Time dilation for the beginner.

Time Dilation:

What is it? We have all heard of it, although to most, I suspect it is little more than a sound bite that we associate with Relativity. 

Let us diagram it and see how it works. We will use Einstein's Light Clock that has a light source that emits a flash of light that is reflected back to the source by a mirror a set distance away. On its return the flash of light is reflected over the same path by a mirror at the light source. One round trip is counted as one 'tick' of the clock. 

(Fig. 1). 

Now, we will take two Light Clocks installed on spaceships, A & B. These spaceships are passing in space at a relative velocity of 0.6c. These two clocks are identical with mirrors 1 light-second apart, therefore each 'tick' will have a duration of two seconds. Our two clocks are synchronised at the moment that A and B pass by one another.

Fig. 1 is a diagram of a light clock. The vertical scale measures light seconds.  The mirror is 1 light second from the source. Therefore light will take 1 second to travel to the mirror and the position of the light flash may also be used to measure the time since emission.  Fig. 1 shows a light clock after 1 second, when the light flash (Shown in green) has travelled 1 light second to the mirror. This diagram could represent either clock A or Clock B, taken in isolation; for either clock may be considered to be stationary and the other to be moving.

Fig. 2a shows clock B moving at 0.6c to the right of clock A. The green path and green measurements are those made by the observer on board Spaceship A for whom Clock A is stationary. The red path and measurements are those made by the observer on board the Spaceship B for whom Clock B is stationary. The blue path is what is observed from Spaceship A. When the light has travelled 1 light second, Clock B will have travelled 0.6 light seconds on board ship B and 0.8 of the distance to its mirror.

(Our old friend Pythagoras is in evidence here in the shape of a 3,4,5 right angled triangle,  where 12 = 0.62 + 0.82 or 1 = 0.36 + 0.64) Yet this leads to a conundrum; how can the time for the light in clock B take 1 second to reach the mirror as measured by an observer, at rest with the clock, yet 1.25 seconds when measured by an observer relative to whom the clock is moving? The answer of course is that they are measured under different conditions. As measured by the observers on ship A, clock B is moving at 0.6c, a very high speed. Einstein pointed out that the elapsed time between two ticks of a moving clock was greater than that of a stationary clock; which we see here as 1.25 seconds for the moving clock which is greater than the 1 second for the stationary clock. It is reasonable to conclude that this is because the moving clock also needs the time to move 0.6 light seconds displacement from the observer and so the light path is longer when measured in a moving clock, and as light is travelling at c, it must take longer to travel further. Two different elapsed times then, but between the same two events - the light being emitted and the light hitting the mirror! So Einstein realised that time is not absolute, but varies depending on how it is measured. That more time is measured between successive ticks on a moving clock or as he described it: As a consequence of its motion the clock goes more slowly than when at rest. And there we have it - time slows as speed increases.

Twin Paradox Explained. Simples.

The Twin Paradox - Probably the most well known of all the supposed paradoxes associated with Special Relativity.

So what is it?

In 1911, Paul Langevin first proposed it as an attempt to visualize effects of a journey at near Light Speed, utilising the additional twist of making the subjects twins, thereby emphasizing the paradoxical nature of the effect.  It was the tale of a traveller taking a trip at the Lorentz Factor, ɣ = 100. (v = 99.98c) He travels for one year of his time, stops and then reverses direction. On his return, the traveller will have aged two years, while 200 years have passed on Earth.

This was very much in accord with the visualization of Special Relativity, enjoyed by Einstein himself. 

In order to see this effect more easily we will choose a much lower value for ɣ of 1.25 (v = 0.6c)as that will show the relationship more clearly using our new diagram:

Outgoing Journey

Fig. 1 21 Twins Rotated Frames (1)

In Fig 13 we see the path of the travelling twin as seen by his sibling, travelling 6 light years (x axis) in ten years (v = 0.6c) as measured by the Resting Twin in her Frame of Reference; but in the Traveller's green, rotated Frame, we see that she will measure that he has travelled 7.5 light years in 12.5 years (again v= 0.6c).  

Dilated Time	Contracted Length
t' = γt	x = x'/γ
t' = 1.25 * 10	x = 7.5/1.25
t' = 12.5 years	x = 6 light years

Both changes functions of the Lorentz factor of 1.25. 

Turnaround Point

Fig. 2 Twins Rotated Frames (2)

This brings us to the next significant point in the Traveller's journey, the turnaround point, where he slows down to rest relative to the Resting Twin, and therefore joins her in the one Frame of Reference.  v = 0, Lorentz factor = 1.  All measurements will be the same wherever they are measured from.

t' = t

x = x'

No Time dilation, no length contraction.

And note that the angle of rotation β which represents the velocity will also have reduced to zero.

Return Trip

Fig. 3 Twins Rotated Frames (3)

On the return trip we see the moving Frame rotated in the opposite direction as the relative velocity v = -0.6c. 

It is important to recognize that, as a form of Minkowski diagram, in which the current velocity is shown by rotation about the point of departure O; acceleration changes the angle of rotation for that current velocity, i.e. for the whole journey.  There are not, nor ever can be, changes within the length of that line, it is always a simple straight line, no kinks or bends.

So we see, 

1. OP_O', outgoing, v = 0.6c, decelerating to;

2. P, turnaround, v = 0,  then acceleration to; 

3. P_R'D', returning, v = -0.6c, followed by the return to rest; 

4. D, home again, v = 0, 

giving us the journey as OPD in the stationary Twin's Frame and OP_O'PP_R'D'D in the traveller's Frame Transformed to be relative to the stationary Twin.  Returning to D the twins are at rest with one another, in a single Frame, once more.  

Which leads to the inevitable conclusion that there is a fault in Paul Langevin's, and, apparently Einstein's, visualization of the Twin Paradox.  It is quite evident now that there is nothing whatsoever Paradoxical in this thought experiment when taken to it's logical conclusion.  

The apparent difference in ageing can only be evident if the traveller were to continue past his sibling and not return to rest; for the Age Difference, an effect of Time Dilation, only exists while there is a relative velocity; Time Dilation (and of course, Length Contraction) being effects of the relative velocity, only exist while their is a relative velocity.

They are effects of that relative velocity, of the relative velocity that exists at that moment; not of how long that relative velocity existed, nor of what other relative velocities prevailed before that moment.

Again, I reiterate: when v = 0, the Lorentz Factor is 1 and time dilation cannot occur.

What IS the speed of light?

What IS the speed of light?

The speed at which light travels from one event in Spacetime to another Event.

An event being a location at a point in time.

The separation between two events, in space and in time is the same in ANY Frame of Reference - it is only the coordinates that change.

Remember: being a location (a point in space) at a moment in time there is no movement, there can be no movement an event is fixed and occurs in every Frame of Reference.

So as light always travels at the same speed between any two events, and being massless nothing can travel faster; for an observer who is travelling with respect to those events to measure that it has travelled faster is only complicating the measurement by introducing the relative velocity of the observer into the measurement; changing the conditions of the measurement, NOT the speed of the light!

So the only valid measurement of the speed of light is that of the distance between events divided by the time between those same stationary events, within a single Frame of Reference.

Length Contraction and Time dilation are merely effects that change the measurements of space AND time to show how the speed of light is not exceeded. The unit sizes are contracted, while the unit quantities are dilated, both by the factor gamma, thus the total magnitude, unit size x unit quantity, remains constant.

So wherever and however the speed of light is measured, the speed of light remains constant.

Special Relativity for everyone - 7. The Reciprocality of Relativity

The Reciprocality of Relativity.

A Frame of Reference then, maps the whole of Spacetime relative to a single, specific, Spacetime Point, or Event, that we care to define, real or imaginary. And, as a single specific Event in Spacetime is a moment in time, it cannot, therefore, be moving; Spacetime must be fixed relative to that Frame of Reference, i.e. relative to that Event.

Consider Space as we see it from the Earth. It moves in relation to us. The night sky changes throughout the year, while at night the stars, constellations and the moon rise and set. So relative to our Frame of Reference, here on the earth, everything else in the Universe is moving.

Similarly, if we were mapping space from Saturn's moon, Titan, the whole of the universe, including the Earth, would be moving relative to Titan's Frame of Reference.

It follows from this that if the whole of Spacetime can be considered static, relative to any individual Frame of Reference, then the relative movement of two Frames of Reference will each be the reciprocal of the other; i.e. as A moves with respect to B, so B moves with respect to A.  

Imagine two Kings on a chess board, facing one another three spaces apart.  If either one is moved one space forward, the distance between them, will be reduced to two spaces from either perspective; similarly, if white moves one space to the right, Black will be three spaces away and one space to the left; similarly black will now see white three spaces away and one space to its left.  For any such movement, the participants can be reversed and the record of movements will still be valid.

From this it follows that position and movement can only be relative; there can be no absolute position nor movement, for any entity, real or imaginary, may be considered to be stationary. (As it is within its own Frame of Reference. i.e. the Frame of reference centred on that entity).

This raises an interesting conundrum for how may a body possess properties due to its speed or velocity if it is at rest relative to Spacetime? If such attributes or properties can only be measured from within Frames of Reference in which that entity is moving? 

Specifically, those two old phantoms, Time Dilation and Length Contraction, only materialize where the entity affected is measured to be moving. They cannot materialize in that entity's own Frame of Reference in which there is no movement, the entity being at rest.

Suffice it to say that Time Dilation and Length Contraction (which are the same phenomenon viewed from different perspectives) are real enough, yet only when measured by a moving observer. 

Special Relativity for everyman (or woman) - 6. More on Frames of Reference.

More on Frames of Reference

The Origin of the Frame of Reference, has the coordinates 0,0,0,0 and it is the location of the Nominal Observer and his clock.

There are at least three ways of viewing a Frame of Reference:

1. 1. From the perspective of a Nominal Observer (real or imaginary) located at the Origin, holding, or at least adjacent to, a standard clock. This clock will be measuring Proper Time, as the clock is at all times, at the Origin, with that Frame's Nominal Observer and so will be tracing the path of that Frame of Reference.
2. 2. From the perspective of an Observer, elsewhere in that Frame of Reference, carrying a clock synchronized with the Nominal Observer's Clock. Measurements, made using a synchronized clock and standard ruler, will also be Proper Times and Proper Lengths.
3. 3. From the perspective of a remote Observer who is moving with respect to that Frame of Reference, rather than in it. All measurements are taken by the Frame's Nominal Observer and are then converted (Transformed) by that remote observer to cater for the relative velocity. This is done using the Lorentz Transformation Equations. These Transformed measurements are Coordinate Measurements.

It is immediately apparent that an observer, on any body or at any location in Spacetime, will measure time on his local clock and measure lengths, in his frame of Reference, with his standard ruler and that those measurements will be in Proper units.

Spacetime is Homogeneous and Isotropic. It is the same everywhere, in any direction. It obeys the same basic scientific laws throughout. Therefore, if we place an object within a Frame of Reference, its properties will be the same as they would be in any Frame of Reference. This is one way of stating Einstein's First Postulate of his Theory of Special Relativity.

Let us consider what this means by taking an Event, a flash of light, and examining how it appears from different points of view.

If we take the time and location of that flash of light as the origin of a Frame of Reference it will have the coordinates 0,0,0,0. Light travels away from that event at 'c' in every direction.

After 1 second the light emitted will be measured to have travelled 1 light second in every direction and will have traced out a sphere in Spacetime, radius 1 light second. And it will be a sphere, radius 1 light second in each and every Frame of Reference, only the coordinates of that Event will be different.

There is one rather surprising outcome from these considerations however. For when we define our Frame of Reference, Spacetime is fixed and at rest from our perspective, then surely, one would think, Spacetime must be moving for every other Frame of Reference, that is moving with respect to our 'fixed' Frame of Reference.

Yet as soon as one thinks this way, one has fallen into the trap, and failed to grasp the essential meaning of Relativity. Everything is Relative. No Frame of Reference is fixed and at rest absolutely; yet each and every Frame of Reference is fixed and at rest from its own perspective.

No, there is not, nor ever can be any one preferential Frame of Reference. For if Spacetime were at rest in only one Frame that Frame would take precedence having simpler Laws of Science than other Frames of Reference.

Again I say No! For the simple reason that Spacetime is at rest as observed from any Frame of Reference.

Each and every Frame of Reference is a Map of Spacetime, with the origin of that Frame of Reference as the fixed centre of that Map.

From the perspective of any observer, at rest in any Frame of reference, every other entity or Frame of Reference is moving, in Spacetime, relative to that Frame of Reference! That is each and every Frame of reference is moving relative to every other Frame of Reference or Map of Spacetime (for if they are not moving they are different parts of the same Frame of Reference).

I have repeated myself, ad nauseam, in the passage above, because it is describing the fundamental principle of Relativity: Everything is Relative.

This is the most fundamental and I may say surprising facet of Relativity, and one that so many eminent scientists, indeed the whole scientific establishment have, as yet, failed to grasp; determined as they still are to see everything relative to some particular Frame of Reference, thereby failing to recognize that that particular Frame too, must also interact in exactly the same way relative to other Frames as those Frames interact with it.

AS A IS TO B, SO B IS TO A

So let us try and picture this, shall we?

Let us take for an example a train moving along a railway track. A lightning strike hits the track, as the very centre of the train is passing that point on the track. How is the flash of the lightning observed from the track and from the middle of the train. Fig. 1

Fig. 1

This is a practical example of our original event the flash of light and two frames of reference, the track and the train, that are moving relative to one another.

Now as we saw in the earlier discussion, each will see the light travel at 'c' relative to their Frame of Reference, so the observer on the track will measure the light travel equal distances, in the each direction, along the track as the train moves away. Fig. 2

Fig. 2

while the observer on the train will measure the light travel equal distances, in each direction along the train, as the track moves away from the train. That same observer on the train, will see the light reach both ends of the train at the same time; although when that happens, the two ends of the train will no longer be equidistant from the observer on the track. Fig. 3

Fig. 3

So which one is correct, the observer on the train or the one on the track?

As we have just seen, they both are, hence the need for Einstein to explain it!

Think about it! At the moment of the flash of light both B and B' are coincident at the flash of light.

The two Observers are located, one at B on the Track and one at B' on the Train, which coincide when the flash of light occurs. So each Observer, at rest in Spacetime as they Map it, will observe the light travel at 'c' in all directions in their Frame of Reference. The light will, therefore, reach the points, A and C on the track, and A' and C' on the train, at the same time. It is the observers B and B' who are each moving away from the other and so are no longer at the location of the flash of light, AS MEASURED IN THE OTHER FRAME.

The important fact to realize here is that every observer will measure the light expanding evenly from the initial event, the flash of light, within his own Frame of Reference! But that every other Frame of reference, will be moving away from him. Exactly as we see in our 'thought experiment' with the train .

A paradox, or a conundrum at the very least, one might think, yet the answer is a simple one: there is only one expanding sphere of light that is mapped as being at rest in each and every observer's view of Spacetime!

For each and every observer the light expands evenly in their Spacetime, while all other observers are moving through that expanding Sphere of light; thus the stationary observer's inevitable conclusion that the moving observer cannot see the light travelling evenly in both directions.

Note: that it is only in the measurement, relative to a stationary observer, that the space and time of the moving observer, is distorted.

So how is this distortion, of the moving observer's view and measurements of Spacetime, experienced by those concerned, how do we relate the stationary observer's measurements with those of the moving observer?

At which point we ask those two venerable old rogues, Time Dilation and Length Contraction to step forth and take a bow!

Special relativity for everyman (or woman) - 5. Special Relativity

Special Relativity

Before we examine what this is and how it works we must address the subject of how we map Spacetime using Frames of Reference to relate separate entities within Spacetime.

Frames of Reference

We measure Space using the same three axes we are used to using to map any space, length, breadth and height, only in Space, we refer to them as x,y,z. We also imagine a standard clock at the Origin, set to zero, upon which we measure time, t. Thus giving us the 4 axes or dimensions of Spacetime.

Because there is no fixed point in space to base a map upon, we use a Frame of Reference based upon whatever location and time suits our needs. Each and every Frame of Reference will have its own map of Spacetime, in which it is stationary and everything else is moving.

Yes, in Spacetime there is no fixed framework that everything can be measured against; the location, and the is movements of those objects can only be measured relative to other objects in Spacetime.

Yet if that is the case, how do we define two Frames of Reference moving relative to one another? They cannot both be stationary, can they? So which one would we designate as stationary and which one as moving?

It is the one we are taking taking measurements from that we designate as stationary, and the other to be moving.

Confusing isn't it? Well, maybe so at first glance, but that is what Special Relativity is all about. Giving a simple, easily understood answer to the conundrum of how everything in Spacetime is stationary and at the same time everything is moving!

The easiest way to explain that, is to take an example and see how it works.

Light clocks

   

Fig. 1                                                               Fig. 2                                                               

For this 'thought experiment' we will use Einstein's Light Clock. A very simple device. A pulse of light is sent to a distant mirror where it is reflected back to the base of the clock, where it triggers a new pulse of light. So the time in the clock is measured by the speed of light. If we say that in our clock the mirror is one light second away, the light will take one second to reach the mirror and one second to return. It will 'tick' every two seconds. Fig. 1 Imagine two identical, synchronised clocks alone in Space so far away from anything else that there is nothing that will affect them. And imagine the two clocks are moving relative to one another, with a relative velocity of 0.6c. Nominal Observers situated at the base of each clock, will measure their local clock as stationary and the other, their remote clock, to be moving away at 0.6c. We will refrain from identifying the individual clocks and merely refer to the local stationary clock and the remote moving clock. To the diagram of the simple Stationary clock (in blue), we will add the moving clock (in red). Fig. 2 As measured by our stationary observer, the light in the stationary clock travels 1 light second to arrive at the mirror, while the moving clock's light path is 1.25 light seconds, to the mirror.

The configuration of the two clocks and the observers upon them are identical and reciprocal; so, as we draw their positions and measurement from the perspective of the stationary observer, each of the two clocks will be both the stationary one and the travelling one, depending on which observer's view is taken.

The time for the light to reach the mirror in each stationary clock is one second, yet the time when that same clock is moving at 0.6c, is 1.25 seconds. Yes, the light in each clock will take both 1 second to reach the mirror, when measured as the stationary clock AND 1.25 seconds when measured as the travelling clock! Yes both times for the same clock, depending on which observer is measuring! Time and distance are measured differently due to the movement of the remote system, yet the duration when measured as a stationary clock, remains the same. So it has to be the measurement scales that change. Time and distances measured locally, within a Frame of Reference, is Proper Time and Proper Distance, while those measured in a remote, moving system are Coordinate Time and Coordinate Distance. We use the Lorentz Transformation Equations to translate between these two scales of measurement.

This has, unfortunately led to the almost inevitable conclusion that time passes differently and distances measure differently in a system moving at a great proportion of the speed of light.

Whereas, in fact, the times and distances are exactly the same. They do not change. The differences are an effect of the conditions under which the measurements are taken; it is the observer's motion relative to the clock that has to be accounted for, and that is why the measurement scales have to change.

Special Relativity for everyman (or woman) - 4. What is this mysterious thing we call Relativity?

What is this mysterious thing we call relativity?

Relativity is a term almost guaranteed to bring a blank look to the face of most of the population. It also brings a certain uneasiness to many scientists. Something that is acknowledged and respected, yet with an air of foreboding, a foreboding that comes from a lack of confidence in our understanding of it.

Why?

Just what is this thing called Relativity?

It has the reputation of being mathematically complex, and esoteric. That it can only be understood in the abstract realms at the boundaries of science. Despite this, it is in fact very simple.

Imagine two passengers sitting in trains on opposing platforms in a station. Each will see the other as stationary. Until one of the trains starts to move. Then, for a moment, each passenger is convinced that his train is moving. He automatically assumes that what is outside the window is stationary. Only the one train is moving relative to the railway track; yet each train, and the observer within it, is moving relative to the other train.

Take this a step further and imagine two trains passing one another. Observers seated on those trains will each measure the same speed for the other train, relative to themselves for each observer will deem their own train stationary and the other to have all the movement.

In fact we can expand this to say: If, relative to system K, K' is a uniformly moving co-ordinate system devoid of rotation, then natural phenomena run their course with respect to K' according to exactly the same general laws as with respect to K. This statement is called the principle of relativity (in the restricted sense).

(The secret is to imagine one train is stationary, taking all measurements relative to that train. The other train is then moving with velocity v relative to the stationary train.

Yet in the same way, if we were to take the second train as stationary then the first train would no longer be stationary but would be travelling with velocity -v with respect to the now stationary second train.

And it this been with us from the distant realms of history, before being defined in those terms by Galileo. First when man learned to sail the seas, using maps to Navigate by and reckoning in the winds and currents obtaining in the seas. Then similarly in the days of aviation, accounting for winds and weather systems, right up to the modern day, when the need to account for the velocity of the orbiting satellites used in GPS navigation brought new challenges to be catered for. Challenges predicted by the genius of Einstein and his follow pioneers.

In one chapter of his little book, Einstein described: 'the theorem of the addition of velocities employed in Classical Mechanics'. In it he stated that a man walking with velocity w along a train travelling at velocity v, would be travelling with velocity, v + w, relative to the track. He termed this the Galileian Transformation.

Essentially, relativity is about how the locations and measurements from one observer's perspective are transformed to become those of another observer, moving with respect to the first.

At low speeds, simple Relativity, where Relative measurements can be calculated using simple addition and subtraction, Gallilei Transformations, is all we need. Such calculations being sufficiently accurate for all practical purposes. But Einstein went on to point out, that using light shining along the railway track instead of the man, the speed of the light relative to the train would be c ± v. This, of course, is contrary to the many experimental measurements that have all shown that the speed is in fact 'c'.

Thus the need for a new theory of relativity was born and that new theory was Einsteins Theory of Special Relativity.

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).