For someone who does not have a strong mathematical or physics background, you sure do know who to ask ambitious questions. Let's see what we can say about them.
Guy From Alberta said:
I have a number of questions arising; hope it is OK to just spout them out.
You(r questions) are one of the reasons I visit the forum. Please ask all the questions you want, or I'll start to get bored.
Guy From Alberta said:
In this topic, some of us have touched on how gravity does indeed “curve the spatial geometry of the universe.” Can we further define “spatial geometry?”
Gravity curves space-time, as well as space in itself. Geometry is what gets curved. Imagin a volume. This volume is filled with points. Each point has an identity. They form a set. There is also some notion of points being close to each other and points being far away. This notion (called topology, I think, I'm still learning this stuff) is
not sufficient to support a notion of curvature. So the points are endowed with a more specific notion of, not only close or far, but how close or how far. This notion is called geometry. (This notion does not have to arise from the same closeness relationships as the topology, as I understand it, but I might not understand it.) It was inspired by Euclid, but later on, people like Gauss and Riemann came along and decided that Euclid put unnecessary restrictions on his geometry.
You can think of geometry as the collection of distances between all the points together with those points between which the distance is defined.
Spatial geometry implies that these distances are always positive definite for any two distinct points.
Guy From Alberta said:
A “frame of reference;” I presume as meaning a collection of condition, axis, or assumption;
Not exactly. There is a more exact definition in the context of relativity. This is one of the things to which I think Einstein (the man himself) gives a decent portrayal. He gives a working definition of a frame of reference is a set of intersecting planes. The (perpendicular) distance that would be measured to these planes gives the (spatial) coordinates of any point in this frame. If you imagine that two sets of such intersecting planes can exist without mutual interference, and then further imagine that they move wrt each other, then these two sets of intersecting planes give two distinct frames of reference.
A frame of reference is something to which you refer to give meaning to your expression. In relativity, this means a set of intersecting planes that allow you to label points with a set of numbers.
Guy From Alberta said:
From what I see so far, time is not really the fourth dimension of space; but of “spacetime.” It is within the sphere of general or special relativity where we see time, plus three dimensional space being treated together as a single, four dimensional “manifold,” called “space time.”
Exactly
Guy From Alberta said:
spacetime cannot be viewd as a fixed background;
This is a GR notion. In SR, spacetime is fixed for sure. Since you really want to talk about BHs, then you are correct to say that the geometry is dynamical. Whether spacetime itself, on which the geometry is endowed, is dynamical is more of a philosophical debate (as I understand it). Actually, this is another item for which I rather liked the treatment of Einstein. (See
Relativity, 5th ed., pp. 135-57, "Appendix V: Relativity and the Problem of Space", or the first chapter in
The Meaning of Relativity, I forget what it's called.)
Guy From Alberta said:
but rather, a networking and developing of certain evolving relationships.
Very good. When you get right down to it, physics can only go so far as to make epistemic judgements.
Guy From Alberta said:
What exactly are a) the dimensions of space?
b) the dimensions of spacetime?
The dimensions of space are the degrees of freedom that a point particle may utilize. A point particle can go up, right, or out. Any other behavior of a strict point particle is a combination of scaled versions of these behaviors (including negative and vanishing multiples). The dimensions of spacetime are not quite so understandable. They
do not indicate degrees of freedom for a point particle. The reason why spacetime is treated like a 4-D space is that, there are results of a certain type of transformation (Lorentz transformation), which are very much like what rotations would be in 4-D space.
Imagine just a 1-D space, like a line. A particle in this space has 1 degree of freedom. It can go, let's say, to the right. (Going to the left is just a negative scaled multiple thereof.) Now, let us consider the behavior wrt time. This behavior can most readily be considered from a 2-D type construct called the 1+1 D spacetime for this particle. If we further posit onto this structure the rules of SR, then there is a characteristice wedge in this 2-D construct. I will hereafter refer to this wedge as the "lightcone." The interior of the lightcone is allowed, however, the exterior is forbidden. This lightcone follows the particle through the 2-D construct. It would look kind of like a tiny car with its headlights shining out front with a spread beam (though this is
not why it is called a
lightcone). The car is allowed to drive wherever the beam shines. 2 stipulations on the car: 1) Regardless of the direction of the car, the beam always shines in the same direction, 2) the car always moves forward at the same speed. Clearly, this car does not have the freedom to enjoy every point in this 2-D construct, so the time dimension doesn't exactly add a degree of freedom, at least, not like a spatial degree of freedom, not a direction that a particle can
choose.
Guy From Alberta said:
What does it mean to say “a spacetime interval between two events?” How is an “interval” best defined, and what is an example of this kind of “event,” in a case like this?
The only way I really know how to field this one is with some math. An event is a "point" in spacetime. Return again to the 1+1 D spacetime (forget about the car for the moment). Imagine 2 distinct points in this spacetime (any two will do). Imagine the straight line segment connecting them. This line segment represents the interval. In Euclidean geometry, this interval would be quantified by the Pathagorean theorem. In spacetime geometry (which is
not Euclidean), the Pathagorean theorem is generalized to a metric.
Imagine that the time axis is verticle and the space axis is horizontal. Then, qualitatively, the more verticle the interval, the greater it is. Intervals that are not verticle are shorter, to the limit that the interval coincides with the edge of the lightcone (as in the car example, the edge of the head light beam). If the interval is even more horizontal than the edge of the lightcone, then it is negative. This is clearly not possible using the Pathagroean theorem. (Note: you may have heard of a treatment that invokes an
ict axis, and that claims the validity of the Pathagroean theorem therefrom. This is OK if all you ever want is a cursory understanding. The idea of non-Euclidean geometry is mathematically more sophisticated and will take you further. I have never tried to understand GR in terms of the
ict notion, but I don't think that the
ict notion can even be carried that far, so, even for our purposes, it is useless.)
Guy From Alberta said:
How are “coordinate time,” and “proper time,” affected by/related to said “Intervals,” and “events?”
There is a concept called a "worldline." This is the collection of all events that represent the existence of a particle through time. Returning once again to the 1+1 D spacetime, any particle in the 1-D space will actually be a line (or, in general, a curve) in the 1+1 D space time. This line, which can be considered the posisition of the particle at all points in time, is the worldline. According to the rules of relativity, two distinct events on the worldline of a particle are separated by an interval that is
always more horizontal than the edge of the wedge. Therefore this interval is positive. The proper time (between these two events) is one way of quantifying the length of this interval. The proper time can be directly related to the coordinate time interval (Δt) by the time dilation (Lorentz transformation).
The proper time usually implies an interval that does not "violate" the light cone. For intervals that do extend outside the light cone (more horizontal than the edge of the lightcone), the value changes sign (becomes negative), and it is usually denoted as the proper distance instead of the proper time. Fundamentally, proper time and proper distance are the exact same thing (with the occasional exception of a sign convention), and it is just a matter of context that determines the particular label.
Guy From Alberta said:
Do black holes teach us anything about coordinate time, or proper time?
Yes. For instance, "inside" a BH, the distance from the center becomes another time like axis. That is, once a particle enters a BH, it loses another degree of freedom, because it now not only must move forward in time, but its distance from the center of the BH must consistently decrease. At the center, the worldline of the particle terminates.
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