The development of Special Relativity along this path has been unconventional, emphasizing the effects of acceleration. This has the advantage of removing all abstraction from the discussion. To compare the passage of time for two travelers in a non-abstract way, we can either compare the clocks of our two travelers by having them start together and finish together at the same points in space and time, or we can compare the clocks by sending electromagnetic signals between the travelers. These methods of comparing time ensure that the act of comparing time is a physical event occurring at a single point in space at a single instance of time.
An abstract method of comparing time, the method that is traditionally invoked when developing the mathematics of special relativity, is to set up a coordinate system that moves with each of our travelers. In doing this, we assume gravity plays no role. As an example, we can set up a coordinate system by spreading a set of transponders throughout space that each communicate with and maintain a fixed distance with its neighbor. Distance can be determined by measuring the light-travel time to a neighbor. The transponders, once they establish their positions within the grid, can then synchronize their clocks to a single master clock by taking the light-travel time into account. Measurements of time are done locally by each transponders, with the measurement of position following from the position of the transponder. For instance, a transponder can measure the time at which a traveler passes by, and because we know the distance of the transponder from us, we know the position of the traveler at that measured time. In this way, we will have a coordinate system that can measure both length and time.
In the jargon of physics, our coordinate system is an inertial reference frame; as a practical matter, such systems can only be created within the Solar System. This coordinate system is a physical realization of the mathematical coordinate system used to describe motion in special relativity. Much of the terminology of special relativity, such as the concepts of time dilation and length contraction arise in the context of inertial reference frames.
The abstractness comes in through the synchronization process. We infer that the time at a distant transponder is equal to our time by taking into account the light travel time. This is very different from the previous two ways of measuring time. What is interesting about this abstract method is that if one compares the measurements from two grids, one of which is moving relative to the other, we come up with very different results. Each coordinate system measures the passage of time in the other grid to be slower, so we have time dilation; each grid measures the lengths of the other grid to be shorter, so we have length contraction; and each grid measures events as occurring in different sequences, so that events that are measured as simultaneous in one coordinate system are measured as occurring at different times in the other coordinate system. So a rocket in motion has a measured length that is shorter than when the rocket is at rest; the clock on that rocket moves more slowly when the rocket is moving than when it is at rest; and if the rocket has a light on its nose and a light on its tail that flash simultaneously as measured when the rocket is at rest, those lights will be measured to flash at different times when the rocket is in motion.
The word measured is important in this process. What we measure and what we see are very different, because what we see involves the propagation of light to our eyes or instruments. When I measure the time at which a light flashes on a rocket, the time is measured by a transponder in my coordinate system that happens to be at the position of the light when it flashes, but when I see a light flash, the time I measure includes the time required for the light to reach me. If my position is such that I see the lights on the nose and tail of a rocket flash simultaneously, then everyone at this position, regardless of how fast they are moving and regardless of the times they measure with their coordinate systems for when the lights flashed, will see the lights flash simultaneously. For instance, a passenger in the middle of the rocket always sees the lights flashing simultaneously, and if a man at rest just happens to be at the same point in space (or a little to the side so that he doesn't get hit by the rocket), he also will see the lights flash simultaneously at that instant. The abstraction of measurement can be different for the two men, but the physical event at that point in space and time must be identical.
This point is often forgotten, but it cannot be neglected. Many of the faux paradoxes of special relativity arise when the effects of light propagation or of the skewing of simultaneity are neglected. This point is discussed in the contect of length contraction on the next page.
If a man travels to a distant star and back, and his twin brother stays on Earth, which twin will be older at the end of the journey? This is the twins problem, which is solved within one of the pages discussing space travel under constant acceleration. The twin that ages less is the twin who accelerates; acceleration produces the physical time dilation.
The twins problem becomes a paradox when one tries to solve this problem by invoking the abstract time dilation measured in inertial coordinate systems. The point is that a man in a rocket measures the clocks in the rest coordinate system moving more slowly than his own, while the man at rest measures the clocks in the rocket moving more slowly. Each man therefore believes that his brother ages more slowly than himself, leading to the paradox that the man on Earth expects his brother on the spaceship to age less than himself, while the man on the spaceship expects his brother on Earth to age less than himself.
The problem with this analysis is that it neglects how acceleration shifts the synchronization of clocks within an inertial reference frame. If a twin accelerates to a high speed and then coasts, the coordinate system he then sets up will measure events that are simultaneous in Earth's coordinate system as occurring first for objects ahead of him in the direction of his destination and last for objects behind him in the direction of Earth. When he accelerates at his destination, so that now he is returning back to Earth at a high speed, the inertial coordinate system he sets up and calibrates for his new speed finds that events that are simultaneous in Earth's coordinate system now occurs first ahead of him, which is towards Earth, and last behind him, which is towards the place he visited. This means that his acceleration has forced him to shift the synchronization of his clocks so that the clocks in his system in the direction of Earth are shifted forward in time. The leap in time comes solely from recalibration, and it more than offsets the slow passage of time he measures for Earth's clocks. So even in the abstract approach to the problem, acceleration is the key in determining who ages more during the trip.