We present OGRe, a modern Mathematica package for tensor calculus, designed to be both powerful and user-friendly. The package can be used in a variety of contexts where tensor calculations are needed, in both mathematics and physics, but it is especially suitable for general relativity. By implementing an object-oriented design paradigm, OGRe allows calculating arbitrarily complicated tensor formulas easily, and automatically transforms between index configurations and coordinate systems behind the scenes as needed, eliminating user errors by making it impossible for the user to combine tensors in inconsistent ways. Other features include displaying tensors in various forms, automatic calculation of curvature tensors and geodesic equations, easy importing and exporting of tensors between sessions, optimized algorithms and parallelization for improved performance, and more.
We perform a rigorous piecewise-flat discretization of classical general relativity in the first-order formulation, in both 2+1 and 3+1 dimensions, carefully keeping track of curvature and torsion via holonomies. We show that the resulting phase space is precisely that of spin networks, the quantum states of discrete spacetime in loop quantum gravity, with additional degrees of freedom called edge modes, which control the gluing between cells. This work establishes, for the first time, a rigorous proof of the equivalence between spin networks and piecewise-flat geometries with curvature and torsion degrees of freedom. In addition, it demonstrates that careful consideration of edge modes is crucial both for the purpose of this proof and for future work in the field of loop quantum gravity. It also shows that spin networks have a dual description related to teleparallel gravity, where gravity is encoded in torsion instead of curvature degrees of freedom. Finally, it sets the stage for collaboration between the loop quantum gravity community and theoretical physicists working on edge modes from other perspectives, such as quantum electrodynamics, non-abelian gauge theories, and classical gravity.
If time travel is possible, it seems to inevitably lead to paradoxes. These include consistency paradoxes, such as the famous grandfather paradox, and bootstrap paradoxes, where something is created out of nothing. One proposed class of resolutions to these paradoxes allows for multiple histories (or timelines), such that any changes to the past occur in a new history, independent of the one where the time traveler originated. We introduce a simple mathematical model for a spacetime with a time machine, and suggest two possible multiple-histories models, making use of branching spacetimes and covering spaces respectively. We use these models to construct novel and concrete examples of multiple-histories resolutions to time travel paradoxes, and we explore questions such as whether one can ever come back to a previously visited history and whether a finite or infinite number of histories is required. Interestingly, we find that the histories may be finite and cyclic under certain assumptions, in a way which extends the Novikov self-consistency conjecture to multiple histories and exhibits hybrid behavior combining the two. Investigating these cyclic histories, we rigorously determine how many histories are needed to fully resolve time travel paradoxes for particular laws of physics. Finally, we discuss how observers may experimentally distinguish between multiple histories and the Hawking and Novikov conjectures.
Spin networks, the quantum states of discrete geometry in loop quantum gravity, are directed graphs whose links are labeled by irreducible representations of SU(2), or spins. Cosmic strings are 1-dimensional topological defects carrying distributional curvature in an otherwise flat spacetime. In this paper we prove that the classical phase space of spin networks coupled to cosmic strings may obtained as a straightforward discretization of general relativity in 3+1 spacetime dimensions. We decompose the continuous spatial geometry into 3-dimensional cells, which are dual to a spin network graph in a unique and well-defined way. Assuming that the geometry may only be probed by holonomies (or Wilson loops) located on the spin network, we truncate the geometry such that the cells become flat and the curvature is concentrated at the edges of the cells, which we then interpret as a network of cosmic strings. The discrete phase space thus describes a spin network coupled to cosmic strings. This work proves that the relation between gravity and spin networks exists not only at the quantum level, but already at the classical level. Two appendices provide detailed derivations of the Ashtekar formulation of gravity as a Yang-Mills theory and the distributional geometry of cosmic strings in this formulation.
These lecture notes were prepared for a 25-hour course for advanced undergraduate students participating in Perimeter Institutes Undergraduate Summer Program. The lectures cover some of what is currently known about the possibility of superluminal travel and time travel within the context of established science, that is, general relativity and quantum field theory. Previous knowledge of general relativity at the level of a standard undergraduate-level introductory course is recommended, but all the relevant material is included for completion and reference. No previous knowledge of quantum field theory, or anything else beyond the standard undergraduate curriculum, is required. Advanced topics in relativity, such as causal structures, the Raychaudhuri equation, and the energy conditions are presented in detail. Once the required background is covered, concepts related to faster-than-light travel and time travel are discussed. After introducing tachyons in special relativity as a warm-up, exotic spacetime geometries in general relativity such as warp drives and wormholes are discussed and analyzed, including their limitations. Time travel paradoxes are also discussed in detail, including some of their proposed resolutions.
In a recent paper, we introduced a new discretization scheme for gravity in 2+1 dimensions. Starting from the continuum theory, this new scheme allowed us to rigorously obtain the discrete phase space of loop gravity, coupled to particle-like edge mode degrees of freedom. In this work, we expand on that result by considering the most general choice of integration during the discretization process. We obtain a family of polarizations of the discrete phase space. In particular, one member of this family corresponds to the usual loop gravity phase space, while another corresponds to a new polarization, dual to the usual one in several ways. We study its properties, including the relevant constraints and the symmetries they generate. Furthermore, we motivate a relation between the dual polarization and teleparallel gravity.
We present a conceptually clear introduction to quantum theory at a level suitable for exceptional high-school students. It is entirely self-contained and no university-level background knowledge is required. The lectures were given over four days, four hours each day, as part of the International Summer School for Young Physicists (ISSYP) at Perimeter Institute, Waterloo, Ontario, Canada. On the first day the students were given all the relevant mathematical background from linear algebra and probability theory. On the second day, we used the acquired mathematical tools to define the full quantum theory in the case of a finite Hilbert space and discuss some consequences such as entanglement, Bells theorem and the uncertainty principle. Finally, on days three and four we presented an overview of advanced topics related to infinite-dimensional Hilbert spaces, including canonical and path integral quantization, the quantum harmonic oscillator, quantum field theory, the Standard Model, and quantum gravity.
