I was thinking about the idea that time equals distance, and why you would be able to "move back" in distance but not in time. Then I realized that in a sense, you can. You just can't do it on a very large scale...
A simple example of how “relative time” implies that time-travel is effectively impossible.
As an example let us consider an event involving 2 particles A and B, moving away from a point or planet P, perhaps after an explosion. Let us consider it from the perspective of A moving relative to P, from which we observe that B is also moving away from P.
The notion of “universal time” suggests the idea that if time were to be “reversed”, then every process involving time would be reversed. A would move back toward P, as would B, and they would do so consistently along a single “time line”.
However, we know that universal time is not real, and that time is in fact relative. The aspect of that which is important in this example is that time according to A is not the same as time according to B.
Suppose that A did in fact reverse direction and began moving back toward P. Suppose that it's possible to consider this in a way where we can't distinguish between the reversal of time between A and P, vs a simple reversal of direction of travel of A relative to P. For all intents and purposes, a simple enough particle A moving back toward a simple enough particle P might be considered time-travel backwards.
However, the time defined by A and P is independent of the time between B and P. What is done to affect the former does not necessarily affect the latter. So while time can be considered going in reverse for A and P, particle B is continuing to move away from P, which we would call “forward in time”.
The same applies to any particles C, D, etc. So suppose we define a clock at P between particles P and C (or any set of particles that we wish). Manipulation of the relative time between A and P would not affect the relative time measured by P and C etc. So while A can be considered moving back in time toward P, that doesn't affect the time measured by the clock at P. A can move back in time and return to a state of P relative to A that is identical to a former state of P relative to A, yet it cannot simply return to a former state of P relative to B, C, etc. As observed by A, P has continued moving forward in time relative to everything else, including its own clocks.
Thus the effect of any sort of time-travel involving A and P will have no noticeable effect in a complex enough system involving multiple particles, or particles with their own internal time-related processes.
In conclusion, I submit that effective time-travel would not involve manipulation of a single variable called “time”; it would require manipulation of countless variables of time defined between all of the particles involved. In other words, time-travel is possible, but only relatively, not universally.
Addendum: Due to relativity of simultaneity, it would be impossible to choose an instant at which to begin a reversal of time, that would be agreed upon by all observers. So even if you could somehow time-reverse all of a complex system, you would lack the notion of a universal time-reversal along a single time-line. While from one perspective it may seem that everything suddenly reversed, from another perspective it may seem that some things reversed while others continued "forward", with various parts beginning to reverse at different time.
Time-travel as it is commonly understood is a notion that is tied to the classical idea of universal time, which people still use to define and understand their world, even though universal time is known to be incorrect.
To get more complicated, we might say that time is related to entropy in this way: When you have any 2 particles split from a single location, you introduce distance between them, which effectively defines a measure of time between them. The greater number of independent locations of particles relative to each other that you have, and the greater the distances between them, the harder it is to get everything back to the way it was previously.
We could say that time is simply the measure of distance between everything.
I've talked about the idea that time and distance are simply perceptual side-effects of the consistency of the underlying physical nature of the universe.
If this is so, then the mysterious imaginary spherical surface that you can describe around any point, which is defined by geometry or perhaps even defines geometry, may be a reflection of entropy. As time (radius) increases, the possible configurations that can fit on the sphere, ie entropy (sphere's surface), increases. This could be why at distances r from you, more "stuff" in the universe can fit around you the bigger r is.
If you have a strictly expanding universe (where no energy reverses outward direction, ie "goes back in time") then its entropy would be proportional to the area of the surface of a sphere of radius t (the age of the universe).
I'm sure that this is related to the holographic principle, which suggests something similar... but I'm not sure if it's a meaningful idea or not.
I'd like to think so, though. :) Time relativity has suggested that time and distance are illusory, and because I don't know of any need for additional dimensions, I've had this hunch-like feeling that the universe can be described completely in 2 dimensions. I've also figured that the universe is exactly like a black hole (and looks like a singularity from the outside), and so I've taken the holographic principle to heart as "something else that also suggests the universe might be 2 dimensional". It seems believable or somehow right -- I have "faith" in it -- yet I don't understand it enough to say why. It would certainly be nice if everything came together.
It certainly feels like a simplification of everything. Working with time relativity is "weird" and confusing, but in the end if it yields other interesting results everything might disappear in a puff of equivalence.
What is the universe? A singularity, which appears to be more due to geometry.
What is geometry? An effect of entropy.
What is entropy? An effect of distance, which is equivalent to time.
What is time? Nothing.
Could it be that the universe came from nothing, and all along, it has remained nothing?
...
Possibly, but until I can make sense of the meaning of that, I'm not going to claim it is so.
The ideas don't stop.
Perhaps then, a two dimensional universe with perceptual side-effects of time and length appears to us consistently as a 3-dimensional thing.
But, since entropy is proportional to the surface area (square of r) around any volume, yet the possible amount of matter in the volume is proportional to the cube of r, the same "magic" that allows us to consistently see a 2D universe in 3D would require that the more matter you pack into a volume, the smaller r must appear.
The result: space-time curvature.
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