Nickelodeon said:
So in practicality, we have two time variables, the significant one due to an illusion and the other real and due to relative acceleration (presumably this includes gravitational effects)?
Proper time is a useful term. It's a property of a
timelike curve (also called a world line) through spacetime. A timelike curve represents the possible path of an object through spacetime. If you think of points in spacetime as
events, the proper time between two events on the curve is the spacetime analogy of
arc length in space. It represents the amount of time experienced by an object "travelling" through spacetime along the curve from one of those events to the other.
A
geodesic is the most direct path between two points; in spacetime between two events. Examples of geodesics in space: a straight line in Euclidean space, a segment of a
great circle on a sphere (such as the equator). A material object on a spacetime geodesic is in
freefall.
With no gravity, we have what's called
flat spacetime. This is the geometry that special relativity describes. In flat spacetime, you can have curves that are straight lines. These are the geodesics of flat spacetime. In space, a geodesic is the shortest path between two points; it's the curve with
least arc length. But in spacetime, between a given pair of events, if it's possible for a material object to travel between them, the geodesic path between those events is the one with
most proper time. In flat spacetime, straight world lines represent motion with constant velocity: clocks record most time along these. Curved world lines represent accelerated motion: clocks record less time along these, less and less the more acceleration there is. Velocity is relative, since it depends on an arbitrary choice of coordinate system, but acceleration, in this sense, is absolute, since it doesn't depend on the choice of coordinates. There is a special, "favoured" state of acceleration, a natural choice of what to call zero acceleration (inertial motion, constant velocity).
Gravity curves spacetime itself. It causes geodesics to be curved too. This makes the definitions of velocity and acceleration more complicated, and there are people here far better qualified than me to talk about this. In curved spacetime, something on a geodesic is still in freefall. There's still something special about geodesic motion. Locally (meaning in the limit, as you examine smaller and smaller regions of spacetime) an object in freefall experiences less time between two events than an object taking a non-geodesic (i.e. a more roundabout) path between those same events. But on a large scale, the curvature of spacetime itself also contributes to the amount of proper time along a world line, so this has to be taken into account too, as well as the amount of deviation from freefall.
Nickelodeon said:
Could it be possible that if you accelerate time slows down but when you decelerate it speeds up again, ie. reversing the process?
I think this might be open to more than one interpretation. One way to answer might be that deceleration is acceleration by another name. Taken in isolation, there's no natural way to say which of two accelerations is "speeding up" and which "slowing down"; you'd have to first make an arbitrary choice of velocity as your baseline, zero velocity. It's not as if there's a special absolute velocity (with respect to time dilation) that you move further and further away from the more you accelerate, and then have to decelerate to get back down to. All velocities, except c, are physically equivalent.
