The Clock Riddle and Einstein’s Third Postulate of Special Relativity

The present work calls attention to an undeclared assumption made by Einstein in his landmark paper [Ann. Physik 322 (10), 891 (1905)] in which he introduced the Special Theory of Relativity (STR). The emphasis in textbooks and periodicals is always on his two postulates of relativity [the Relativity Principle (RP) and the constancy of the speed of light in free space]. Yet, the well-known results of his theory such as Fitzgerald-Lorentz length contraction (FLC) and the symmetry of time dilation (two clocks in motion each running slower than the other) are based exclusively on this third postulate. It is shown that an alternative assumption of clock-rate proportionality (Newtonian Simultaneity) is also consistent with Einstein’s first two postulates and with the Relativistic Velocity Transformation (RVT), but that it leads to a fundamentally different space-time transformation than the Lorentz Transformation (LT) of STR. It is referred to as the Newton-Voigt Transformation (NVT). Its predictions regarding length and time measurements by moving observers differ sharply from those of the LT. A “clock riddle,” distinct from the well-known “clock paradox,” is presented to underscore the differences between these two versions of the relativistic space-time transformation. It is shown that the NVT is consistent with remote simultaneity and the impossibility of time inversion, and therefore does not rule out the existence of faster-than-c particles under the condition that they have null proper mass

Rational Reflection and the Notorious Unmarked Clock

This chapter concerns a reflection principle discussed by David Christensen and Adam Elga according to which a rational agent’s credence in ϕ‎ ought to be his or her expectation of the ideal credence for the agent, which is the degree to which the evidence ideally supports ϕ‎. The principle seems to have paradoxical consequences, illustrated in its application to an example to which the title of this chapter refers, but it is argued that the paradox arises from mistaken assumptions about what determines the agent’s epistemic situation. The principle may still need qualification, and it needs to be reconciled with the possibility of epistemic modesty (uncertainty about whether one’s credences are ideal), but the clock paradox can be dissolved, and doing so helps to clarify issues concerning higher-order evidence.

The Three Relativities

Galileo’s and Einstein’s Special and General Relativity are introduced. So is the equivalence principle between gravity and acceleration, the basic assumption of general relativity. A first encounter with the twin paradox of Einstein’s relativity, also called the clock paradox. The twins are further discussed in a subsequent chapter (16).

On the Clock Paradox

Special Relativity presents before us several thought challenging paradoxes. The most famous one is the clock or twin paradox which arises from the well-known time dilation phenomenon. In this paper, we will give a brief but pedagogical treatment of this paradox. The crucial point is that the relativity of simultaneity, for events happening at different places, plays a fundamental role in understanding the apparent disagreement between the rest and the moving observers. Once this has been taken care of, the disagreement and the paradox would disappear.