Resource request: textbook on classical mechanics from a relativistic viewpoint

Question

[Edit: I haven't been fully satisfied with the books I found so far, so I did it myself, still in progress.]

I'm looking for an undergraduate textbook that covers classical mechanics with all the standard subtopics and applications (conservations laws, gravity, Hookean springs, friction, and similar), but approaching them from a point of view and notions that are closer to relativity theory (special and general). I do not mean the full maths of covariance, coordinate transformations, and curvature, but basic physical notions as we understand them today owing to relativity.

Examples – not a complete list – of what I mean by such an approach:

• Mass is presented as the total energy content of a body, so it changes if we for example heat up the body. It is explained that the differences brought by such energy exchanges are so small compared with the total energy content, that we can consider the latter as constant; hence mass conservation (example exceptions: nuclear physics & nuclear energy).

• Momentum is introduced as a notion in its own right, not defined as "mass times acceleration" or mass flow, and it is not linked to material bodies. It is explained that, in everyday situations, we can associate to a small body a momentum roughly equal to its total energy content times its velocity. But it is briefly pointed out that momentum is generally not exactly collinear with velocity, and that it also comes with other things, such as light and electromagnetic waves. For example it can be briefly explained that for a body with total mass-energy $m$, velocity $\pmb{v}$, and emitting a heat flux $\pmb{q}$, its momentum is $\pmb{p}=m\pmb{v}+\pmb{q}/c^2$. In many situations, however, the contribution $\pmb{q}/c^2$ is so incredibly small that we can simply take $\pmb{p} \approx m\pmb{v}$ as an excellent approximation.

• It is explained at the outset that time lapse is in principle different for all bodies, depending on their motion. So if two clocks are originally put side-by-side and synchronized, then moved along different trajectories, then brought together again, they will turn out to be unsynchronized. It is explained that such time differences are so small that in many practical applications we can consider clocks to be always synchronized (example exceptions: GPS applications).

• It is pointed out that gravitational and inertial (centrifugal, Coriolis, etc) forces are effects of the motion of a body with respect to spacetime.

Again, the maths doesn't need to be different from that of standard undergraduate mechanics textbooks, but the physical notions are consistently and recurringly presented in a different way that, besides being more modern, makes the transition to relativity and nuclear physics easier.

I have looked at references coming from other questions, most of which are compiled into this answer, but none of them are what I'm looking for. Bondi's book Relativity and Common Sense comes closest, but it's still a mostly a qualitative book.

[Please note: I don't want to start a necessarily subjective debate about whether mechanics should be taught differently. I'm just asking for resources & references. So from now on I won't reply to comments or answers of the "why would/wouldn't you do that" kind.]

Answer 1

To the best of my knowledge, there is no such text -- for good reasons as I'll briefly explain below. A modern text that comes somewhat close to what the OP is looking for may be Kogut's "Special Relativity, Electrodynamics, and General Relativity. From Newton to Einstein." In this book he begins with Newtonian mechanics and its shortcomings and immediately (in the second chapter) moves on to special relativity. Crucially, in Chapter 7, he discusses the "Death of Newton's Third Law, and Static Forces: The Birth of Fields". There the author convincingly (and in fairly basic terms!) argues that a consistent formulation of relativistic mechanics requires the introduction of fields carrying energy and momentum. In this way, energy-momentum conservation is guaranteed without action being reaction (Newton's Third Law), which can no longer hold due to retardation effects. Note that this rules out (or complicates) all of statics and the notion of constraint forces. There is also no such thing as a relativistic solid body, and even a relativistic harmonic oscillator is a rather subtle concept as retardation is in conflict with potential forces. As a result, most texts discussing relativistic mechanics just consider point charges coupled to electromagnetic fields. Even in this case, one ultimately runs into trouble when one tries to eliminate the field from the equations of motion. This leads to the relativistic Lorentz-Abraham-Dirac equation, the pathologies of which have plagued generations of physicists (with no end in sight).

Beginning with early work by Dirac there have been many attempts (mostly in the 1980s) to formulate a relativistic action-at-a-distance theory of interacting particles without field mediation. While some progress has been made, possibly culminating in the "non-interaction theorem", these attempts have never reached a fully satisfactory status and thus have remained on the fringes of physics research. The latest review article I'm aware of can be found in the journal Quantum Reports here (open access).

Answer 2

One textbook that may meet your criteria is "Classical Mechanics: A Modern Perspective" by Vernon Barger and Martin Olsson. This book introduces the basic principles of classical mechanics in a way that emphasizes their connection to relativity theory. It starts with a discussion of the principles of relativity and the role of the observer in physics, before moving on to topics like Newton's laws, conservation laws, and central forces. The authors also introduce the Lagrangian and Hamiltonian formulations of mechanics, and use these to derive the equations of motion for a variety of systems.

Throughout the book, the authors emphasize the importance of considering the relativistic properties of physical systems, even in situations where the effects of relativity are small. They also discuss the limitations of classical mechanics in situations where relativistic or quantum effects are important. The book includes a number of worked examples and problems, as well as a brief introduction to special relativity and some of its applications in mechanics.

Overall, "Classical Mechanics: A Modern Perspective" provides a modern and relativistic approach to classical mechanics that may be of interest to you. The book assumes some familiarity with calculus and basic physics concepts, but does not assume any prior knowledge of relativity theory.

Answer 3

Many courses in mechanics, including the one I took in the university and Feynman's famous "Lectures on Physics volume 1", treat special relativity as a part of mechanics, and not a separate course. However, almost always the order is still a few lessons on "Newtonian" mechanics and all the concepts you mentioned - force, acceleration, gravity, rotation, springs, etc., followed by what is changed by relativity. You seem to be hoping for a book which starts the other way around - start with relativity as the more correct physics, and then look at special cases with low energies and speeds and what you can learn with them.

I see several pedagogical problems with that "reversed" approach:

• First, it means the students would need to learn the more difficult and "unnatural" formulas before they can be introduced to stuff they have more natural understanding of (everyone is familiar with, or can be shown, inertia, acceleration, harmonic oscillation, rotation, etc - but Lorentz transformation, time dilation, mass change, etc., doesn't come naturally).
• Secondly, some of the things you mentioned, like friction, springs, etc., involve energy so does involve miniscule changes to the mass of the objects involved, so you'd need to explicitly "neglect" these issues to come up with the familiar and simple formulas. Even simple stuff like addition of velocities is different in special relativity. Instead of spending half of each lesson saying what is neglected, it's easier not to introduce these neglected things yet.
• Thirdly, as you also noted, mechanics isn't just about fixed velocity, all the interesting phenomena, including gravity, forces, springs, rotation, and so on, involve acceleration. When you have acceleration, you are no longer talking about special relativity, but about general relativity. And although I guess it is theoretically possible to start learning mechanics with GR, this would require extraordinary discipline from the students and probably too much math. Learning Newton's approximation of gravity - instantaneous action at a distance, potential, etc. - may not give the students the right tools to calculate to orbit of Mercury (or understand black holes), but will leave the students understading something, which will later help them when they are ready for GR. By the way, you gave a good example with GPS - to implement GPS correctly, it's not enough to correct for special relativity, you also need to correct for general relativity (gravitational time dilation).