ASTRONOMY 9: HISTORY OF COSMOLOGY

Assignment #14--Solutions

2000 March 24

1.

A man, charged with going through a red light, comes before a traffic court. He argues that the Doppler shift made the red light appear green to him. If red has a wavelength of 700 nm (nm = nanometer = 10^-9 m) and green has a wavelength of 550 nm, what was his speed as a fraction of the speed of light? If you were the judge, what would your verdict be?

The Newtonian (non-relativistic) Doppler formula is

$$\frac{v}{c} = \frac{\Delta\lambda}{\lambda}$$

(see Hawley & Holcomb, p. 96). The change in wavelength is $\Delta\lambda = 700$ nm - 550 nm= 150 nm, while the original (emitted) wavelength is 700 nm. Thus $\Delta\lambda/\lambda = 150/700 = 0.21$. The car had to be moving at about 21% of the speed of light, which would be 60,000 km every second! Not very likely; the verdict would be guilty! (Note: if you use the formula from p. 96, you will actually get a negative answer for the speed, since the $\lambda$'s are flipped around. This just means that the car is approaching the light; a positive speed would correspond to receding from the light.) Looking ahead to relativity, we see that the Doppler formula is actually modified when speeds are close to c (see p. 183; you didn't have to do this part to get credit for the problem!). After doing some algebra on equation 7.9 to solve for v/c, you can show that the correct formula is

$$\frac{v}{c} = \frac{x-1}{x+1}$$

where $x = (1+z)^2 = (\lambda_{\mathrm{rec}} / \lambda_{\mathrm{em}})^2$. For this problem, the value of x is (550/700)² = 0.617, which gives a speed of 0.237c (again negative because it is approaching). So even at a speed as high as about 0.2c, the Newtonian approximation is not too bad.

2.

Define a blackbody. In what way is the radiation of a blackbody very simple?

A blackbody is any object that is a perfect absorber, meaning that it absorbs all the radiation that falls on it. Nothing is reflected, hence the word ``black''. But a blackbody is also a perfect emitter: when it reaches a state of equilibrium, it is both absorbing and emitting the same amount of energy.

The spectrum of radiation from a blackbody is remarkable because it only depends on the temperature and not anything else (composition, size, shape, etc.). Moreover, the shape of the spectrum is always the same. As the temperature increases, more radiation of all wavelengths is emitted, and the spectrum also shifts to shorter wavelengths (higher frequencies), but it still has the same shape.

3.

Life evolves into more and more complex, ordered systems. Does this disprove the second law of thermodynamics? Why or why not?

The second law is that entropy (a measure of randomness or disorder) tends to increase in a closed system. However, the Earth is not a closed system, because there is lots of energy coming in from the sun. The increase in entropy due to the nuclear reactions in the sun is much greater than the decreasing entropy on Earth. So if you consider the Earth and Sun together as a closed system, the total entropy increases.

4.

Airplane: The Sequel.

(a)

You wake to find yourself in an airplane with all its windows covered. Is there any experiment you can perform inside the airplane to determine whether you are flying with a uniform velocity, or at rest on the runway? (Ignore effects such as engine noise, which might be simulated as park of a diabolical plot to trick you, and don't look out the windows.) If there is, give an example of an experiment you might perform that could detect the velocity.

There is no way to tell--this is the old idea of Galilean relativity. Motion at a constant velocity is relative. If I am moving past you at 10 mph, I could say that I am at rest and you are moving. Or you could say that you are at rest and I am moving. There is no way to tell who is right!

(b)

If the airplane changed velocity, could an experiment inside the airplane show this? If so, give an example of an experiment you might perform that could detect the acceleration.

Unlike motion at constant velocity, accelerated motion can be detected. One thing that you could do is to hang a weight from a string and see what happens to it (if you are driving, hang something from the rear-view mirror and think about what happens to it when you change velocity). If the airplane speeds up, the weight no longer hangs vertically; it hangs towards the back of the plane. If the plane slows down, it hangs forward. If the plane accelerates sideways, it hangs to the opposite side. What about accelerating up or down? You could take a scale along with you: you know that your ``weight'' will seem greater if the plane accelerates up and smaller if it accelerates down.

(c)

Based on the above, is motion with uniform velocity relative or absolute? Is acceleration relative or absolute?

Motion at constant velocity is relative, but acceleration is absolute! If I am acclerating past you, we can both do experiments and agree on who is doing the accelerating, and who is not. So it's not all relative.

5.

Why did the appearance of the speed of light in Maxwell's equations create a problem for Galilean relativity? If a train is moving with velocity v and shines a headlight ahead of itself, Galilean relativity would tell us that a passenger on the train would see the light moving at c, but an observer at rest on the ground would see the light moving at c+v. Which speed should we use in Maxwell's equations: c or c+v? There is no way to know! So, either Maxwell is wrong, or Newtonian mechanics is wrong. Part of Einstein's genius was in questioning the latter, even though it had been around much longer than the former.

6.

If it is impossible to exceed the speed of light, why is it nevertheless possible to get to $\alpha$ Centauri, over four light years away, in less than four years time as measured by the space traveler?

Here one uses the idea of length contraction, which is implied by special relativity. Rulers moving past you are shorter in the direction of their motion than are rulers at rest. So when you take off from Earth and start moving at speeds close to (but not over!) c, the distance ahead of you seems to shrink, and so you can get there in less than four years as measured by your own clock. But according to someone who stayed on Earth, it took you over four years to get there! When you get back, you will find that you have aged less than the people you left behind.

7.

Explain why we are unaware of the effects of special relativity in our everyday lives.

The strange effects of special relativity only occur when you are talking about very high speeds: close to that of light, 300,000 km/s! Compare this to the speed of a supersonic airplane, which might be only 0.5 km/s. Space and time are stretched by only a factor of 1+10^-12, which is hardly different from 1 at all! So relativistic effects are almost always miniscule in our daily lives.

8.

Romulan and Klingon ships are approaching you onboard the starship Enterprise at 9/10 the speed of light (0.9c) from opposite directions. Before you can say ``he's dead, Jim'', both ships fire their laser-light cannons at you.

(a)

What is the speed you measure for the light waves from the two ships? The second postulate of special relativity is that everybody always measures the same speed of light, c!

(b)

What is the speed of the light from the Klingon ship as measured by the Romulans?

Again, the speed of light is always c.

(c)

What speed does the Romulan ship measure for your motion? For the motion of the Klingon ship? Just as you see the Romulans approaching you at 0.9c, they see you approaching at 0.9c. For the Klingons, you have to be a little careful! Galilean relativity would say the Romulans see the Klingons approaching at u+v=0.9c+0.9c=1.8c, but that is wrong! The correct formula for adding velocities (p. 182) gives

$$\frac{u+v}{1+uv/c^2} = \frac{1.8c}{1 + (0.9)^2} = \frac{1.8}{1.81} = 0.9945c.$$

(d)

Whose clock (the Romulan's, Klingon's, or Enterprise's) runs slowest as measured by the Romulans? They see their own clock running fastest (``proper time''), and the clock with the highest relative speed running slowest, so the answer is the Klingon's.

(e)

If all ships have the same length in their own rest frame, which ship do the Romulans measure to be shortest? They measure their own ship to be longest (``proper length''), and the ship with the highest relative speed shortest, so the answer is again the Klingon's.

(f)

Does the light from the Klingon laser appear blueshifted or redshifted to the Romulans?

The Klingon ship is approaching, so its light will clearly be blueshifted.

9.

Describe two quantities considered invariant in Newtonian physics that are relative in special relativity. Describe two new quantities that are now known to be invariant. Intervals of space ($\Delta x$) and intervals of time ($\Delta t$) were both considered invariant in Newtonian physics, meaning that all observers would agree on how big they were. But in special relativity, these quantities are relative: different observers moving at different speeds will measure different lengths and different time intervals!

The speed of light is now known to be invariant: everyone will always measure the same value c. Another new invariant is the spacetime interval, $(\Delta s)^2 = (c \Delta t)^2 - (\Delta x)^2$. The invariance of this quantity, with the funny - sign, leads to a lot of strange behavior!

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