Chapter 4, The Theory of Special Relativity Video Solutions, An Introduction to Modern Astrophysics

Problem 1

Use Eqs. (14) and (15) to derive the Lorentz transformation equations from Eqs. $(10-13).$
$$\begin{array}{l}
x^{\prime}=a_{11}(x-u t) \\
y^{\prime}=y \\
z^{\prime}=z \\
t^{\prime}=a_{41} x+a_{44} t
\end{array}$$
$$\begin{array}{c}
x^{2}+y^{2}+z^{2}=(c t)^{2} \\
x^{\prime 2}+y^{\prime 2}+z^{\prime 2}=\left(c t^{\prime}\right)^{2}
\end{array}$$

Linda Winkler

Numerade Educator

11:04

Problem 2

Because there is no such thing as absolute simultaneity, two observers in relative motion may disagree on which of two events $A$ and $B$ occurred first. Suppose, however, that an observer in reference frame $S$ measures that event $A$ occurred first and caused event $B$. For example, event $A$ might be pushing a light switch, and event $B$ might be a light bulb turning on. Prove that an observer in another frame $S^{\prime}$ cannot measure event $B$ (the effect) occurring before event $A$ (the cause). The temporal order of cause and effect is preserved by the Lorentz transformation equations. Hint: For event $A$ to cause event $B$, information must have traveled from $A$ to $B$, and the fastest that anything can travel is the speed of light.

Linda Winkler

Numerade Educator

02:43

Problem 3

Consider the special light clock shown in Fig. $12 .$ The light clock is at rest in frame $S^{\prime}$ and consists of two perfectly reflecting mirrors separated by a vertical distance $d$. As measured by an observer in frame $S^{\prime},$ a light pulse bounces vertically back and forth between the two mirrors; the time interval between the pulse leaving and subsequently returning to the bottom mirror is
$\Delta t^{\prime} .$ However, an observer in frame $S$ sees a moving clock and determines that the time interval between the light pulse leaving and returning to the bottom mirror is $\Delta t .$ Use the fact that both observers must measure that the light pulse moves with speed $c,$ plus some simple geometry, to derive the time-dilation equation (27).

Narayan Hari

Numerade Educator

02:24

Problem 4

A rod moving relative to an observer is measured to have its length $L_{\text {moving }}$ contracted to one-half of its length when measured at rest. Find the value of $u / c$ for the rod's rest frame relative to the observer's frame of reference.

Linda Winkler

Numerade Educator

03:45

Problem 5

An observer $P$ stands on a train station platform as a high-speed train passes by at $u / c=0.8$ The observer $P$, who measures the platform to be $60 \mathrm{m}$ long, notices that the front and back ends of the train line up exactly with the ends of the platform at the same time.
(a) How long does it take the train to pass $P$ as he stands on the platform, as measured by his watch?
(b) According to a rider $T$ on the train, how long is the train?
(c) According to a rider $T$ on the train, what is the length of the train station platform?
(d) According to a rider $T$ on the train, how much time does it take for the train to pass observer $P$ standing on the train station platform?
(e) According to a rider $T$ on the train, the ends of the train will not simultaneously line up with the ends of the platform. What time interval does $T$ measure between when the front end of the train lines up with the front end of the platform, and when the back end of the train lines up with the back end of the platform?

Dading Chen

Numerade Educator

12:24

Problem 6

An astronaut in a starship travels to $\alpha$ Centauri, a distance of approximately 4 ly as measured from Earth, at a speed of $u / c=0.8.$
(a) How long does the trip to $\alpha$ Centauri take, as measured by a clock on Earth?
(b) How long does the trip to $\alpha$ Centauri take, as measured by the starship pilot?
(c) What is the distance between Earth and $\alpha$ Centauri, as measured by the starship pilot?
(d) A radio signal is sent from Earth to the starship every 6 months, as measured by a clock on Earth. What is the time interval between reception of one of these signals and reception of the next signal aboard the starship?
(e) A radio signal is sent from the starship to Earth every 6 months, as measured by a clock aboard the starship. What is the time interval between reception of one of these signals and reception of the next signal on Earth?
(f) If the wavelength of the radio signal sent from Earth is $\lambda=15 \mathrm{cm},$ to what wavelength must the starship's receiver be tuned?

Lucas Finney

Numerade Educator

12:24

Problem 7

Upon reaching $\alpha$ Centauri, the starship in Problem 6 immediately reverses direction and travels back to Earth at a speed of $u / c=0.8 .$ (Assume that the turnaround itself takes zero time.) Both Earth and the starship continue to emit radio signals at 6 -month intervals, as measured by their respective clocks. Make a table for the entire trip showing at what times Earth receives the signals from the starship. Do the same for the times when the starship receives the signals from Earth. Thus an Earth observer and the starship pilot will agree that the pilot has aged 4 years less than the Earth observer during the round-trip voyage to $\alpha$ Centauri.

Lucas Finney

Numerade Educator

05:25

Problem 8

In its rest frame, quasar $Q 2203+29$ produces a hydrogen emission line of wavelength $121.6 \mathrm{nm}$ Astronomers on Earth measure a wavelength of $656.8 \mathrm{nm}$ for this line. Determine the redshift parameter and the apparent speed of recession for this quasar. (For more information about this quasar, see McCarthy et al. $1988 .$

Linda Winkler

Numerade Educator

01:07

Problem 9

Quasar 3C 446 is violently variable; its luminosity at optical wavelengths has been observed to change by a factor of 40 in as little as 10 days. Using the redshift parameter $z=1.404$ measured for $3 \mathrm{C}$ 446, determine the time for the luminosity variation as measured in the quasar's rest frame. (For more details, see Bregman et al. $1988 .$ )

Lizandra Chagas

Numerade Educator