# The Idea of Space

While 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 ‘*physica*l’ 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
[p_{1},p_{2}] in physical space might be defined as
follows:

[1]

In the context of [1], we are simply quantifying the position of
particles [p_{1},p_{2}] 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. [x_{0},y_{0},z_{0},t_{0}].
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]

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]

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
[p_{x},p_{y},p_{z}] 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]

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 mode*l, 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.