Abstract
After sketching the nature of the central problem in rapid spacetime transport – the manipulation of inertia – Mach’s ideas on the topic are mentioned. The origins of the concept of inertia and the principles of relativity and equivalence in the seventeenth century are outlined. But they did not lead to the theory of relativity, in no small part because of Newton’s adoption of absolute space and time. Special relativity theory is investigated, leading to Einstein’s discovery of the relationship between energy and inertial mass: \( m = E/{{c}^2} \), where E is the total non-gravitational energy of an isolated object at rest and c the speed of light in vacuum. How general relativity theory bears on this definition of inertial mass is then explored. The role of the Equivalence Principle – particularly, the prohibition of the localization of gravitational potential energy and the nature and role of “fictitious” forces – is examined, preparing the way for a discussion of Mach’s principle in Chap. 2. The behavior of light in the vicinity of negative mass matter is mentioned in anticipation of the third section of the book.
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- 1.
The simple case analyzed by Einstein in his first paper on special relativity theory – titled “On the Electrodynamics of Moving Bodies” – is the motion of a magnet with respect to a loop of wire. If the relative motion of the magnet and wire causes the “flux” of the magnetic field through the loop of wire to change, a current flows in the loop while the flux of the magnetic field passing through the loop is changing. It makes no difference to the current in the loop whether you take the loop as at rest with the magnet moving, or vice versa. The rest of the paper consists of Einstein’s demonstration that the mathematical machinery that gets you from the frame of reference where the magnet is at rest to the frame where the loop is at rest requires that the speed of light measured in both frames is the same, or “constant.” This is only possible if space and time are inextricably interlinked, destroying Newton’s absolute notions of space and time as physically distinct, independent entities. The concept underlying the full equivalence of the two frames of reference is the principle of relativity: that all inertial frames of reference are equally fundamental and no one of them can be singled out as more fundamental by any experiment that can be conducted locally.
- 2.
Galileo himself was guilty of this failing. When Kepler sent him his work on astronomy (the first two laws of planetary motion anyway), work that was incompatible with the compounded circular motions used by Copernicus, Galileo, a convinced Copernican, ignored it.
- 3.
Most historians of science would probably name Newton the greatest physicist of all time. Most physicists would likely pick Einstein for this honor (as did Lev Landau, a brilliant Russian theoretical physicist in the mid-twentieth century). Getting this right is complicated by the fact that Newton spent most of his life doing alchemy, biblical studies, pursuing a “patent” of nobility, and running the government’s mint after the mid-1690s. Physics and mathematics were sidelines for him. Einstein, on the other hand, aside from some womanizing, spent most of his life doing physics, albeit out of the mainstream after the late 1920s. It’s complicated.
- 4.
The period of a pendulum depends only on its length if the Equivalence Principle is true, so you can put masses of all different weights and compositions on a pendulum of some fixed length, and its period should remain the same. Newton did this using a standard comparison pendulum and found that Galileo was right, at least to about a part in a thousand. Very much fancier experiments that test this principle have been (and continue to be) done to exquisitely high accuracy.
- 5.
The technical definition of a “fictitious” force is one that produces the same acceleration irrespective of the mass of the object on which it acts. It has nothing to do with whether the force is “real” or not.
- 6.
This is a common problem encountered in teaching Newtonian mechanics, and evidently part of the reason for the program of rejecting the idea that inertia involves forces in physics pedagogy. That program is misguided at best.
- 7.
Named for their inventor, Rene Descartes, these axes are chosen so that they are (all) mutually perpendicular to each other. He got the idea lying in bed contemplating the location of objects in his room and noting that their places could be specified by measuring how far from a corner of the room they were along the intersections of the floor and walls.
- 8.
Since \( d\mathbf{p}/dt = d\left( {m \mathbf{v}} \right)/dt = m\mathbf{a} + \mathbf{v}dm/dt \), we see that force is a bit more subtle than m a. Indeed, if you aren’t careful, serious mistakes are possible. Tempting as it is to explore one or two in some detail, we resist and turn to issues with greater import.
- 9.
When something doesn’t change when it is operated upon (in this case, moved around), it is said to possess symmetry. Note that this is related to the fact that momentum is “conserved.” In 1918 Emmy Noether, while working for Einstein, proved a very general and powerful theorem (now known as “Noether’s theorem”) showing that whenever a symmetry is present, there is an associated conservation law. Noether, as a woman, couldn’t get a regular academic appointment in Germany, notwithstanding that she was a brilliant mathematician. When the faculty of Gottingen University considered her for an appointment, David Hilbert, one of the leading mathematicians of the day, chided his colleagues for their intolerance regarding Noether by allowing as how the faculty were not the members of a “bathing establishment.”
- 10.
As mentioned earlier, the term “proper” is always used when referring to a quantity measured in the instantaneous frame of rest of the object measured. The most common quantity, after time, designated as proper is mass – the restmass of an object is its proper mass.
- 11.
We continue to use Frank Wilczek’s enumeration of Einstein’s laws.
- 12.
Minkowski characterized Einstein the undergraduate student as a “lazy dog.”
- 13.
Changes in internal energies of accelerating objects may take place if the objects are extended and not rigid. As we will see later, this complication leads to the prediction of interesting transient effects.
- 14.
Einstein’s critics, it appears, have been quite happy to use unrealizable conditions when it suited their purposes in other situations. For example, they are quite content to assume that spacetime is Minkowskian at “asymptotic” infinity, or that spacetime in the absence of “matter” is globally Minkowskian. Actually, neither of these conditions can be realized. Their assumption is the merest speculation. Just because you can write down equations that model such conditions does not mean that reality actually is, or would be, that way. What we do know is that at cosmic scale, spacetime is spatially flat. And that condition corresponds to a mean “matter” density that, while small, is not zero. In fact, in Friedmann-Robertson-Walker cosmologies (which are homogeneous and isotropic) spatial flatness results from the presence of “crititcal” cosmic “matter” (everything that gravitates) density – about 2 × 10−29 g/cm3. That’s about one electron per 50 cm3. Not very much stuff, to say the least.
Reference
Misner C, Thorne K, Wheeler JA (1973) Gravitation. Freeman, San Francisco
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The Misner, Thorne, and Wheeler discussion of localization of gravitational energy in their comprehensive textbook, Gravitation:
Gravitation by Misner, Thorne, and Wheeler. © 1973 by W.H. Freeman and Company. Used with permission of W.H. Freeman and Company
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© 2013 James F. Woodward
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Woodward, J.F. (2013). The Principle of Relativity and the Origin of Inertia. In: Making Starships and Stargates. Springer Praxis Books(). Springer, New York, NY. https://doi.org/10.1007/978-1-4614-5623-0_1
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