Explore the universe
 

Welcome to our rocketry Archive. Have fun browsing!

 

Article #421: The Metaontology of Universe

(Browse for more articles)
 
Euclid's parallel postulate, in its (corresponding to space).
modern reformulation, holds that, on a Intrinsic curvature, which was introduced
plane, given a line and a point not on by Riemann to explain how straight lines
the line, only one line can be drawn could have the properties associated with
through the point parallel to the line. curvature without being curved in the
Gerolamo Saccheri (1667-1733) brilliantly ordinary sense, is now used to explain
attempted to prove this through a how something which is obviously curved,
reductio ad absurdum argument. There were e.g. the orbit of a planet, is really
two ways to contradict the postulate: straight. Something has gotten turned
space could have 1) no parallel lines around. If the curvature of spacetime is
(straight lines in a plane will always evident to us in extrinsically curved
meet if extended far enough), or 2) lines in three dimensional space, then
multiple straight lines through a given the form of the analogy forces us to
point parallel to a given line in the posit the "higher" or extrinsic
plane. These become non-Euclidean axioms. dimension, into which the straight lines
Saccheri convincingly achieved his are curved, as a spatial one, not the
reductio for the first possibility with temporal one. If three dimensional space
the innocent assumption that straight is not extrinsically curved into time
lines are infinite [cf. Jeremy Gray, according to the axiom of open
Ideas of Space Euclidean, Non-Euclidean, ortho-curvature, then it must be time
and Relativistic, Oxford, 1989; p. 64]. that is extrinsically curved into the
Later David Hilbert (1862-1953) would dimensions of space. In the model, where
point out that the same reductio proof before the surface of the sphere was
could be achieved by assuming that given analogous to solid space, now the surface
three points on a line only one can be must be analogous to two dimensions of
between the other two [David Hilbert and space plus time, with the third dimension
S. Cohn-Vossen Geometry and the of space as that into which the geodesics
Imagination (Anschauliche of spacetime are extrinsically curved.
Geometrie--better translated Intuitive Switching the role of time suddenly makes
Geometry), Chelsea Publishing Company, the model very non-intuitive, but it is
1952; p. 240]. For the second compelled by the feature of the model
possibility, however, Saccheri did not that the geodesic is on the surface of
achieve a good proof. And it was using the sphere. It does not help the
just such an axiom that the first philosophical issue to eject the
complete non-Euclidean geometries were complications of the axiom of open
achieved by Bolyai (1802-1860) and ortho-curvature and simply take the four
Lobachevskii (1792-1856). dimensions of spacetime as satisfying
If by "flat" we mean a plane of straight hetero-curvature; for this loses sight of
lines as understood by Euclid, then true Kant's Antinomy of Space, which we hope
non-Euclidean manifolds (i.e. areas, to answer, and of the circumstance that
volumes, spacetimes, etc.), in order to even in Relativity the dimension of time
really contradict Euclid, who was talking is not exactly the same as the dimensions
about straight lines, would have to be of space. That is the most intuitively
flat. They could not be curved. Straight obvious in the "separation" formula: s2 =
lines would be Euclidean straight, but t2 - (x2 + y2 + z2)/c2. Here the
the properties specified by non-Euclidean Pythagorean formula for changes in
axioms would be satisfied. Nevertheless, spatial location, divided by the velocity
since Bernhard Riemann (1826-1866), of light squared, is subtracted from the
non-Euclidean manifolds are said to be change in time squared, to give the
"curved," and only Euclidean space itself spacetime "separation" in units of time.
is called "flat." Contradiction #1 above Thus time is not treated as simply
produces "positively" curved space another spatial dimension. Thus we must
("spherical" or "elliptical" geometry, consider the differences between space
first described by Riemann himself), and and time, and the axiom of open
contradiction #2 "negatively" curved ortho-curvature alone allows for this.
space ("hyperbolic" or Lobachevskian The result of attributing extrinsic
geometry). To Euclid, this doubtlessly curvature to time is also suggested by
would seem to prove his point: the the peculiarity of using "curved space"
parallel postulate is about straight alone to explain gravity, as is common in
lines, so using curved lines hardly museums and textbooks around the world;
produces an honest non-Euclidean for curved space conjures up images of
geometry. "Curvature" in this respect, hills and valleys through which moving
however, is used in an unusual sense. objects describe curved paths. However,
Euclidean geodesics "straight" and those images presuppose motion, and
generalized straight lines "geodesics". motion is the very thing to be explained.
"Flat" spaces of more than three Gravity does not just direct motion; it
dimensions may be called "Euclidean" causes it. An object passing by the earth
because of their lack of curvature; but is accelerated towards the earth and
this is an extension of geometry that thereby acquires a velocity along a
would have very much been news to Euclid, vector where it previously may have had
and I wish to retain the historical no velocity at all. An object placed at
connection between "Euclidean" and rest with respect to the earth, with no
Euclid]. What "curvature" would have initial velocity in any direction, will
meant to Euclid is now "extrinsic" be accelerated with a velocity towards
curvature: that for a line or a plane or the earth. If there are no "forces"
a space to be "curved" it must occupy a acting on the body, as Einstein says,
space of higher dimension, i.e. that a then the only change that takes place is
curved line requires a plane, a curved the body's movement along the temporal
plane requires a volume, a curved volume axis; and if the body is thereby
requires some fourth dimension, etc. Now displaced in space, it must be displaced
"intrinsic" curvature has nothing to do by its movement along that axis. The
with any higher dimension. But how did temporal axis can displace the object if
this happen? Why did "curvature" come to the axis is itself curved; so the
have this unusual meaning? Why should we curvature of spacetime in a gravitational
confuse ourselves by saying that field must result from the curvature of
"intrinsic" straight lines, geodesics, in time, not of space. The extrinsic
non-Euclidean spaces have curvature? This dimension of ortho-curvature, into which
happened because non-Euclidean planes can the straight lines curve, is a dimension
be modeled as extrinsically curved of ordinary Euclidean space. This can be
surfaces within Euclidean space. Thus the intuitively shown, not so much in our
surface of a sphere is the classic model non-Euclidean models, but simply in a
of a two-dimensional, positively curved graph plotting time (t) against one
Riemannian space; but while great circles dimension of space (r). An accelerating
are the straight lines (geodesics) body will describe a curved line that
according to the intrinsic properties of changes its coordinate in the r axis as
that surface, we see the surface as its coordinate in the t axis changes. If
itself curved into the third dimension of the acceleration comes from spacetime
Euclidean space. A sphere is such a good itself, then the coordinate grid will
representation of a non-Euclidean itself be curved: the t axis lines will
surface, and spherical trigonometry was curve, displacing themselves against the
so well developed at the time, that it r axis (spatial location), while the r
now is a little surprising that it was axis lines will not curve. The curvature
not the basis of the first non-Euclidean of time itself is hidden from us because,
geometry developed [cf. Gray ibid. indeed, we intersect only one point on
p.171]. However, as noted, such a the temporal axis. Consequently, how do
geometry does contradict other axioms we know we are being accelerated by
that can easily be posited for geometry. gravity? In free fall we are being
Accepting positively curved spaces means displaced with space itself, and so we
that those axioms must be rejected. Also, move with our entire frame of reference
and more importantly, these models in and would not be able to detect that
Euclidean space are not always locally. Indeed, we cannot. It is
successful.with Lobachevskian space. A Einstein's own "equivalence" principle of
saddle shaped surface is a Lobachevskian General Relativity that we cannot tell
space at the center of the saddle, but a the difference between free fall in a
true Lobachevskian space does not have a gravitational field and free floating in
center. Other Lobachevskian models the absence of a gravitational field. The
distort shapes and sizes. There is no motion induced in us by the curvature of
representation of a Lobachevskian surface time is evident only because we can
that shares the virtues of a sphere in observe distant objects that are not
having no center, no singularities (i.e. subject to our local acceleration. When
points that do not belong to the space), we are not in free fall, e.g. standing on
and in allowing figures to be moved the surface of the earth, we feel weight,
around without distortion in shape or just as according to the equivalence
size. Three dimensional non-Euclidean principle when we are being accelerated
spaces of course cannot be modeled at all by a force (e.g. a rocket engine) in the
using Euclidean space. absence of a gravitational field. These
This raises two questions: 1) what can we are indeed equivalent because in each
spatially visualize? (a question of case we are moving relative to space
psychology) And 2) what can exist in according to our own frame of reference.
reality? (a question of ontology). We When we are accelerated by a rocket we
cannot visualize any true Lobachevskian say that we move in the stationary
spaces or any non-Euclidean spaces at all reference of external space; but when we
with more than two dimensions--or any are accelerated standing on the surface
spaces at all with more than three of the earth, it is space itself that is
dimensions. Also we can only visualize a displaced (by time) relative to us.
positively curved surface if this is Either we move through space, or space
embedded in a Euclidean volume with an moves through us. That is the experience
explicit extrinsic curvature. "Curvature" of weight.
was thus a natural term for intrinsic A question remains about the global
properties because there always was character of spacetime. Gravitational
extrinsic curvature for any model that fields are locally positively curved, but
could be visualized. Why are there these Einstein and his philosophical successors
limits on what we can visualize? Why is evidently expected that spacetime as a
our visual imagination confined to three whole would be positively curved, since a
Euclidean dimensions? It is now common to finite but unbounded universe is
say that computer graphics are breaking aesthetically more satisfying--and it
through these limitations, but such answers Kant's Antinomy of Space. Now,
references are always to projections of however, the geometry of cosmological
non-Euclidean or multi-dimensional spaces spacetime is usually tied to the
onto two dimensional computer screens. dynamical fate of the expanding universe.
Such projections could be done, Open, ever expanding universes, are
laboriously, long before computers; but regarded as having Lobachevskian or even
they never produced more, and can produce Euclidean geometry and only closed
no more, than flat Euclidean drawings of universes, headed for ultimate collapse,
curves. If such graphics are expected to positive Riemannian curvature. The
alter our minds so that we can see things observational evidence at the moment is
differently, this is no more than a for an open universe, and "inflationary"
prediction, or a hope, not a fact. And models even have reasons to prefer a
considering that non-Euclidean geometries Euclidean over a Lobachevskian geometry.
have been conceived for almost two These possibilities, however, introduce
centuries, the transformation of our considerable trouble; for Euclidean and
imagination seems a bit tardy, however Lobachevskian spaces are both infinite,
much help computers can now give to it. and it is a much different proposition to
Mathematicians don't have to worry about say that an infinitely dense Big Bang
these questions of visualization because starts at a finite singularity, into
visualization is not necessary for the which a finite positively curved space
analytic formulas that describe the can be packed, than it is to say that an
spaces. The formulas gave meaningfulness infinite homogeneous and isotropic
to non-Euclidean geometry as common sense universe, which must have begun infinite,
never could. starts from an infinitely dense Big Bang.
The Euclidean nature of our imagination An infinitely dense singularity can have
led Kant to say that although the denial a finite mass, but an extended infinite
of the axioms of Euclid could be density, even in a small finite region of
conceived without contradiction, our space, cannot.
intuition is limited by the form of space In a recent cosmological article in
imposed by our own minds on the world. Scientific American, "Textures and Cosmic
While it is not uncommon to find claims Structure" (March 1992), the authors,
that the very existence of non-Euclidean Spergel and Turok, speak of the universe
geometry refutes Kant's theory, such a (they do not say "the observable
view fails to take into account the universe") starting from an
meaning of the term "synthetic," which is "infinitesimally small point" or of the
that a synthetic proposition can be universe being at one time the size of a
denied without contradiction. Leonard "grapefruit," as though that would hold
Nelson realized that Kant's theory true for all model universes. The
implies a prediction of non-Euclidean infinite universes are not even
geometry, not a denial of it, and that considered, and so the questions about
the existence of non-Euclidean geometry density can be happily ignored. The
vindicates Kant's claim that the axioms problem is compounded here because there
of geometry are synthetic. The are actually two infinities competing
intelligibility of non-Euclidean geometry with each other: there is the infinite
for Kantian theory is neither a volume of space, and there is the
psychological nor an ontological infinite shrinkage, or compression,
question, but simply a logical one--using represented by the big bang singularity.
Hume's criterion of possibility as However much you shrink an infinite
logically consistent conceivability. Kant space, it is still infinite. On the other
does not say non-Euclidean geometry is hand, any finite region within infinite
logically impossible, but that is only space, however large, can be compressed
because he does not claim that any to a single point at the big bang. There
geometry is logically true; geometry in is no conflict between the two infinities
his view is synthetic, not analytic. And so long as you specify just what it is
Kant's belief that Euclidean geometry was that you are talking about.
true, because our intuitions tell us so, The problem here, however, is not
seems to me to be either unintelligible visualization, it is the hard logical
or wrong. truth that an infinite space remains
If we are unable to visualize infinite and that the big bang for an
non-Euclidean geometries without using infinite space, although it can be
extrinsically curved lines, however, the described as a singularity in relation to
intelligibility of Kant's theory is not any finite region of space, cannot be a
hard to find. The sense of the truth of finite singularity.
Euclidean geometry for Kant is no more or Einstein himself introduced his
less than the confidence that centuries Cosmological Constant to preserve a
of geometers had in the parallel static universe, before Hubble's evidence
postulate, a confidence based on our very of the red shift. He thus seems to have
real spatial imagination. If Kant's claim been thinking that a global positively
is "unintelligible," then Gray has not curved geometry for spacetime was not
reflected on why everyone in history necessarily tied to some dynamical
until the 19th century believed that the evolution of the universe. This is still
parallel postulate was true. That is the a possibility. Three dimensional space
psychological question, not the logical can still be conceived as having an
or ontological one. The sense of ancient inherent hetero-curvature apart from the
confidence can be recovered at any time gravitational fate of the universe:
today simply by trying to explain non-Euclidean without the need to regard
non-Euclidean geometry to undergraduate time or anything else as a fourth
students who have never heard of it dimension into which space needs to be
before. We might say that attempts to extrinsically curved. This makes for a
prove the postulate show that people were finite Big Bang regardless of the
uneasy about it; but the universal dynamical fate of the universe, where
expectation was that the postulate was that fate is tied to the effect of the
really a theorem, and no one cashed in curvature of time, locally positively
their unease by trying to construct curved but globally possibly
geometry with a denial of it. Saccheri Lobachevskian or Euclidean. However, a
denied it, but only because he was theory of global hetero-curvature then
constructing reductio ad absurdum proofs. stands separate from the mathematical
Non-Euclidean geometry did not change our Relativistic theory of gravity and
spatial imagination, it only proved what becomes a theory in metaphysical
Kant had already implicitly claimed: the cosmology more than a theory in physical
synthetic and axiomatically independent cosmology.
character of the first principles of A positively hetero-curved universe
geometry. It could well be the case that happens to suit the most commonly used
Kant is right and that we will never be cosmological model of all: the inflating
able to imagine the appearance of balloon, where motion is added to our
Lobachevskian or multi-dimensional spherical model of non-Euclidean
non-Euclidean spaces, or to model them geometry. The surface of the balloon
without extrinsic curvature, however well remains spherical regardless of whether
we understand the analytic equations. the balloon is blown up forever or
This is purely a question of psychology whether it eventually is allowed to
and not at all one of logic, mathematics, deflate. As a model the balloon therefore
physics, or ontology. Mathematicians are actually posits five dimensions, with the
free to ignore the limitations of our surface representing the three dimensions
imagination, although they then run the of space, time as the fourth, but as a
risk of wandering so far from common fifth the third spatial dimension into
sense that the frontiers of mathematics which the surface is curved and through
will never be intelligible to even which the surface moves in time. A
well-informed persons of general positively hetero-curved universe,
knowledge. Furthermore, since Kant however, does not need that fifth
believed that space was a form imposed by dimension. Space would be non-Euclidean
our minds on the world, he did not without higher dimensions, even while it
believe that space actually existed apart moves along a temporal axis that is
from our experience. This leads us to the locally ortho-curved into an apparently
ontological question: what can exist in hetero-curved spacetime because of the
reality? Non-Euclidean geometry was no curvature of time. The balloon model
more than a mathematical curiosity until therefore can represent a different kind
Einstein applied it to physics. Now the of theory than it was intended to, but a
whole issue seems much deeper and complex most suggestive one, where the global
than it did in Kant's day, or Riemann's. structure of the isotropic and
If our imagination is necessarily homogeneous universe may allow us to
Euclidean, hard-wired into the brain as avoid an infinite Big Bang independent of
we might now think by analogy with the dynamical fate of the universe and
computers, but Einstein found a way to fulfill the hope of the philosophers that
apply non-Euclidean geometry to the Einstein answered Kant's Antinomy of
world, then we might think that space Space.
does have a reality and a genuine ยง4. Conclusion
structure in the world however we are Just because the math works doesn't mean
able to visually imagine it. that we understand what is happening in
In light of the distinction between nature. Every physical theory has a
intrinsic and extrinsic curvature, we mathematical component and a conceptual
must consider all the kinds of component, but these two are often
ontological axioms that will cover all confused. Many speak as though the
the possible spaces that Euclidean and mathematical component confers
non-Euclidean geometries can describe. If understanding, this even after decades of
the limitations imposed by our the beautiful mathematics of quantum
imaginations present us with features of mechanics obviously conferring little
real space, we would have to say that understanding. The mathematics of
intrinsic curvature, despite being Newton's theory of gravity were beautiful
analytically independent of extrinsic and successful for two centuries, but it
curvature, can only exist in conjunction conferred no understanding about what
with extrinsic curvature and so with an gravity was. Now we actually have two
embedding in higher dimensions. This competing ways of understanding gravity,
could be called the axiom of either through Einstein's geometrical
ortho-curvature, according to which there method or through the interaction of
would actually be no true non-Euclidean virtual particles in quantum mechanics.
geometry, for non-Euclidean geodesics Nevertheless, there is often still a kind
would necessarily have extrinsic of deliberate know-nothing-ism that the
curvature and so would never be the mathematics is the explanation. It isn't.
actual straight lines that we need ex Instead, each theory contains a
hypothese to contradict Euclid. The conceptual interpretation that assigns
geometry of the surface of a sphere would meaning to its mathematical expressions.
thus involve ortho-curvature because its In non-Euclidean geometry and its
intrinsic straight lines, the great application by Einstein, the most
circles, must be simultaneously important conceptual question is over the
visualized and understood to be curved meaning of "curvature" and the
lines in three dimensional Euclidean ontological status of the dimensions of
space. On the other hand, it may be that space, time, or whatever. The most
intrinsically curved spaces can exist in important point is that the ontological
reality without extrinsic curvature and status of the dimensions involved with
so without being embedded in a higher the distinction between intrinsic and
dimension. This could be called the axiom extrinsic curvature is a question
of hetero-curvature, and it would make entirely separate from the mathematics.
true non-Euclidean geometry possible, It is also, to an extent, a question that
since lines with non-Euclidean relations is separate from science--since a
to each other would be straight in the scientific theory may work quite well
common meaning of the term understood by without out needing to decide what all is
Euclid or Kant. going on ontologically. Some realization
A further ontological distinction can be of this, unfortunately, leads people more
made. Even if the ortho-curvature axiom easily to the conclusion that science is
is true, a functionally non-Euclidean conventionalistic or a social
geometry would be possible if a higher construction than to the more difficult
dimension that allows for extrinsic truth that much remains to be understood
curvature exists but is hidden from us. about reality and that philosophical
We must consider whether only the three questions and perspectives are not always
dimensions of space exist or whether useless or without meaning. Philosophy
there may be additional dimensions which usually does a poor job of preparing the
somehow we do not experience but which way for science, but it never hurts to
can produce an intrinsic curvature whose ask questions. The worst thing that can
extrinsic properties cannot be visualized ever happen for philosophy, and for
or imaginatively inspected by us. Thus we science, is that people are so overawed
should distinguish between an axiom of by the conventional wisdom in areas where
closed ortho-curvature, which says that they feel inadequate (like math) that
three dimensional space is all there is, they are actually afraid to ask questions
and an axiom of open ortho-curvature, that may imply criticism, skepticism, or,
which says that higher dimensions can heaven help them, ignorance.
exist. This gives us three possibilities: These observations about Einstein's
That, with the axiom of closed Relativity are not definitive answers to
ortho-curvature, there are no true any questions; they are just an attempt
non-Euclidean geometries (and no spatial to ask the questions which have not been
dimensions beyond three), but only asked. Those questions become possible
pseudo-geometries consisting of curves in with a clearer understanding of the
Eu