Sciences

Space Time Theory

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: L₁² + L₂² = L_H² where L₁ and L₂ 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 (X₁,Y₁) = (1,6), and the bottom point has the coordinates (X₂,Y₂) = (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 (X₁,Y₁) and (X₂,Y₂) as: L₁₂² = (X₂ - X₁)² + (Y₂ - Y₁)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.)