Chapter 2: Special Relativity

Consider Anna, Bob and Carl in the twin paradox.

(a) According to Anna, when Planet X passes her, clocks on Planet X and Earth tick simultaneously. What is the time interval between these two events in the Earth-Planet X frame?

(b) According to Carl, when Planet X passes, clocks on Planet X and Earth tick simultaneously. What is the time interval between these two events in the Earth-Planet X frame?

(c) What does the clock on Planet X read when Carl and Anna reach it? Show how your results from part (a) and (b) agree with Figure 2.20.

You stand at the center of your $100m$ spaceship and watch Anna's identical ship pass at $0.6c$. At $t=0$ on your wristwatch, Anna, at the center of her ship, is directly across you and her wristwatch also reads $0$.

(a) A friend on your ship,$24m$ from you in a direction towards the tail of the ship, looks at a clock directly across from him on Anna's ship. What does it read?

(b) Your friend now steps onto Anna's ship. By this very act he moves from a frame where Anna is one age to a frame where she is another. What is the difference in these ages? Explain.

(c) Answer parts (a) and (b) for a friend $24m$ from you but in a direction toward the front of Anna's passing ship.

(d) What happens to the reading on a clock when you accelerate toward it? Away from it?

From a standstill. you begin jogging at$5m/s$ directly toward the galaxy Centaurus A. which is on the horizon $2\times {10}^{23}m$away.

(a) There is a clock in Centaurus A. According to you. How Will readings on this clock differ before and after you begin jogging? (Remember: You change
frames.)

(b) The planet Neptune is between Earth and Centaurus $4.5\times {10}^{9}m$is from Earth- How much readings a clock there differ?

(c)What would the time differences if had instead begun jogging in the opposite direction?

(d) What these results tell you about the observations of a traveling twin who accelerates toward his Earth-bound twin? How would these observation depend on the distance the twins?

Question: You are gliding over Earth's surface at a high speed, carrying your high-precision clock. At points and on the ground are similar clocks, synchronized in the ground frame of reference. As you pass overclock. it and your clock both read . (a) According to you, do clocksand advance slower or faster than yours? (b) When you pass overclock , does it read the same time. an earlier time, or later time than yours? (Make sure your answer agrees with what ground observers should sec.) (c) Reconcile any seeming contradictions between your answers to parts (a) and (b).

A meterstick is glued to the wall with its 100 cm end farther to the right, in the positive direction. It has a clock at its center and one on each end. You walk by the meterstick in the positive direction at speed v . (a) When you reach the center clock, it reads 0. What do the other two read at this instant in your frame. (b) You instantly reverse direction. The clock at the center is still reading 0 and so is yours. What do the others read? (c) How does this relate to the twin paradox?

Show that for a source moving towards an observer equation (2-17) becomes$f_{obs}=f_{source}\sqrt{\frac{1+v/c}{1-v/c}}$

By what factor would a star's characteristic wavelengths of light be shifted if it were moving away from Earth at $\mathbf{0}\mathbf{.}\mathbf{9}\mathit{c}$?

At rest, a light source emits $\mathbf{532}\mathit{n}\mathit{m}$light. (a) As it moves along the line connecting it and Earth. observers on Earth see$\mathbf{412}\mathit{n}\mathit{m}$ . What is the source's velocity (magnitude and direction)? (b) Were it to move in the opposite direction at the same speed. what wavelength would be seen? (c) Were it to circle Earth at the same speed. what wavelength would be seen?

The light from galaxy NGC 221 consists of a recognizable spectrum of wavelengths. However, all are shifted towards the shorter-wavelength end of the spectrum. In particular, the calcium “line” ordinarily observed at $\mathbf{396}\mathbf{.}\mathbf{85}\mathit{n}\mathit{m}$is observed at $\mathbf{396}\mathbf{.}\mathbf{58}\mathit{n}\mathit{m}$. Is this galaxy moving toward or away from Earth? At what speed?

A space probe has a powerful light beacon that emits 500 nm light in its own rest frame. Relative to Earth, the space probe is moving at 0.8c . An observer on Earth is viewing the light arriving from the distant beacon and detects a wavelength of 500nm. Is this possible? Explain.