Section 27

# Exact Formulation Of General Relativity

March 8, 2022

Our provisional formulation was=

All bodies of reference K, K', etc., are equivalent for the description of natural phenomena (formulation of the general laws of nature), whatever may be their state of motion

This now cannot be maintained because it required rigid reference-bodies as followed in Special Relativity (SR).

In Section 27, we replaced it with Gauss co-ordinates. This creates a new statement for General Relativity (GR):

All Gaussian co-ordinate systems are essentially equivalent for the formulation of the general laws of nature.

According to SR, the equations for the general laws of nature pass over into equations of the same form when we replace the space-time variables `x, y, z, t`, of a Galileian reference-body `K` by the space-time variables `x', y', z', t'`, of a new reference-body `K'` using the Lorentz transformation.

According to GR, on the other hand, by application of arbitrary substitutions of the Gauss variables `x1, x2, x3, x4,` the equations must pass over into equations of the same form. This is because every transformation (not only the Lorentz transformation) corresponds to the transition of one Gauss co-ordinate system into another.

If we want to adhere to our old-school 3D view of things, then we can characterise the development which is being undergone by GR as= Special relativity works where no gravitational fields exist.

In this way, a Galileian reference-body serves as body of reference, i.e. a rigid body the state of motion of which is so chosen that the Galileian law of the uniform rectilinear motion of “isolated” material points holds relatively to it.

Certain considerations suggest that we should refer the same Galileian domains to non-Galileian reference-bodies also. A gravitational field of a special kind is then present with respect to these bodies (cf. Sections 20 and 23).

In gravitational fields, there are no such things as rigid bodies with Euclidean properties.

Thus, the fictitious rigid body of reference is useless in General Relativity.

The motion of clocks is also influenced by gravitational fields so that a physical definition of time which is made directly with the aid of clocks has by no means the same degree of plausibility as in SR.

This is why non-rigid reference-bodies are used which are as a whole not only moving in any way whatsoever, but which also suffer alterations in form ad lib. during their motion.

Clocks, for which the law of motion is of any kind, however irregular, serve for the definition of time.

Each of these clocks is fixed at a point on the non-rigid reference-body. These clocks are the “readings” which are observed simultaneously on adjacent clocks in space differ from each other by an indefinitely small amount.

This non-rigid reference-body, called a “reference-mollusk,” is the main equivalent to a Gaussian 4D co-ordinate system chosen arbitrarily.

The “mollusk” gets a comprehensibleness compared to the Gauss co-ordinate system because of the (really unjustified) formal retention of the separate existence of the space co-ordinates as opposed to the time co-ordinate.

• Every point on the mollusk is treated as a space-point
• Every material point which is at rest relatively to it is also at rest, so long as the mollusk is considered as reference-body.

GR requires that all these mollusks can be used as reference-bodies with equal right and equal success in the formulation of the general laws of nature.

The laws themselves must be independent of the choice of mollusk. The great power of General Relativity is in the comprehensive limitation imposed on the laws of nature.