Cosimo Bambi - Introduction to General Relativity

Undergraduate Lecture Notes in Physics (ULNP) publishes authoritative texts covering topics throughout pure and applied physics. Each title in the series is suitable as a basis for undergraduate instruction, typically containing practice problems, worked examples, chapter summaries, and suggestions for further reading.

ULNP especially encourages new, original, and idiosyncratic approaches to physics teaching at the undergraduate level.

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Neil Ashby

University of Colorado, Boulder, CO, USA

William Brantley

Department of Physics, Furman University, Greenville, SC, USA

Matthew Deady

Physics Program, Bard College, Annandale-on-Hudson, NY, USA

Michael Fowler

Department of Physics, University of Virginia, Charlottesville, VA, USA

Morten Hjorth-Jensen

Department of Physics, University of Oslo, Oslo, Norway

Michael Inglis

Department of Physical Sciences, SUNY Suffolk County Community College, Selden, NY, USA

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The formulations of the theories of special and general relativity and of the theory of quantum mechanics in the first decades of the twentieth century are a fundamental milestone in science, not only for their profound implications in physics but also for the research methodology. In the same way, the courses of special and general relativity and of quantum mechanics represent an important milestone for every student of physics. These courses introduce a different approach to investigate physical phenomena, and students need some time to digest such a radical change.

In Newtonian mechanics and in Maxwells theory of electrodynamics, the approach is quite empirical and natural. First, we infer a few fundamental laws from observations (e.g., Newtons Laws) and then we construct the whole theory (e.g., Newtonian mechanics). In modern physics, starting from special and general relativity and quantum mechanics, this approach may not be always possible. Observations and formulation of the theory may change order. This is because we may not have direct access to the basic laws governing a certain physical phenomenon. In such a case, we can formulate a number of theories, or we can introduce a number of ansatzes to explain a specific physical phenomenon within a certain theory if we already have the theory, and then we compare the predictions of the different solutions to check which one, if any, is consistent with observations.

For example, Newtons First, Second, and Third Laws can be directly inferred from experiments. Einsteins equations are instead obtained by imposing some reasonable requirements and they are then confirmed by comparing their predictions with the results of experiments. In modern physics, it is common that theorists develop theoretical models on the basis of guesses (motivated by theoretical arguments but without any experimental support), with the hope that it is possible to find predictions that can later be tested by experiments.

At the beginning, a student may be disappointed by this new approach and may not understand the introduction of ad hoc assumptions. In part, this is because we are condensing in a course the efforts of many physicists and many experiments, without discussing all the unsuccessfulbut nevertheless necessary and importantattempts that eventually led to a theory in its final form. Moreover, different students may have different backgrounds, not only because they are students of different disciplines (e.g., theoretical physics, experimental physics, astrophysics, and mathematical physics) but also because undergraduate programs in different countries can be very different. Additionally, some textbooks may follow approaches appreciated by some students and not by others, who may instead prefer different textbooks. This point is quite important when we study for the first time the theories of special and general relativity and the theory of quantum mechanics, because there are some concepts that at the beginning are difficult to understand, and a different approach may make it easier or harder.

In the present textbook, the theories of special and general relativity are introduced with the help of the Lagrangian formalism. This is the approach employed in the famous textbook by Landau and Lifshitz. Here, we have tried to have a book more accessible to a larger number of students, starting from a short review of Newtonian mechanics, reducing the mathematics, presenting all the steps of most calculations, and considering some (hopefully illuminating) examples. The present textbook dedicates quite a lot of space to the astrophysical applications, discussing Solar System tests, black holes, cosmological models, and gravitational waves at a level adequate for an introductory course of general relativity. These lines of research have become very active in the past couple of decades and have attracted an increasing number of students. In the last chapter, students can get a quick overview of the problems of Einsteins gravity and current lines of research in theoretical physics.