Special Sub-Topic: Special Relativity and the Space Patrol
|Our tale begins as Officers Albert, Max and Hendrik depart the police station that orbits Earth. As the patrol ship accelerates to relativistic speeds, Officer Hendrik's thoughts turn to his twin brother, Stephen, who has never left Earth. They were born by Caesarean section at exactly the same time. Will they still be the same age when Hendrik returns from his patrol?|
No. Stephen will be older than Hendrik.. This is the celebrated twin paradox! One of the more bizarre effects of relativity is that there is no universal clock. Elapsed time depends on speed, and a moving clock runs slow. Hendrik, whose body clock is moving at relativistic speeds, will age more slowly on his trip than Stephen will at home; it's a small effect, but it will add up over time. (But wait! From Hendrik's perspective, isn't Stephen the one who's moving, so shouldn't he be younger? It isn't quite that simple -- to get to relativistic speeds, Hendrik must accelerate, and then decelerate to come back to rest. It's the acceleration and deceleration that makes their "world lines," and thus their proper times, asymmetric. The twin "paradox" isn't really a paradox at all.)
By the way, Albert, Max, Hendrik and Stephen are named for four famous physicists who have worked out the principles of special and general relativity: Albert Einstein, Max Planck, Hendrik Lorentz, and Stephen Hawking.
|After rescuing a hapless space cat from an asteroid, our heroes find themselves traveling behind a prefabricated space station being moved to Alpha Centauri. It's a very wide load which seems to occupy three whole lanes, but there are no "WIDE LOAD" stickers or hazard lights, so Officer Max puts on the siren and pulls it over. The driver is insistent he's done nothing wrong: "It's all relative, Officer," he says. "I'm going at the speed limit, 0.7 c, so my load is length-contracted. When I'm going so fast and the road is at rest, the load fits perfectly within the lane." Does Officer Max write him a ticket?|
Yes. Length is only contracted in the direction of motion, so the load is just as wide at 0.7c as it is at rest.. Here we encounter the important quantity gamma = 1 / sqrt(1 - v^2/c^2), where sqrt = square root. In the phenomenon of length contraction, a fast-moving object appears to be foreshortened in the direction of motion. If L is the object's rest length, then an observer watching the object move at relative speed v will measure a length of L/gamma.
But only the length in the direction of motion is shortened! The driver of the wide load is moving forward in his three lanes, but the width of the load is perpendicular to the forward direction and isn't contracted at all.
When the officers were following behind, by the way, they would not have perceived any length contraction at all since they were moving at about the same speed. The patrol car and the space station were at rest relative to each other, in the same way that cars you pass on the highway often appear to be standing still.
|As the patrol ship travels along at the brisk pace of 0.7c (that's seven-tenths the speed of light!), Officer Albert (who is new at this) notices that the headlights only seem to reach very nearby sections of road. How fast is the light from the headlights traveling relative to the road?|
c. This is the basic defining principle of special relativity: light in a vacuum travels at c, no more and no less, in absolutely any reference frame you choose to measure it! This seems counterintuitive to us, living in a basically Newtonian world where a tennis ball thrown at 20 mph from a train moving 80 mph is going at 100 mph relative to the ground, but the invariance of the speed of light is absolutely necessary so that the laws of physics are the same everywhere and at any speed.
|Officer Max is operating the radar gun to look for speeders. He points it at a ship in the next lane over; the gun gives a measurement of 0.6 c relative to the patrol ship, which is moving at 0.7 c relative to the road. If the speed limit is 0.7 c relative to the road, should the officers pull the other ship over and give it a ticket?|
They should pull it over: it's speeding at 0.915 c.. Luckily for the rule of law (but unluckily for the speeding pilot), Officer Max remembers the rule of velocity addition in special relativity: V' = (Va + Vb)/(1 + Va Vb / c^2). Here, Va = 0.6 c is the velocity of the speeder relative to the radar gun, and Vb = 0.7 c is the velocity of the radar gun relative to the road, so V' = 0.915 c is the velocity of the speeder relative to the road. Slower-moving police officers (like the ones on Earth highways today) can get away with using the low-speed approximation V' = Va + Vb, but this gives nonsensical answers at relativistic speeds: no matter how hard that pilot is pressing the gas, he certainly can't fly a ship at 1.3 c. That's faster than light -- not allowed!
|Our heroes are running late for the next leg of their patrol, so Officer Hendrik, who's driving, applies a constant acceleration. The officers feel this acceleration as 1 g, pointing downward toward the floor. Officers Albert and Max head to the back of the patrol ship for a relaxing game of tennis. What makes the game on the spaceship different from a tennis game on Earth?|
Nothing. The ship's uniform acceleration acts the same way as the Earth's gravitational field.. This is Einstein's famous equivalence principle: There is no experiment (not even a tennis game!) that can distinguish a uniform acceleration from a uniform gravitational field. The formulation of the equivalence principle led Einstein to discover general relativity, so it made sense that he would later describe it as "'glueckischste Gedanke meines Lebens,' the happiest thought of my life."
|Officer Max, looking out a window, notices another ship running a red light. The officers immediately turn on their siren and pull the ship over. The pilot protests the ticket: "But officers, I was moving at relativistic speeds toward the light, so the red light was Doppler-shifted to green. I didn't know it was supposed to be red!" Is this a reasonable argument?|
Yes. An observer moving toward a red light at just over 0.1 c will see it as green.. The Doppler shift is well-known to anyone who has listened to a train or an ambulance as it passes by. As the source approaches you, the sound appears to have a higher frequency ("blue shift"); as it moves away from you, it appears to have a lower frequency ("red shift"). The same effect happens for light (which is, after all, a wave) when the source or observer is moving at relativistic speeds. We can use the relativistic Doppler shift formula. Let v be the speed of the observer, w be the wavelength of the source, and w' be the observed wavelength of the source. Then:
Z = (w' - w)/w = sqrt((1 + v/c)/(1 - v/c)) - 1
where a positive velocity means that the observer is moving away from the source. The shortest wavelength in the red range is 625 nm and the longest wavelength in the green range is 565 nm, so we can solve for v and determine that this Doppler shift could have happened if the pilot was traveling toward the red light at 0.101 c. Clearly we'll need to make some refinements before traffic lights are ready for use on relativistic highways!
|The patrol has been traveling about an hour without encountering trouble; it's definitely a slow stretch. Officer Albert, driving now, decides to engage cruise control; the ship is now moving at a constant velocity of 0.6 c relative to the road. What phenomenon do the officers experience? Their ship has no artificial gravity.|
Weightlessness. The sensation of weight is due to a mass experiencing some acceleration. On the surface of the Earth, our weight is due to the force of gravity, which accelerates us downward at 9.8 m/s^2. On the surface of the Moon, the force of gravity is only a sixth as much as on Earth, so we only feel a sixth our normal weight. Non-gravitational acceleration has the same effect -- this is why you're pushed back into your seat when you accelerate your car or as your plane speeds up before taking off. If there's no acceleration at all -- no gravity, constant speed -- then there's no weight either. Hopefully the officers will find it relaxing!
|The next item on the patrol's to-do list is a random check of sleep logs at a randomly chosen truck stop. Officer Albert figures he's got a live one when he finds a trucker whose last entry on the sleep log is dated twenty hours ago (station time): truckers are required to sleep at least every 16 hours! But the trucker insists that he's been traveling at 0.6 c until stopping just now, so his time has moved more slowly than station time and he hasn't driven too long at all. Should Officer Albert write him a ticket?|
No. Only 16 hours have passed for the trucker since his last nap, so as long as he sleeps before leaving the truck stop, he's fine.. Officer Albert has just encountered the amazing phenomenon of time dilation, perhaps most famous for the twin paradox described in Question 1. The clock inside the moving truck measures an elapsed time equal to the elapsed station time, times a factor of gamma. (We defined gamma in Question 2; for a velocity of 0.6 c, gamma is equal to 5/4.) Thus, the trucker experiences only 4 hours for every 5 that pass on the road. Moving clocks run slow!
|Officer Hendrik is also roaming the truck stop, weighing the trucks at rest. This section of highway has a relativistic mass limit of 15 million tons (there are some wormholes that could be damaged by excess mass). The speed limit for trucks is 0.6 c. What is the rest mass limit for trucks on this highway, assuming that they travel at the posted speed limit?|
12 million tons. If we define the rest mass as m0, then the relativistic mass is m0 times gamma (the quantity defined in Question 2; at a speed of 0.6 c, gamma is 5/4). The rest mass limit is then the relativistic mass limit divided by 5/4, or 12 million tons.
The idea of relativistic mass was introduced to make clear that objects pick up more and more inertia as they move faster and faster; that is, the closer you are to the speed of light, the harder it is to accelerate! Modern textbooks are phasing this concept out in favor of the idea of relativistic momentum (p = m0 gamma v instead of m v), but the concept is still useful. I'm sure some of the truckers whom Officer Hendrik tickets wish that relativistic mass had been phased out a little sooner!
|"Well, we're almost home," says Officer Max as the patrol draws to an end. They're cruising at 0.5c. When they reach the corner of Einstein and Lorentz, which is eight light-minutes away on the map, they'll need to start slowing down to pull into the station parking lot, but until then they'll maintain their speed. How long, according to the station clock, will it take them to get to Einstein and Lorentz?|
16 minutes. Elapsed time = distance / speed. The distance is 8 light-minutes, which means the distance that light travels in 8 minutes. Since our heroes are traveling at only half the speed of light, it will take them twice as long to traverse the distance: 16 minutes.
The trip will take even less time as measured by the shipboard clock due to the effects of length contraction. From the ship's perspective, the station is moving towards it at 0.5 c, but has a shortened distance to travel -- namely 8 light minutes divided by gamma (defined in Question 2 and here equal to the square root of 4/3). From the perspective of the officers, it only takes a little over four and a half minutes to get to the intersection. We can get the same answer a different way by using the principle of time dilation.
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