Special Relativity

Special Relativity

Constant Acceleration

We live our life subject to the constant acceleration of gravity. A central tenant of general relativity is that this acceleration is no different than the acceleration experienced in a rocket. We can therefore develop some understanding of gravity by examining constant acceleration in special relativity. Three effects in particular are important to consider for an object undergoing constant acceleration: the dilation of time, Doppler shift and bending of light, and the existence of an event horizon. These effects are identical to those that appear just above the event horizon of a black hole, so a traveler creates his own artificial black hole when he accelerates at a constant rate.

We can imagine the world of the passengers on a spacecraft that is accelerating at a constant rate. The travelers feel as though they are being pulled to the back of the spacecraft by a pseudo-gravitational field. To them, this artificial gravity pushes them in the direction opposite to the direction of acceleration of the spacecraft. To the passengers, the structure of the spacecraft is unchanging in time, and provides them with a frame of reference for judging distance.

Time Dilation

Among the most striking results of special relativity is time dilation; by accelerating, I can slow the passage of time for myself relative to some one who is not accelerating. In fact, if I could travel in a rocket that can accelerate at one Earth gravity in a round-trip to our nearest neighboring galaxy, the Large Magellanic Cloud, which is 50 kpc away, I would find at the end of the trip that I had aged by 47 years, while the Earth will have aged by 326,000 years. In a round trip to the Andromeda Galaxy (M31), which is 730 kpc away, I will have aged by 57 years, and Earth will have aged by 4.8 million years. This striking effect sound like pure science fiction, and it is the foundation of several science fiction stories, such as the Charlton Heston version of The Planet of the Apes, but it is a scientifically-verified effect: unstable particles accelerated to nearly the speed of light decay at a slower rate than particles at rest, as expected in special relativity.

The amount of time dilation experienced within our hypothetical spacecraft depends on our position within the spacecraft. In special relativity, one finds that when a spacecraft maintains its structural integrity, so that the distance from tail to nose is constant for passengers in the spacecraft, it experiences different rates of acceleration throughout its structure. The nose of the spacecraft accelerates at a lower rate than the tail of the spacecraft; the amount of acceleration depends solely on position along the direction of acceleration. The acceleration increases as one moves back. Because time dilation is determined by the rate of acceleration, a passenger in the spacecraft finds that a clock placed in the nose of the spacecraft moves faster than a clock placed in the tail.

Doppler Shift and Light Bending

The frequency of light emitted by a light source is a type of clock. This implies that the frequency one sees for light emitted in one part of the spacecraft will shift in frequency as it propagates to another part of the spacecraft. Because time passes more slowly in the back of the spacecraft than in the front, the frequency of light emitted from the back will fall as the light travels to the front of the spacecraft. For our passengers, the light is red-shifted. Conversely, the frequency of light emitted from the nose of the craft rises as the light travels to the back; the light is blue shifted.

Our passengers can interpret this in terms of a pseudo-gravitational potential. They could say that the light from the tail appear red in the nose, because the light lost energy as it overcame the spacecraft's artificial gravitational field. Conversely, they could say that the light emitted at the nose appears blue at the tail because the light gained energy as it fell deeper into the artificial gravitational potential well. This point of view is mathematically valid, and cannot be distinguished from the time-dilation interpretation of the light's Doppler shift.

For our travelers, light no longer travels in a straight line; it moves in a path that bends in the direction of the pseudo-gravitational field. The travelers can interpret this as from the light increasing its momentum in the direction of the pseudo-gravitational field as it falls into the potential well; the light's momentum perpendicular to the pseudo-gravitational field is constant, so the direction of the momentum vector rotates to align itself with the pseudo-gravitational field. Of course, it is the rocket, not the light, that is accelerating, and the apparent bending is a consequence of the rocket acquiring more velocity, which means that it more rapidly sweeps past light moving to some extent perpendicular to the rocket's motion.

The Event Horizon

One of the more interesting effects that our passengers will notice is the event horizon following is spacecraft. From the passengers' point of view, this occurs because the acceleration required to keep a constant distance from the back of the spacecraft goes to infinity at a finite distance. If our passengers drag a cord behind the spaceship, they would find that the cord would be pulled apart just above the event horizon, because the cord does not have the strength to produce the acceleration required to maintain a constant distance from the spacecraft. Time at the end of this cord would slow until time passed imperceptibly, and light emitted from the cord end would be so severely redshifted that it would appear as low intensity radio waves. To the passengers, objects dropped from the spacecraft appear to come to a stop as they approach the event horizon. The event horizon is therefore a place where time stops and objects go dim; it is a wall of darkness following the spacecraft. Acceleration equal to the average gravitational acceleration at Earth's surface places this wall about 0.3 parsecs behind our travelers.

From the standpoint of an observer at rest, the event horizon is simply a mathematical abstraction. The observer sees nothing unusual at the event horizon of the spacecraft. To our observer, the event horizon marks the point where the accelerating spacecraft can outrun light. At the event horizon. light requires an infinite amount of time to reach the spacecraft; above the event horizon, light reaches the spacecraft in a finite amount of time, and below the event horizon, light closes to a fixed distance behind the spacecraft.

An object dropped from the spacecraft is equivalent to an observer at rest. The event horizon passes by him in a finite amount of time. If the object is emitting light, only the light emitted before the event horizon sweeps over the object can reach the spacecraft.