Teleparallel Gravity: An Introduction (Fundamental Theories of Physics Book 173) 2013th Edition by Ruben Aldrovandi (PDF)

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Teleparallel Gravity (TG) is an alternative theory for gravitation, which is equivalent to General Relativity (GR). However, it is conceptually different. For example in GR geometry replaces the concept of force, and the trajectories are determined by geodesics. TG attributes gravitation to torsion, which accounts for gravitation by acting as a force. TG has already solved some old problems of gravitation (like the energy-momentum density of the gravitational field). The interest in TG has grown in the last few years. The book here proposed will be the first one dedicated exclusively to TG, and will include the foundations of the theory, as well as applications to specific problems to illustrate how the theory works.

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⭐This book on a very important topic suffers from some deep mistakes.CHAPTER 17: EINSTEIN-CARTAN THEORY1. The authors define the momentum tensor as the variation of the Lagrangian with respect to the metric tensor (page 172). This yields a “fake” momentum tensor that is symmetric.The momentum tensors should be defined by variation of the Lagrangian with respect to affine connection coefficients.• Variation with the frame field (which consists of the translational connection coefficients) yields a non-symmetric linear momentum tensor.• Variation with the rotational connection coefficients yields the law of conservation of angular momentum in Einstein-Cartan theory. This enables Einstein-Cartan theory to effortlessly include exchange of intrinsic and orbital angular momentum – which general relativity cannot do because of its symmetric momentum tensor, which is caused by varying with respect to the metric.Einstein-Cartan theory is an affine theory, not a metric theory. If you treat it as a metric theory, and vary the Lagrangian with the metric, you are likely to end up with the wrong momentum tensor. The equations lose their simple interpretation, though you might be able to retain the correct tensor equations.It is likely that the authors’ tensor equations are correct, but the conclusions they draw are wrong. The authors claim that:2. Einstein-Cartan theory violates the principle of general covariance (page 176).3. Einstein-Cartan theory violates the gauge invariance of the electromagnetic field (page 176-177).The root cause of these three problems is that the authors do not discuss (and appear not to understand) the most basic concept of affine differential geometry: affine connections. (See comments below on chapter 1 which deals with foundations.)All three errors have a more proximate cause.• Fiber indices (Roman indices) should be differentiated with the full connection (including torsion).• Spacetime indices (Greek indices) should be differentiated with the torsion-free Levi-Civita connection.This strange-sounding rule for spacetime indices results from a deeper issue. The spacetime indices are not really covariant differentiated at all. They always appear in antisymmetric groupings that represent spacetime flux boxes. They are always (covariant-) exterior differentiated without any connection; or their tensors are Hodge_dualed (a ‘k’ tensor goes to dim-k tensor density) and then a divergence is applied. The Levi-Civita connection is a computational device for including the terms that arise when div(Hodge_dual(tensor)) appears.CHAPTER 1: BASIC NOTIONSThis chapter omits the most basic concepts in E. Cartan’s theory of affine differential geometry: an affine connection. (I skip the technicalities.) The authors elide over the differences among an affine connection, a linear connection and a metric connection. (page 6)Their tensor formulations in this basic chapter appear to be correct. However, the omission of affine connections and the incorrect treatment of derivatives lies behind the errors discussed above, and undermines he geometric interpretation of the theory.CHAPTER 4: FUNDAMENTALS OF TELEPARALLEL GRAVITATIONI haven’t read this chapter yet, and I tremble at the thought of relying on this book to guide me.Impact of the errorsFor the second half of the twentieth century, Einstein-Cartan theory got unjustly labeled as nut-ball stuff. This has partly been turned around, for example in the book by Blagojević and Hehl (2013). However, very few physicists are equipped or interested in evaluating the theory, so these errors could have a serious impact on people who might consider it.

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