Time Dilation Help (page 2)

Introduction

The relative location of an observer in space affects the relative readings of clocks located at different points. Similarly, relative motion in space affects the apparent rate at which time “flows.” Isaac Newton hypothesized that time flows in an absolute way and that it constitutes a fundamental constant in the universe. Einstein showed that this is not the case; it is the speed of light, not time, that is constant. In order to understand why relativistic time dilation occurs based on Einstein’s hypothesis, let’s conduct a “mind experiment.”

A Laser Clock

Suppose that we have a space ship equipped with a laser/sensor on one wall and a mirror on the opposite wall (Fig. 20-2). Imagine that the laser/sensor and the mirror are positioned so that the light ray from the laser must travel perpendicular to the axis of the ship, perpendicular to its walls, and (once we get it moving) perpendicular to its direction of motion. The laser and mirror are adjusted so that they are separated by 3.00 m. Because the speed of light in air is approximately 3.00 × 10 ⁸ m/s, it takes 1.00 × 10 ⁻⁸ s, or 10.0 nanoseconds (10.0 ns), for the light ray to get across the ship from the laser to the mirror and another 10.0 ns for the ray to return to the sensor. The ray therefore requires 20.0 ns to make one round trip from the laser/sensor to the mirror and back again.

Fig. 20-2 . A space ship equipped with a laser clock. This is what an observer in the ship always sees.

Our laser emits pulses of extremely brief duration, far shorter than the time required for the beam to get across the ship. We might even suppose that the beam emits just a few photons in each burst! We measure the time increment using an extremely sophisticated oscilloscope so that we can observe the pulses going out and coming back and measure the time lag between them. This is a special clock; its timekeeping ability is based on the speed of light, which Einstein proposed is constant no matter from what point of view it is observed. There is no better way to keep time.

Clock Stationary

Suppose that we start up the ship’s engines and get moving. We accelerate with the eventual goal of reaching nearly the speed of light. Suppose that we manage to accelerate to a sizable fraction of the speed of light, and then we shut off the engines so that we are coasting through space. You ask, “Relative to what are we moving?” This, as we shall see, is an important question! For now, suppose that we measure speed with respect to the Earth.

We measure the time it takes for the laser to go across the ship and back again. We are riding along with the laser, the mirror, and all the luxuries of a small spacecraft. We find that the time lag is still exactly the same as it was when the ship was not moving relative to Earth; the oscilloscope still shows a delay of 20.0 ns. This follows directly from Einstein’s axiom. The speed of light has not changed because it cannot. The distance between the laser and the mirror has not changed either. Therefore, the round trip takes the same length of time as it did before we got the ship moving.

If we accelerate so that the ship is going 60 percent, then 70 percent, and ultimately 99 percent of the speed of light, the time lag will always be 20.0 ns as measured from a reference frame , or point of view, inside the ship.

At this point, let’s add another axiom to Einstein’s: In free space, light beams always follow the shortest possible distance between two points. Normally, this is a straight line. You ask, “How can the shortest path between two points in space be anything other than a straight line?” This is another good question. We’ll deal with it later in this chapter. For now, note that light beams appear to follow straight lines through free space as long as the observer is not accelerating relative to the light source.

Time Dilation Practice Problems

Problem 1

Why don’t we notice relativistic time dilation on short trips by car, train, or airplane? When we land, the clocks are still synchronized (except for time-zone differences in some cases). Why isn’t the traveling clock behind the earthbased ones?

Solution 1

Problem 2

What speed is necessary to produce a time dilation factor of k = 2.00?

Solution 2

Practice problems of these concepts can be found at: Relativity Theory Practice Test