**Foundation**

Space
includes three mutually perpendicular
dimensions
that we might think of as forward-back, left-right and up-down. For the sake of
brevity I will call forward-back the x dimension, left-right the y dimension and
up-down the z dimension. To locate an
object
in space we need to choose a reference point along each dimension from which to
start our measurements. We will call that reference point zero. We get to choose
the reference point because space is symmetrical with regard to location and
orientation. If I move my apparatus to another lab table and turn it around to
face a different way to repeat an experiment, I get the same results.

Next picture three
real number
lines, one along each spatial dimension, placed such that their zero points
coincide. We will call the common intersection the origin. This arrangement
allows us to identify any point in space by a set of three numbers, each
representing the
distance
from the origin along a dimension. The three lines are called the x-axis, y-axis
and z-axis each named according to the dimension it spans. The three numbers,
written as (x, y, z), called coordinates, specify a point in space. The
arrangement of three lines is called a
reference frame.
Each observer of events in spacetime may have her own reference frame. These
frames may overlap, coincide or be in motion relative to one another without
interference. If a frame is not rotating and not accelerated, it is called
inertial. In inertial frames
particles
not subject to external forces, called free particles, move with constant
velocity (speed and direction).

Points in space may be identified by (x, y, z) but events require additional
specification. We must say not only where (location) but also when (time) in
referring to events. Prior to the advent of Einsteinian Relativity people thought
of space and time as two independent aspects of nature. In ordinary human
experience the unity of space and time is not obvious. In fact events, which are
the things that really matter in understanding the universe and predicting the
future, take place in a unified spacetime framework.

So, we need to include the t-axis in the discussion. The t-axis intersects the
space axes with the chosen time zero at the origin. Since we move along the
t-axis at a speed dictated by Nature, without necessarily moving along any of the
spatial axes, the t-axis must be perpendicular to all three spatial axes. We
cannot visualize a fourth line perpendicular to three others but it surely is there.

With this unification, space and time are on an equal footing. That does not
mean there is no difference between space and time but they may be treated in
the same way and combined in mathematical expressions. One distinction between
time and space is that time seems to have had a definite beginning (more on this
later) whereas space has no natural limit in any direction. The second
significant difference is that to use time in meaningful mathematical expressions
in exactly the way that space is used; one must replace the time coordinate with
the time coordinate multiplied by the
square root of -1.
No amount of mathematical finagling will get rid of the pesky minus sign that
appears on
squaring
the time coordinate.

The location of an event is specified by (x, y, z, t) where t is the time
coordinate of the event. Distances along the spatial axes are commonly reckoned
in length (meters for example) while distances along the t-axis are commonly
reckoned in units of time (seconds for example). In spacetime calculations it
is often convenient to have common units on all axes. Fortunately Nature has
provided us with the
conversion factor
between seconds and meters. One second is approximately 300,000,000 meters.
We know this universal and constant conversion factor as the speed of light,
300,000,000 meters per second.

It may help to forget about two of the space dimensions for the moment and focus
on the x-axis and the t-axis. It turns out that limiting our displacements to
forward and back does not hide any of the essential features of spacetime and
does avoid a lot of complicated mathematical expressions. Also we can show the
axes on a two dimensional surface like a computer screen or a sheet of paper.

Figure 1

**Spacetime Diagram**
Above is a spacetime diagram showing two events, one of which marks a commonly
used zero point of the t-axis, the other of possibly lesser significance.
Any event, idealized as a point in spacetime, may be plotted on such a diagram.

Two points in space are separated by a distance which may be calculated from
the six coordinates of the two points. In accordance with the Pythagorean Theorem,
distance

D=√(x^{2}+y^{2}+z^{2})

where √ represents the square root of the parenthetical quantity that
follows and x, y, and z are the differences in the corresponding coordinates of
the two points. Distance is always positive since the squares of negative real
numbers are positive.
Two events in spacetime are separated by an
interval
which may be calculated from the eight coordinates of the two events. In
accordance with Einstein’s Relativity, interval

I=√(t^{2}-D^{2})

where t is the separation in time between the two events. An interval may be
positive, negative or zero, depending on the relation between the distance and
time contributions to the interval.
As predicted by special relativity, different observers whose reference frames
are in motion relative to one another will disagree on the time and distance
between events and they will both be equally correct. It is the interval between
events, which is fixed for all observers in inertial reference frames. How much
of the interval is distance and how much time is a matter of perspective.
Different observers may not agree if two events are simultaneous or if they
happen at the same place. This is bad news for a person wondering about the
nature of **now** but we will fearlessly plunge onward.