The Space–time Continuum Of General Relativity Is Not A Euclidean ContinuumMarch 10, 2022
We used space-time coordinates which allowed a simple and direct physical interpretation as 4D Cartesian co-ordinates in Section 26. This was possible through the constant speed of light [of Special Relativity].
But according to Section 22, General Relativity (GR) cannot retain this law because, to GR, the speed of light must always depend on the coordinates when a gravitational field is present.
Section 23 explained that the presence of a gravitational field invalidates the definition of the co-ordinates and the time of SR.
According to GR, the space-time continuum cannot be Euclidean like the marble slab (with a hot center and cool sides) as an example of a 2D continuum. It was there impossible to construct a Cartesian co-ordinate system from equal rods. Likewise here, it is impossible to build up a system of rigid bodies and clocks to indicate position and time directly. This was our difficulty in Section 23.
General Relatvity Creates Arbitrary Spacetimes
But Sections 25 and 26 show us how to go around this by using the arbitrary Gauss co-ordinates in a 4D space-time continuum.
We assign to every point or event of the continuum 4 numbers,
x1, x2, x3, x4 as co-ordinates which have definite but arbitrary numbers. This arrangement does not even need for
x1, x2, x3 as “space” co-ordinates and
x4 as a “time” coordinate.
Why do we need to assign the co-ordinates
x1, x2, x3, x4 to an event if these co-ordinates have no significance?
If a moving mass had only a momentary existence without time, then it would be described in spacetime by a single system of values
x1, x2, x3, x4. Its permanent existence would be denoted by an infinitely large number of such values [new coordinate systems created one after the other]. These co-ordinate values are so close together as to give continuity. Thus the moving mass becomes a 1D line in the 4D continuum.
In the same way, any such lines in our continuum corresponds to many points in [the path of] motion. The encounters of those points are the only real things. In my math, such an encounter is expressed as the motions of those points having a particular system of co-ordinate values,
x1, x2, x3, x4, in common.
When we were describing the motion of a material point relative to a body of reference, we only stated that the encounters of this point with particular points of the reference-body. We can also determine the corresponding values of the time by the observation of encounters of the body with clocks, in conjunction with the observation of the encounter of the hands of clocks with particular points on the dials.
It is just the same in the case of space-measurements by means of measuring-rods, as a little consideration will show.
The following statements hold generally:
“Every physical description resolves itself into a number of statements, each of which refers to the spacetime coincidence of two events A and B”
In terms of Gaussian co-ordinates, every such statement is expressed by the agreement of their four co-ordinates
x1, x2, x3, x4. Thus in reality, the description of the time-space continuum by means of Gauss co-ordinates completely replaces the description with the aid of a body of reference, without suffering from the defects of the latter mode of description; it is not tied down to the Euclidean character of the continuum which has to be represented.