###

Sciences

## What is "spacetime"?

### Geometry is "measuring the Earth"

How did human beings come to
use mathematics to describe the world around them? One of the early
motivators for humans to perfect a language for communicating about the
world in terms of numbers came from the need to measure the Earth.
People were learning to build large temples and cultivate large fields.
These people had spiritual and practical needs for understanding how to
measure and describe the space around them.

The word

**geometry**
reflects this need.

**Geo** is Greek for

**Earth**,
from the very ancient Greek Earth Goddess

**Gaia**.

**Meter**
is related to

**measure** and also to

**mother**.
But although the ancient Greeks succeeded in naming most of geometry,
they were not the first people to discover much of what they've been
given credit for. The ancient Mesopotamians figured out much of what
the Greeks wrote down a millenium later, including what became known as
the

**Pythagorean Rule**:

**L**_{1}^{2}
+ L_{2}^{2} = L_{H}^{2}
where L

_{1} and L

_{2} are the
lengths of the two legs of the right triangle shown in the figure, and L

_{H}
is the length of the hypotenuse of that right triangle.

The Mesopotamians discovered
this rule by observation, not by formal derivation from abstract
mathematical priciples. They measured things like clay tablets and
fields of wheat. They were discovering something important about
mathematics and Nature simultaneously.

As far as we know, it wasn't
until ancient Greece that a system of abstract principles describing
geometry emerged.

The Pythagorean Rule became
a

**theorem** provable from completely abstract
arguments (based on a few key assumptions that, as we shall see later,
Nature doesn't actually respect), independent from observations made by
measuring things. This development marked what we now know as

**Euclidean
geometry**, named after the Greek mathematician Euclid who
wrote the first known geometry book, known today as

*Euclid's
Elements*, which gathered together the accumulated
understanding of his time.

#### Who was Pythagoras?

The Pythagorean Rule was not
described by Pythagoras himself. Pythagoras is remembered for having
observed and eloquently described (for his time) the numerical
relationships between musical tone scales and the length scales of the
physical objects producing them, such as the lengths of the strings on
stringed instruments and the diameters of bells.

The Pythagorean Rule itself
probably came from followers of his school of philosophy, whom we call
Pythagoreans.

### The Euclidean model for space

Euclid had a special
obsession with parallel lines. He found out that systems of parallel
lines were powerful tools in proving abstract geometrical truths.

Many centuries later
Descartes improved on the abstract power of Euclid's systems of
parallel lines by inventing

**analytic geometry**. In
analytic geometry, systems of parallel lines are used to build the

**Euclidean
coordinate system** (sometimes also called the

**Cartesian
coordinate system**).

In anayltic geometry,
geometrical shapes are described using equations made from the
variables representing distances along the parallel lines.

An example of this is shown
in the top figure. Each point in the top figure can be described by its
location in the grid of parallel lines that determines this Euclidean
coordinate system. The top point in the figure is represented by the
coordinate pair

**(X**_{1},Y_{1})
= (1,6), and the bottom point has the coordinates

**(X**_{2},Y_{2})
= (7,3).

In the bottom figure, we see
how the old

**Pythagorean Rule** fits nicely with the
Euclidean coordinate system. It turns out that when points on a plane
are described using this Euclidean coordinate system, the distance
between any two points on that plane can be calculated using the
Pythagorean Rule. In the language of analytic geometry, we can write
the distance between any two points

**(X**_{1},Y_{1})
and

**(X**_{2},Y_{2})
as:

**L**_{12}^{2}
= (X_{2} - X_{1})^{2}
+ (Y_{2} - Y_{1})^{2}
The Pythagorean Rule turns
out to be the

**distance function** on what
mathematicians call the

**Euclidean plane**. Another
word for a distance function is a

**metric**. The
Pythagorean Rule can also be called the

**Euclidean metric**
on the two-dimensional plane.

It was Einstein who
theorized that there is a physical relationship in Nature between the
distance function on spacetime and the distribution of mass and energy
in spacetime. This is his model for the gravitational force, called

**General
Relativity**, about which we'll hear more later.

Even though the ancient
Mesopotamians measured the Pythagorean Rule as an effective distance
function for their needs -- according to Einstein's model, in Nature,
matter and energy change distance relationships so that under the right
conditions, the Pythagorean Rule will stop working.

A fancier way of saying that
is that in general, it's okay to model the space around us using the
Euclidean metric.

**But the Euclidean model stops working when
gravity becomes strong**, as we'll see later.

### How Newton modeled time

By Newton's time, people
were getting pretty good at modelling space using the Euclidean
distance function. But what about time? Can we model space and time
together using some version of the

**Euclidean coordinate system**
and

**distance function** adapted from

**space**
to

**space-time**? Isaac Newton thought about this and
decided:

*Absolute, true and mathematical time, of
itself, and from its own nature, flows equably, without relation to
anything external.*
Let's examine some system in
motion and see what Newton meant by this. For example, the three cars
to the left move differently. Let's say that the red one is going 60
mph, the blue one is travelling at 30 mph and of course, the green car
is going 0 mph. What do their paths look like if we try to extend

**space**
to

**spacetime**?

Using Newton's model for
time as flowing exactly the same for all observers, we could draw a
coordinate system with time on one axis and space on the other. Then
the path of the car in space can be plotted against time, as was done
in the figure below:

If we measure

**time**
on the vertical axis and

**space** on the horizontal
axis, the paths of the cars appear as shown to the left. Notice that
the green car's path is just a line parallel to the time axis itself.
This means the green car is staying at the same place in space but
moving through time.

A car that stayed at the
same moment in time but moved through space would follow a path
parallel to the horizontal axis.

We know from observing
Nature that such paths are not found. Yet in the Newtonian model for
spacetime there seems nothing to prevent such a path from existing.

This is where

**Einstein**
and

**Special Relativity** come in, to give us a
mathematical model for spacetime that reflects the observed behavior of
Nature in that nothing can go faster than the speed of light. (At least
not anything observed in a laboratory or in outer space as of yet.)