Appendix

pace and Time

Could there be physical theories of Experience? Our modern understanding of the physical world does indeed allow weird geometries, such as things being laid out in space and time but also at a point, to exist. Geometry is not as simple as we are taught at school as the following example will show. The example is not intended to be an explanation of our observation but is included to open our eyes to the fact that paradoxes such as Experience being "out there" in our brains and also at a point may not be paradoxes at all.

The fundamental theorem of space is Pythagoras’ Theorem. This states that for a right angled triangle the square on the hypotenuse is the sum of the squares on the other two sides.

On a plane Pythagoras’ Theorem gives:

This was the first real law of physics to be discovered.

This theorem takes on huge significance when it is combined with Cartesian Geometry. Decartes, also known as Cartesius, developed an algebraic method of handling geometry called Cartesian Geometry. This uses three independent axes (up-down, left-right, forward-back – abbreviated to (x,y,z) to plot the positions of any object in space.

In Cartesian Geometry Pythagoras’ Theorem is used to relate the three independent directions for arranging things in space (but see note 1 below).

In a three dimensional space:

In English this means that the length of anything projecting straight out from a point can be calculated from its projections along the 3 independent axes.

Suppose we have a plastic ruler and move it around in space, the projections of this ruler on the axes of our three dimensional coordinate system will change but as one projection gets smaller another will get larger. The value of "r", the length of the ruler in the formula above, will stay the same. In mathematics it is said that the length of the ruler is "invariant". The length, "r", is the real thing and and x, y and z are just ways of together calculating "r".

Einstein's great discovery was that the spatial lengths of objects were not actually invariant, if objects are moved at high speed they appear to get shorter. Einstein asked whether this shortening of objects was related to how we measure objects. A similar thing would happen if we only used projections on a two dimensional plane to calculate the length of a ruler, if we tilted the ruler out of the plane its projection on the plane would shrink. Einstein realised that the missing dimension was time. Include time as a fourth dimension and we can always get the correct value for the length of a ruler no matter how fast it is moving.

At the turn of the 20th century it was discovered that an extra independent axis was needed to describe space and time accurately. This axis is dimensional time and each second of dimensional time is 300,000 km long (300,000 km in a second is “c”, the speed of light). The new form of Pythagoras’ Theorem is:

"s" is called the "spacetime interval". It is invariant and is the real length of objects such as the ruler discussed above. The real length of an object consists of a combination of how long it extends in time as well as how far it extends in space.

A consequence of this is that things can be at a single point in four dimensional “spacetime” but separated in space and time.

If an object obeyed this equation it would be zero distance in spacetime from a point but also spread out in time and space. The time dimension, (ct), is sometimes called a "negative dimension". This equation shows how something could conceivably be at a point but also out there, in Experience.

If we were going to explain the time extension of Experience with world geometry we would also need to add a fifth coordinate axis (a fifth dimension) because the brain is too small to host time extensions of more than about a tenth of a nanosecond. Perhaps there is a fifth dimension and the problem of consciousness is partly due to our failure to detect it in the physical world or perhaps Experience is due to some other physical phenomenon. Experience does indeed involve geometry but there are many possible geometries known to mathematicians so maybe Experience is not the geometry discussed here or maybe it is.

This discussion does NOT mean that the observation point is necessarily described by the equation given above, it just shows that a point such as the observation point is not impossible according to modern physics. An observation point is conceivable.

If you are amused by maths you can use simple algebra to get from the equations for the spacetime interval given above to the basic equations of Relativity.

Suppose John travels past Bill at v metres per second for t seconds on Bill’s clocks, so y=0 and z=0, and x is the distance travelled by John which is vt metres long according to Bill. John thinks he has been travelling for T seconds when Bill’s clocks read t seconds.

John and Bill are both describing the same thing, namely John’s motion. John thinks he has only travelled in time but Bill thinks John has travelled in space and time. The overall projection in space and time “s” from Bill to John and vice versa is the same for both of them. In Bill’s system of measurements:

Bill describes John's motion by:

In John’s system this interval is described by:

The interval “s” is the same for both so:

so

Which is the Relativistic Lorentz Transformation for time between two observers and one of the major discoveries of the twentieth century.

**Note 1:**

Cartesian Geometry is simple for mathematicians but horribly complicated for people who do not like maths. What Descartes discovered is that the positions and lengths of objects can be represented by their displacements within a three dimensional coordinate system. The bottom of the red line below would be given the coordinates X1,Y1,Z1 and the top would have coordinates X2,Y2,Z2.

The length of the line is then:

This was abbreviated in the text above as: