The Length of a Moving Object in SR


      "The length of a moving object is contracted" is a conclusion drawn from any attempt to determine the length of a moving object using Einstein's Theory of Special Relativity. All books on Special Relativity give examples of how the length can and should be determined. After reading several, I have concluded that they are all derived from the same source and all contain the same confusing wording. So, at the risk of adding still another, I am offering my version here.

      First I will describe a process for determining the length of a moving object using Newtonian Physics. You probably know that this is redundant. In Newtonian Physics the length of an object is absolute. So if I know the length when it is at rest with respect to me, I know the length when it is moving (at constant speed) with respect to me. But what if I wanted to verify this? Here is one way to do that.

      Suppose you and I are on a train moving at a constant 1 meter per second with respect to the ground. I wish to determine the length of the car in which we are riding. We climb on top and I lay out a measuring tape along its length. You mark off the location of one end of the car on the tape. I mark off the location of the other end. I then subtract the numbers on the marks. Let's say that you made a mark at 1 meter on the tape and I made a mark at 11 meters. I determine that the train car is 10 meters long. I now know the length of the car when it is at rest with respect to me. Now suppose I get off the train and I try to verify that a measuring tape in my new reference frame will indicate the same length. I lay out another tape along the train tracks but the train is moving too fast for me to mark both the front and back of the train car. I ask you to do it as the ground tape goes by. I plan to take the difference in readings on the ground tape and compare it to the difference in readings on the train tape. So you line yourself up with the back end of the train car and swipe at the ground tape. Then you line yourself up with the front end of the train car and swipe at the ground tape again. I locate the two swipes. Let's say that one is at 101 meters and the other is at 171 meters. Simple subtraction would result in the conclusion that the train car was 70 meters long. What happened? Well, you did not make the swipes at the same time and the train moved between swiping. I ask you to repeat the marking this time making both swipes at once. But you cannot do that. So I make one last request. I ask you to record the time when you make each swipe and tell me later. You repeat the swiping. Then you tell me that you made the first swipe at exactly noon and the second exactly 60 seconds after noon. I locate the new swipes and combine them with the time data. Let's say the first swipe is now at (601 meters, 12:00:00Noon) and the second is (671 meters, 12:00:60PM). In 60 seconds the train moved 60 meters. Now I again have 70 meters between swipes, but I subtract out the 60 meters the train moved leaving 10 meters for the length of the car.

      This may seem like a cumbersome way to validate a measurement, but nothing is easy when measuring a moving object. I have a reason for using the above process. In Special Relativity, pairs of positions with the same time in one reference frame don't transform to the same time in another reference frame. And I need to know how to make adjustments to determine the length of moving objects when I get front and back positions with different times. I will explain.

      Suppose the train is moving really fast, 60% the speed of light relative to the ground. That's too fast to perform the swiping I described. But I can predict what the process would show by using the Lorentz Transformation. The Lorentz Transformation is a set of equations that relate position and time between reference frames for those doing relativistic physics. You on the train are in one reference frame. I, on the ground, am in another. In the reference frame of the train, one end of the target car is at (1 meter, zero seconds). The other end is at (11 meters, zero seconds). Actually, I could put any value in for the time even different values for the two. With respect to the tape on the train, the two ends of the car will always be at the same location, regardless of the time. But the Lorentz Transformation requires me to input position and time values and a time of zero for both positions makes for an easier calculation. When I transform those values to my ground reference frame I get (1.25 meters, -0.0000000250 seconds) for one end of the car and (13.75 meters, -0.0000000275 seconds) for the other. Just as with the Newtonian transformation process, I now have position values that are not simultaneous in time. And, just as with the Newtonian process, I determine the length of the car by subtracting out the movement of the train over the time difference. Let's see. The initial position difference is (13.75 - 1.25) = 12.5 meters. The time difference is 0.0000000025 seconds. At 60% the speed of light, the train traveled 4.5 meters in that time. The actual length of the train car in the ground reference frame is (12.5 - 4.5) = 8 meters. The Lorentz Transformation says that the train car is only 8 meters long for anyone in the ground reference frame. It turns out that any transformation of the length of the train car from the reference frame in which it is at rest to another reference frame will result in a smaller length.

      This shrinkage in length predicted by the Lorentz Transformation is the Length Contraction mentioned in the Special Relativity textbooks.