(Very draught, please tell me if there are any errors).

Suppose there is a line in a space. The length of this line can be expressed in terms of a coordinate system. A short length of line Ds in a two dimensional space may be expressed in terms of Pythagoras' theorem as:

Ds² = Dx ² + Dy ²

Suppose there is another coordinate system with two axes: x₁, x₂, how can the length of the line be expressed in terms of these new coordinates? Gauss tackled this problem and his analysis is quite straightforward for two coordinate axes:

Therefore:

DY = Dx₁ dY/dx₁ + Dx₂ dY/dx₂

DY² = (Dx₁ dY/dx₁ + Dx₂ dY/dx₂)²

DY² = Dx₁ dY/dx₁ * Dx₁ dY/dx₁ + Dx₁ dY/dx₁ * Dx₂ dY/dx₂ + Dx₁ dY/dx₁ * Dx₂ dY/dx₂ + Dx₂ dY/dx₂ * Dx₂ dY/dx₂

DY² = Dx₁Dx₁ dY/dx₁ dY/dx₁ + Dx₁Dx₂ dY/dx₁ dY/dx₂ + Dx₁Dx₂ dY/dx₁ dY/dx₂ + Dx₂Dx₂ dY/dx₂ dY/dx₂

And

DZ = Dx₁ dZ/dx₁ + Dx₂ dZ/dx₂

DZ² = (Dx₁ dZ/dx₁ + Dx₂ dZ/dx₂)²

DZ² = Dx₁ dZ/dx₁ * Dx₁ dZ/dx₁ + Dx₁ dZ/dx₁ * Dx₂ dZ/dx₂ + Dx₁ dZ/dx₁ * Dx₂ dZ/dx₂ + Dx₂ dZ/dx₂ * Dx₂ dZ/dx₂

DZ² = Dx₁Dx₁ dZ/dx₁ dZ/dx₁ + Dx₁Dx₂ dZ/dx₁ dZ/dx₂ + Dx₁Dx₂ dZ/dx₁ dZ/dx₂ + Dx₂Dx₂ dZ/dx₂ dZ/dx₂

Therefore:

DY² + DZ² =

(dY/dx₁dY/dx₁ + dZ/dx₁dZ/dx₁)Dx₁ Dx₁

+ (dY/dx₂Y/dx₁ + dZ/dx₂dZ/dx₁)Dx₂Dx₁

+ (dY/dx₁dY/dx₂ + dZ/dx₁dZ/dx₂)Dx₁ Dx₂

+ (dY/dx₂dY/dx₂ + dZ/dx₂dZ/dx₂)Dx₂ Dx₂

For a flat surface dY= dx₂ and dZ= dx₁ so dY/dx₂ = 1 and dZ/dx₁ = 1 also dY/dx₁ = 0 and dZ/dx₂ = 0.

Ds² = DY² + DZ² =

(0 + 1)Dx₁ Dx₁

+ (0 + 0)Dx₂Dx₁

+ (0 + 0)Dx₁ Dx₂

+ (1 + 0)Dx₂ Dx₂

so Ds² = DY² + DZ² = Dx₁² + Dx₂²

Which recovers Pythagoras' theorem. However in the most general case the small intervals may not be related by Pythagoras' theorem:

Suppose

Ds² = -DY² + DZ²

This type of analysis can be extended to any number of dimensions. In 3D:

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, using indicial notation (see maths.htm):

Ds² = g_ikDxⁱDx^k

Where

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

If the coordinates are not merged then Ds is dependent on both sets of coordinates. In matrix notation:

Ds² = gDxDx

becomes:

(Dx₁a + Dx₂c) Dx₁ + (Dx₁b + Dx₂d) Dx₂ = Dx₁²a + 2Dx₁Dx₂(c + b) + Dx₂²d

So:

Ds² = Dx₁²a + 2Dx₁Dx₂(c + b) + Dx₂²d

Ds² is a bilinear form that depends on both Dx₁ and Dx₂. It can be written in matrix notation as:

 Where A is the matrix containing the values in g_ik. This is a special case of the bilinear form known as the quadratic form because the same matrix (Dx) appears twice; in the generalised bilinear form B = x^TAy (the matrices x and y are different).

If the surface is a Euclidean plane then the values of g_ik are:

Which become:

So the matrix A is the unit matrix I and:

and:

Ds² = Dx₁² + Dx₂²

Which recovers Pythagoras' theorem.

If the surface is derived from Ds² = -DY² + DZ² then the values of g_ik are:

Which becomes:

Which allows the original 'rule' to be recovered ie: Ds² = -Dx₁² + Dx₂²

The fundamental assumption of modern relativity theory is that the space-time interval is invariant. The space-time interval is given by the following equation rather than Pythagoras' theorem:

Ds² = - Dt² + Dx₁² + Dx₂² + Dx₃²

The origin of the negative sign in front of Dt is of considerable interest.

Suppose that Pythagoras theorem applied to the space-time interval and:

Ds² = D t ² + Dx₁² + Dx₂² + Dx₃²

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

For a flat surface dt/dx⁰ = dx/dx¹ = dy/dx² = dz/dx³ = 1 and all other coefficients are zero therefore:

'g' =

Which means that the time interval must be imaginary if the assumption of relativity is to be supported ie: Dt ² = (DT Ö -1) ²

So that Ds² = Dt² + Dx₁² + Dx₂² + Dx₃² becomes Ds² = - DT² + Dx₁² + Dx₂² + Dx₃²

This form of time is not supported in General Relativity Theory

If real time is used then the expressions for each displacement along each coordinate axis remain the same eg:

DT = Dx₁ dT/dx₁ + Dx₂ dT/dx₂ + Dx₃ dT/dx₃ + Dx₄ dT/dx₄

etc.

But when they are combined the formula Ds² = - DT² + Dx₁² + Dx₂² + Dx₃² is used instead of Pythagoras' theorem (see above for a fully worked example in 2D). This results in the following metric tensor:

'g' =

Where g₀₀ is given by -1 times (dt/dx⁰) ².

There is a third possibility that is not generally discussed. The 'plane' in figure 1 is a plane in the observer's coordinate system and the surface has its own coordinate system. If the time coordinate on the plane were 'imaginary' and that on the plane were real then using Pythagoras' theorem:

Ds² = Dt² + Dx₁² + Dx₂² + Dx₃²

where t equals (kt), k being a constant that is yet to be determined. In flat space-time g₀₀ is given by (dt/dx⁰) ². But t is imaginary so g₀₀ equals -1. This then gives exactly the same metric tensor as the assumption of real time.

'g' =

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

s² = g_mn xⁿx^m

where the values of s, x represent tiny displacements in each of the four coordinate axes 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:

(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:

Which resolves to: s² = x₁² + x₂² + x₃² - t²

Which is the 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 (!).

ds² = dx₁² + dx₂² + dx₃² - dt²

Or, equivalently:

ds² = dt² - dx₁² - dx₂² - dx₃²

(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.)