Hence, the Minkowski formulation seems somewhat unhelpful in clarifying this aspect of time. A limitation on the idea of treating time as a fourth Euclidean-type dimension, though, is that it does not appear possible, at least in the macroscopic world, to freely travel in the time dimension as is possible with space dimensions. This formulation appears to imply that time is a dimension, orthogonal to the other three space dimensions and so naturally leads to the concept of time being a fourth dimension. We can also attempt to draw further insight from the representation of time in the Minkowski spacetime framework, where t is the time coordinate.

Observations of distant cosmic objects appear to confirm that the laws of nature and the fundamental constants are essentially invariant throughout the universe, and only some small variations have been claimed. We would expect observers throughout the universe observing fundamental particles, such as electrons, in this way, would all observe the same frequency and thus provide a universal clock. So in this frame of reference, exactly what sort of clock can we then utilize? One possible candidate is the de Broglie frequency that is assumed to be associated with every particle of energy E, which in its own rest frame, is given by ν = E/ h = mc 2/ h, where m is the invariant mass of the particle and h is Planck's constant 1.

So, in order to retain Newton's definition, we are required to add the proviso that it applies to an inertial observer in the rest frame of the system under observation, and to a region of space near the observer so that differential gravitational fields become negligible. Newton's apparent conceptualization of a universal and perfect clock is significantly modified by relativity theory, which finds that the observed rate of clocks depends on relative motion as well as the relative strength of the gravitational field, between source and observer. One of the most influential ideas regarding time was published in 1686 by Sir Isaac Newton, Principia Book 1, “ Absolute, true, and mathematical time, of itself, and from its own nature, flows equably without relation to anything external, …”. Historically, there have been many attempts to understand the nature of time and provide a rigorous definition. Our main result is that time is the scalar component of a Clifford space and can be viewed as an intrinsic geometric property of three-dimensional space without the need for the specific addition of a fourth dimension. This description using Clifford algebra is found to provide a natural alternative to the Minkowski formulation as well as providing new insights into the nature of time. In order to answer this question we begin by exploring the geometric properties of three-dimensional space that we model using Clifford geometric algebra, which is found to contain sufficient complexity to provide a natural description of spacetime. The Minkowski framework is indeed found to provide a versatile formalism for describing the relationship between space and time in accordance with Einstein's relativistic principles, but nevertheless fails to provide more fundamental insights into the nature of time itself. Minkowski, though, rejected the formalism of quaternions suggested by Hamilton and adopted an approach using four-vectors. With the publishing of Einstein's theory of special relativity these ideas then lead to the generally accepted Minkowski spacetime formulation of 1908. Newton conceived of time as absolute and universal which “ flows equably without relation to anything external.” In the nineteenth century, the four-dimensional algebraic structure of the quaternions developed by Hamilton, inspired him to suggest that he could provide a unified representation of space and time. The proper description of time remains a key unsolved problem in science.

3School of Mathematical Sciences, University of Nottingham, London, UK.2School of Physical Sciences, Institute for Photonics and Advanced Sensing, University of Adelaide, Adelaide, SA, Australia.1School of Electrical and Electronic Engineering, University of Adelaide, Adelaide, SA, Australia.Hartnett 2, Nicolangelo Iannella 3, Azhar Iqbal 1 and Derek Abbott 1

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