The theory of relativity states that a concept known as *curved space*
is responsible for the action of gravity. 'Curved space' can be described
as the physical warping of the surface surrounding a body due to the body's
mass. The mass of the body causes the body to 'sink down' into the surface,
surrounding the body with a downhill 'curved sag' that brings the body down to a level below the
level of the rest of the surface. The stack-diagram below to the right - portraying
the 2-dimensional spherical universe that we are familiar with - displays a
2-dimensional planet surrounded by the distinct 'curved sag' that we call curved
space. This curved sag shown spread across the slices of this stack-diagram represents
*a 3-dimensional curvature of a 2-dimensional surface*. The curvature is 3-dimensional
in the sense that the task of physically
warping ('curving') a 2-dimensional surface involves 3 dimensions: the first 2 dimensions
of the arrangement involve the 2 dimensions associated with the surface itself. The
extra dimension - the third dimension - allows the surface to be bent, shaped, and
distorted in ways not expressable as of the 2 dimensions within the surface. Because a curved
sag formed by such a means is, as stated, a curvature of a 2-dimensional surface into
the third dimension, the curved sag itself can be labeled appropriately as a
*3-dimensional entity*.

As you can see, our view of this 3-dimensional
curved sag is somewhat limited, given the fact that we can only observe it in the form of its
cross-sections (the 2-dimensionally curved sags that exist within the individual
1-dimensional surfaces of the slices of the stack-diagram). This limitation, however, is not
a problem for us: upon moving the slices of the 2-dimensional universe over to their
3-dimensional positions behind and in front of the central slice, we can then mentally
"combine" these divided units into a single entity. Afterward, we can easily picture
the 3-dimensional contour of the shape of the curved sag. Doing so allows us to see the
3-dimensional curved sag for the element of 'pure curvature' that it is - an element existing
beyond 'cross-sections' and 'slices'. Like the curved sag, the 2-dimensional planet lying at
the bottom of the curved sag is divided into
cross-sections: as you can see, the 2-dimensional planet is itself curved *along with*
the 3-dimensional curvature of the curved sag. The outer surface of the 2-dimensional planet,
furthermore, existing within a curved 2-dimensional surface in the way that it does, is
surrounded by the curved sag *on all sides*. The manner in which the cross-sections
of the planet increase and then decrease in size, in turn, is a direct result of the
*circular shape* of the planet.

The stack-diagram below to the right - portraying a
3-dimensional hyperspherical universe - extends our
knowledge of the concept of 'curved space' by attempting to portray the very *4-dimensional
curvature* associated with the theory of relativity. The 'curved sag' shown spread across
the slices of this stack-diagram represents *a 4-dimensional curvature of a 3-dimensional
surface*. The curvature is 4-dimensional in the sense that the task of physically
warping ('curving') a 3-dimensional surface involves 4 dimensions: the first 3 dimensions
of the arrangement involve the 3 dimensions associated with the surface itself. The
extra dimension - the fourth dimension - allows the surface to be bent, shaped, and
distorted in ways not expressable as of the 3 dimensions within the surface. Because a curved
sag formed by such a means is, as stated, a curvature of a 3-dimensional surface into
the fourth dimension, the curved sag itself can be labeled appropriately as a
*4-dimensional entity*.

As you can see, our
view of this 4-dimensional curved sag is somewhat limited, given the fact that
we can only observe it in the form of its cross-sections (the 3-dimensionally curved sags that exist within the individual 2-dimensional surfaces of the slices of the stack-diagram). This
limitation occurs for the same reason that we cannot experience a **hypersphere** as a
*single entity*, but only as "chopped up" into an array of spheres. There is,
however, a means by which we can attempt to overcome this obstacle: we are to picture
moving all of the slices of the 3-dimensional universe over onto the central slice, so that all
of these slices occupy the 'same space' as the central slice. We are then to imagine the
cross-sections of the 4-dimensional curved sag to themselves "overlap" in the same manner that
the slices of the 'compressed hypersphere' in a previous section were considered to "overlap".
Once in their positions, we are to imagine the individual, separate
2-dimensional surfaces of these 3-dimensional curved sags to "combine" into a single entity
(in the same manner that the slices of the 'compressed hypersphere' were observed to
"combine" into a single entity). The result is an undivided 4-dimensional curved sag that
we are to observe as an element of 'pure curvature' - an element existing beyond
'cross-sections' and
'slices'. Like the curved sag, the 3-dimensional planet lying at the bottom of the
curved sag is divided into cross-sections: as you can see, the 3-dimensional planet is itself curved *along with* the 4-dimensional curvature of the curved sag. The outer surface of the 3-dimensional planet, furthermore, existing within a curved 3-dimensional surface in the
way that it does, is surrounded by the curved sag *on all sides*. The manner in which the cross-sections of the planet increase and then decrease in size, in turn, is a direct result of the *spherical shape* of the planet.

Take note that though this 'curvature' of space is a very real 'warping' of
the shape of a surface, its physical presence is not in fact detectable from
*within* that
surface. The illustration to the right demonstrates this notion by displaying
a representation of a 1-dimensional creature (the short light blue segment)
standing on a 1-dimensional planet (the larger curved segment).
As you can see, the situation takes place from within a curved surface. The
illustration, it so happens, presents the 1-dimensional creature's conception
of his placement within that surface - which, as you can see, lacks the
curvature that we observe. The same could be said to apply to us: just as
the creature shown here experiences his world as being completely 1-dimensional
when in reality his world happens to be curved into the second dimension, we
experience our world as being completely 3-dimensional when in reality our
world happens to be curved into the fourth dimension.

Why do objects fall? The typical explanation states that the objects
that are falling do so because they are "attracted" to another object in
exactly the same way *magnets* are attracted to each other: gravity
is a mysterious 'pulling force' that is "built in" to the way objects work,
and cannot be explained beyond that. It is this approach to gravity that
Newton is famous for and represents our basic concept of gravity. Though
this approach accurately predicts the behavior of gravity, it does not
explain what gravity is or why gravity does what it does. The purpose of
this section is to address these issues, and in the process provide an
explanation as to *why objects fall*.

'Curved space' is not the only thing that the theory of relativity associates
with the action of gravity. The theory of relativity associates **motion**
with the action of gravity. In fact, the theory of relativity states that
the action of gravity and the effects of motion are 2 ways of thinking
of the same thing! Einstein's famous *elevator thought experiment*
addresses the issue of how gravity and motion are related. In this thought
experiment there is an elevator travelling though outer space at a high
speed. A person in the elevator is pressed to the floor of the elevator
as a result of the physical force of the motion, in the same way that we
would imagine gravity to 'pull' someone to a floor. The conclusion of this
thought experiment is that *nothing actually falls*: a ball 'dropped'
within the elevator *appears* to fall, but in reality what happens
is the floor of the elevator *moves up to* a ball that itself was
stationary the whole time! What this means is that when *we* drop
a ball and it falls to the ground, it wasn't *the ball* that moved
(down to the ground), but *the ground* that moved (up to the ball)!
Does this mean that the earth is expanding?

This would seem to be the logical conclusion given the reasoning of the
*elevator thought experiment*. Though this assumption is not entirely
accurate, it is a step in the right direction: it is not an 'expanding earth',
but rather the expansion of **the universe**, you see, that provides the
motion necessary for the earth to come up to meet the ball - the earth
is *carried along* with the universe as the universe expands! The concept of an
"expanding earth" as the source of the motion behind the action of gravity, though
based on the reasoning of a valid argument, is in reality *3-dimensionally
impossible*: to be able to *move up to* a ball dropped *anywhere* on
the earth's surface,
the earth would have to be expanding outward in *all directions* - a feat that
upon being attempted would rip the earth apart! The question that we must ask, then, is that
of how the earth could travel in a *single direction*, and yet at the same time address
*all points* on its 2-dimensional spherical surface. There exists no 3-dimensional means
by which
such a feat can be performed. To
**physically envision** the process of the earth *moving up to* a ball dropped
onto the ground anywhere on its surface, we must approach the situation from a
*4-dimensional* point of view.

The stack-diagram of a 3-dimensional hyperspherical universe that is presented below
to the right represents such an approach. The expansion of this universe - the very
expansion that would make it possible for the earth to *move up to* a dropped ball -
is indicated by means of the arrows surrounding the hyperspherical slices. Our best means
of understanding how the 2-dimensional spherical
surface of the earth could *move up to* a ball dropped onto any point on its surface
exists in the form of what we will refer to as the *4-dimensional curvature* of the
earth. Take note, firstly, that we cannot **directly** visualize a 4-dimensionally
curved earth. That is, we cannot experience a 4-dimensionally curved earth as a *single
entity* (in the same way that we cannot experience a **hypersphere** as a 'single
entity' but only as "chopped up" into an array of spheres). What we *can* comprehend,
however, is a 4-dimensionally curved earth as it exists spread across multiple 3-dimensional
planes. To visualize a 4-dimensionally curved earth, we divide the earth into *circular
cross-sections*, and then assign each cross-section to its own 3-dimensional
plane (as has already been done in the stack-diagram below to the right). We are then to
modify the cross-sections by '3-dimensionally curving' the circle on each and every
3-dimensional plane: we curve the circle
upward so that the 1-dimensional circular edge of the circle is raised up above the rest
of the circle (putting the centerpoint of the circle at the very bottom). The resulting
formation will resemble a bowl - a 3-dimensionally curved circle. This curvature of circular
cross-sections, like the cross-sections themselves, is observable within the
stack-diagram.

This presents to us
our best conception of a 4-dimensionally curved sphere - an **array** of 3-dimensionally
curved circles. Because the 1-dimensional circular edge of the curved circle making up each
cross-section is "raised up" above what is understood to be that circle's centerpoint
(which lies at the very bottom), we in turn
assume that when these circular cross-sections "combine" to form a 4-dimensionally
curved earth, the 2-dimensional spherical surface of the earth is equally considered
to be "raised up" above its own 3-dimensional centerpoint, which, like the centerpoint
of the curved circle described, is a centerpoint that lies at the very bottom of the
curved entity of which
it is a part. The description of a 4-dimensionally curved earth just given represents our
basic understanding as to the **shape** of an earth as it would lie at the bottom of a
4-dimensional curved sag. The placement of a 4-dimensionally curved earth at the bottom of
such a curved sag is displayed, as you can see, in the stack-diagram. Assuming that
the earth is at the bottom of
such a curved sag, we would observe the expansion of the universe to always bring the
surface of the earth up to meet a dropped ball as it should, no matter where along the
4-dimensional curved sag the ball is dropped. In the material to be addressed next, we
will cover topics that will allow us to better understand the details of how the earth
*moves up to* objects dropped onto its surface. The entire 3-dimensional surface
of our universe, then, is in fact the 'elevator' we've been speaking of
- the 'elevator' that provides the very motion that makes gravity possible.
How do 'curved space' and the effects of motion work together to produce
the effect we call gravity?

Displayed to the left is a stack-diagram of the hyperspherical 3-dimensional
universe we have been working with. Included are the cross-sections of a spherical
planet and the 4-dimensional 'curved sag' that surrounds it. On the left side of the
3-dimensional universe, divided into cross-sections in the same manner as the planet
is divided into cross-sections, is an
object we are labeling as a floating 'space rock'. Just as our own universe
is in a state of constant expansion, we are to imagine the hyperspherical
universe in this stack-diagram to be in a state of constant expansion. This
expansion is represented in the stack-diagram by means of the arrows surrounding
the slices. As a result of this constant expansion, both the planet and the space
rock are carried outward along
with the hyperspherical surface as it expands. This in fact puts the planet
in the 'motion' required of it to *move up to* objects that are dropped
onto its surface. The space rock, itself also being carried outward by
the expansion of the hyperspherical surface, can be considered to be in
a state of motion equal to that of the planet. The space rock, though in motion
in terms of the outward expansion of the hyperspherical surface (motion of the
surface itself), is not affected by
the effects of gravity (motion that occurs *within* the surface).
Why not? In order for gravity to occur, the effects of motion must take
place *in the presence of curved space*. Because the space rock is
not in the presence of curved space, it is in a state of what we call
**weightlessness**:
any and all motion of the space rock within the hyperspherical surface
will always be the result of a force *other than gravity*. The bottom illustration
to the left shows a *simplified illustration* of how being "pressed down" upon by
the effects of motion has no effect on an object not in the presence of curved
space.

In the stack-diagram to the left, the space rock that was far from the planet in the stack-diagram of the hyperspherical universe shown above has now been placed in the very 'curved space' that surrounds the planet. There exists essentially 2 ways, take note, that an object can come into a state of being overtaken by curved space: firstly, an object can drift into the curved space from weightlessness. Secondly, an object can simply be dropped. Once within the grip of the curved space (and unaware of the rising planet at the bottom of the 4-dimensional curved sag) a second element comes into action: the outward motion of the hyperspherical universe. As a direct result of the physical force of expansion "pressing down" upon the space rock, the space rock is 'pushed' into the curvature of the curved sag as the planet rises toward it. In response, the space rock "slides" toward the planet as a result of being pushed in that direction by the rising curvature.

This is the process we call "falling" - gravity-related
motion toward a massive body. Its occurrence in the stack-diagram is designated
by the red arrow. Unless affected by forces other than gravity, the space
rock will continue along its path until met from below by the planet. The
bottom illustration to the left shows a *simplified illustration*
of the process of gravity as it has been described, showing a simple portrayal
of how the effects of motion take action in the presence of curved space. The
illustration makes clear the manner in which an object "slides" toward a massive
body as a result of being 'pushed' into the curvature surrounding the body.
As you can see by what was
described here, gravity is not a mysterious 'pulling force', but is rather
a clearly defined process that can be understood in terms of very simple,
straightforward concepts. In the next section we will see how **time**
and the expansion of our hyperspherical universe are related.

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