Let us take a trip to a distant star in a spacecraft that can accelerate at a constant rate. What do we see? If we look at Earth, we can watch its clocks by watching the Doppler shift of radio transmissions; the carrier frequencies of the radio transmitters on Earth are clocks that display Earth's time.
On the outward bound part of the journey, the radio transmissions are redshifted, while on the return part of the journey, they are blueshifted, so as far as passengers on our spaceship are concerned, time is passing more slowly for Earth during the outbound leg, and it is passing more quickly on the inbound leg. So far, this point has nothing to do with special relativity. Where special relativity comes into play is when we add these times over the course of a journey. Under Newtonian mechanics, the time lost on the outbound journey is made up on the inbound journey, so the the amount of time that has passed during the trip is equal for those on Earth and those on the spaceship. But under special relativity, the passage of time for the passengers is much shorter than for those on Earth, so the increased clock rate associated with the blueshift of the radio signal on the inbound portion of the journey dramatically exceeds the slowed clock rate of the redshifted radio signal. The live figure on this page shows the apparent passage of time on Earth with the passage of time on our spacecraft.
This live figure shows the apparent passage of time on Earth during a spacecraft's round trip under constant acceleration. The blue vertical line marks the destination point, and the vertical black line marks the return to Earth. The reader can change both the acceleration, which is given in units of 980 cm s-2, which is the average gravitational acceleration at Earth's surface, and the distance to the destination, which is given in parsecs. The acceleration values can range from 0.25 to 50, and the distances can range from 1 parsec to 1 million parsecs. The destination distance can be set to the distance to one of several prominent astronomical objects by selecting one of the radio buttons to the left. Control of the applet from the keyboard is described in the Applet Control Guide.
Regardless of the acceleration rate, as long as the distance to the destination is large enough to drive time dilation on the spacecraft, the apparent passage of time on Earth will slow almost to a standstill. This can be interpreted in terms of the event horizon following the spaceship. For constant acceleration at the average gravitational acceleration at Earth's surface, we have an event horizon trailing 0.30 parsecs behind the spacecraft. This means that as we travel away from Earth, it drops through the event horizon after one year as measured on Earth. Our communications from Earth slow until only highly redshifted signals from the last moments before Earth's plunge through the event horizon are received. As long as our spacecraft is accelerating away from Earth, our passengers will never see Earth pass through the event horizon; they only see the passage of time for Earth become slower and slower,and the radiation redshift to lower and lower frequencies.
When our spacecraft starts accelerating towards Earth, the event horizon flips to the opposite side of the spacecraft, and Earth is again able to communicate with the spacecraft. The passage of time on Earth gradually returns to a rate equal to the passage of time on the spacecraft, with equality occurring when the spacecraft reaches its destination.
On the return journey, time appears to speed up for Earth. This increase is dramatic, reaching a peak at the half-way point in the journey home. For a journey to a destination 1 kpc away from Earth, 44 minutes of Earth time appears to pass with each second of spacecraft time; with a destination 1 Mpc away from Earth, 25 days of Earth time appears to pass with each second of spacecraft time. In this way our passengers see themselves flung into Earth's future.
On this return journey, once our spacecraft has reached the half-way mark and has swung its engines around to accelerate away from Earth to slow the spacecraft to a stop at Earth, our spacecraft's event horizon displays a surprising property: radio waves from Earth, which is beyond the event horizon, are able to reach our spacecraft. We often think that only radiation from objects above the event horizon is able to reach us, but in fact, this is an overinterpretation of the event horizon property that once an object falls through the event horizon, it can no longer communicate with us. This is what happened when our spacecraft was accelerating away from Earth, which went from being above the event horizon to below the event horizon.
If an object has never previously been above the event horizon, however, it is possible, although not always possible, for its radiation and it to rise through the event horizon. When this happens, from the standpoint of our travelers, the radiation from the object, this case Earth, is blueshifted despite rising out of a potential well, and the object rises away from the event horizon toward the observer. The travelers in our spacecraft see Earth above the event horizon from the time of the event horizon's formation, but those on Earth see Earth pass through the event horizon. Not until the object rises to its maximum altitude above the event horizon does its radiation becomes redshifted. From this point the object falls back to the event horizon. For our travelers, Earth reaches its maximum altitude above the event horizon when they arrive at Earth.