Special Relativity for everyman (or woman) - 2. setting the scene; the Fourth Dimension

Setting the scene; a fourth dimension.

Space the Final Frontier

Before we start, let us begin by considering where we are. Yes, we ought to define what just what we mean by Space in terms of relativity.

Firstly, there is one space; we know that it is all around us; that it stretches away to infinity, in every direction; that it is the same everywhere.

That there is no privileged point in space from which Space may be observed.

All movement in Space is relative as there is no fixed point to which absolute movements may be referred. Everything in Space may be considered to be moving as movement may only be measured relative to another body.

Spacetime.

We are all familiar with the concept of 3 dimensional space. It is the way we think and the way that we view our world. Everything we relate to can be seen as having height, width and depth. And position, for example, a location anywhere on the Earth, can be defined by its three coordinates, Latitude and Longitude and its height above Sea Level. But Einstein and Minkowski insisted that we should consider time as a fourth dimension, by adding a fourth coordinate, t, to our three familiar coordinates, x,y,z.

These three are the scientists way of referring to any physical location, relative to a given point. By adding the time coordinate we can define not only an individual point in Space (x,y,z) but that point at a particular time.

This combination of a point in Space at a specific moment in Time, is unique and is labelled as an Event. This means that we can follow how the content or properties of any location in Space change over time. Yet for for those of us who think in pictures, visual thinkers, the question remains: how do we conceptualize or visualize this fourth dimension?

In fact it can be considered in a very straightforward way.

Fig. 1

Start with a single dimension; all the points in that dimension, taken together as a continuum, comprise a straight line; let us term this the x axis. Any point on this axis can be defined by a single coordinate (x) denoting how far it is from the start of the line.

Adding a second dimension, normal, that is at right angles, to our x axis means that at each and every point on this y axis we may imagine straight lines, parallel to the x axis, giving us the y coordinate. Thus any point on such a plane can be defined or referenced by our two coordinates (x,y).

Now we will add a third dimension, normal to our two dimensional x,y plane. At each and every point on this z axis we can imagine another plane, parallel to our x,y plane, representing our z coordinate. This addition gives us a three dimensional volume, any point in which may be defined or referenced as x,y,z, our old familiar Cartesian Coordinates.

Now, to follow this pattern, the fourth axis must be normal to each of the existing axes, x,y and z. The only construction that will achieve this is a sphere, centred on the given point at the Origin of our 3D space, or Frame of Reference. For the surface of a sphere always approximates to a plane, normal to any radius of that sphere.

Our t coordinate may be measured along any radius of that sphere; this is compatible with the fact that time has no spatial direction. As a consequence of this a moment in time will exist throughout the whole of our sphere, and include every location within that sphere.

If a single coordinate defines a point on a line in one dimension; and two coordinates define a point on a plane in two dimensions; and three coordinates define a point in a volume in three dimensions; then four coordinates are required to define a point in space and time, three defining the point in space and the fourth the moment in time.

Each moment in time will represent the whole content of that space, at that moment in time. Then that a point on this t axis will be the time coordinate for the whole of that volume of Space, within what we might term the Time Sphere, being the the whole of space at a moment in time.

But, hold hard a moment, surely each moment in time is no more than a representation of the 3D space, as we would have with a simple Cartesian diagram or indeed with a photograph of the space. Each successive moment would be no more than a copy of that representation including any changes occurring in the very brief time period between one moment and the next.

In real time this could be no more than watching the hands move round a clock face, or indeed whatever we are looking at!

Thus a Four Dimensional diagram could be effected by the (relatively) simple device of an animated 3D diagram. But then, are we not saying that by adding the fourth dimension, Time, to a 3D representation of space, such as a photograph, is merely progressing from still photography to a moving film?

That is essentially what we are doing when we see the world changing around us, we are seeing the same representation of three dimensional space changing moment by moment.

When we look at something we are not viewing it in 3D we are viewing it in 4d! Time included. All we need now is a way to map all those changes against time. Mapping all of space at each moment of time to give us a map of Spacetime.

Try visualizing this. Picture time as the colour of the Universe; as if we were observing it through a coloured glass. Let us say that it began as a single wavelength in the red part of the spectrum and that, for each successive instant of time, another wavelength is used or added. Then the colour of Spacetime would progress through the rainbow as time increases. An instant at a point in space, which we would call an event, could be seen as 3 spatial coordinates for the position and a colour coordinate for the time. (Note: this is merely a way to visualize it).

I picture the Big Bang as a spherical wave of light coming from one point that is the Universe at the beginning. That sphere being the limit of the universe at any one moment, expanding at the speed of light and the colour changing continuously.

So if time is represented as an expanding sphere encompassing more and more space moment by moment, (Why does the Big Bang constantly intrude upon my thoughts?) where does the Time axis lay? In which direction should we draw it?

The time axis exists everywhere in every direction running from the origin of whatever Frame of reference we are using. So, to be mathematical for a moment, Pythagoras tells us that the radius of a sphere equals √(x²+y²+z²) giving us t² = x²+y²+z² or t²-x²-y²-z² = 0, the signature of Spacetime.

Fig. 2

It is interesting to note that this formula, arrived at by reasoning and simple logic, is identical to that found by the best mathematicians; giving this visualization a level of credibility.

(If the other axes are also scaled in light second times, any of the three space axes may also represent, or be replaced by, a time axis if one is appropriate to what is depicted, e.g. in a diagram using only one or two spatial dimensions).

(The surface of the Future Time Sphere denotes the limits of Space, that could be affected, by an Event at the Origin; or, for a negative value, the Past Time Sphere is, for that time, the volume of Space in which a change, at that moment in Time, could affect an event at the Origin).

(It is important to understand that the time coordinate, that is the 'now' of an event does not apply only to the surface of the sphere, but to everywhere within it).