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Invariance under coordinate transformation.
...Here is a key concept of General Relativity, which is not easy to explain. We said that seeking a "cosmological solution," whether stationary or non-stationary, amounts to constructing a four-dimensional hypersurface that is a "solution to the field equation."
...Take, for example, a sheet-metal object with the topology of a sphere. It's "a sheet-metal sphere." Again, it's easy to imagine that one could deform this surface by heating and cooling it in different places. For instance, by heating at one point and cooling the antipodal region, one could transform this sphere into an egg. An egg is an object with the topology of a sphere, but it is a surface with variable curvature.
...By heating in one place and cooling in another, one creates stresses in the metal. Of course, since this material is conductive, if heating and cooling were stopped, the temperature would eventually homogenize and the object would return to its spherical shape. What matters is that we can create a stationary situation with a non-uniform temperature field. This field generates stresses, which we could concretize as a mathematical object T, called a tensor.
Something describes the geometry of the object. This is called a metric. From this second mathematical object, we can:
- Calculate the geometric tensor S — Calculate the geodesics of the surface.
The geometry of this surface could be computed from an equation analogous to Einstein's equation, of the form:
S = a T
where a is a constant. Knowing the temperature field in the sheet metal a priori — that is, the stress tensor — we could deduce its geometry. The best way to "read" this geometry would be to analyze the system of geodesics. We know the geodesics of the sphere (its "great circles"). The geodesics of an egg are different.
...To describe these geodesics, we will need to define a coordinate system on the surface. For the sphere, we could use the standard azimuthal-zenith system.
...In this particular coordinate system, the geodesics of the sphere correspond to certain equations.
On this sphere, the curves q = constant represent the family of geodesics passing through two points. However, the curves j = constant (parallels) are not geodesics of the surface.
...We could also define an analogous coordinate system and write the equations of the geodesics of the "egg" surface. But we immediately notice an essential fact: the geodesics of the surface are independent of the coordinates chosen to describe them, just as the points on a sphere or on an egg exist independently of the coordinate system used to locate them.
...Similarly, on a plane, we can represent points using Cartesian or polar coordinates. The straight lines of the plane are geodesics.
A straight line can be described in two different coordinate systems:
...It is the same geodesic, described in two completely different ways. The straight lines on the plane exist independently of how we describe them, independently of the choice of coordinates. And we can imagine... an infinite number of them.
...So what is intrinsic? Answer: the length s measured along a straight line (or along any arbitrary curved contour). Between two points M1 and M2 on a surface, the shortest path is a geodesic.
...Likewise, the distance separating two points along a geodesic on either a "sphere" or an "egg" is also a quantity independent of the chosen coordinate system. If we take two points M1 and M2 on a surface and draw the geodesic arc connecting them, the length s measured along this arc will be the same, regardless of the coordinate system used to locate the points.
...The same applies to the four-dimensional hypersurface we call "universe." It has its own system of geodesics, also invariant under coordinate transformations. We do not live in a space (x, y, z, t) with position coordinates and a time coordinate, but in a four-dimensional hypersurface that can be entirely described by its network of geodesics. On these geodesics, there exists a length s that is also invariant under coordinate changes. The points of this hypersurface are no longer points in space, but points in a space-time hypersurface. We call them events. Two distinct events are therefore separated by something called s. But what is this?
It is the proper time.
...A geodesic trajectory in this space-time hypersurface separates two events M1 and M2. All I can say is that if I had used a vehicle to travel this path through space-time, a time interval s would have elapsed on my onboard clock.
A choice of coordinates consists in locating space-time points using spatial coordinates (x, y, z) and a time coordinate t. But since this choice is arbitrary, space and time do not have intrinsic existence. They are merely ways of "reading" the surface, of traversing it. The only constraint is that, depending on the assumptions made, we can only move along geodesics, and on these, the only reliable quantity we can rely on is the "proper time elapsed" s, not the time t, which is merely a chronological marker.
For each choice of coordinate system, we have a different way of reading events, different ways of observing phenomena.
...Physicists therefore sought a formalism independent of the choice of coordinates. This is the essence of the tensorial formalism. We cannot say more on this topic without entering into relatively complex technical details.
The problem of singularities.
On a sphere, the standard angular coordinate choice introduces two polar singularities.
It is impossible to map a sphere without introducing such polar singularities.
...Note that one can map a sphere using a single singularity. We create on the sphere a first family of curves (circles) by slicing it with planes, as shown below:
Then a second family:
Except for this single singularity, there is no problem. If we looked at the sphere from the other side, we would see this:
...Except for the single singularity S, points can be located without difficulty. But the values of the parameters a and b defining this mesh singularity S are... arbitrary...
...Yet a sphere is not geometrically or intrinsically singular. Turn a billiard ball or an egg in any direction — you will not find any singular point.
These singularities were therefore created by the choice of coordinates.
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