Consciousness Studies/The Philosophical Problem/Appendix

Gauss's Analysis of Curved Surfaces - The Origin of the Metric Tensor

It became apparent at the start of the nineteenth century that issues such as Euclid's parallel postulate required the development of a new type of geometry that could deal with curved surfaces and real and imaginary planes. At the foundation of this approach is Gauss's analysis of curved surfaces which allows us to work with a variety of coordinate systems and displacements on any type of surface.

Suppose there is a line on a surface. 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 + Dy² Suppose there is another coordinate system on the surface with two axes: x₁, x₂, how can the length of the line be expressed in terms of these coordinates? Gauss tackled this problem and his analysis is quite straightforward for two coordinate axes:

Figure 1:

It is possible to use elementary differential geometry to describe displacements along the plane in terms of displacements on the curved surfaces:

DY = Dx₁dY/dx₁ + Dx₂dY/dx₂

DZ = Dx₁dZ/dx₁ + Dx₂dZ/dx₂

The displacement of a short line is then assumed to be given by a formula, called a metric, such as Pythagoras' theorem

DS² = DY ² + DZ²

Or some other metric such as the metric of a 4D Minkowskian space:

DS² = -DT ² + DX² + DY² + DZ²

This type of analysis can be extended to any number of dimensions. It is then possible to express the short length, Ds, in terms of the coordinates. The full algebraic analysis is given at the end of this appendix. In 3D the expression for the length is:

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:

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:

Dx1 Dx2	Times	a	b	times	Dx1
		c	d		Dx2

 

Where a, b, c, d stand for the values of g_ik.

Dx1a + Dx2c	Dx1b + Dx2d	Times	Dx1
			Dx2

Which is:

(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:

Ds² = Dx^T A Dx

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:

dY/dx1dY/dx1 + dZ/dx1dZ/dx1	dY/dx2Y/dx1 + dZ/dx2dZ/dx1
dY/dx2Y/dx1 + dZ/dx2dZ/dx1	dY/dx2dY/dx2 + dZ/dx2dZ/dx2

Which become:

1	0
0	1

So the matrix A is the unit matrix I and:

Ds² = Dx^T I Dxand:

Ds² = Dx₁² + Dx₂²

Which recovers Pythagoras' theorem.

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

-(dY/dx1dY/dx1) + dZ/dx1dZ/dx1	-(dY/dx2Y/dx1) + dZ/dx2dZ/dx1
-(dY/dx2Y/dx1) + dZ/dx2dZ/dx1	-(dY/dx2dY/dx2) + dZ/dx2dZ/dx2

Which becomes:

-1	0
0	1

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

The Space-Time Interval

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. It could originate from an assumption that time is imaginary, that time is real and the metric has a negative sign for time, or that time is mixed real and imaginary with a Pythagorean metric.

Imaginary Time

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' =

1	0	0	0
0	1	0	0
0	0	1	0
0	0	0	1

Which means that the time interval must be imaginary if the assumption of relativity is to be supported i.e.: 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

Real Time

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

DT = Dx₁dT/dx₁ + Dx₂dT/dx₂ + Dx₃dT/dx₃ + Dx₄dT/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' =

-1	0	0	0
0	1	0	0
0	0	1	0
0	0	0	1

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

Mixed Real and Imaginary Time

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 surface were 'imaginary' and that on the plane were real (or vice versa) 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' =

-1	0	0	0
0	1	0	0
0	0	1	0
0	0	0	1

The Modern Formulation of the Metric Tensor

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. It is assumed that g₀₀ is opposite in sign to the other constants on the principle diagonal (i.e.: real or mixed real and imaginary time are assumed). With this assumption the expression becomes:

s² = x₁² + x₂² + x₃² - x₄²

The expansion is shown below. In matrix notation this is:

T X1 X2 X3	-1	0	0	0	times	T
	0	1	0	0		X1
	0	0	1	0		X2
	0	0	0	1		X3

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

Evaluating the first matrix multiplication this becomes:

-T x1 x2 x3	times	T
		X1
		X2
		X3

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 (!).

The metric is normally expressed in differential form so that it can be used with a curved space-time and with displacements that are not measured relative to the origin.

ds² = dx₁² + dx₂² + dx₃² - dt²

Or, equivalently:

ds² = dt² - dx₁² - dx₂² - dx₃²

Full analysis of the constants in Gauss' analysis

DY = Dx₁dY/dx₁ + Dx₂dY/dx₂

DY² = (Dx₁dY/dx₁ + Dx₂dY/dx₂)²

DY² = Dx₁dY/dx₁ * Dx₁dY/dx₁ + Dx₁dY/dx₁ * Dx₂dY/dx₂ + Dx₁dY/dx₁ * Dx₂dY/dx₂ + Dx₂dY/dx₂ * Dx₂dY/dx₂

DY² = Dx₁Dx₁dY/dx₁ dY/dx₁ + Dx₁Dx₂dY/dx₁dY/dx₂ + Dx₁Dx₂dY/dx₁dY/dx₂ + Dx₂Dx₂dY/dx₂dY/dx₂

And

DZ = Dx₁dZ/dx₁ + Dx₂dZ/dx₂

DZ² = (Dx₁dZ/dx₁ + Dx₂dZ/dx₂)²

DZ² = Dx₁dZ/dx₁ * Dx₁dZ/dx₁ + Dx₁dZ/dx₁ * Dx₂dZ/dx₂ + Dx₁dZ/dx₁ * Dx₂dZ/dx₂ + Dx₂dZ/dx₂ * Dx₂dZ/dx₂

DZ² = Dx₁Dx₁dZ/dx₁ dZ/dx₁ + Dx₁Dx₂dZ/dx₁dZ/dx₂ + Dx₁Dx₂dZ/dx₁dZ/dx₂ + Dx₂Dx₂dZ/dx₂dZ/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

So, as before:

DY = Dx₁dY/dx₁ + Dx₂dY/dx₂

DY² = Dx₁Dx₁dY/dx₁ dY/dx₁ + Dx₁Dx₂dY/dx₁dY/dx₂ + Dx₁Dx₂dY/dx₁dY/dx₂ + Dx₂Dx₂dY/dx₂dY/dx₂

DZ = Dx₁dZ/dx₁ + Dx₂dZ/dx₂

DZ² = Dx₁Dx₁dZ/dx₁ dZ/dx₁ + Dx₁Dx₂dZ/dx₁dZ/dx₂ + Dx₁Dx₂dZ/dx₁dZ/dx₂ + Dx₂Dx₂dZ/dx₂dZ/dx₂

So:

-DY² + DZ² =

(-(dY/dx₁dY/dx₁) + dZ/dx₁dZ/dx₁)Dx₁ Dx₁

+ (-(dY/dx₂Y/dx) + dZ/dx₂dZ/dx₁)Dx₂Dx₁

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

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