So here it goes...
CTC's or Closed Timelike Curves:
If CTCs exist, their existence would seem to imply at least the theoretical possibility of time travel backwards in time
, raising the spectre of the grandfather paradox, although the Novikov self-consistency principle seems to show that such paradoxes could be avoided.
In "simple" examples of spacetime metrics the light cone is directed forward in time. This corresponds to the common case that an object cannot be in two places at once, or alternately that it cannot move instantly to another location. In these spacetimes, the worldlines of physical objects are, by definition, timelike. However this orientation is only true of "locally flat" spacetimes. In curved spacetimes the light cone will be "tilted"
along the spacetime's geodesic.
For instance, while moving in the vicinity of a star, the star's gravity will "pull" on the object
, affecting its worldline, so its possible future positions lie closer to the star. This appears as a slightly tilted lightcone on the corresponding spacetime diagram. An object in free fall in this circumstance continues to move along its local axis, but to an external observer it appears it is accelerating in space as well—a common situation if the object is in orbit, for instance.
A closed timelike curve can be created if a series of such light cones are set up so as to loop back on themselves, so it would be possible for an object to move around this loop and return to the same place and time that it started.
An object in such an orbit would repeatedly return to the same point in spacetime if it stays in free fall.
Returning to the original spacetime location would be only one possibility; the object's future light cone would include spacetime points both forwards and backwards in time, and so it should be possible for the object to engage in time travel under these conditions.
Okay then...I've highlighted in Blue
the important things to recognize about CTC's.
Now I know for fact that I have mentioned this in this thread. But I failed to recognize its significance.
1) The n-body problem
In physics, the n-body
problem is the problem of predicting the individual motions of a group of celestial objects interacting with each other gravitationally.
Solving this problem has been motivated by the desire to understand the motions of the Sun, Moon, planets and the visible stars. In the 20th century, understanding the dynamics of globular cluster star systems became an important n-body problem.
The classical physical problem can be informally stated as:
Given the quasi-steady orbital properties (instantaneous position, velocity and time) of a group of celestial bodies, predict their interactive forces; and consequently, predict their true orbital motions for all future times.
orbital positions of a planet's orbit, he was able to produce an equation by straightforward analytical geometry, to predict a planet's motion; i.e., to give its orbital properties: position, orbital diameter, period and orbital velocity. Having done so, he and others soon discovered over the course of a few years, those equations of motion did not predict some orbits very well or even correctly. Newton realized it was because gravitational interactive forces amongst all the planets was affecting all their orbits.
The above discovery goes right to the heart of the matter as to what exactly the n-body problem is physically
: as Newton realized, it is not sufficient to just specify the initial position and velocity, or three orbital positions either, to determine a planet's true orbit: the gravitational interactive forces have to be known too.
Thus came the awareness and rise of the n-body "problem" in the early 17th century. These gravitational attractive forces do conform to Newton's Laws of Motion and to his Law of Universal Gravitation, but the many multiple ( n-body) interactions have historically made any exact solution intractable. Ironically, this conformity led to the wrong approach.
Newton concluded via his third law of motion that
"according to this Law all bodies must attract each other." The existence of gravitational interactive forces, is key.
The problem of finding the general solution of the n-body
problem was considered very important and challenging.
In case the problem could not be solved, any other important contribution to classical mechanics would then be considered to be prizeworthy. The prize was awarded to Poincaré, even though he did not solve the original problem. (The first version of his contribution even contained a serious error). The version finally printed contained many important ideas which led to the development of chaos theory.
So why is this important? It brings you to:
2) The Two-body problem
Any discussion of planetary interactive forces has always started historically with the two-body problem.
In classical mechanics, the two-body problem is to determine the motion of two point particles that interact only with each other. Common examples include a satellite orbiting a planet, a planet orbiting a star, two stars orbiting each other (a binary star), and a classical electron orbiting an atomic nucleus (although to solve the electron/nucleus 2-body system correctly a quantum mechanical approach must be used).
The two-body problem can be re-formulated as two one-body problems
, a trivial one and one that involves solving for the motion of one particle in an external potential. Since many one-body problems can be solved exactly, the corresponding two-body problem can also be solved. By contrast, the three-body problem
(and, more generally, the n-body problem for n ≥ 3) cannot be solved in terms of first integrals, except in special cases.
Which brings us to the:
3) The Three-body Problem
In physics and classical mechanics, the three-body problem
is the problem of taking an initial set of data that specifies the positions, masses, and velocities of three bodies for some particular point in time and then determining the motions of the three bodies,
in accordance with Newton's laws of motion and of universal gravitation, which are the laws of classical mechanics. The three-body problem is a special case of the n-body problem. Unlike two-body problems, there is no general closed-form solution for every condition, and numerical methods are needed to solve these problems.
Historically, the first specific three-body problem to receive extended study was the one involving the Moon, the Earth, and the Sun. In an extended modern sense, a three-body problem is a class of problems in classical or quantum mechanics that models the motion of three particles.
In 1911, United States scientist William Duncan MacMillan found one special solution. In 1961, Russian mathematician Sitnikov improved this solution.
The Sitnikov problem is a restricted version of the three-body
problem named after Russian mathematician Kirill Alexandrovitch Sitnikov that attempts to describe the movement of three celestial bodies due to their mutual gravitational attraction. A special case of the Sitnikov problem was first discovered by the American scientist William Duncan MacMillan in 1911, but the problem as it currently stands wasn't discovered until 1961 by Sitnikov.
Thus...Configuration of the Sitnikov problem
Look familiar yet? Let me help you...
But that is not all...more to come with this equation and CTC's