An Essay: it’s about time

—John Gideon Hartnett and James Chappell

What is time? This seems to be a particularly intractable question!

Also, while space appears quite tangible and easy to visualise, time on the other hand seems an extremely elusive concept.  Some theorists have thereby concluded that time is perhaps merely a human invention and that only space is real.  However, many authors have also concluded¹  that all mathematical concepts are freely created and so it is therefore invalid to single out time in this way.

Similarly, some scientists say that while time is not real the laws of physics are perfect and unchanging. This is part of what is referred to as the Newtonian paradigm.  Time becomes merely a parameter to calculate kinematical and dynamical processes. However, how can we say that the laws of physics do not change with time when we do not have a rigorous definition of time?

Sir Isaac Newton said that he was searching for the way God made the Universe, in the laws of physics.

“The True God is a living, intelligent, and powerful being: His duration reaches from eternity to eternity.”²

He believed, as I do also, that the Creator superintended His own creation, and that He was not part of it, but extant to it and transcended our notion of time. Newton also held that those fundamental and perfect laws of nature are real and immutable, i.e. they do not change with time. In that sense time is not real. However, some theorists propose that time is real³ and external to everything in the Universe—in an eternal universe. They also add that the laws of physics, by extension change, but that is more wild conjecture than anything else.⁴

Part of the Newtonian paradigm was an absolute external time “flowing equably without relation to anything external.” However, such a concept of time was superseded by Einstein when he showed that it can only be defined locally for each observer.

So, could it be that time is not real? We only experience the present. We can observe simultaneously the three spatial dimensions—length, breadth and depth—but we cannot experience the past or the future. We are bound in time.

The Creator God on the other hand is not bound, but can see the future from the past. He is extant and transcendent. He does not occupy this material realm. Thus it could be that, because we do, we cannot experience anything more than the present. We might be able to calculate God’s laws, with mathematical formalisms that are inherently intrinsic to nature,⁵ and predict future actions, but we always only experience the present.

Unification of space and time

A vital part of the history of development of our understanding of time was indeed this recognition of the mutual interdependence of space and time by Einstein, in the special theory of relativity.  The modification of space and time being caused by relative motion in this theory.  This relationship between space and time was later extended by Einstein to include gravity in the general theory of relativity.  In this theory, the presence of mass (or energy) produces a distortion of the space and time components creating an effect called gravity.  Both relativity theories leading directly to a concept of a unified spacetime continuum. Using the conventional Minkowski four-vector formalism we have a spacetime event S = [t, x, y, z].  However, in order for this to be consistent, clearly both t and x, y, z need to be measured in the same units.  Indeed the international standards CIPM,⁶ since 1983, define both space and time in units of seconds.  Alternatively, multiplying by the fixed constant c, the canonical speed of light in vacuum, we can define both space and time in units of metres. This appears to be an important step forward in understanding spacetime, as having separate units (of seconds and metres) obviously obfuscates their unified nature.

However, if we move a suitable distance from strong gravitational fields and are at rest with respect to the object being observed we can isolate for separate measurement the space and the time components.

Hence, before trying to establish precisely what time is, it is perhaps wise to first measure this property as accurately as possible in order to determine its attributes.  Indeed, modern-day time and frequency metrologists know very well how to measure the rate at which time flows—the ‘tick’ of a clock—with both classical and atomic clocks.

So, while we can precisely measure the change in time and accurately determine the length of a second with an accuracy now approaching one part in 10¹⁸, how does this inform us about what time is?  We appear to be able to determine that time is infinitely divisible and flows steadily if isolated from all outside influences? However, due to the presence of a unified spacetime perhaps our question should really be changed though to “What is spacetime?”  That is, before we can answer the question of what is time, surely we need to know what spacetime is, from which the concept is derived.

Clifford Algebra

Now, in terms of physical variables defined within our spacetime, we find that they can be classified into four

types:

1. Scalars such as energy,
2. Polar vectors such as linear momentum,
3. Axial vectors or pseudovectors such as angular momentum, and
4. Pseudoscalars.

These four quantities are also closely related to the four types of geometric quantities commonly described as points, lines, areas and volumes found in three-dimensional space.  In fact, our assumption of a three-dimensional space is well supported experimentally with the inverse square laws being verified to high accuracy as well as the existence of exactly five regular solids. Hence these four types of quantities provide a well-founded initial insight into the nature of spacetime.

So the question now is, how can we algebraically describe these four aspects of spacetime?  This can be achieved using the Clifford algebra Cl(R³), a system of geometric algebra developed by William Clifford at the end of the nineteenth century.  The algebra begins for three-dimensional space by defining three basis elements i,j,k  that define the three dimensions of space.  We then form the bivectors ij, ki, jk and the trivector τ =ijk, where the basis elements anti-commute and i² = j² = k² = +1.  Combining these elements we form a multivector

S = a + x + τ n + τ b.

This multivector indeed contains real scalars a and b, a three component polar vector x, a three component axial vector τ n and a pseudoscalar τ b.  This thus encapsulates algebraically the four geometric components in three dimensions and is thus proposed as a mathematical description of spacetime.

Comparing this with the Minkowski formulation of spacetime, as x is a polar vector it therefore describes what is conventionally called space.  For time, however, we have three choices, the scalar, the bivector or the trivector.  If we select the bivector then we will have a three-dimensional time coordinate τ n.  While this initially appears inappropriate, in fact, multi-time theories have indeed been proposed by Itzhak Bars and others.  Restricting ourselves though to a single time coordinate we are left with either the scalar or the trivector.  The main difference between these two is that the trivector encodes the idea of handedness, that is ijk = -ikj.  Indeed, in fundamental particle interactions a positron is viewed as an electron with opposite handedness moving backwards in time and so the idea appears to have merit.  However for macroscopic classical mechanics the scalar appears more appropriate as those equations are time invariant.

Hence, in order to correspond with common definitions of time we are therefore constrained to have a spacetime event t + x and defining an involution S^* = t – x we have SS* = t² – x² which is the correct invariant spacetime distance.  Using Clifford geometric algebra, a spacetime event t + x is thus a geometrical union of points and lines. However we wish to note that both the bivector and trivector components of the multivector have some justification as time variables.  Hence these other choices for time variable from spacetime perhaps explains some of the confusion about time.  That is, we could define time more generally as: those components of spacetime that are not space.

Thus time is simply a parameterisation of the variation of the four geometric quantities found in spacetime.  That is for a spacetime multivector we can write

S = t + x + τ n + τ b = (1 + v + τ w + τ h) t

where v = x/t linear velocity, w = n/t an angular velocity and h the changing torsion. Thus time indeed provides a natural parameterisation of the unified spacetime multivector.

Interestingly, using this definition of time, we can see that time as a scalar is therefore geometrically a point, which obviously cannot be ‘seen’ the way spatial extensions can.  Hence this perhaps explains why time cannot be visualised like space can.

Clock rates and flow of time

A further observation about time is that all types of clocks, both classical and atomic, after adjusting for differences in gravitational potential, appear to ‘tick’ at the same rate.  This leads us to suppose that time runs at the same rate throughout the Universe, for all inertial observers, under the same conditions. However if all of the fundamental constants changed simultaneously in a particular way could the rate of time change without us being able to detect it? For example, could the assumed spatial expansion of the Universe affect clock rates and in turn result in the galaxy-redshift phenomenon? Normally, that is assumed to be the case. Atomic clocks based on different atomic species are used to calculate the ‘drift’ in dimensionless constants, like the fine structure constant, for example. But if one detected a real drift, could that be due to a difference in the way different clocks measure the flow of time? The normal assumption is that it is not, and all atomic clocks fundamentally tick off a ‘second’ exactly the same. It is believed, that any real ‘drift’ in dimensionless constants highlights new physics.

A further question that is often asked is why we cannot travel in time like we can in space. If we imagine an animation produced by a stack of cards that uses a slowly changing image imprinted on each card in sequence.  Then, if we flick through the cards we appear to see motion on the cards.  Also, we can flick through the cards in either direction—that is, travel back in time.  However in order to do this all cards must simultaneously exist.  However a particular material object does not produce copies of itself as it moves through space.  Hence, essentially only one copy of an object is available, or one card available to flick, and hence we cannot therefore travel in time.  This distinct nature of time and space also reflected in their distinct geometrical representation as a point and a line respectively.

Also, as we previously noted, time cannot really be separated from space and so we need to think in terms of a unified spacetime.  Hence, we just observe the changing configuration of the eight-dimensional Clifford space Cl(R³)  and as a result we only really can perceive the present in this space.  This is because time is only defined within this space as a scalar attribute.  An all-knowing divine being on the other hand, presumably through foreknowledge using precise prediction can see the future from the past. He is typically assumed to be extant and transcendent, not occupying this material realm.  However if we can contact this higher reality with some sense then this would create a sense of an eternal now, the ‘present’.  This is clearly consonant with experience.

Newton, in fact, may have been more motivated by scientific requirements than religious dogma, since by accepting the role of a divine Creator, he appears to have solved several conundrums about time, such as how time began and the arrow of time. That is, with an initial creative act by a first cause outside of the normal laws of physics, the Universe would have obviously have started in a very special state (such as low entropy) from which it/we have steadily degenerated (according to the arrows of time).  The behaviour of the universe right now therefore can tell us a lot about what a creative act looks like, after the Universe has been left to itself, to operate according to apparently fixed laws.  In other words, the concept of a Creator may be an important foundational principle in properly understanding time.  Scientists, in particular though, do not like to introduce the idea of an external creative being as this is then beyond the predictive power of the laws of physics.

With infinitesimal translations of spatial entities from one present moment to the next present moment we can observe what we perceive as the flow of time. This might be manifest in the frequency we record from a clock signal which is measured as the successive changes in voltage on some measuring device. The existence of such devices then is necessary for time to be perceived and from this it follows then that if we were to empty the universe of all matter there would be no way to perceive time. This seems at first sight to support of the conjecture that time is not as real as space.  However, we now contradict this idea as we have demonstrated that time is a geometrical aspect of three-dimensional space. So if space exists then time also intrinsically exists. Conversely, if space does not exist neither can time.

Speculation on higher dimensions

With the stated benefits to the concept of time with the inclusion of a Creator and an initial creative act (not to mention our own existence!), we might seek to understand spacetime from a more general perspective.  That is, if our spacetime is described as Cl(R³), which is a mathematical description, then if time is not real then perhaps the mathematics is what is real – and a natural creation of the mind of God.

Did God first create this mathematics before creating the Universe?

Could there be a parallel spiritual universe to our three-dimensional material universe?

Indeed this concept could perhaps be represented as Cl(R⁴), which thus doubles the size of the space to sixteen dimensions and so creates a copy of the eight dimensional material Clifford space.  Our three-dimensional space then becomes simply a subspace of a higher dimensional Clifford space.  Therefore higher (spiritual?) dimensions and perhaps even the supposed mind of God itself is perhaps typified by Cl(Rⁿ) a 2ⁿ dimensional space.  Interestingly, as all these spaces have a scalar component they all have a definable time variable. What would that mean for God? Is His mind and the spirit realm representable by a mathematically definable space?

In summary, we conclude that time can be defined as the scalar/point-like component of any n-dimensional space described with the 2ⁿ-dimensional Clifford multivector Cl(Rⁿ). But does that help us understand what time is?

References

1. D. Abbott, “The Reasonable Ineffectiveness of Mathematics”, Proc. IEEE, vol 101 (10), 2013.
2. I. Newton, Principia Book III, General Scholium, Paragraph 4, 1687.
3. L. Smolin, Time Reborn, From the Crisis in Physics to the Future of the Universe, Mariner Books, Boston New York, 2013.
4. J.G. Hartnett, The Unreality of Time, November 10, 2015.
5. J.G. Hartnett, Is mathematics intrinsic to the Universe?, May 11, 2015.
6. http://www.bipm.org/en/committees/cipm/