Amateurs’ Physics – Special Relativity

Part 1

When Joe checked the clock again, there were still several hours to go before his spaceship arrived at the Frontier space station. Joe was a little bored since his job for now was basically to   fly  straight, and then wake up the pilots of the next shift for approach and landing. At the moment, the instruments indicated a speed of 200 Mmps (our sci-fi unit, Million meters per second) AWAY FROM the sun. Joe seemed to remember that the speed of light was about 300 Mmps, but he forgot a lot of the stuff he learned in school. Then he thought: “300-200=100, so if I measured the speed of sun light, I should get a reading of 100Mmps. How clever I am!” Out of curiosity, Joe measured the speed of sun light, and got a reading of 300 Mmps relative to the spaceship. He was confused.

Several minutes later, another spaceship entered the communication range. Joe soon recognized that it’s the ship his friend Peter  flew . Peter was on his way home from the Frontier station, and his instruments indicated a speed of 200 Mmps TOWARDS the sun. Upon Joe’s request, Peter also measured the speed of sun light, and he got a reading of 300 Mmps as well. Now Joe was even more confused.

After the flight, Joe talked with his friend Nicole, a physician (not physicist) in the Frontier station, about this puzzle in his mind. Nicole spent a few moments to think about it and said: “I know that you went to many different elementary schools since your family moved so often back then. And let me ask you a question: Did people teach you 1+1=2 in one school, and say that 1+1=2 is not true in another school?”

Joe: “Now this is confusing. What are you talking about?”

Nicole: “Alright, alright. I just thought some examples might help. Anyways, a special property of light is that regardless of how fast the observer is moving relative to the light source, the observer always observes the same speed of light*. And I think it’s called the invariance of the speed of light.”

Joe: “That’s strange. But it sounds familiar.”

Nicole: “I guess it is a little strange. As you told me, from our daily experiences and some simple arithmetic, you thought you would get a reading of 100Mmps, and you also thought Peter would get a reading of 500 Mmps.”

Joe: “So what’s wrong with our daily experiences?”

Nicole: “When you measured the speed, you actually measured a distance and a time. Then you divided the distance by the time and got the speed, right?”

Joe: “I guess that’s what the instrument did.”

Nicole: “But the measurements of time and lengths change with the observer’s speed relative to the things to be measured. At a high relative speed, things look very different. For example, when Peter’s ship passed by your ship, did you notice that his ship looked shorter compared to what we usually see at the terminal?”

Joe: “Yeah, his ship did look shorter. I thought that was just my illusion.”

Nicole: “When the relative speeds are high, not too slow compared to the speed of light, these effects become noticeable. And under the circumstance, we can no longer use simple arithmetic to calculate the relative speeds. I thought they taught you these in the Space Academy.”

Joe: “Uh… they probably did.”

Nicole: “You better review your lessons if you still want to be a pilot. And I think I’ll take someone else’s ship home next time.”

Part 2

Three days later, Joe and his ship were on the way back to the Earth Orbit Station. When his shift was done, he left the cockpit and went to the ship’s lounge with his old text books. His off-duty colleague, Jill, was there, too. Joe wanted to have some coffee even though he is still not used to drinking coffee in a low gravity environment.

Joe: “Hey Jill, can you show me again how to use this coffee machine?”

Jill: “Just push that red button, but don’t open the lid. We don’t want to have coffee drops floating all over the place.”

So Joe got some coffee and fixed himself on the wall close to a reading lamb.

Jill: “What are you reading?”

Joe: “Just our old textbooks. I am reading the part about Special Relativity. I guess I never really understood it, and it still doesn’t make sense to me now. Wanna help me out?”

Jill: “Well, I haven’t studied that part for a while, but I’ll see what I still remember.”

Joe: “That’ll be great. Thanks.”

Jill: “Let me see the book.”

So she took a few minutes to scan through several pages, and said: “This chapter talks about the Maxwell’s equations. I think the equations describe the behaviors of electric fields, magnetic fields, some relationships between them, and so on. The equations can also be used to describe how electromagnetic waves, such as light, traveling through space. On the other hand, do you still remember the term: invariance of the speed of light?”

Joe: “I think so. I just talked with Nicole about that a few days ago.”

Jill: “Okay. Any particular part you want to review?”

Joe: “How about the Lorentz transformation.”

Jill: “Let me think about how to explain this… Let’s talk about the Galilean transformation first. For example, assume I am on a train going east at 200 km/h, and you are standing beside the tracks 4 km east of me. You and I are two different inertial reference systems. Assume at that moment, you see another train 3 km east of you, heading west at 150 km/h. Then you can conclude that, if I can see that train at that moment, I should see it 7 km east of me, moving at 350 km/h relative to me. Make sense?”

Joe: “Yeah.”

Jill: “This is basically an example of the Galilean Transformation.”

Joe: “This one is easier to understand.”

Jill: “Galilean transformation and Lorentz transformation are two examples of reference system transformations. If you are reference system A, and I am reference system B, then a transformation between you and me basically describes the relationships between your measurements and mine. So if you take some measurements from a physical event, and know the correct transformation between you and me, then you should be able to know what measurements I will get from the same event, just like the example I just mentioned.”

Joe: “That makes sense.”

Jill: “Now let’s talk about conditions at high relative speeds. We are  flying  at 67% of the speed of light away from the Frontier station. If now we measure the properties of the navigation laser beam shot out from the Earth Orbit Station, such as its speed and wave length, do you think the measurements will be consistent with the Maxwell’s equations and the invariance of the speed of light?”

Joe: “I believe so. Physics rules should work regardless of where we are.”*

Jill: “I think so, too. By the way, do you know Mike, the engineer at the Frontier station? He is in charge of examining these navigation signals.”

Joe: “Yeah, I know him, a boring guy.”

Jill: “No he’s not. Anyways, if you ask him to measure the properties of the same laser beam now, some of his measurements will be different from yours. The speed of light should be still the same, but the wave length he sees should be different. Anyways, do you think his measurements will be still consistent with the Maxwell’s equations and the invariance of the speed of light?”

Joe: “Sure, I think any physics rules valid for me should be valid for others as well*. I am not so special in the universe.”

Jill: “No offense but I think you are right. On the other hand, assume now you measured the laser beam and got a speed of 300 Mmps relative to our ship, and you also know that we are moving at 200 Mmps away from the Frontier station. If you apply the Galilean transformation in this case, you would expect Mike to see the laser beam traveling at 100 Mmps towards him. But then he will tell you: ‘That’s not right. It contradicts the results from the Maxwell’s equations and the invariance of the speed of light.'”

Joe: “I guess that the Maxwell’s equations and the invariance of the speed of light are still right, but the Galilean transformation is no longer valid in this high relative speed condition.”

Jill: “I think that’s right, and if Mike is really to measure the properties of that laser beam, his results should support this point.”

Joe: “You haven’t really talked about the Lorentz transformation.”

Jill: “Okay, okay. Then the Lorantz transformation was derived. At a low relative speed, the Lorentz transformation works just like the Galilean transformation. But the Lorantz transformation is valid for both low and high speed conditions. If you want to predict Mike’s measurements of that laser beam, use your measurements and the Lorentz transformation, and this time I believe he will tell you the prediction is consistent with both the physics rules and the measurements.”

Joe: “Sounds good.”

Jill: “Oh, do you remember last time there was an accident of a cargo ship, and the crew had to abandon the ship. The Frontier station issued a warning to all ships in that area. The warning message included the calculated trajectory of the abandoned ship relative to the Frontier station. I input the data into the Lorentz transformation computer along with my location and velocity relative to the Frontier station, and told the computer to detect the time reporting signals from the station. Then the computer came up with the trajectory of the abandoned ship relative to me. That was how I managed to stay away from that ship. Was your ship around that route as well?”

Joe: “Yeah… so that’s the procedure I was supposed to follow?”

Jill: “What!? You didn’t? I am going to tell Chief about this.”

* This discussion is for inertial reference systems only.

Source by Steve Wu