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).

Special Relativity for everyman (or woman) - 3. Time

Time - What is time? Time is, strictly speaking, not a dimension; so much as it is the rational ordering of events; It is an effect, an effect produced by the process of change.

Time is a mysterious amorphous entity whose presence is everywhere, yet its definition has eluded man ever since he started to question his understanding of the world.

So what is time? To determine that, we could start by defining what we know, the Fundamentals of time. And yet, how can we determine the fundamentals of time, without defining what time is?

Newton wrote:

"Absolute, true and mathematical time, of itself, and from its own nature flows equably without regard to anything external, and by another name is called duration: relative, apparent and common time, is some sensible and external (whether accurate or inequable) measure of duration by the means of motion, which is commonly used instead of true time ..."

Which seems to be a very sensible and concrete definition. It is something that agrees well with our own perception. That time flows equably, without regard to anything external, all we can do is to choose the background, or coordinate scale, against which to measure time. We can change the units we measure by, we can change their dimension but time moves inexorably on.

It is said, in a somewhat light-hearted way, that: "Time is what stops everything happening at once"; yet I would venture to declare that it is, in fact, the very opposite, that Time is created because changes don't happen 'in an instant'. Time is an effect of change.

Time Happens. It accumulates.

A Time Interval, is the temporal separation of two Events, that may, or may not, be at the same location.

A System is a set of interacting or interdependent components forming an integrated whole.

Any System has an Absolute time. That is the time measured from the start of that system. It stops at the end of that system. The Universe is one such system, where time began at the Big Bang.

Then time is being created by many systems at the same time, I hear you say, so what happens to it all?

The time created by each system is independent and created in parallel but it can only be measured by comparison to other generated times.

And how is it absolute? Because, whatever scale we use, all times occur at individual points on that timescale.

Because it can only be measured against another time. All times are measured against whatever timescale we choose.

And one time scale, the life of the Universe, encompasses all others. Every event occurs at one individual specific point on that scale.

So again What is time? A good question. For, as I say, it is not easy to determine the fundamentals of time without defining what time is. Let us begin by examining what we do know about this strange entity, that has a presence, but no physical form, yet is described as the fourth dimension.

How do we detect time? How does it interact with the physical world as we know it and with the other three dimensions?

Anything but an instantaneous change (if such a thing is even possible) has a duration. A period of time that lasts from the start of the process of change until the end of that change. Time is the label that we give to the interval between the start and the end of a change, or to the interval between Events.

(Remember; an Event is a specific point in space at a particular instant in time).

Whenever and however we measure time, we are measuring change. How long it takes to change from a 'Before' state to an 'After' state. Time is the Duration of that change.

If there existed a volume of Spacetime, or a unique Spacetime, in which there wes no change: a completely empty vacuum, not affected by any kind of radiation, for example; no time could pass within that volume.

(Think about it; if that space were visited at 1,000 year intervals, as measured by an outside observer, within that space nothing could have changed so, within that space, no time could be measured to have passed).

I believe it is valid therefore, to aver that time is generated by change. Intervals between Events are measured within a global or absolute time that has existed from the Big Bang since which there has been a single ongoing change - the expansion of the Universe.

If we keep two identical clocks, in identical systems, or in the same system, they would keep identical time. Viz. Einsteins First Postulate.

If we imagined them as the finest Swiss mechanical watches, or indeed as atomic clocks, it would mean that when synchronized, they would always read the same time but if one were to be set running slow then the time created by that clock would be less; e.g. it might read 59 secs when the other watch read a full minute. Or it could be stated just as correctly that one watch was running fast, reaching 60 seconds while the other read only 59 seconds. Each would be creating its own local time.

The one thing that we can say is that if two identical clocks keep different times, that the conditions under which those clocks operate are different.

Durations

Time: that constitutes the temporal interval between events.

Event: a location in Spacetime; i.e. a point in space at an instant in time.

Interval:

As Processes progress, so time accumulates.

The past cannot be changed, as it has already happened. To change it every particle of mass or energy would have to returned to its state at that past time.

The present is where we are and it always will be, just where we are.

Future time does not exist, it has yet to be created.

Can time's rate of progression vary?

No, the duration of the processes measured may vary due to the conditions under which those processes operate, OR because of the conditions under which those measurements are made.

Time is an effect created by change; It is no more than a measurement, against a scale.

It is change that has a rate, not time; time is the scale against which a rate of change is measured. To say that time can be fast or slow is to measure time against itself!

Processes may be fast or slow and this may affect the duration of an event; that is the time that has passed between the start and the end of a process, measured against some scale, but that does not mean that more or less time has passed, only how it is measured.

We must not only count the units of time, for the total amount of time measured is also dependent on the magnitude of those units.

Time is absolute. It may be considered to have started with the Big Bang and can, logically, be measured against the expansion of the Universe. Every event has happened at some point in that expansion of the Universe. At some point on a simple, single, straight-line scale. And each point marks how long after the Big-Bang that event occurred. If two events have the same separation from the Big-Bang then they must be simultaneous measured against the life of the Universe! It is as simple as that.

Different observers may perceive Time differently, but that is only because of the conditions under which they measure Time.

If we say that time passes more slowly when observed by a speeding observer, it is the movement of the observer that is affecting the measurement of the time passing, rather than the passage of time. It cannot be the passage of time, for, when measured in the clock's own Frame of Reference, rate that time passing is unchanged; and again, when measured against a wider scope, such as the expanding Universe, the same time will have passed.

Special Relativity for everyman - (or woman) - 1. Introduction.

The speed of light. An innocuous phrase. Stating the seemingly obvious fact that light moves at a particular speed. In the same way that anything that moves has a speed. We all know that nothing can travel faster than light and that the speed of light, in a vacuum, is 'c'.

Yet that one simple property, that Light travelling through a vacuum has the speed 'c' wherever and whenever it is measured, threw the whole of Physics into a turmoil..

Why the turmoil?

What is the Speed of Light, but the time it takes for light to travel from A to B, two points fixed in space and time? Or to be more precise, between two events in Spacetime, as scientists refer to it now.

So now we have the time it takes light to travel, in a vacuum, between fixed point A at time T_A and fixed point B at time T_B

Simple enough surely, for any man in the street to understand, so why does it cause a problem for the scientists.

It is only when one examines what that means, that its apparently paradoxical nature is apparent.

Let me illustrate that:

Light from the sun, and all the stars and galaxies in Space, is travelling towards the Earth at 'c'. Fair enough, everyone knows that.

Yet think about it; as our Earth orbits the sun, it is travelling at around 70,000 mph, sometimes towards a star and sometimes away from it. The stars are moving in vast orbits round the centres of their Galaxies, and indeed the Galaxies are moving vast distances on their paths through the heavens. Yet, despite all of this and contrary to all common sense and expectations, the speed of the light from any star relative to the Earth is always measured to be 'c' wherever in the Earth's orbit we measure it, whether the Earth is travelling at 70,000mph toward that light, or 70,000mph away from that light.

Again, imagine two trains speeding along a straight track in opposite directions; imagine that as the trains pass, an observer on each train measures the speed of light from the sun rising at one end of the track. One train is speeding towards the sun and one is speeding away from it yet each observer will still measure the light to be coming towards them, at the same speed.

To the average layman this is not a problem, as the speed of light, 'c', is how fast it travels between two points, the sun and the point where the trains are passing. Indeed the time it takes to do so is the same; yet to the scientist that is anathema, for to him it is the speed that the light is measured to be travelling relative to the observer who is making the measurements on each train. They have to factor in the velocity of the train, +/- v. So, for those observers, it would seem reasonable to claim that the speed of the light would be c + v or c - v, depending on the direction of the train.

Yet, as stated above, it has been found that light is measured to travel at 'c' by any observer; whether that observer is stationary with respect to the points between which the light travels, or is moving with respect to one or both of them.

Einstein's genius led him to understand this; led to the adoption of his two Postulates, that the general Laws of Physics are the same everywhere, in any frame of reference and the constancy of the speed of light from which he developed his Theories of Special and, subsequently, General Relativity.

New(?) Theory of Gravity

An epiphanic moment of realisation this pm. driving home from Peterhead. Gravity is a relativistic effect. It is due to inter-dimensional rotation into a 5th dimension. (I am still unsure whether this should be, or could be, described as a 'pseudo'-dimension). The well known effects of Length Contraction and Time Dilation are due to such inter-dimensional rotations as observed in bodies travelling at near light speeds. Rotated into another dimension time and space become foreshortened causing the said effects. Acceleration may be equated to gravity … or to centrifugal force; which may be considered a pseudo gravity. Then we have the body on the edge of Einstein's rotating flat disc, experiencing both the pseudo gravity of centrifugal force AND the inter-dimensional rotation due to its tangential velocity (that which causes the time dilation and length contraction of the body). It is but a small step there-from to say that another such body – on its own flat rotating disc would experience a similar effect and if those two forces were acting toward one another we have gravitational effects upon the two bodies in proportion to their inertial masses (remember - inertial mass = gravitational mass) and, being considered as motion on flat discs, inversely proportional to their displacement one from the other. Or Newtons theory of Gravity. As I say this only came to me this afternoon and the details need some more work but it seems to fit well with the rest of relativity, General relativity in this case, as the said foreshortening due to the inter-dimensional rotation fits with Einstein's space curvature explanation of gravity. Of such things dreams are made ...

The Twin Paradox explained; what it is and how it works. Simples!

Twin Paradox
The Twin Paradox

This is possibly the most well known of all the supposed paradoxes associated with Special Relativity.

In 1911, Paul Langevin first proposed it as an attempt to visualise effects of a journey at near Light Speed, utilising the additional twist of making the subjects twins, thereby emphasising the paradoxical nature of the effect. It was the tale of a traveller taking a trip at a  Lorentz Factor of ɣ = 100. 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, paradoxical effect of each seeing their sibling age more than him/her-self, is how I, and I think many others, first visualise Special Relativity; and right from the start it seemed obvious to me that there is nothing in the least paradoxical about each seeing the other age, anymore than it is paradoxical for two people separated by a distance to see that the other is much smaller.
We might even call it Relativistic Perspective.

There have been many attempts to explain why it is not a paradox and even to say that one twin does in fact age more than the other and to claim that that is the real paradox.

Yet, as so often happens, we find that when we take a step back; to dispense with assumption, presumption and biased thinking, we find that it is, as is generally the case with scientific truths, simple and easily explained.

We will now do just that.

First appearances indicate that we are looking at a two way process: i.e. the journey away and the return; or even that we should consider three parts: the journey, the turn around and the return.

Yet on close examination there are four states: travel at v, rest v=zero, return at -v, and finally rest v=zero, once more. Just those four and the transitions between them.

And if that were not complicated enough, each state can be viewed from three different perspectives: that of the outgoing twin; that of the home-alone twin, using his own measurements made with his own rulers and clocks; and the view of the outgoing twin's measurements, relayed to her sibling and transformed by the Lorentz transformations to take account of their relative velocities.

We will now examine each of those states from the three perspectives in order to demonstrate how it all fits together into one coherent whole.

Stage 1; Outward Bound.


Fig. 10

After 4 units of time have passed for each of the Twins, each being in an inertial Frame of Reference and therefore being subject to identical laws of physics, they will both agree that 4 units of Proper Time have passed along the black ct axis.

The third perspective is different though. For in this view the homebound twin will measure, by applying the Lorentz transformations, to the measurements of his travelling Twin, giving 5 units of time passing. But even so this rotated axis will be projected down onto the home Twin's ct axis and measure 4 units.


Stage 2; The Turnaround.


Fig. 11

The traveller now reduces speed and comes to a halt (let us say for the pedants who may read this that such accelerations and decelerations only take a day so will not show on these scales).

Having now come to a stop the rotation is zero degrees, with a relative velocity of zero there will be no requirement for Lorentz Transformation as those transformed units of time will be unchanged and equal with those measured directly by each twin.

Stage 3; Inward Bound.


Fig. 12

Now on the return journey the travellers time axis, as observed by the Stay-at-home Twin is rotated downwards to represent the relative velocity v = -0.6c.

Once again the black ct axis depicts what each of the Twins will measure within their own Inertial Frames of Reference; while the rotated ct' axis of the traveller's measurements, Transformed to be relative to the home-bound Twin, shows the ct' axis projected back onto the ct axis; 10 Lorentz units becoming equal to 8 Proper units.




Stage 4; The Arrival.


Fig. 13

Stage four, the Return, has two possible scenarios depending on whether the traveller slows down and comes back to rest by her sibling.

If she continues past her stay-at-home Twin the conditions described in Stage 3 will continue; giving the well know result that each Twin will calculate, via the Lorentz Transformations, that their sibling has aged differently.

However, if the Traveller returns to be stationary beside her Twin, then once again, the Lorentz Transformations become irrelevant, as they are once again in the same Inertial Frame of Reference and, with a relative velocity of zero, will measure the same times.

It is important to remember that these are not Proper Units. No they are Lorentz Units (where as stated above) 1 proper unit = ɣ Lorentz units.

Points to consider:

1. each twin may be considered to be stationary in Spacetime.
2. each, being within an Inertial Frame of Reference, will be subject to identical scientific laws and principles - courtesy of Einstein's First Postulate.
3. Therefore, the units of time and space will be the same in both Frames of Reference.
4. each twin will observe, only those measurements taken from the other twin's Frame of Reference, to be transformed, in order to make them relative to his/her own Frame of Reference.
5. such transformations, Lorentz Transformations, being a function of the velocity of one twin relative to the other will be the same for each twin.
6. therefore, such an effect will be reciprocal, and either twin could be considered to be the traveller.
7. the Lorentz factor, ɣ = 100, is that for a relative velocity of 0.9999c
8. the distances measured - 1 light-year/year (proper units) = 100 light-years/years (Lorentz units, when ɣ = 100)


Relativity is about transforming measurements from one Frame of Reference to make them Relative to another Frame of Reference. It is the transformation of measurements that exist at that one point in time; transforming them to make them relative to another, moving Frame of Reference at that same point in time.

That is it.

Just that.

And as the relative velocity is the same for each then the transformations will also be the same.

As defined by the Lorentz Transformations.

Note that we are dealing with a particular case here. We are not merely looking at measuring the path or size of a moving object. No, we are dealing with how measurements from one Frame of Reference may be represented relative to an observer in another Frame of Reference; one that is moving with a constant, non-rotational velocity, with respect to the first Frame of Reference.

So in the case of the twins it is how the measurements, taken with respect to one twin from within that twin's Frame of Reference, will be transformed from the perspective of the other twin as a function of the velocity between their Frames of Reference.

I.e. How will time and distance be affected such that those measurements will still conform with Einstein's two postulates.

Now, having said all that we can dispense with some of the more outrageous claims and conclusions that have been pinned on this particular donkey!

And those include:

1. that the Contraction/Dilation effects are physical changes undergone by the travelling Twin;
but: 1.1 they are only observed by a 'stationary' observer.
1.2 it is only the unit size and quantity that change. The Absolute magnitudes are unaltered.
1.3 that means that they would have to be an infinite number of different sizes at the same time to reflect their change in size with respect to every other body in Minkowski Spacetime.

2. One twin really does age differently from the other;
but: 2.1 that cannot be. Relativity is relative. Reciprocal. Each would measure the
other as having an equal relative velocity.
2.2 this would disprove the First Postulate and therefore Special Relativity!

3. The difference is caused by a lack of symmetry because only one twin accelerates;
but: 3.1 special Relativity makes no reference to acceleration. It is not in any such formula.
3.2 relative movement requires no knowledge of the cause of the movement.
3.3 like movement, acceleration is relative i.e. each accelerates relative to the other.

4. The difference is an effect of changing or reversing Frames of Reference;
but: 4.1 this is tied in with the idea that currently holds about the Relativity of simultaneity; that two events can only be observed as simultaneous by one, Privileged Frame of Reference.
4.2 a moving observer will experience a different simultaneity, yet all observers are, to their own view, stationary. And if two events are simultaneous to a stationary observer midway between those events, they must be simultaneous to any stationary observer midway between those events. It is only in another's Frame of Reference that they move, so they will be seen by others to have a different appreciation of Simultaneity, while holding the same simultaneity within their own purview, from their own stationary Frame of Reference. This is exactly the same description of Relative Simultaneity described by Einstein with his trains.

Length contraction and Time dilation

Length Contraction and Time Dilation

The first thing to say is that these two phenomenon do occur; yet not, in any way, as described by the current scientific gobbledygook.

Bodies in motion do not become physically shorter, nor does time run at a different speed!

As a body moves faster relative to an observer the absolute length (unit size x unit quantity, the magnitude), does not change. The units of that measurement do change however, the faster it goes the smaller the measured units become, yet as they become smaller their quantity increases in proportion to maintain that absolute length.

Time is similarly affected.

Current Relativity tells us that when two bodies have a relative velocity that is approaching the speed of light, their lengths will be contracted to near zero, in the direction of their travel. That this is a genuine, real reduction in their lengths, yet one that can only be measured by from the other body!

Pardon me! But what absolute nonsense! Surely it is not difficult to work out that this has to be that the measurements are what have been affected not what they are measuring?

And a second point that I have to make here; if a body has a number of other bodies moving at different speeds it will have to have a different actual physical length according to measurements taken from each of those bodies! Many different lengths! At the same time!

Ah but no! I have been told! That is because each of those bodies own measures change, which is why one body has different lengths as measured from each body moving at a different speed!

Seems very similar to the ouzelum bird!

These effects depend solely on the observer's perspective. And only when the moving frames coordinates are transformed and plotted onto resting Frame of Reference's coordinates. i.e. Only when the measurements from the moving frame are used.

But if the observer in the resting frame were to make his own measurements with his own 'rulers' and 'clocks' would he still observe dilation or contraction.

Let me demonstrate with a simple thought experiment:

Two observer's at A and B, 1 light year apart with synchronised clocks.
A space ship passes from A to B at 0.5c.

Space ship —————————————————————————————————>
A < - - - 1 light year - - - > B

Fig. 7


A and B record the time when the space ship passes them:
How long is their time interval?
Simple arithmetic tells us! 1 light year at 0.5c will take 2 years as measured on the stationary observers own standard clocks and rulers.

How long does is it for an observer on the ship?
Just the same, 2 years as all the measurements are local.

Summary

The important thing to be assured of here is that time dilation and length contraction only occur when the moving observer's measurements have to be transformed to give the stationary body the measurements it would have had if it had measured them using the moving body's own rulers and clocks at their relative velocity and vice versa.