreover, there is a similar quantity for generic difference equations of one parameter arising from a self-adjoint operator.”
Recommended citation: D. Cartin (2012). "Conserved quantities in isotropic loop quantum cosmology." Europhysics Letters 98(3), 30007.
Exploration of the local solar neighbourhood I: Fixed number of probes
Published in International Journal of Astrobiology, 2013
Abstract:“Previous work in studying interstellar exploration by one or several probes has focused primarily either on engineering models for a spacecraft targeting a single star system, or large-scale simulations to ascertain the time required for a civilization to completely explore the Milky Way Galaxy. In this paper, a simulated annealing algorithm is used to numerically model the exploration of the local interstellar neighbourhood (i.e. of the order of ten parsecs of the Solar System) by a fixed number of probes launched from the Solar System; these simulations use the observed masses, positions and spectral classes of targeted stars. Each probe visits a pre-determined list of target systems, maintains a constant cruise speed, and only changes the direction from gravitational deflection at each target. From these simulations, it is examined how varying design choices - differing the maximum cruise speed, number of probes launched, number of target stars to be explored, and probability of avoiding catastrophic system failure per parsec - change the completion time of the exploration programme and the expected number of stars successfully visited. In addition, it is shown that improving this success probability per parsec has diminishing returns beyond a certain point. Future improvements to the model and possible implications are discussed.”
Recommended citation: Daniel Cartin (2013). "Exploration of the local solar neighbourhood I: Fixed number of probes." International Journal of Astrobiology 12(4), 271-281.
Quantifying the Fermi Paradox in the Local Solar Neighbourhood
Published in Journal of the British Interplanetary Society, 2014
Abstract:“The Fermi paradox highlights the dichotomy between the lack of physical contact with other civilizations and the expectation that technological civilizations are assumed likely to evolve in many locations in the Milky Way galaxy, given the large number of planetary systems within this galaxy. Work by Landis and others has modeled this question in terms of percolation theory and cellular automata, using this method to parametrize our ignorance about possible other civilizations as a function of the probability of one system to colonize another, and the maximum number of systems reachable from each starting location (i.e. the degree in the network used for percolation). These models used a fixed lattice of sites to represent a stellar region, so the degree of all sites were identical. In this paper, the question is examined again, but instead of using a pre-determined lattice, the actual physical positions of all known star systems within 40 parsecs of the Solar System are used as percolation sites; in addition, the number of sites accessible for further colonization from a given system is determined by a choice of maximum distance such efforts can travel across. The resulting simulations show that extraterrestrial colonization efforts may reach the Solar System, but only for certain values of the maximum travel distance and probability of an occupied system further colonizing other systems. Indeed, large numbers of systems may be colonized with either vessels that lack insufficient travel distance to reach the Solar System or else have a colonization probability where they are statistically unlikely to reach us.”
Recommended citation: D. Cartin (2014). "Quantifying the Fermi Paradox in the Local Solar Neighbourhood." Journal of the British Interplanetary Society 67, 119-126.
Braids as a representation space of SU(5)
Published in Journal of Mathematical Physics, 2015
Abstract:“The standard model of particle physics provides very accurate predictions of phenomena occurring at the sub-atomic level, but the reason for the choice of symmetry group and the large number of particles considered elementary is still unknown. Along the lines of previous preon models positing a substructure to explain these aspects, Bilson-Thompson showed how the first family of elementary particles is realized as the crossings of braids made of three strands, with charges resulting from twists of those strands with certain conditions; in this topological model, there are only two distinct neutrino states. Modeling the particles as braids implies these braids must be the representation space of a Lie algebra, giving the symmetries of the standard model. In this paper, this representation is made explicit, obtaining the raising operators associated with the Lie algebra of $SU(5)$, one of the earliest grand unified theories. Because the braids form a group, the action of these operators are braids themselves, leading to their identification as gauge bosons. Possible choices for the other two families are also given. Although this realization of particles as braids is lacking a dynamical framework, it is very suggestive, especially when considered as a natural method of adding matter to loop quantum gravity.”
Recommended citation: Daniel Cartin (2015). "Braids as a representation space of SU(5)." Journal of Mathematical Physics 56(6), 061703.
Upper limits on the probability of an interstellar civilization arising in the local Solar neighbourhood
Published in International Journal of Astrobiology, 2015
Abstract:“At this point in time, there is very little empirical evidence on the likelihood of a space-faring species originating in the biosphere of a habitable world. However, there is a tension between the expectation that such a probability is relatively high (given our own origins on Earth), and the lack of any basis for believing the Solar System has ever been visited by an extraterrestrial colonization effort. From the latter observational fact, this paper seeks to place upper limits on the probability of an interstellar civilization arising on a habitable planet in its stellar system, using a percolation model to simulate the progress of such a hypothetical civilization’s colonization efforts in the local Solar neighbourhood. To be as realistic as possible, the actual physical positions and characteristics of all stars within 40 parsecs of the Solar System are used as possible colony sites in the percolation process. If an interstellar civilization is very likely to have such colonization programmes, and they can travel over large distances, then the upper bound on the likelihood of such a species arising per habitable world is of the order of 10$^{-3}$ on the other hand, if civilizations are not prone to colonize their neighbours, or do not travel very far, then the upper limiting probability is much larger, even of order one.”
Recommended citation: Daniel Cartin (2015). "Upper limits on the probability of an interstellar civilization arising in the local Solar neighbourhood." International Journal of Astrobiology 14(4), 571-575.
Knotted 4-regular graphs: Polynomial invariants and the Pachner moves
Published in Journal of Mathematical Physics, 2022
Abstract:“In loop quantum gravity, states of quantum geometry are represented by classes of knotted graphs, equivalent under diffeomorphisms. Thus, it is worthwhile to enumerate and distinguish these classes. This paper looks at the case of 4-regular graphs, which have an interpretation as objects dual to triangulations of three-dimensional manifolds. Two different polynomial invariants are developed to characterize these graphs\textemdashone inspired by the Kauffman bracket relations and the other based on quandles. How the latter invariant changes under the Pachner moves acting on the graphs is then studied.”
Recommended citation: Daniel Cartin (2022). "Knotted 4-regular graphs: Polynomial invariants and the Pachner moves." Journal of Mathematical Physics 63(6), 063502.
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Knotted 4-regular graphs. II. Consistent application of the Pachner moves
Published in Journal of Mathematical Physics, 2024
Abstract:“A common choice for the evolution of the knotted graphs in loop quantum gravity is to use the Pachner moves, adapted to graphs from their dual triangulations. Here, we show that the natural way to consistently use these moves is on framed graphs with edge twists, where the Pachner moves can only be performed when the twists, and the vertices the edges are incident on, meet certain criteria. For other twists, one can introduce an algebraic object, which allow any knotted graph with framed edges to be written in terms of a generalized braid group.”
Recommended citation: Daniel Cartin (2024). "Knotted 4-regular graphs. II. Consistent application of the Pachner moves." Journal of Mathematical Physics 65(6), 061701.
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