Patterner said:
The planet causes a spacetime curvature, as planets tend to do. That curvature has a certain size and shape, which is the result of the planet's composition, size, and whatever else affects the size and shape of spacetime curvatures. If we could inject die into this
I'm not sure I understand your question, but as far as your image goes, you might like
http://www.eftaylor.com/pub/chapter2.pdf, an excerpt from Taylor's book, "Exploring Black Holes".
Taylor said:
Nothing is more distressing on first contact with the idea of curved space-
time than the fear that every simple means of measurement has lost its
power in this unfamiliar context. One thinks of oneself as confronted with
the task of measuring the shape of a gigantic and fantastically sculptured
iceberg as one stands with a meterstick in a tossing rowboat on the surface
of a heaving ocean.
Were it the rowboat itself whose shape were to be measured, the proce-
dure would be simple enough (Figure 1). Draw it up on shore, turn it
upside down, and lightly drive in nails at strategic points here and there
on the surface. The measurement of distances from nail to nail would
record and reveal the shape of the surface. Using only the table of these
distances between each nail and other nearby nails, someone else can
reconstruct the shape of the rowboat. The precision of reproduction can be
made arbitrarily great by making the number of nails arbitrarily large.
In space-time, the nails are replaced by events. I'm not sure if you're familiar with events, they aren't terribly complicated. Events have a location and a time of occurence. For example, if one was writing a police report about a crime, one might give the location (a street address, say), and a time and date of the crime. The street address is the location of the event, the crime, that tells where it occurred in space. Specifying the location of the event is not sufficient, however, we also need to know the time at which it occured. Events are one of the most fundamental elements of space-time, replacing the "nails" in Taylor's rowboat, which only have a location in space, but do not have a time of occurence.
The tricky part here is understanding what replaces the notion of the "distance between nails" on the rowboat. What is the equivalent "distantce" between events in space-time? The needed concept here is called the Lorentz interval. The Lorentz interval says that given two events in space-time, there is a single number that is simliar to a "distance", a number that is independent of the observer. A key part of why we need the Lorentz interval is a feature of special relativity called "the relativity of simultaneity". This feature says that whether or not two events occur "at the same time" depends on the observer, specifically the observer's state of motion, the observer's velocity. The end result is that we do not have a separate "spatial distance" and a "time distance" between two events that is the same for all observers. The spatial distance can change due to Lorentz contraction, the time "distance" can change due to the relativity of simultaneity. The Lorentz interval, however, does not change. It's the same for everyone, and it makes talking about what happens much, much easier.
I would guess offhand from what you write that you are not already familiar with these concepts (the Lorentz interval and the relativity of simultaneity). They both occurs in special relativity, which is much simpler mathematically than General relativity, though people still stumble over some of the needed concepts.
Myl recommendation to everyone is to understand special relativity first, before tackling General relativity. But I'd settle for people realizing that they should at least study special relativity in addition to general relativity.
