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vm6502q / OpenRelativity

An open source framework to add the effects of traveling at relativistic speeds to visualizations or games
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Lemaître coordinates instead of Schwarzschild #4

Closed WrathfulSpatula closed 5 years ago

WrathfulSpatula commented 5 years ago

The "Schwarzschild" "ConformalMap" object should use comoving coordinates, due to Lemaître. (https://en.wikipedia.org/wiki/Lema%C3%AEtre_coordinates) I should make some general points about the "ConformalMap" class, here.

First of all, I'm not sure if the name "ConformalMap" is even appropriate, in retrospect. I had some motivation when I named it, but, informally, what I called a "conformal map" could also semi-informally be called "the metric with mixed raised/lowered indices."

The idea is, by Einstein equivalence principle, limiting to a free-falling point, space is assumed to always appear locally flat. Any test point, in a locally flat neighborhood, is embedded in some geometry which may be curved, intended to be represented by the "ConformalMap" object. Additionally, player acceleration gives the appearance of gravitational fields, basically per Rindler coordinates, as is expected by the "twins paradox." The point here is that, starting from locally flat space, we want to combine the geometric effects of applied background curvature with the apparent gravitational field due to player acceleration. My thinking was that we could do this by taking displacement "vectors" (as specifically opposed to "covectors") from a locally flat neighborhood through successively encompassing metric transformations, by successive "left-multiplication" of "the (local) metric with mixed indices." (Pardon, if my language or reasoning is semi-introductory.)

If a RelativisticObject is in free-fall in a curved space, its "viw" ("velocity-in-world") parameter will be constant, by convention. (Say that, for example, it's 0.) The object appears to move, though, with viw = 0, in Schwarzschild metric. For this to work, the coordinates used probably always have to be comoving, and Lemaître coordinates specifically are the comoving coordinates we want for a Schwarzschild black hole. To get the correct comoving behavior, with 0 (or constant) viw, we transform the coordinate time interval the RelativisticObject passes in a FixedUpdate with the "metric with mixed indices," AKA "ConformalMap," to get an apparent displacement in "world coordinates," which will mix displacement in both apparent time and space, telling us how the object (co-)"moves".

The point is, the motion is relative, but we need to pick appropriately convenient and informative coordinates. Following the very well-chosen coordinate conventions of the original OpenRelativity project, I think it is in the same spirit, to require that "viw" is always constant in coordinates that comove with any applied background geometry, that "viw" is constant in free-fall.

WrathfulSpatula commented 5 years ago

After I write this and play with coordinate transformations, something might have just become apparent to me: this "metric with mixed indices" mouthful I learned from certain introductory textbooks might simply be AKA "the Jacobian," AKA the coordinate transformation "matrix."

WrathfulSpatula commented 5 years ago

Point being, the misnomer is to think of it as any kind of "metric"-like object at all. It is not an intrinsic measure of distance. Because of the mixing of the indices, it exactly takes a "vector" to a "vector." It's a coordinate transformation, which is exactly what we wanted in the first place.