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

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:

Ds^{2} = Dx ^{2} + Dy ^{2}

Suppose there is another coordinate system on the surface with two axes: x_{1}, x_{2}, 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 = Dx_{1} dY/dx_{1} + Dx_{2} dY/dx_{2}

DZ = Dx_{1} dZ/dx_{1} + Dx_{2} dZ/dx_{2}

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

DS^{2} = DY ^{2} + DZ^{2}

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

DS^{2} = -DT ^{2} + DX^{2} + DY^{2} + DZ^{2}

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:

Ds^{2} = SS (dX/dx^{i} dX/dx^{k} + dY/dx^{i} dY/dx^{k} + dZ/dx^{i} dZ/dx^{k}) Dx^{i}Dx^{k}

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

and so, using indicial notation:

Ds^{2} = g_{ik}Dx^{i}Dx^{k}

Where

g_{ik} = (dx/dx^{i} dx/dx^{k} + dy/dx^{i} dy/dx^{k} + dz/dx^{i} dz/dx^{k})

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

Ds^{2} = **gDxDx**

becomes:

Dx_{1} Dx_{2} |
Times | a | b | times | Dx_{1} |

c | d | Dx_{2} |

Where a, b, c, d stand for the values of g_{ik}.

Dx_{1}a + Dx_{2}c |
Dx_{1}b + Dx_{2}d |
Times | Dx_{1} |

Dx_{2} |

Which is:

(Dx_{1}a + Dx_{2}c) Dx_{1} + (Dx_{1}b + Dx_{2}d) Dx_{2} = Dx_{1}^{2}a + 2Dx_{1}Dx_{2}(c + b) + Dx_{2}^{2}d

So:

Ds^{2} = Dx_{1}^{2}a + 2Dx_{1}Dx_{2}(c + b) + Dx_{2}^{2}d

Ds^{2} is a **bilinear form** that depends on both Dx_{1} and Dx_{2}. It can be written in matrix notation as:

**Ds ^{2} = 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 ^{T}Ay** (the matrices

**x**and

**y**are different).

If the surface is a Euclidean plane then the values of g_{ik} are:

dY/dx_{1}dY/dx_{1} + dZ/dx_{1}dZ/dx_{1} |
dY/dx_{2}Y/dx_{1} + dZ/dx_{2}dZ/dx_{1} |

dY/dx_{2}Y/dx_{1} + dZ/dx_{2}dZ/dx_{1} |
dY/dx_{2}dY/dx_{2} + dZ/dx_{2}dZ/dx_{2} |

Which become:

1 | 0 |

0 | 1 |

So the matrix **A** is the unit matrix **I** and:

**Ds ^{2} = Dx^{T} I Dx**

and:

Ds^{2} = Dx_{1}^{2} + Dx_{2}^{2}

Which recovers Pythagoras' theorem.

If the surface is derived from Ds^{2} = -DY^{2} + DZ^{2} then the values of g_{ik} are:

-(dY/dx_{1}dY/dx_{1}) + dZ/dx_{1}dZ/dx_{1} |
-(dY/dx_{2}Y/dx_{1}) + dZ/dx_{2}dZ/dx_{1} |

-(dY/dx_{2}Y/dx_{1}) + dZ/dx_{2}dZ/dx_{1} |
-(dY/dx_{2}dY/dx_{2}) + dZ/dx_{2}dZ/dx_{2} |

Which becomes:

-1 | 0 |

0 | 1 |

Which allows the original 'rule' to be recovered i.e.: Ds^{2} = -Dx_{1}^{2} + Dx_{2}^{2}

### The Space-Time IntervalEdit

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:

Ds^{2} = - Dt^{2} + Dx_{1}^{2} + Dx_{2}^{2} + Dx_{3}^{2}

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 TimeEdit

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

Ds^{2} = D t ^{2} + Dx_{1}^{2} + Dx_{2}^{2} + Dx_{3}^{2}

g_{ik} = (dt/dx^{i} dt/dx^{k} + dx/dx^{i} dx/dx^{k} + dy/dx^{i} dy/dx^{k} + dz/dx^{i} dz/dx^{k})

For a flat surface dt/dx^{0} = dx/dx^{1} = dy/dx^{2} = dz/dx^{3} = 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 Ds^{2} = Dt^{2} + Dx_{1}^{2} + Dx_{2}^{2} + Dx_{3}^{2} becomes Ds^{2} = - DT^{2} + Dx_{1}^{2} + Dx_{2}^{2} + Dx_{3}^{2}

This form of time is not supported in General Relativity Theory

#### Real TimeEdit

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

DT = Dx_{1} dT/dx_{1} + Dx_{2} dT/dx_{2} + Dx_{3} dT/dx_{3} + Dx_{4} dT/dx_{4}

etc.

But when they are combined the formula Ds^{2} = - DT^{2} + Dx_{1}^{2} + Dx_{2}^{2} + Dx_{3}^{2} 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 g_{00} is given by -1 times (dt/dx^{0}) ^{2}.

#### Mixed Real and Imaginary TimeEdit

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:

Ds^{2} = Dt^{2} + Dx_{1}^{2} + Dx_{2}^{2} + Dx_{3}^{2}

where t equals (kt), k being a constant that is yet to be determined. In flat space-time g_{00} is given by (dt/dx^{0}) ^{2}. But t is imaginary so g_{00} 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 TensorEdit

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

s^{2} = g_{mn} x^{n}x^{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 g_{00} 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:

s^{2} = x_{1}^{2} + x_{2}^{2} + x_{3}^{2} - x_{4}^{2}

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

T X_{1} X_{2} X_{3} |
-1 | 0 | 0 | 0 | times | T |

0 | 1 | 0 | 0 | X_{1} |
||

0 | 0 | 1 | 0 | X_{2} |
||

0 | 0 | 0 | 1 | X_{3} |

(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 cofficients that were described above.

Evaluating the first matrix multiplication this becomes:

-T x_{1} x_{2} x_{3} |
times | T |

X_{1} |
||

X_{2} |
||

X_{3} |

Which resolves to: s^{2} = x_{1}^{2} + x_{2}^{2} + x_{3}^{2} - t^{2}

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.

ds^{2} = dx_{1}^{2} + dx_{2}^{2} + dx_{3}^{2} - dt^{2}

Or, equivalently:

ds^{2} = dt^{2} - dx_{1}^{2} - dx_{2}^{2} - dx_{3}^{2}

## Full analysis of the constants in Gauss' analysisEdit

DY = Dx_{1} dY/dx_{1} + Dx_{2} dY/dx_{2}

DY^{2} = (Dx_{1} dY/dx_{1} + Dx_{2} dY/dx_{2})^{2}

DY^{2} = Dx_{1} dY/dx_{1} * Dx_{1} dY/dx_{1} + Dx_{1} dY/dx_{1} * Dx_{2} dY/dx_{2} + Dx_{1} dY/dx_{1} * Dx_{2} dY/dx_{2} + Dx_{2} dY/dx_{2} * Dx_{2} dY/dx_{2}

DY^{2} = Dx_{1}Dx_{1} dY/dx_{1} dY/dx_{1} + Dx_{1}Dx_{2} dY/dx_{1} dY/dx_{2} + Dx_{1}Dx_{2} dY/dx_{1} dY/dx_{2} + Dx_{2}Dx_{2} dY/dx_{2} dY/dx_{2}

And

DZ = Dx_{1} dZ/dx_{1} + Dx_{2} dZ/dx_{2}

DZ^{2} = (Dx_{1} dZ/dx_{1} + Dx_{2} dZ/dx_{2})^{2}

DZ^{2} = Dx_{1} dZ/dx_{1} * Dx_{1} dZ/dx_{1} + Dx_{1} dZ/dx_{1} * Dx_{2} dZ/dx_{2} + Dx_{1} dZ/dx_{1} * Dx_{2} dZ/dx_{2} + Dx_{2} dZ/dx_{2} * Dx_{2} dZ/dx_{2}

DZ^{2} = Dx_{1}Dx_{1} dZ/dx_{1} dZ/dx_{1} + Dx_{1}Dx_{2} dZ/dx_{1} dZ/dx_{2} + Dx_{1}Dx_{2} dZ/dx_{1} dZ/dx_{2} + Dx_{2}Dx_{2} dZ/dx_{2} dZ/dx_{2}

Therefore:

DY^{2} + DZ^{2} =

(dY/dx_{1}dY/dx_{1} + dZ/dx_{1}dZ/dx_{1})Dx_{1} Dx_{1}

+ (dY/dx_{2}Y/dx_{1} + dZ/dx_{2}dZ/dx_{1})Dx_{2}Dx_{1}

+ (dY/dx_{1}dY/dx_{2} + dZ/dx_{1}dZ/dx_{2})Dx_{1} Dx_{2}

+ (dY/dx_{2}dY/dx_{2} + dZ/dx_{2}dZ/dx_{2})Dx_{2} Dx_{2}

For a flat surface

dY= dx_{2} and dZ= dx_{1} so dY/dx_{2} = 1 and dZ/dx_{1} = 1 also dY/dx_{1} = 0 and dZ/dx_{2} = 0.

Ds^{2} = DY^{2} + DZ^{2} = (0 + 1)Dx_{1} Dx_{1}+ (0 + 0)Dx_{2}Dx_{1} + (0 + 0)Dx_{1} Dx_{2}+ (1 + 0)Dx_{2} Dx_{2}

so Ds^{2} = DY^{2} + DZ^{2} = Dx_{1}^{2} + Dx_{2}^{2}

Which recovers Pythagoras' theorem.

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

Suppose

Ds^{2} = -DY^{2} + DZ^{2}

So, as before:

DY = Dx_{1} dY/dx_{1} + Dx_{2} dY/dx_{2}

DY^{2} = Dx_{1}Dx_{1} dY/dx_{1} dY/dx_{1} + Dx_{1}Dx_{2} dY/dx_{1} dY/dx_{2} + Dx_{1}Dx_{2} dY/dx_{1} dY/dx_{2} + Dx_{2}Dx_{2} dY/dx_{2} dY/dx_{2}

DZ = Dx_{1} dZ/dx_{1} + Dx_{2} dZ/dx_{2}

DZ^{2} = Dx_{1}Dx_{1} dZ/dx_{1} dZ/dx_{1} + Dx_{1}Dx_{2} dZ/dx_{1} dZ/dx_{2} + Dx_{1}Dx_{2} dZ/dx_{1} dZ/dx_{2} + Dx_{2}Dx_{2} dZ/dx_{2} dZ/dx_{2}

So:

-DY^{2} + DZ^{2} =

(-(dY/dx_{1}dY/dx_{1}) + dZ/dx_{1}dZ/dx_{1})Dx_{1} Dx_{1}

+ (-(dY/dx_{2}Y/dx) + dZ/dx_{2}dZ/dx_{1})Dx_{2}Dx_{1}

+ (-(dY/dx_{1}dY/dx_{2}) - dZ/dx_{1}dZ/dx_{2})Dx_{1} Dx_{2}

+ (-(dY/dx_{2}dY/dx_{2}) - dZ/dx_{2}dZ/dx_{2})Dx_{2} Dx_{2}