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It is often said that Newtonian physics is adequate for an understanding of the physical world. This is wrong. Newtonian physics provides a set of approximations that give almost right answers for all the wrong reasons. Most people hope that 'modern' physics in the form of Relativity and Quantum Theory only applies at high speeds or when describing things that are very small. However, contrary to popular belief, Special Relativity applies at all velocities and even the simplest physical formulae, such as those for kinetic energy, depend upon the simple assumption of space-time (see Kinetic energy and electromagnetism below). It is also the case that Quantum Theory is responsible for effects at all scales as can be seen from the covalent bonding that holds together much of the environment around us. Without some understanding of Relativity and Quantum Theory the student is confined to a nineteenth century level of scientific inference. The following brief excursion into physics will take the reader to the beginning of modern physics with occasional forays beyond.

Much of modern physics is based on the concept of symmetry and invariance (Note 1A). A simple example of invariance is shown below. It can be seen that a straight rod remains the same length on a plane so long as it is rotated on the plane.

The example also shows that a two dimensional coordinate system cannot be used to determine the length of a rod once it is rotated out of the plane. Once the rod is rotated out of the plane a third coordinate must be used to specify its position. This new, three-dimensional coordinate system seems to fully specify the position and length of an object no matter how it is rotated or where the coordinates or the object are positioned. (See Note 1 for a derivation of the three dimensional version of Pythagoras' theorem).

Any scientific description of consciousness must use a coordinate system that has sufficient coordinate axes to describe our experience. If too few axes are being used it will appear as if we are holding up a coordinate system to the world of our experience and artificially projecting things into it rather than describing the experience in itself. It is often assumed that a three dimensional coordinate system is adequate for all descriptions in neuroscience but, as will be shown below, such an assumption is equivalent to insisting upon an elementary, nineteenth century level of physical analysis.

'Views' are an example of the limited applicability of three dimensional coordinate systems. A view seems to require the movement of data from one place to another and needs either a succession of three dimensional snapshots or one four dimensional coordinate system with a coordinate axis for time:

The need for a time coordinate to describe events had been appreciated since the introduction of Cartesian Coordinates by Descartes in 1637, but it was only in 1908 that Minkowski discovered that the lengths of vectors in a four dimensional coordinate system were invariant. This means that lengths in a limited, three dimensional coordinate system are properties of the external observer as much as properties of things in themselves. We should be profoundly shocked by this discovery because it means that what we previously understood to be properties of the world were often properties of our own measuring systems and this affects our understanding of every area of science from simple mechanics to magnetic fields. The entire ontology of science has been changed.

Minkowski's discovery of a four dimensional universe was based on Einstein's (1905) work on Special Relativity. The new fourth coordinate axis, which will be called 'physical time', is arranged at right angles to the normal space coordinates and it is found that simple lengths are no longer invariant because objects can extend through time as well as space. The new invariant property of objects that replaces simple length is the 'space-time' interval.

One end of the object is positioned at the origin, 't' is the time difference from one end to another of the object and 'c' is a constant that converts the time difference to meters (units are metres per second). It is assumed that no accelerations and strong gravitational fields are present§. See Note 4B for a discussion of the differential form of this equation.

The time dimension in Relativity has an opposite sign to that of the three space dimensions. The reason for this is discussed below. The four dimensional space-time is usually described as (3+1)D to emphasise the opposite sign of physical time (see Note 4).

The conventional view in physics is that, for a given object, the space-time interval from one end of the object to the other is invariant between coordinate systems. The space-time intervals describing different features of an object are therefore true properties of the object itself rather than properties of the coordinate system used to describe the object.

The proposal that the world is four dimensional is testable by experiment. The (3+1)D theory predicts that there is a particular velocity that is a universal constant, that clocks should go more slowly on objects that are moving quickly and that people travelling at different speeds will differ about what they think are simultaneous events in the world around them. If it can be shown that these effects occur then the theory of a (3+1)D universe is supported.

Pythagoras' theorem is the fundamental formula that describes the space-time length of an object (r), in a four dimensional universe:

r² = x ² + y² + z² - (ct)²

For simplicity it is assumed that the object has one end at the origin and the space-time is uniform§. The conversion constant in the equation, 'c', converts seconds to metres at the rate of approximately 300,000,000 metres to the second.

Notice that 'c', the conversion factor from seconds to metres, has the units of velocity (metres per second). If something travelled at this speed it would have a space-time interval of zero along its path:

As a result of invariance the space-time interval of something moving at c metres per second would be zero length for all other observers and all the observers would measure the velocity as 'c'. Ordinary velocities, on the other hand, are not constant between observers; if something moves at 'v' the space-time interval of its path is given by: r² = (vt)² - (ct)² and this same interval can be fitted by a wide range of values of v and t. In other words, a space time interval of zero only occurs when (vt)² = (ct)² which only happens when v=c and all observers, no matter where they set up their coordinates or how fast they are going obtain the same value for c. This universally constant velocity has been experimentally verified and is known, for historical reasons, as the 'speed of light'.

The prediction that clocks go more slowly on fast moving objects is relatively easy to work out. For clarity the example will be about planets in space but it could use things as simple as the electric charges in a coil. Suppose there are two people, John and Bill, sitting on different planets. People on a planet think that they are stationary even though their planets are moving relative to other things. John and Bill can see each other through powerful telescopes. They notice that the planets are moving away from each other at high speed. John draws a diagram of Bill's motion:

Bill sits on his planet and looks at his watch. He thinks he is only moving through time but John sees Bill and his watch move through both time and space. John suddenly has a shock of realisation, Bill's time axis is leaning over. If Bill draws any graphs of distance against time he is going to use the sloping line as if it were an upright time axis. From John's point of view, what Bill regards as a second of time is a mixture of both time and distance.

John knows about invariance and realises that, in a (3+1)D universe Bill's space-time path must be the same length no matter who measures it. When John measures Bill's path it is t seconds through time and x metres through space but when Bill measures his own path it is T seconds through time and no distance through space. John converts the time measurements to space measurements by multiplying them by 'c' the number of metres in a second. He then writes out the Pythagoras' theorem for the lengths:

( y and z are zero in this example). John knows how fast Bill is going so, from velocity (v) times time equals distance, he can write:

He also realises that Bill will measure the squared length of his path as -(cT)² metres through time, T being the time elapsed on Bill's watch. ( Bill calculates the path length as r² = X ² - (cT)² but Bill does not think he has travelled any distance so X is 0 and the square of the invariant space-time interval is -(cT)²)

According to Noether invariance these two squared lengths are equal because they describe the same space-time interval so:

John can now perform experiments to check whether the hypothesis that the world is (3+1)D is supported. All he need do is compare the elapsed time on Bill's watch (T) with the elapsed time on his own (t), if the two are related by T = t Ö (1 - v²/c²) then the hypothesis is supported.

Both Bill and John get the same equation describing the relationship between time intervals on their respective planets. Both think that the other person's clocks are running slow. This slowing of clocks on moving objects is known as 'time dilation'. See Note 2.

John discovers that his equation for the rate of clocks and watches is supported by his experiments and wants further evidence. He calculates that there will be a relationship between length intervals that is similar to that for time intervals, this relationship for comparing lengths in the direction of travel is given by:

X = x Ö (1 - v²/c²)

So distances between two points according to Bill are simple lengths in space (X) whereas John sees Bill's measurement of distance as a combination of a distance (x) and a time interval:

But from X = x Ö (1 - v²/c²)

So: t = (v/c²) x

This time interval (t) is known as 'phase' and varies linearly along the direction of motion. It is now possible to draw a graph of the way Bill's coordinate system appears to John.

It has been possible to predict a whole range of effects that would result from a four dimensional universe ranging from the constancy of the speed of light to the existence of phase. These effects have all been confirmed in various experiments over the last century and strongly support the concept of a four dimensional universe.

Phase is interesting because it means that events that are simultaneous for one person are not necessarily simultaneous for another person. If Bill set ten clocks to the same time and placed them at different positions on his planet he would see them all reading the same time*. John however, would see them all reading different times depending on how far along the direction of motion they were placed.

The discussion so far has involved the comparison of interval measurements (time intervals and space intervals) between two observers. The observers might also want to compare more general sorts of measurement such as the time and position of a single event that is recorded by both of them. The equations that describe how each observer describes the other's recordings in this circumstance are known as the Lorentz Transformation Equations.

The table below shows the Lorentz Transformation Equations.

Notice how the phase ( (v/c²)x ) is important and how these formulae for absolute time and position of a joint event differ from the formulae for intervals.

When Bill and John measure lengths with a ruler they both believe that they are applying the ruler to both ends of the thing they are measuring at the same instant; everyone knows that lengths are measured by applying a ruler to both ends of an object simultaneously. Suppose Bill and John are both in rockets. As they pass each other they both agree to measure an object on John's rocket. When John looks at Bill measuring lengths he is shocked because Bill seems to have a different idea of 'simultaneous' from him. John notices that Bill always measures lengths by putting his ruler to the two ends of the object at different times and, because Bill is moving this seems bound to lead to wrong measurements. Bill's ruler seems to be tilted into physical time by an amount given by the phase (vx/c²) so that Bill always seems to make measurements that are too short.

It is often said that Relativity is just about measurement and that things don't really contract, they just 'seem' to do so. This statement embeds the idea that spatial length and not the space-time interval is invariant. However, it is the space-time interval occupied by an object that is an intrinsic property of the object and the spatial length is merely a projection of this onto an observer's three dimensional coordinate system. Spatial length is observer dependent.

The sad fact is that we do not understand the intrinsic nature of length whether spatial or temporal or spatio-temporal, all that mathematics and science is telling us is that things can be arranged in at least four independent directions and that displacements in these directions are related by Pythagoras' theorem.

The twin 'paradox' is a famous example of just how complicated the relationship between two sets of coordinate systems can become. The failure to accept the twin 'paradox' is often the cause of students' rejections of modern physics. In this problem two twins meet on Earth then one, called Bill, leaves, travels out in space at high velocity to Mars and then returns. Bill ages less than John who stayed at home because of time dilation. This appears to be a paradox because it is alleged that, if all motion is 'relative' Bill should see John appearing to leave, along with the whole planet Earth, and then return hence John should also experience time dilation. In fact the problem is not symmetrical. The key to the asymmetry is that Bill goes off in one direction and Earth AND Mars appear to go off in the other direction.

The reason this asymmetry is important is that whilst Mars and Earth travel together the phase difference between clocks on the two planets is zero whereas anyone travelling rapidly between the planets, such as Bill, will see a phase difference between them. The problem would be symmetrical if both Bill and John went out in spaceships to an equal distance on either side of the earth then returned. In this case a phase difference would appear between both Bill and John and their respective destinations.

Suppose, for simplicity, Mars is 10 light seconds away (it's actually a few light minutes) and Bill travels at 0.8 times the speed of light to Mars and back. John measures the trip to Mars as 12.5 seconds (ie: x/v) and the return journey as 12.5 seconds making 25 seconds for the round trip. John will observe time dilation affecting Bill's clocks. He will see that these measure 7.5 seconds on the outbound journey and 7.5 seconds on the return journey hence measure 15 seconds for the round trip (7.5 seconds is 12.5 times Ö (1 - v²/c²) which equals 12.5*Ö (1 - 0.8²) = 12.5*0.6 = 7.5).

Bill observes an entirely different scene. To Bill the distance to Mars is length contracted and is only 6 light seconds (ie: 10*Ö (1 - 0.8²) ). His outward journey takes 7.5 seconds (x/v equals 6/0.8 = 7.5) which corresponds to John's observation of his clock. He calculates that John's clock will be reading 4.5 seconds when he reaches Mars (7.5*Ö (1 - 0.8²) = 7.5*.6 = 4.5 secs). He also calculates that the phase difference between the clocks on Earth and those on Mars will be 8 seconds (xv/ c², with x=10). This means that the clocks on Mars will read 12.5 seconds further forward than those on Earth when the journey began (4.5+8). When Bill turns around to do the return journey he obtains the same 7.5 seconds for his own time of travel and 12.5 seconds for John's.

What is especially interesting in this analysis is that when Bill gets to Mars he calculates the elapsed time on the clocks on Earth to be 4.5 seconds. He turns round and goes back to Earth where he arrives 25 seconds after originally leaving but calculates the elapsed time for the journey from Mars on clocks on Earth as 4.5 seconds. This means that Bill believes that the elapsed time on the clocks on Earth only accounted for 9 seconds of his journey. The other 16 seconds that make the total time into 25 seconds are accounted for by phase differences. The 9 seconds elapsed on Earth is the time that would be expected if John experienced time dilation ie: 9 = 15*Ö (1 - v²/c²) = 15*0.6.

(The calculation given here was greatly expedited by Dr John Simonetti's Relativity Pages - see Bibliography).

This explanation of the Twin 'Paradox' in terms of John's two periods of steady timing and a sudden time gap due to phase is known as the 'Time Gap Explanation'. It is illustrated below using figures from the example:

And the effects of phase are analysed in more detail below.

There are other explanations of the Twin 'Paradox' that are all consistent with the Time Gap explanation - see Note 3. The Time Gap explanation shows that an understanding of phase, the relativity of simultaneity, is essential for an understanding of Relativity.

Things that move at the speed of light in our four dimensional universe have surprising properties. If something travels at the speed of light along the x-axis and covers x meters from the origin in t seconds the space-time interval of its path is zero.

r ² = x ² - (ct)²

but x = ct so:

r ² = (ct) ² - (ct)² = 0

Similarly, if something travels at the speed of light in any direction into or out from the origin it has a space-time interval of 0 because the distance travelled is Ö (x ² + y² + z²) which equals ct:

This equation is known as the Minkowski Light Cone Equation. If light were travelling towards the origin the Light Cone Equation would describe position and time of emission of all those photons that could be at the origin at a particular instant. If light were travelling away from the origin the equation would describe the position of the photons emitted at a particular instant at any future time 't'.

At the superficial level the light cone is easy to interpret. It's backward surface represents the path of light rays that strike a point observer at an instant and it's forward surface represents the possible paths of rays emitted from the point observer at an instant (assuming the conditions appropriate to a special relativistic treatment prevail).

Events that lie outside the cones are said to be "space-like" or, better still "space separated" because their space time interval from the observer has the same sign as space (positive according to the convention used here). Events that lie within the cones are said to be "time-like" or "time separated" because their space-time interval has the same sign as time.

However, there is more to the light cone than the propagation of light. If the added assumption is made that the speed of light is the maximum possible velocity then events that are space separated cannot affect the observer directly. Events within the backward cone can have affected the observer so the backward cone is known as the "affective past" and the observer can affect events in the forward cone hence the forward cone is known as the "affective future".

The assumption that the speed of light is the maximum velocity for all communications is neither inherent in nor required for Special Relativity although the speed of light does seem to be the maximum velocity for massive objects. Special Relativity only requires that the speed of light is a constant for all observers.

It has been shown above that physical experiment supports the concept of the universe as a four dimensional space-time. The discussion was limited to simple space-time where the time and space directions are always linear and homogenous and it used Pythagoras' theorem to relate space and time intervals. The concept of space-time can also be generalised to cases where time and space are curved (General Relativity). The mathematics of this amendment is quite complicated because it involves the introduction of the actual intervals between events rather than the squares of the intervals.

The complexity arises because the expression for the intrinsic length of an object in 4D space time (r) involves a dimension, physical time, that has an opposite sign to the space dimensions ie:

r² = x ² + y² + z² - (ct)²

The most straightforward approach to this problem is to propose that time is an imaginary coordinate axis measured in units of the square root of minus one (i). This gets rid of the minus sign in Pythagoras' theorem:

r² = x ² + y² + z² + (ict)²

where i = Ö -1

Much of the physics of the early half of the twentieth century used this 'space-time formalism' either implicitly or explicitly. It has been said that many of the major advances in our physical knowledge in the twentieth century implicitly used this approach (Walter 1999).

However, the introduction of imaginary time has a major drawback because it does not account for causality; imaginary time has no in-built direction because it is much like a length in space. Modern physicists use 'real' time and a 'metric tensor' to explain the opposite sign of time in Pythagoras' theorem. This is equivalent to deriving - (ct)² by multiplying -ct by +ct and, according to Zeeman(1964) is partially consistent with causality. See note 4 for a more complete treatment.

In Note 4 it can be seen that the metric tensor is a combination of differential coefficients each of which is a rate of change of one coordinate axis with another. These differential coefficients can be used to derive the values of a new set of coordinates in a new coordinate system from an existing coordinate system and are evaluated at the point under observation whose position is being calculated. This property is known as covariance. An important consequence of the way that points in space appear to be encoded with transformations from one coordinate system to another is that there no longer seems to be a requirement for space or time as extended entities. The impact of this was noticed by Einstein as early as 1916:

"That the requirement of general covariance, which takes away from space and time the last remnant of physical objectivity, is a natural one, will be seen from the following reflection. All our space-time verifications invariably amount to a determination of space-time coincidences. If, for example, events consisted merely in the motion of material points, then ultimately nothing would be observable but the meetings of two or more of these points. Moreover, the results of our measurings are nothing but verifications of such meetings of the material points of our measuring instruments with other material points, coincidences between the hands of the a clock and points on the clock dial, and observed point-events happening at the same place at the same time. The introduction of a system of reference serves no other purpose than to facilitate the description of the totality of such coincidences". (Einstein 1916a).

The simultaneity of the things in conscious experience contradicts Einstein's uneasy assertion that "If, for example, events consisted merely in the motion of material points, then ultimately nothing would be observable but the meetings of two or more of these points." We do indeed observe more than the meetings of material points. Clearly the description of space and time based on (3+1)D metric tensors may be incomplete, furthermore Einstein has focussed on precisely the problem that arises if conscious experience is modelled using processes (see chapter 1). Processes are sets of material points in motion and Einstein's insight could be reworded as: if, for example, conscious experience consisted merely in the motion of material points, then ultimately nothing would be observable but the meetings of two or more of these points.

If it is accepted that space has an objective existence then the discussion of invariance given above shows that time must also exist, as Einstein put it:

"With the discovery of the relativity of simultaneity, space and time were merged in a single continuum in a way similar to that in which the three dimensions of space had previously been merged into a single continuum. Physical space was thus extended to a four dimensional space which also included the dimension of time" Einstein (1934).

Or as Roger Penrose wrote in his introduction to Six Not So Easy Pieces by Richard Feynman: "It was the Russian/German geometer Hermann Minkowski, who had been a teacher of Einstein's at the Zurich Polytechnic, who first put forward the idea of four dimensional space-time in 1908, a few years after Poincare and Einstein had formulated special relativity theory. In a famous lecture in 1908 Minkowski asserted : 'Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of unity between the two will preserve an independent reality.'

Feynman's most influential discoveries, the ones that I have referred to above, stemmed from his own space-time approach to quantum mechanics. There is thus no question about the importance of space-time to Feynman's work and to modern physics generally. It is not surprising, therefore that Feynman is forceful in his promotion of space-time ideas, stressing their physical significance. Relativity is not airy-fairy philosophy, nor is space-time mere mathematical formalism. It is the foundational ingredient of the very universe in which we live." Penrose (1997)

People find it difficult to accept the reality of space-time despite the views of Einstein, Feynman and Penrose. This may be because we are taught in school science that we live at an instantaneous present and can neither observe the future nor the past except by prediction and recording and this apparent certainty then becomes used as an argument against the existence of coordinate time. This argument is dispatched as we read it because at the instantaneous present we could not know the argument. There can be no 'knowing' at an instant. This "Presentism", where only the durationless instant exists, is the conventional wisdom in many scientific disciplines even though it flies in the face of physical theory, experiment and personal observation. Every time we measure a length on earth anyone in orbit could look down and say that what we considered to be a length was actually a combination of space and time. Space and physical time are known to have no independent existence.

The interrelationship of space and time is critical to consciousness science because it is evident that our conscious experience is a 'view' that is a continuous field of simultaneous things that can change rapidly. Such a phenomenon is difficult if not impossible to describe using a three dimensional coordinate system. Furthermore the use of only three coordinates preempts the analysis because it only permits the consideration of a succession of stationary states viewed in some unknown way by an external observer. The inevitability of external observers when three coordinates are used to describe experience has long been known to philosophers and is summarised as the 'Homunculus Argument' (see chapter 1). It is only when four coordinates are used that our descriptions have a chance of being coordinate independent and independent of external observers. (See "4. A Theory of Consciousness").

Kinetic energy and electromagnetism

QUANTUM PHYSICS