f110
| 10 |
|---|
- *Representation space.
...It was shown that a cylinder is a developable surface. Now take a sheet of paper. It is a flat, Euclidean surface. Its geodesics are straight lines. Draw a few straight lines on this sheet, then crumple it.
...If you could rigidify this crumpled flat surface, you would realize that this operation has not altered the distribution of its geodesics at all—you could once again trace them with your adhesive tape. You have merely played with the way this plane is represented in its three-dimensional embedding space.
A less complicated way to proceed consists in transforming a flat metal sheet into a...corrugated sheet:
Geodesics: unchanged.
...Geometric objects exist independently of the way we represent them, independently of their representation space.
...We are supposed to inhabit a "four-dimensional hypersurface": spacetime. General Relativity consists in trying to construct its geometry as a solution to a field equation, then "reading" this geometry by analyzing the geodesics of the hypersurface. It is clear that the concept of representation space no longer applies here. To do so would require a five-dimensional vision, which we do not possess.
...In practice, we use coordinates corresponding to Euclidean space, projections. Imagine we are seeking a geometric solution suitable for describing spacetime near a massive body and within it. We will assume the system has spherical symmetry. Furthermore, we assume the system is stationary (or quasi-stationary).
...We will then use spherical coordinates (r, q, j). In two dimensions, we will have only two coordinates, and our symmetry will be circular. We will then use the polar coordinate system of the plane:
...This model of the flattened body is a 2D didactic image of a stationary solution that actually exists in General Relativity and was invented by the Austrian Schwarzschild in 1917 as a particular solution to "Einstein's equation":
S = c T
already presented earlier. This solution is clever and subtle. Computationally, it is not simple to construct. This clarification is intended to dispel a myth: that of Einstein, a genius isolated in his time, surrounded by ignoramuses.
...From this solution, it is then shown that there exist, around a spherically symmetric mass, geodesics in planes, lying within planes, and their shape can be calculated: r = f(q). These trajectories (or at least their projection in our mental Euclidean representation space) are "quasi-Keplerian," and Kepler's laws then appear as approximations when the mass creating this geometry (in Newtonian terms, this "force") remains moderate, that is, when the local curvature within this mass remains small.
...This solution is one of the pillars of General Relativity. Although this cannot be conveyed through simple didactic images such as those we offer the reader, it is precisely this solution that allows us to predict and calculate, for example, the advance of Mercury's perihelion. Einstein used this solution to explain this effect, already known, and in doing so, collected all the laurels of what was henceforth called "Einstein's theory." Why did Schwarzschild not exploit his own discovery? Because he insisted on joining the front lines, where he was gassed and died shortly afterward.
...In fact, we are not entirely certain that this famous Einstein equation is truly his. Apparently, it was suggested to him by the great mathematician Hilbert. Einstein also did not enthusiastically welcome the later discovery by the Russian Friedmann, who found the non-stationary solution of the field equation allowing the description of the universe's evolution. The same applies to the work of the young mathematician Kaluza in 1921, whose work, rediscovered, now forms the foundation of superstring theory. These matters are scientifically of little interest and in no way diminish Einstein's value, but they show that sporting spirit does not necessarily correlate with scientific merit.
In the solution developed by Schwarzschild, technically, space is divided into two parts. Inside the body, matter density r is assumed constant. The energy-matter tensor T, which depends on it, is also non-zero. Outside, r and T are zero.
...This composite geometry is thus a solution to two different equations, with and without a source term. Matter density exhibits a discontinuity at the body's surface (the same applies to the Schwarzschild interior and exterior solutions: in this case, the body is a sphere of constant density, which abruptly drops to zero at the surface). However, geodesic continuity can still be ensured through mathematical conditions whose image was shown above (cone frustum-spherical cap matching).
...When mass becomes large and curvature effects are pronounced, trajectories deviate more noticeably from the Keplerian model—for example, near a neutron star. Below is the advance of the perihelion around such a body (around the Sun, Mercury's orbital ellipse advances by 0.15 degrees per century).
...The formula and program enabling the calculation of these trajectories are in fact not complicated at all. We will provide them one day on this site, for the curious.
...At present, we are laying down a few geometric foundations in preparation for later discussions, while reminding the reader that the models indicated are merely illustrative.
../../../bons_commande/bon_global.htm
Table of Contents
article Table of Contents
Science Home Page
Number of views of this page since July 1, 2004: