Run a rocket engine without stopping, at a steady ( m/s²). Everyone aboard feels the same weight as on the ground the whole time.
The speed measured by ground clocks is after year, after , and only after . However long the acceleration continues, the speed approaches without passing it.
The point in the figure stops at , , and years on the ground clock in turn. The closer the speed is to , the less it grows in each year.
Starting from rest at and accelerating, the world line is the following hyperbola. This is called hyperbolic motion.
The asymptotes are the two lines , the world lines of light through the origin. The hyperbola approaches them without touching, and light chasing the ship from the origin never reaches it.
At , is about light years, and this length sets how sharply the hyperbola bends. The lower half of the figure is the part where, with the same acceleration, the ship came in from far away while slowing down.
Aboard the ship the acceleration stays at . The given by is called the rapidity, and with the time on the ship's clock, grows in proportion to time.
The speed is , which never exceeds however large becomes. The point in the figure stops at , and years on the ship's clock, where is , and and is , and .
The time on the ground clocks is , so while years pass on the ship, about years pass on the ground.
As long as the ship keeps accelerating, light emitted above and to the left of the asymptote never reaches it. This asymptote is called the Rindler horizon.
Light emitted on the near side of the horizon catches up with the ship; light emitted beyond it keeps its distance and never arrives. The ship keeps drawing closer to the speed of light, so the light never catches it.
If the engine is cut, the ship moves on at constant velocity, and light from beyond the horizon eventually arrives too. The horizon is not a wall at some place but a boundary set by the condition of accelerating forever.
In the ship's frame, the horizon is always a distance behind, about light year at . In that events beyond it can never reach the ship, it shares its character with the event horizon of a black hole.
Let the traveller turn around gently, changing velocity from to with a steady acceleration. The figure draws the world line and the line of simultaneity at each moment.
During the turn the slope of the line of simultaneity swings from rising to falling, and the point where it meets the Earth's world line moves quickly upward. The Earth time that seemed to jump in an instantaneous turnaround becomes this rapid motion when the turn is gentle.
In a frame accelerating at , a clock a distance ahead in the direction of the acceleration runs faster by the factor . The turning traveller accelerates toward the Earth, so the distant Earth clock runs faster by this factor.
In a gravitational field too, clocks higher up run faster. That clocks behave the same way in an accelerating frame and in gravity is the starting point of the idea called the equivalence principle.