The Idea of Space

1While not really the focus of this specific discussion, the current cosmological model is grounded in the idea that the space, we now called the universe, has expanded from a singularity, i.e. of near zero volume. While there are many speculative ideas as to the nature, cause and scope of this expansion, it is probably fair to say that nobody really understands why or even if the perceived expansion of space might be explained in another way. See following links to earlier discussions, although the last is possibly the most appropriate introduction to the ‘scope’ of speculation now being discussed:

From a historical perspective, the development of the current cosmological model also had its roots in general relativity, which suggested that the cosmos could not be a static system and that space was subject to curvature in the presence of strong gravitational fields. Again, the link below is simply a point of reference to an earlier discussion.

Given the scope of the debate surrounding these issues, it would seem that the idea of space might remain open to other speculative ideas; especially as it is unclear that theoretical physics has a model that has been subject to any reasonable level of verification. This said, mathematics has still formulated many models of space for the purposes of calculation. Therefore, building on the previous discussion of models, this discussion will now focus on the idea of space in terms of its physical reality and the scope of the mathematical models used to describe it. The initial question to be tabled is:

Do any of the mathematical models of space disprove the idea of physical space only consisting of just 3 spatial directions plus time?

Of course, anybody already aware of the idea of special relativity might immediately suggest that the concept of space and time has already been merged to become four dimensional spacetime, as declared by Hermann Minkowski as early as 1908:

"Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality." 

We might also cite the many extended descriptions of space developed within the field of mathematics, which are now used extensively to model the ‘physical’ reality of space. For example, Wikipedia lists many types of space, e.g.

  • Linear and topological spaces
  • Affine and projective spaces
  • Metric and uniform spaces
  • Normed, Banach, inner product and Hilbert spaces
  • Smooth and Riemannian manifolds spaces
  • Measurable, measured and probability spaces

Clearly, there is a lot of scope for abstract semantics in the list above, which may not really help in the current discussion. However, in an earlier discussion of quantum physics, an attempt was made to define a smaller subset of ideas in terms of Euclidean 3D space and  Minkowski 4D spacetime, which might also be extended to include the mathematical description of configuration and phase space. While it is not the purpose of this discussion to address the mathematical details of these different descriptions of space, it might be worth simply outlining some basic concepts that define the ‘dimensionality’ of physical, phase and configuration space.

In classical mechanics, a phase space is often described in terms of the space of all possible states of a physical system, where a ‘state’ does not just include the positions [q] of all the objects in the system, as per physical space or configuration space, but also the momentum [p] of each object, which could be said to define a momentum space.

Fair enough, but let us try to be a little clearer about the number of physical ‘dimensions’ involved as opposed to the ‘dimensionality’ of a given mathematical model of space. Let us start  with what most people intuitively understand by space and time, i.e. a concept grounded in 3 spatial dimensions plus time, such that the position of 2 particles [p1,p2] in physical space might be defined as follows:

[1]      1

In the context of [1], we are simply quantifying the position of particles [p1,p2] in terms of their Cartesian [xyz] coordinates at some instant in time [t]. Of course, we might realise that there is some ambiguity in this description until we define the origin of this arbitrary frame of reference, e.g. [x0,y0,z0,t0]. While it is accepted that the introduction of relativity will add some complexity to this description, in terms of inertial frames of reference and gravitational curvature, it might still be argued that the concept of time can still be quantified as a separate variable, distinct from the 3 orthogonal dimensions of space. As such, we might define the ‘dimensionality’ of physical space-time as R[3+1]. However, from the description above, it might be seen that any description of a system of particles might adopt an extended notation that leads to the definition of a configurations space.

[2]      2

In part, there is no real difference in the description supplied by [1] and [2], although in mathematical terms the ‘dimensionality’ of this configuration space is now defined  as R[3N+1], where [N] equals the number of particles in the system. As such, a configuration space consisting of 7 particles would have a dimensionality of [3*7+1=22], while it might still be argued that the dimensionality of physical space-time remains independent of the number of particles, i.e. it does not change even when no particles are present. Therefore, we need to clarify the distinction between a  mathematical model, e.g. string theory, which might speculate the existence of additional physical dimensions, e.g. 10 or 11, from the mathematical description of ‘dimensionality’. In this context, phase space only appears to change the definition of dimensionality of a mathematical space, not the dimensions of physical space:

[3]      3

In the form shown in [3], it might be said that phase space has a ‘dimensionality’ of 6 for each point-particle, where the idea of time is subsumed into the definition of the momentum vector components [px,py,pz] for each spatial direction [xyz]. However, based on a system of particles, as per [2], then a 7 particle system would have a dimensionality of [6N=42]. However, while there are many mathematical benefits for adopting this mathematical model of space, see Hamiltonian mechanics, it would seem that the basic dimensions of space-time are still unchanged, i.e. R[3+1]. While it is not immediately obvious from the definition in [3], the idea of momentum space also requires the inclusion of the mass of a particle, while still requiring an independent measure of  time [dt].

[4]      4

While the idea of mass and energy will be expanded in the next discussion, the idea of time in the equations above remains consistent within a single frame of reference, i.e. as measured by an inertial observer assumed to be at rest. In this context, it is not clear that Minkowski assertion that “space by itself, and time by itself, are doomed to fade away” need necessarily be true. However, what may well need some further examination is the criteria by which science has come to define the measure of space and time.

So what conclusions, if any, might be drawn about the mathematical and physical reality of space and time?

While most people might agree that mathematics definitely has something to say about the physical world, not everybody agrees that the mathematical description of space and time is the same thing as physical space and time. While many of the ‘shut up and calculate’ school of thought might reasonably argue that only the mathematical description is relevant to theoretical progress; the counter-argument is that while the evidence remains inconclusive, mathematics should still be considered as a model of physical reality, not its replacement. Of course, both of these positions might be subject to change in the context of a wider philosophical debate characterised by the following quote by John Briggs:

"The question is, shall we inhabit a world shaped, as we have long believed,  by lifeless mechanically interacting fragments driven by mechanical laws and awaiting our reassembly and control? Or shall we inhabit a world - the one suggested by fractals and chaos - that is alive, creative, and diversified because its parts are unified, inseparable, and born of an unpredictability ultimately beyond our control?"

Even if we put such issues to one side, we still need to question why quantum physics cannot do without the idea of a mathematical space. For example, Feynman’s QED model, also known as the path integral, proceeds to calculate the probable destination of a particle based on the probabilistic sum of all possible paths available to a particle, irrespective of how physically improbable most of these paths may appear. However, from the perspective of a theoretical model, each paths is said to have a mathematical existence, even though this existence cannot be reconciled with any intuitive understanding of physical reality. As such, we might recognise why theoretical physics might come to reject all previous ideas of physical reality in much the same way as it rejected the need for a deity.

But does the rejection of such basic ideas require more than just an unverified  mathematical premise?

Of course, supporters of this model will cite that what emerges out of this mathematical model is a prediction that is verifiable in observation. As such, this appears to be enough justification for some to assert that the process itself must reflect the underlying reality, i.e. it constitutes the new ‘physical’ reality of space and time. Therefore, in many respects, the only way that this conclusion might be seriously questioned is for somebody to forward another model, which leads to the same verifiable results, while possibly providing a more ‘realistic’ description. If so, the question which then has to be considered is:

Is this idea completely impossible in the face of established science?

In part, much of the previous discussion is orientated towards the mathematical description of space. However, this introduction should also make some reference to the development of ideas within physics. For example, after the acceptance of Maxwell’s equations, in the 1860’s, the accepted position of science swung in favour of a wave model, rather than Newton’s corpuscular model. As a consequence, the idea emerged that the vacuum of space might also be an influencing medium for the propagation of EM waves, e.g. light from the stars. In this context, the vacuum of space was often described in terms of the ‘luminiferous aether’ against which the speed of light [c] might be measured. However, the Michelson–Morley experiment, first carried out in 1887, started to throw serious doubts on the existence of the ‘aether’ as a type of physical media involved in the propagation of light waves. The subsequent publishing of Einstein’s papers on special relativity and the photoelectric effect, in 1904, would also lead to another step-change in thinking that appeared to relegate the idea of the ‘aether’ to a footnote in history. While the later publication of Einstein's ideas on general relativity, in 1915, would not change this position, it did start to highlight a dichotomy in the description of space in as much as space now appear to be subject to curvature. As such, we might table the question:

How can the ‘nothing’ of the vacuum of space be subject to curvature?

Over time, it was realised that physical space may not just conform to a Euclidean mathematical description, if it had the physical property of curvature, i.e. it was something that could be distorted. However, it was recognised that such an idea should probably not be linked with the rejected concept of the ‘aether’ and so the idea re-emerged as the ‘fabric’ of space. However, the description of the ‘fabric’ of space needed to be reconciled with two coexisting, but somewhat conflicting, worldviews of science, i.e. relativity and quantum physics. However, the existence of the fabric of space appeared to be substantiated in terms of both the macroscopic view of relativity and the microscopic view of quantum physics via two experiments:

As both these experiments are well-documented in other sources, as per the Wikipedia links above, no attempt will be made to detail these experiments, other to say that both appear to support the idea that the fabric of space has a physical existence. In terms of general relativity, the presence of mass-energy moving in space is said to cause space to be distorted, while in contrast, quantum physics seems to suggest that space might account for a huge reservoir of energy, i.e. vacuum energy. Clearly, both these ideas are pertinent to many speculative ideas in cosmology involving the expansion of space within the universe; although some ideas appear to open the door to far more speculative scope, which challenges the very notion of 3-dimensional space. For example, based on the known physics of a holographic 2-dimensional plate to reconstruct a 3-dimensional image, some are now suggesting that our perception of a 3-dimensional universe may only be a projection, i.e. an illusion, derived from a 2-dimensional membrane. Of course, even if any tangible evidence in support of this suggestion could be found, it may only transfer our more profound questions about creation and existence to a 2-dimensional membrane rather than actually answer them.