Text reference: Section 29.1-3.

We'll define the symbol gamma as  g = (1 - v²/c²)^-½.  This definition is common usage which, for unknown reasons, isn't used in the Walker text.  With this definition, the formulas of special relativity are written compactly as Dt = Dt_og  and DL = DL_o/g.

Velocities of 0.6c and 0.8c show up frequently in relativity problems.  That's because you can calculate the value of gamma in your head for these velocities. For illustration, though, we'll do a calculation step-by-step for v = 0.6c.

 g = (1 - v²/c²)^-½ = [1 - (0.6c)^²/c^²]^-½ = [1 - 0.36]^-½ = (0.64)^-½ = (0.8)^-1 = 5/4

A common usage when dealing with astronomical distances is to use the unit of the light-year (ly) and to express velocities as fractions of c.  This simplifies calculations.  For example, the time for a rocket to travel a distance of 8 ly at a velocity of 0.4c is t = d/v = (8 ly)/(0.4c) = 20 y.  There's no need to convert ly to meters and c to meters/second.

The formula v = d/t applies to Special Relativity as well as to everyday relativity (where we ignore relativistic effects). When applying it to relativistic situations, however, you have to be very careful that the distance and time that you substitute are measured from the same reference frame.  The terminology proper length and proper time has been developed to help determine whether a particular time (or length) plays the role of Dt_o (or DL_o) in the time dilation and length contraction formulas.  You should review the definitions of proper time and length in the text. In order to avoid confusion and keep you cognizant of the reference frames in which distances and times are measured, always subscript distances and times with the acronym representing the observer (for example, E for Earth-based observer and R for rocket-based observer).

Note that in the formula v = d/t, distance and time cannot both be proper. That's because the proper length is always measured from a different reference frame than the proper time.

While the situation described below is fanciful, the physics of relativity has been verified experimentally in real-world situations.

Here's the problem situation.  (Click here to open a diagram in a new window.)  The distance to the star, Alpha Centauri, as measured by an Earth-bound observer (EO), is 4.0 ly. In an experiment to test the theory of relativity, a rocket ship travels to Alpha Centauri from the Earth at a speed of V_RE = 0.8c relative to the Earth. (Recall that the notation V_RE means speed of the rocket with respect to the Earth.)

1. Why must gamma always be greater than 1?

2. Calculate the value of gamma for v/c = 0.8.  Express the result as a reduced ratio of two integers.

3. The distance L_E indicated in the diagram is determined by EO.  Is L_E the proper length?  Explain using the textbook definition of the term.

4. Let t_E represent the time that EO measures for the RO's trip from the Earth to Alpha Centauri.  Is t_E the proper time?  Explain using the textbook definition of the term.

5. Calculate the value of t_E in units of years.  (The time dilation formula won't help for this.)

6. Now let's examine the situation from RO's point of view.  Click here for a diagram of that situation.  RO interprets the situation as this:  The Earth moves away from the rocket at a speed of 0.8c while AC moves toward the rocket at a speed of 0.8c. Let t_R represent the time between the departure of the Earth and the arrival of AC, as measured by RO.  Use the textbook definition to explain why t_R is the proper time.

7. Use the time dilation formula to calculate t_R in units of years.

8. RO says that the trip takes 2 years less than EO says the trip takes.  How about distances. EO says the rocket travels 4 ly. How far does RO say that the Earth and AC travel? Use the length contraction formula to calculate L_R to the nearest 0.1 light year.

9. Let's review what has been found. EO says that it takes 5 years for the rocket to travel 4 light years at a speed of 0.8c. RO, says that it takes 3 years for the Earth (and AC) to travel a distance of 2.4 light years.  A check on these results is to verify that V_ER, the speed of the earth with respect to RO, is 0.8c. Substitute previously determined values into V_ER = L_R/t_R. Note that all quantities are measured from the same reference frame.