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6. Cathy has a “yardstick" that she thinks is 1 unit of distance long, which she holds in what she thinks of as a stationary position in the direction that the spaceship is moving. How long does the same yardstick appear to Bob as it moves along?

6. Cathy has a “yardstick" that she thinks is 1 unit of distance long, which she holds in what she thinks of as a stationary position in the direction that the spaceship is moving. How long does the same yardstick appear to Bob as it moves along?

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Bob and Cathy are in spaceships that pass by each other. Bob gives space and time B-coordinates and Cathy gives space and time C-coordinates : Suppose we change from C to B by multiplication by the Lorentz matrix - [; :] a B where a? – 32 = 1. So for any event e, if E = [e]g are its B-coordinates and if E' = [e]c are its C-coordinates, then PE' = E. 1. Verify that a = 2/3/3 - 1.16, ß = v3/3 z 0.58 give a Lorentz matrix. For Bob in the B-coordinate system, how fast does Cathy's spaceship appear to be passing him? For the next five problems, use the numbers from problem 1. 2. Consider the “unit square" Sc in Cathy's C coordinate system whose vertices are at Draw the parallelogram in the B system corresponding to Sc. Now draw the parallelogram in the C system corresponding to the unit square Sg in Bob's B coordinate system. 3. Suppose we consider the event whose C-coordinates are given by E' = . What are its B-coordinates? Does Cathy think this event happens is in the future or the past? Does Bob think this event happens in the future or the past? Do the same thing with the event whose C-coordinates are E' = 3 4. The event whose C-coordinates are E' represents the position of a photon of light that started at the origin and moved in a positive direction 1 unit of time (in C-coordinates). Verify that E = PE' also represents the position of a point of light after a certain amount of time in the B-coordinate system. Do the same for E' = 5. Cathy considers herself to be stationary with a' = 0. She looks down at her watch after 3/2 units of time in her coordinates. How many units of time does Bob think (in the B-coordinate system) have passed by at the event of her looking? More generally, how does Cathy's watch appear to Bob? 6. Cathy has a “yardstick" that she thinks is 1 unit of distance long, which she holds in what she thinks of as a stationary position in the direction that the spaceship is moving. How long does the same yardstick appear to Bob as it moves along?
Transcribed Image Text:Bob and Cathy are in spaceships that pass by each other. Bob gives space and time B-coordinates and Cathy gives space and time C-coordinates : Suppose we change from C to B by multiplication by the Lorentz matrix - [; :] a B where a? – 32 = 1. So for any event e, if E = [e]g are its B-coordinates and if E' = [e]c are its C-coordinates, then PE' = E. 1. Verify that a = 2/3/3 - 1.16, ß = v3/3 z 0.58 give a Lorentz matrix. For Bob in the B-coordinate system, how fast does Cathy's spaceship appear to be passing him? For the next five problems, use the numbers from problem 1. 2. Consider the “unit square" Sc in Cathy's C coordinate system whose vertices are at Draw the parallelogram in the B system corresponding to Sc. Now draw the parallelogram in the C system corresponding to the unit square Sg in Bob's B coordinate system. 3. Suppose we consider the event whose C-coordinates are given by E' = . What are its B-coordinates? Does Cathy think this event happens is in the future or the past? Does Bob think this event happens in the future or the past? Do the same thing with the event whose C-coordinates are E' = 3 4. The event whose C-coordinates are E' represents the position of a photon of light that started at the origin and moved in a positive direction 1 unit of time (in C-coordinates). Verify that E = PE' also represents the position of a point of light after a certain amount of time in the B-coordinate system. Do the same for E' = 5. Cathy considers herself to be stationary with a' = 0. She looks down at her watch after 3/2 units of time in her coordinates. How many units of time does Bob think (in the B-coordinate system) have passed by at the event of her looking? More generally, how does Cathy's watch appear to Bob? 6. Cathy has a “yardstick" that she thinks is 1 unit of distance long, which she holds in what she thinks of as a stationary position in the direction that the spaceship is moving. How long does the same yardstick appear to Bob as it moves along?
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