Dy = Dx¹ dy/dx¹ + Dx² dy/dx²+ Dx³ dy/dx³

Dz = Dx¹ dz/dx¹ + Dx² dz/dx²+ Dx³ dz/dx³

Therefore:

Dx² = (Dx¹ dx/dx¹ + Dx² dx/dx²+ Dx³ dx/dx³)²

and so on for y and z.

Substituting into the pythagorean formula:

Ds² = SS (dx/dxⁱ dx/dx^k + dy/dxⁱ dy/dx^k + dz/dxⁱ dz/dx^k) DxⁱDx^k

(for i=1 to 3 and k=1 to 3)

and so:

Ds² = g_ikDxⁱDx^k

Where

g_ik = (dx/dxⁱ dx/dx^k + dy/dxⁱ dy/dx^k + dz/dxⁱ dz/dx^k)

Notice that the substitutions for Dx etc result in a new formula for the length being squared (ie: (Dx¹ dx/dx¹ + Dx² dx/dx²+ Dx³ dx/dx³)² etc.).

This analysis can be extended to any number of dimensions. The new concept introduced in the standard formulation of relativity is that the substitution for time is not simply squared but is the product of two lengths:

-(Dx¹ dt/dx¹ + Dx² dt/dx²+ Dx³ dt/dx³ + Dx⁴ dt/dx⁴) (Dx¹ dt/dx¹ + Dx² dt/dx²+ Dx³ dt/dx³ + Dx⁴ dt/dx⁴)

One of which is negative and the other positive.

The differential coefficients can be collated into a 4X4 table of numbers known as a metric tensor (g_mn). The metric tensor is a set of differential coefficients such as rate of change of time interval with displacement on the x axis, rate of change of displacement on the y axis with displacement on the z axis etc. The differential coefficients are determined at the location of the point in space-time that is being observed. The process of deriving a new coordinate from an old coordinate from the differential coefficents of the old coordinates with respect to the new is known as covariance.

The modern formulation uses the following mathematical expression for the space-time interval:

s² = g_mn xⁿx^m

where the values of x represent tiny displacements in each of the four coordinate axes measured from the origin and 'g' is the metric of the space. This expression becomes s² = x₁² + x₂² + x₃² - x₄² when expanded as is shown below. In matrix notation this is:

s² = t x₁ x₂ x₃ times -1 0 0 0 times t

(Where t is time in metres, ie:c times time in secs). The numbers -1,1,1,1 are the values of the combinations of differential cofficients that were described above.

Evaluating the first matrix multiplication this becomes:

s² = -t x₁ x₂ x₃ times t

Which resolves to: x₁² + x₂² + x₃² - t²

Which is the Pythagorean metric of space-time and applies to quite large values of s,x and t in the absence of accelerations and strong gravitational fields. Notice how the computation is more like a squared norm than a simple square and carries with it the physical implication of a product of a vector with its reflection (!).

Note 4B: The metric is generally expressed as differentials rather than crude intervals ie:

ds² = dx₁² + dx₂² + dx₃² - dt²

Or, equivalently:

ds² = dt² - dx₁² - dx₂² - dx₃²

The use of intervals, s² = x₁² + x₂² + x₃² - t², might be thought to embody a Kantian prejudice about the nature of space whereas the refusal to accept intervals is a Leibnizian prejudice. The fact that we experience entire separations in our experience of the world as a 'view' makes the Kantian prejudice credible. The use of the term "Pythagoras' Theorem" to describe the metric comes from Weyl's statement that: "Einstein = Newton + Pythagoras" and from the derivation of the metric tensor from Gauss' analysis based on Pythagoras' theorem.

Note 5: In general, if the number of states is N then the number of bits required to represent them is log₂N and if the number of bits is 'b' the number of states that can be represented is N = 2^b. Information theory is useful for calculating the properties of communication channels and information systems (see Shannon(1948)).

Note 6: Wave Equations.

Waves are repetitive events that propagate. The most basic repetitive event in physics occurs in objects that act like springs and is known as 'simple harmonic motion'. If the object is stretched the force increases linearly with the amount of stretch, when the object is released it springs back to its original position, its kinetic energy increasing all the way. The object overshoots its original position and decelerates as the kinetic energy is converted to potential energy. When the object springs back the potential energy is converted to kinetic energy again and so on. The defining feature of simple harmonic motion is that the force restoring the object to its mean central position is proportional to the displacement from this position:

In other words, the acceleration of the displacement is proportional to the displacement. A mathematical solution to this type of equation is:

Displacement at time t is: y(t) = y₀e^iωt

Where y₀ is the displacement at time zero, i is the square root of minus one and ω is a constant equal to sqrt(k/m) in this case. The constant 'e' is a constant that occurs whenever a rate of change of a thing is proportional to the amount of thing that is present (for instance population growth depends on the number of breeding pairs at any instant).

Waves are often described as "sine waves". Euler's equation relates exponential to trigonometric quantities so that, in general, e^ix = cos x + i sin x so:

e^iωt = cos ωt + i sin ωt

Now suppose the object is a wire, membrane or some other extended thing. When it is pulled at right angles to its normal position force is developed along it. This force has a component along the wire or membrane etc in both directions, either side of where it is pulled and a component back to the original position. When the string etc. is released two disturbances or 'waves' travel in opposite directions away from the point where it was pulled. In the case of a string the amplitude of the wave going to the left has the equation: amplitude = some function of (x - ct):

ie: y = f(x -ct)

Where c is a characteristic velocity due to the material of the string, t is time after release of the string and x is distance along the string from the point of release. For the wave going to the right y = f(x + ct).

Suppose the string is pulled with a simple harmonic motion then at the point where it is being pulled: y = Ae^iωt where A is some constant. The wave is symmetrical and maintains its shape as it propagates, at position x and time t the displacement of the string for the left hand wave is Ae^ik(ct-x) and for the right hand wave it is Be^ik(ct+x) where k is ω/c.

The overall equation for the displacement of the string is:

y(x,t) = Ae^ik(ct-x) + Be^ik(ct+x)

If the string were fixed at one end this would become:

y(x,t) = Ae^ik(ct-x)

Another way of looking at waves is to combine the differential equation of the disturbance that creates the wave d²y/dt² = - Ky with the way this varies with distance d²y/dx². This results in a wave equation expressed as a partial differential equation:

d²y/dt² = c² (d²y/dx²)

This is the standard mathematical form of the wave equation in one dimension. It has solutions of the form given above (y(x,t) = Ae^{i(ω t-Kx)}). In 'n' dimensions it has the form:

d²y/dt² = c^{26 2}y

Where 6 ² = δ²/δx² + δ²/δy² + δ²/δz².

The energy delivered to a particular volume of space in a given period of time depends on the square of the amplitude of the wave. This is evident from the way that the displacement, y, is usually accompanied by an increase in potential energy (in the case of strings, water waves, EM waves etc). The derivation of the intensity formula is quite complicated see (8, 10) in the bibliography.

Note 7: Eigenvalues and Eigenvectors: see http://www.users.globalnet.co.uk/~lka/maths.htm)

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