I couldn’t read from the prose how the difference between lengths A and B get established.
I think I’ve corrected this; I failed to note that A is closer to the planet’s surface than B, which was more obvious in the original version where I had a picture. Or do you mean I don’t justify it in terms of the new abstraction I’ve established?
I too have wondered about how solid the “no boosting to infinity” restrictions are. If one takes a unit de-sitter Space that should cover all the directions. From one point of intuition at first glance it would feel that all the directions should be “locally adjacent” ie one should be able to travel on all points of a sphere by sliding the angle the radius is off-set from the origin.
I’m not 100% certain what you mean by “no boosting to infinity” here, so it’s hard to interpret the rest of the statement.
When drawn as a in a space-time diagram one gets an upwards opening parabola and a downwards opening parabola. But then it is puzzling on why they seem to be disconnected regions. I think I have something that speaks about the same are as your discussion 45 degree and 90 degree angles. The parabolas have asymptose at lightspeed ie 45 degree. However if one focuses on positive distance then the region from 45 to 135 doesn’t have anything, the elsewhere region where 90 degrees would lie is never visited. If one wants to include minus distances then there are going to be sidewyas parabola like strctures but the stark magnitude flip from positive to negative doesn’t seem like those would actually be in smooth contact. That is by having a constant radius and letting the angle be undetermined one gets the analog of the unit-sphere, the unit de-sitter space, and this does NOT include the “side-areas”. But it could be interesting if there was a smooth way to slide from the upregion to the downregion. Whether this is a cheating or allowed move is a bit beyond me.
I think we’re talking about the space “inside” a singularity; if I’m mistaken, let me know.
So the 45 and 90 degree explanation is built on an assumption of orthogonality between time and the three spacial dimensions that I’m not sure is entirely accurate. Consider a mass; now consider a point a meter away from that mass. I think the “natural” spacial dimensions here are: Distance (the line directly into/away from the mass), and then four cardinal directions relative to Distance. Is Distance actually meaningfully distinct from Time? If we take simultaneity seriously, a change in Distance is equivalent, in a specific sense, to a change in Time.
In your response to another post, you write “If you are thinking in euclidean terms it might seems that x,y,z,t can be relabeled into each other. However with relativity there is an “odd signature” going either (+---) or (-+++).”
So time is inverted, relative to the other dimensions. More, given that Time and Distance are in a specific sense equivalent, I’d suggest Time is not actually orthogonal to Distance, but parallel, and “pointed” in the opposite direction. So we have something like this:
<--------------------------------------->
+ Distance -
<--------------------------------------->
- Time +
So, if we are pointed in a positive direction in distance, we’re pointed in a negative direction in time; that is, moving away from an object is moving towards its history (because time is emitted, not kept). Considering an arrow pointed into a singularity, we expect it to have the following vector: (-+) - moving closer to the object, and also towards its future. Rotate it 90 degrees, so its orientation in distance becomes its orientation in time, and we get (+-). (Which okay, looks more like a 360 degree rotation, but I will insist it’s a 90 degree rotation for difficult-to-explain reasons.) That is, after the ninety degree motion, we’re now pointed away from the singularity, and also into its history.
So I think the apparently unconnected parabolas of the upregion and downregion are actually the same region.
Updated with a link to the prior post, thanks!