Budgetary Constraints

The Multiplicative Property

In number theory a function f(n) of positive integers n is said to be multiplicative if f(1) = 1 and f(xy) = f(x)f(y) for every co-prime positive integers x,y. For example, Euler’s totient function is multiplicative in this sense. A totally multiplicative function is one that removes the restriction for the arguments to be co-prime. For example, f(x) = x² is totally multiplicative, even over the real numbers.

The analytic function f(z) = z^1/q for some integer q is actually a multi-valued function, since there are in general q values w such that w^q = z. For convenience, we sometimes designate one particular root as the “principal value” of the function. By convention, noting that any non-zero complex number z can be expressed uniquely in the form z = r e^iθ for positive real r and real θ in the range –π < θ ≤ π, the principle value of z^1/q is often defined as S_q(z) = S_q(r) e^i(θ/q). Of course, S_q(r) is the unique real value whose qth power is r. From this we see that (for example) the principle value of the square root of a complex number is not generally multiplicative, as shown by S₂((−1)(−1)) = 1 and S₂(−1)S₂(−1) = −1.

Non-multiplicative functions are commonplace, but even such functions may be multiplicative over a restricted domain or restricted factorizations. For example, F(x) = x+1 is obviously not multiplicative, because F(xy) = xy+1 whereas F(x)F(y) = (x+1)(y+1) = xy+x+y+1. These quantities are equal just if x = −y. So, for any complex z, a factorization z = xy satisfies the relation F(xy) = F(x)F(y) if x = S₂(z)i and y = −S₂(z)i.

Framing the Argument

One of the most basic and common misconceptions about dynamics is the idea of an “accelerating frame”. It's true that, for each event of any given material object there is a standard system of inertial coordinates in terms of which the object is momentarily stationary, and the equations of physics take the same form in terms of each of those systems. However, when students imagine an "accelerating frame", they have in mind something that can be applied over extended regions and time intervals as if it was an inertial coordinate system, which is not true, for the following reason:

In the context of Newtonian theory, there could exist perfectly rigid bodies of indefinite extent, so we could imagine a perfectly rigid grid of rulers and clocks, all undergoing identical acceleration, and this would be an "accelerating frame", such as the student imagines. This assumes the laws of physics are Galilean invariant, and that, therefore, there could be instantaneous conveyance of forces, instantaneous signaling, and so on.

However, if all the laws of physics are Lorentz invariant, conveyances of energy-momentum are not instantaneous, and there can be no such thing as perfectly rigid bodies over indefinite regions. This is not just a practical difficulty, it is a theoretical impossibility. A grid of rulers and clocks in inertial motion would get torn apart at some point when subjected to acceleration. True, there is such a thing as Born rigid motion, but that entails different acceleration profiles for different parts of the object, so it isn't a coherent "frame" in the naive sense, and even this can only be carried out over a limited extent (to the “fulcrum”), beyond which it too gets torn apart. So, there is no such thing as an accelerating "frame" in the sense that the student imagines. Yes, over a some finite region and with sufficiently low acceleration we can construct a serviceable system of coordinates in which an accelerating object is continuously stationary for some period of time, but the equations of physics do not take their usual simple homogeneous and isotropic form in terms of such systems.

Now, at this point, the student may try to deploy the equivalence principle, which he has heard asserts equivalence between the laws of physics in terms of an accelerating system in the absence of gravity and a stationary system in a so-called homogeneous gravitational field. But this does not support the student’s notions, for two reasons. First, the equivalence principle applies only in infinitesimal regions of space and time, and hence can't in general be applied over huge regions encompassing, e/g., the path of an interstellar rocket. Second, the very same time dilation that appears as a consequence of motion in the absence of gravity appears as a consequence of gravity for the stationary objects in the corresponding gravitational field. Einstein himself explained this explicitly in his 1918 paper on the twins paradox in the context of general relativity. (That paper unfortunately didn’t address the directional limitation, since it considered only acceleration toward a distant location, rather than away from that location, the latter having a much shorter range of applicability before exceeding the fulcrum.)

Fundamentally the student is focused on purely kinematical concept of motion, without accounting for the fact that the dynamical aspects of motion are crucial for the understanding of physical phenomena.  We can obviously establish a kinematic description for which an accelerating object is stationary, such as describing the solar system in a Ptolemaic system of coordinates in which the earth is permanently at rest, but in terms of those coordinates the simple dynamical equations of Newtonian physics do not apply (even approximately), and the other planets move in spiraling patterns, etc. Yes, if we are determined to make things difficult, we can describe things in terms of accelerating coordinate systems, but that doesn't change any of the objective facts, it just produces convoluted descriptions. All the invariants, such as the elapsed proper times along every trajectory, are unchanged.

Much of the misunderstanding about relativity comes from the popular slogan that all reference systems are equivalent, and indeed some students arrive with the pre-conceived notion that this actually is the principle of relativity. Of course, that is not the principle of relativity, it’s the cartoonish "everything is relative" juvenile description of relativity. The actual principle of relativity is defined very clearly in Einstein's 1905 paper and all subsequent papers and books:

The laws by which the states of physical systems undergo change are not affected, whether these changes of state be referred to the one or the other of two systems of coordinates in uniform translatory motion.

The student may have heard that special relativity “doesn’t apply in accelerating coordinate systems, but that of course is not true.  Special relativity applies perfectly well in terms of accelerating coordinate systems in the absence of gravity (or in regions where the changes in gravitational potential are negligible), and according to the equivalence principle it also applies perfectly well in the presence of gravity, but then only locally. As explained above, this does not constitute an "accelerating frame", which doesn’t even exist in the naive sense that the student imagines, and, moreover, to the extent that an accelerating coordinate system in flat spacetime can be mapped to a stationary system in curved spacetime, the very same time dilation effects occur… it is just two different ways of describing the same thing. This is nicely illustrated by the example of a clock revolving around a central inertial clock in flat spacetime, noting that the revolving clock runs slower in a secular sense than the central clock due to its motion. Now, we could also view this in terms of a rotating system of coordinates in which the revolving clock is stationary, but in that case there is a pseudo-gravitational field that results in exactly the same time dilation difference between the clocks.

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