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Special Relativity

Constant-Acceleration Simulator

The world for a traveler accelerating at a constant rate is indistinguishable from the world of someone motionless in a gravitational field of a sufficiently large scale. In a previous page, I discuss how a traveler accelerating at a constant rate is followed by an event horizon. The event horizon is an abstract boundary that separates object than can reach our traveler from objects than can never reach him. The event horizon in an accelerated reference frame appears no different to our traveler than does the event horizon of a black hole to an observer hovering just above it. Two previous pages on this survey path discuss the time dilation and the Doppler shift of radiation in the accelerated world. This page and its simulator present the motion of objects and of light in this world.

Close to our traveler, the motion of objects in this accelerated world is the same as the motion we see at Earth's surface: throw an object, and it will trace a parabola. But motion over a larger scale deviates strikingly from our common experience on Earth. The light from an object takes a finite amount of time to reach us, and the path of this light is not a straight line in the accelerated world, but is instead curved in the direction of the event horizon; this causes the world to appear wrapped upward to our traveler, as though he were at the bottom of a bowl.

The combination of Doppler shift, light travel time, and time dilation gives the motion of objects a peculiar asymmetry. Objects moving away from the event horizon jump way in an instant. This is connected to the rapid passage of time that a traveler sees occurring for Earth as he approaches it. In contrast, objects falling towards the event horizon move more and more slowly; this is associated with the strong redshift of Earth's radiation seen by our traveler as he heads away from Earth.

The simulator on this page shows how objects in free-fall move in an accelerated world. Our traveler is located at the green circle, accelerating at 980 cm s-2, which is Earth's average surface gravitational acceleration. His event horizon is represented by the thick green line at the bottom of the simulator. The simulator image plane is 2 parsecs wide, and the traveler is 0.3 parsecs above the event horizon. Each second of simulator time corresponds to one year of time to our traveler. To start the simulator, select two points on the simulator plane above the event horizon by clicking once at each point with the right mouse button and then press the Start button. The results you see are discussed below the simulator.

To define the object's path of motion, select two points by clicking the left mouse button once at each position. Once a path is defined, run the simulator by pressing the Start button. More details on the simulator are given below. The distance as measured by our observer was redefined in this simulator on May 23, 2005 to be the distance derived from the object's projected width perpendicular to the event horizon; the previous version defined the observed distance as the coordinate distance traveled.

Simulator Notes

Simulator Results

The simulator shows two paths for the motion of an object. The path in black shows the motion in an abstract Cartesian coordinate system, and the path in gray is the path as seen by the traveler. The light path from the object to the observer is given by orange lines, with the bright-orange line showing the path in the Cartesian coordinate system, and the pale-orange line showing the path that light appears to follow. The apparent light path is set by the light's direction of travel at the traveler. The boundary between the white background and pale-green shaded background shows the position of the event horizon as seen by our traveler.

Physics of the Simulator

A distinction must be made between what is seen and what is measured. Our traveler measures positions in terms of a Cartesian grid, but he sees objects at the position set by the curvature of the light path as it reaches our traveler. In the Cartesian grid, the event horizon is a plane, and each plane above and parallel to the event horizon defines a unique rate of acceleration, with the acceleration decreasing with altitude above the event horizon. At the event horizon, the acceleration is infinite. As light from objects propagates to our traveler, it bends towards the event horizon, so that the farthest objects appear lift upward. This causes our traveler to sees himself as in a bowl with a surface set by the event horizon.

Our traveler encounters a problem in defining the distance to an object: distance can be defined in several different ways. He can define a distance based on brightness, he can define a distance based on the projected area of the object, he can define a distance based on the projected width or height of the object, or he can define a distance based on the time it takes for light to travel to the object and back. If our observer knows the object's size, he can deduce a distance by measuring the angle it subtends on the sky. In the simulator, distance is defined as D = A/2 tan ( α/2 ), where A is the known width of the object, and α is the angle subtended by the object parallel to the event horizon.

The accelerated world has many clocks, each of which moves at a different rate. The two most important are the clock with our traveler, and the clock on the object in motion. In the simulator, the clock used as our standard is the clock with our traveler. The time associated with a particular position of the object in motion is the arrival time at our traveler of the light emitted at that that position.

Usage

A path can be defined whenever the simulator is stopped. Two points are required to define a path: a starting point and a point of maximum altitude. The order in which these points are defined determines whether the object is started upwards towards the maximum altitude or downward away from it. If the starting point is defined first, then the object starts by moving towards the maximum; if the start point is defined second, then the object moves away from the maximum. The lower-altitude point is always defined to be the starting point. The starting point and the maximum are each selected with a single click of the left mouse button at the desired point above the event horizon. The simulator does not allow the reader to select a point on or below the event horizon, and it does not allow the reader to select the second point at the same altitude as the first.

The reader can start, stop, and reset the simulator with the selected points with the three buttons below the simulator image plane. The Start button starts the simulator from its current position. The Stop button stops the simulator at its current position. The Reset button stops the simulator and sets the object in the simulator to its initial position. Each of these buttons can be controlled from the keyboard. The button with the focus is selected by pressing the space bar. The Start button can be selected by hitting the Enter or Return key, regardless of whether the button has the focus. The behavior is described in the Applet Usage Guide.

I would appreciate hearing from you if you encounter an error while running the simulator or if you have suggestions for improvement. Send your e-mail to the editor of the website.

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