Abstract
When addressing the notion of proper time in the theory of relativity, it is usually taken for granted that the time read by an accelerated clock is given by the Minkowski proper time. However, there are authors like Harvey Brown that consider necessary an extra assumption to arrive at this result, the so-called clock hypothesis. In opposition to Brown, Richard TW Arthur takes the clock hypothesis to be already implicit in the theory. In this paper I will present a view different from these authors by taking into account Einstein’s notion of natural clock and showing its relevance to the debate.
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Notes
In this work it is adopted Fock (1959) terminology. Instead of adopting special and general relativity to name Einstein’s two theories, one refers to the 1905 theory of relativity and the 1915 theory of gravitation. The discussion on natural clocks, proper time and the clock hypothesis is made only in reference to the theory of relativity. It is beyond the scope of this work to address these issues in the context of Einstein’s theory of gravitation.
As such, even if Einstein does not address this issue, we must take the clock as ultimately calibrated to sidereal time, unless there is an argument to do otherwise; Einstein himself makes explicit in later writings the notion of natural clock, which I see as anticipating in part the later adoption of the atomic time scale based on atomic clocks.
Einstein early philosophy is usually considered as a form of empiricism, to some close to what later become known as operationalism, even if there is disagreement regarding Einstein’s operationalist sympathies (see, e.g., Paty 1993, 349–365, Scheibe 2001, 72–73; Dieks 2010, 228–35). On first sight Einstein’s view of rods and clocks as independent conceptual elements of the theory might seem to be an example of this philosophical position. That is not the case. To Einstein, ideally, a physical theory must conform to Poincaré’s conventionalist thesis (see, e.g., Einstein 1921, 212; Paty 1992, 7–8), which Einstein takes to mean that only the conjunction of a mathematical structure G with a physical part P is tested experimentally. This implies in particular that concepts like length and duration would not have a direct operational meaning (see, e.g., Dieks 2010, 229). However, Einstein also recognizes that in the present stage of development of physics, his theory of relativity does not conform to Poincaré’s conventionalism (see, e.g., Paty 1993, 304–5). In the context of his philosophy of geometry, Einstein concludes that this situation implies in particular the above-mentioned necessity of taking rods and clocks as independent conceptual elements of the theory. The clocks and rods Einstein refers to are above all concepts of the theory, which in the present stage have a very direct link to instrumentation and also to the mathematical structure (this is also the case with Einstein’s theory of gravitation; see, e.g., Einstein 1955, 63–4). As conceptual elements, the clocks (and associated notions of duration/time interval/passage, etc.) and rods (and associated notion of length) – and linked as they are with the geometrical structure of the theory – are not used in the context of the theory simply as a representation of particular instruments; they are part of a conceptual-mathematical ‘definition’ of length and duration, which has a clear counterpart in experimentation via rods and clocks as measuring instruments. As instruments, rods and clocks do not define length and duration, they enable the measurement of length and duration. In conclusion, Einstein’s view on rods and clocks as independent conceptual elements of the theory – crucial to the view developed in this work – was developed not in the context of Einstein’s early eventual operationalism, but is related to Einstein’s later views of geometry as practical/physical geometry. In this way it does not fell prey to any operationalist philosophical position, which as Brown remarks “got such a bad name in philosophy, [that] it has been fashionable for some time in the philosophical literature to discuss space-time structure without any reference at all to such base elements as rods and clocks” (Brown 2005, 95).
When in inertial motion the atomic clocks give/read a time that is equal to the inertial time. We can say that experimentally the atomic time scale and the inertial time scale are identical (Ohanian 1976, 187–8).
Einstein also refers to the assumption as the “independence of measuring rods and clocks from their past history” (Einstein 1920, 127).
Brown refers to this assumption as the boostability of rods and clocks (Brown 2005, 30). Brown is clear in the dynamical aspect of this assumption (Brown 2005, 28). However the rods and clocks as ‘primitive’ conceptual elements are not dynamically described in the theory. This means that any dynamical considerations being made, e.g. by reference to applied forces (Brown 2005, 95), are not rigorous, having more a heuristic role.
As a dynamical assumption, even if specific to the particular situation of a boosted reference frame, this assumption does not follows from the principle of relativity of inertial motion, according to which “the laws governing natural phenomena are independent of the state of motion of the [inertial reference frame] with respect to which the phenomena are observed” (Einstein 1910, 123). The principle is circumscribed to a statement of the equivalence of inertial frames for the description of phenomena (in more ‘modern’ terms it states that the laws of the theory are Lorentz covariant; see, e.g. Norton 1993, 796); as such it does not bear on the issue of the boosting of the inertial reference frame itself. In this way, the boosts of rods and clocks as primitive entities inscribed in the notion of inertial reference frame are outside the implications of the principle of relativity.
Torretti associates a different notion of natural clock to Einstein, the light clock (Torretti 1983, 52–4; Ohanian 1976, 192–3). In this context Torretti refers to Einstein’s time scale (Torretti 1983, 54). This time scale cannot be seen as independent from the inertial time scale; as Ohanian calls the attention to, the time variable that appears in the equations of electrodynamics corresponds to an inertial time (Ohanian 1976, 195). The situation with the atomic time scale is different. It is correct that we cannot completely separate conceptually the atomic time scale provided by an atom from the fact that, as a ‘clock’ of an inertial reference frame, its state of motion is taken into account. However it is clear that there is an ‘internal’ physics of the atom leading to the atomic time that, at the present stage of development of theoretical physics, is taken to be independent from the inertial time. The fact that experimentally the two time scales coincide (Ohanian 1976, 187–8) does not mean that conceptually and in experimentation they are not distinguished.
The exact meaning of a clock’s rate not being affected in a direct way by its acceleration is given on pages 6–8 of this article.
In a couple of places Brown uses the term inertial clock instead of ideal clock to refer to a (conceptual) clock in inertial motion giving the inertial time (Brown 2005, 19, 97). To simplify the terminology adopted in this work, and avoid confusion with Arthur’s terminology, I will simply refer to conceptual clocks.
As it is well known, Minkowski presented the notion of proper time in 1908. By definition the proper time is associated just to substantial material systems to which one can associate time-like worldlines. For this particular case one considers the invariant infinitesimal interval along the worldline at the position of the material system c2dτ2 = c2dt2 – dx2 – dy2 – dz2. According to Minkowski, “The integral ∫dτ = τ of this amount, taken along the world-line from any fixed starting-point P0 to the variable endpoint P, we call the “proper-time” of the substantial point in P” (Minkowski 1908, 85). Immediately after defining proper time, Minkowski determines the motion-vector and acceleration-vector using dτ as the infinitesimal time interval gone by the material system.
In Brown’s own words, the clock hypothesis is “the claim that when a clock is accelerated, the effect of motion on the rate of the clock is no more than that associated with its instantaneous velocity – the acceleration adds nothing” (Brown 2005, 9). According to Brown this is the case when “the external forces accelerating the clock are small in relation to the internal ‘restoring’ forces at work inside the clock” (Brown 2005, 115).
This sentence is applicable both in Brown’s as in Arthur’s case; both take into account the clock hypothesis even if disagree on its ‘place’ in the theory. My position is a bit different. For the case of natural clocks, their behaviour under acceleration is an input physical assumption that we ascribe to their conceptual counterpart within the theory, i.e. we do not need any reference to a clock hypothesis in the case of natural clocks. Its ‘utility’ would be restricted only to the case, e.g., of mechanical clocks like the balance wheel clock or the pendulum.
The time coordinate at a particular location in a real/conceptual inertial reference frame can be seen as given by the time reading of a natural/conceptual clock (that we take to have been synchronized with the rest of natural/conceptual clocks of the inertial reference frame). Since an event, like a thunder striking next to the clock, as a space-time point is given by the time reading of the clock and its position in the inertial reference frame, when we talk about the synchronization of distant events we are actually referring to the synchronization of the natural/conceptual clocks in inertial motion. In this way, it is the shared inertial/atomic time of all the clocks of the inertial reference frame that enables the definition of the time coordinate (by synchronizing the clocks), and from this the synchronization of distant events. In this way the bifurcation of time in the theory would ultimately rely on the distinction not of coordinate time and the Minkowski proper time, but of the atomic/inertial time from the Minkowski proper time.
Arthur even contends that “the whole content of relativity theory can now be framed in terms of [the proper time], so that co-ordinates are no longer regarded as primitive” (Arthur 2010, 161–2). It is important to notice that only along the time-like worldline can the coordinates be seen as functions of proper time, when taken to be parameterised by the proper time. It is only in this strict sense that one should read Arthur’s reference to the coordinates as no longer primitive. Minkowski’s presentation is clearer on this. According to him, “on the world-line we regard x, y, z, t … as functions of the proper time τ” (Minkowski 1908, 85 [my emphasis]).
Elsewhere Arthur writes that an ideal clock is “implicitly determined by theory” (Arthur 2010, 168).
As we have seen on pages 3–5 of this article, Einstein only mentions the ‘independence from past history’ in the more specific sense that identical clocks that are again in relative inertial rest have the same rate independently of their past motions.
As mentioned in section 1, the atomic/natural clock and related notion of time (metrologically implemented through the atomic time scale) is independent of the formulation of the theory and in fact is taken into account in input physical assumptions of the theory. It is from measurements made with accelerated atomic clocks that we check that the Minkowski proper time in fact gives the time gone by an atomic system. Not the other way around.
There are however simple physical systems that cannot be seen as constituted by simpler ‘classical atoms’, which in certain cases do not behave as ideal clocks. One example of a theoretical model with this characteristic is a type of flavour oscillation clock that when subjected to a rotational acceleration does not read proper time (Knox 2010). However, in this case, we are considering a theoretical model (with an eventual real counterpart) that is outside the scope of the theory of relativity. Even the idea of worldline cannot be applied in the case of a quantum system like the flavour oscillation clock. The natural clock (not depending on its past history) is conceptualised as an ideal classical clock with a classical worldline. Quantum systems that do not behave as ideal clocks open interesting questions regarding the notion of becoming as something that can be tracked by the Minkowski proper time; however this is outside the scope of the theory of relativity and the views being presented/analysed in this work are only related to this theory.
One should not be mislead by references to atomic clocks in the context of discussions regarding the clock hypothesis (see, e.g., Brown 2005, 94–5, Arthur 2010, 168). For example, in the case of the pendulum one is applying basic dynamics of the theory of relativity (see, e.g., Misner et al. 1973, 394–5). This corresponds to the clock hypothesis as a general (dynamical) rule of thumb expressed in terms of the concept of force (or potential). In the case of the atomic clock we find physical ‘toy models’ in which the clock hypothesis is ‘applied’ in a completely heuristic way (see, e.g., Ohanian 1976, 207–8). As mentioned, when taking the ‘clock’ as a self-sufficient concept to be the counterpart within the theory of the atomic clock that experimentally we find to give a time reading (i.e. to have an empirical proper time) according to the Minkowski proper time, this fact becomes a physical input assumption of the theory that is not explained within the theory of relativity.
It is my view that the quantum mechanical description of atoms does not explain the independence on its past history of natural clocks. Quantum mechanics is ‘built’ on top of the notion of inertial reference frame with its implicit conceptual rods and clocks. As Dickson mentions, “just as classical physics does, quantum physics contains an assumption (usually left implicit) that there is some frame (some system of coordinates) in which the laws are valid” (Dickson 2004, 201). However it is beyond the scope of this work to develop this issue here. Even if that was not the case it does not affect the views being defended here in relation to the clock hypothesis.
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Acknowledgments
I thank anonymous referees for constructive criticisms of earlier and present versions of the work. I did this work as a member of Henrik Zinkernagel’s research project “Tiempo, Modalidad, y Disposiciones en el Cruce de la Física Contemporánea y la Metafísica” (Reference: FFI2011-29834-C03-02).
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Valente, M.B. Proper time and the clock hypothesis in the theory of relativity. Euro Jnl Phil Sci 6, 191–207 (2016). https://doi.org/10.1007/s13194-015-0124-y
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DOI: https://doi.org/10.1007/s13194-015-0124-y