Abstract
In a way similar to classical mechanics where we have the concept of inertial time as expressed in the motions of bodies, in the (special) theory of relativity we can regard the inertial time as the only notion of time at play. The inertial time is expressed also in the propagation of light. This gives rise to a notion of clock—the light clock, which we can regard as a notion derived from the inertial time. The light clock can be seen as a solution of the theory, which complies with the requirement that a clock to be so must have a rate that is independent from its past history. Contrary to Einstein’s view, we do not need the concept of “clock” as an independent concept. This implies, in particular, that we do not need to rely on the notions of atomic clock or atomic time in the theory of relativity.
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Notes
Obviously, the motions of bodies also express the inertial time. However, even for the simplest case of free bodies, the law of inertia in its standard formulation might be conventional (see, e.g., Bacelar Valente 2017). Adopting the light propagation and the light postulate to define the inertial time we avoid any eventual problem arising from a (possibly) conventional definition of inertial time.
Here, we are simplifying the presentation and, implicitly, refer to the value of the inertial time of light propagation (corresponding to one “cycle” of light bouncing between two bodies in relative rest in inertial motion) in terms of an independently measured distance d and an independently measured two-way speed of light c (e.g. by using unit-measuring rods and atomic clocks). We mention below that we do not need to resort to independently measured d and c to have a notion of inertial time, a measure of distance d, and a value of c.
Einstein had noticed early that atoms can be seen as clocks (see, e.g., Einstein 1907a, 232; 1907b, 263; 1910, 134). In his view, “Since the oscillatory phenomena that produce a spectral line must be viewed as intra-atomic phenomena whose frequencies are uniquely determined by the nature of the ions [(atoms)], we can use these ions [(atoms)] as clocks” (Einstein 1910, 124–125). To be rigorous atoms are not yet clocks. We might consider the reference to atoms as clocks as pointing to the possibility of considering atomic clocks.
Since, as is defended in this paper, we can implement a notion of inertial time in the theory of relativity, we might equally consider quantum mechanics or quantum electrodynamics as implementing this notion due to their reference to inertial coordinate systems (see, e.g., Dickson 2004, 201). In this way, we might be tempted to say that, contrary to Einstein's views, we do have a description of clocks in terms of an “underlying” inertial time using quantum theory. This does not have to bear on the views developed in this paper on two accounts: (a) the description of atomic clocks would still be made outside the theory of relativity, (b) while it might rely on a notion of inertial time (associated with the Galilean or Minkowskian space–time), we can, following Einstein, regard the description of clocks in terms of a “solution” of quantum theory as not corresponding to a hypothetical solution made within a single relativistic theory that provides “from within” a description of matter. In this way, in this paper it will not be considered that there is a relativistic description of atoms (in the sense given by Einstein). However, we will move away from Einstein's view, in relation to physical systems that are (in our view) described in the theory of relativity, which we can consider as clocks satisfying the theory's requirement that the clock's rate must be independent of the clock's past history.
While both Geroch (1972) and Synge (1960) present special and general relativity along the lines of Einstein's ideas, i.e. in terms of atomic clocks and the independence of past history assumption, they do not mention explicitly the idea that “clock” is an independent concept. Synge, in particular, mentions the assumption of “the existence of standard clocks” (Synge 1960, 105), which he regards as atoms (Synge 1960, 106). According to him, the time reading of standard/atomic clocks is “the only time of basic importance in relativity” (Synge 1960, 105), and “since all clocks henceforth considered are standard clocks, we shall drop the adjective and call them simply clocks” (Synge 1960, 107). In Synge's approach, “time is the only basic measure. Length (or distance), in so far as it is necessary or desirable to introduce it, is strictly a derived concept” (Synge 1960, 108).
We see that Marzke and Wheeler (1964) actually named the clock “geodesic clock”, not “light clock”. Ohanian (1976) adopted the name “geometrodynamic clock”. The geodesic, geometrodynamic, or light clock is independent of the structure of matter and enables to measure space–time intervals (Marzke and Wheeler 1964, 53–58; Ohanian 1976, 192–200).
As Marzke and Wheeler call the attention to, their construction of parallel worldlines “depends upon the existence of an inertial reference system—a system in which the world lines of all light rays and all free particles appear straight” (Marzke and Wheeler 1964, 52).
To simplify the discussion, we will consider the units of time, length, and velocity (determined in terms of a particular light clock in inertial motion adopted as our standard) to be chosen so that one “tick” of any other light clock, moving inertially, is equal to 2d/c seconds, where d is the distance between the particles and c is the two-way speed of light.
We must recall that Einstein expected that his relativistic theories might be “completed” in the future in the form of a unified theory that besides unifying gravitation and electromagnetism would give a field-theoretic description of matter (see, e.g., Goenner 2004).
In this work, we go a “step further” and define the inertial time scale in terms of the inertial propagation of light by using a light clock.
One might still claim that, in this context, to call a light clock a solution of the theory is stretching the terminology too far (even if taking into account that the theory describes light clocks). Thinking in terms of Lorentz’s or Einstein’s versions of principle theories (Frisch 2011), the kinematic description of a light clock made within special relativity imposes constraints to any dynamical model of a light clock, i.e. a dynamical solution must conform to the kinematic solution of special relativity. To call the kinematic description of a light clock a solution of special relativity seems as rightful as to say regarding a putative solution of a dynamical theory that it describes a light clock. In relation to this, from our point of view, if we were unable to develop a dynamical model/solution of a light clock compatible with the relativistic solution, this does not entail the demise of the light clock as proposed by Marzke and Wheeler; this would point to limitations in the application of the dynamical theories and not in special relativity as a principle theory, neither on light clocks as described within the theory (for an opposite view see Stachel 1983, 257). Regarding the light clock as a dynamical solution, we tend to align with Einstein’s position on clocks and be skeptical regarding the possibility of a consistent dynamical description of light clocks. There seem to be some intrinsic “difficulties” in classical and quantum electrodynamics that lead us to this position (Bacelar Valente 2011). However, further study is necessary.
We must notice that with this approach in terms of inertial time the so-called second postulate (the principle of the constancy of the velocity of light) comes before the first postulate (the principle of relativity). It is only after having completed inertial reference frames with time coordinates that it makes sense to postulate that different inertial reference frames are equivalent for the description of physical phenomena.
The dimension of (1/c of) the length of a timelike worldline (the Minkowski proper time) is \(\left[ {1/{\text{c}}\int {\text{ds}} } \right] = \left[ {\int {{\text{sqrt}}(1 \, {-}\upupsilon^{2} /{\text{c}}^{2} ){\text{dt}}} } \right] = {\text{second}}.\)
Fletcher's theorem applies to the general case of a curved space–time but here, we will just consider it in relation to the Minkowski space–time. According to Fletcher, his work generalizes, in particular, the previous works by Maudlin (2012) and Anderson and Gautreau (1969) regarding “providing an existence proof for sufficiently ideal light clocks” (Fletcher 2013, 1371). From this point onwards, we will follow Fletcher and adopt geometric units, in which c = 1.
In the proof of the theorem, we associate with each γα a sequence of arc lengths of segments of γ[I′] “clocked” by the bounces of light in the worldline γα (Fletcher 2013, 1377–1378). As mentioned, Fletcher also proves in the theorem a result that implies that the light clock can be as regular as we wish. Accordingly, “the second limiting equation, concerning the regularity of the clock, states that the maximum difference in [the arc lengths] between any two ticks over the course of the clock's run will be small” (Fletcher 2013, 1382).
In Bacelar Valente (2016) it is defended that the “empirical proper time” of atomic clocks has a relevant role in the theory. This view is developed by accepting as a premise Einstein's view that the concept of clock must be considered as an independent self-sufficient concept, since, according to him, the theory does not have solutions corresponding to physical systems to which we might ascribe the role of clocks. In this work, we reject Einstein's view. This leads to the view that atomic clocks are not necessary for the foundations of the theory. We can formulate the theory entirely in terms of the inertial time. We can consider “alternative” formulations in which atomic time has a role (see, e.g., Bacelar Valente 2017), but this is not a necessity—it is simply a possibility.
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Acknowledgements
The criticism and commentaries of the anonymous reviewers helped in arriving at a much clearer and coherent manuscript. To both, I want to express my gratitude.
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Bacelar Valente, M. Time in the Theory of Relativity: Inertial Time, Light Clocks, and Proper Time. J Gen Philos Sci 50, 13–27 (2019). https://doi.org/10.1007/s10838-018-9415-2
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DOI: https://doi.org/10.1007/s10838-018-9415-2