 Today, we're gonna be talking about proper length and proper time. So what I have here today is I have a measuring tape that I put in my coordinate system on the ground. And I have a clock, which will also stay with me in my coordinates system. Now, meet my friend Bob. Bob here, he also has a clock and he has a measuring device with him. As long as Bob is not moving relative to me, the time elapsed between two events that Bob is measuring and that I am measuring will be exactly the same. As the distance we measure, for example, we measure both the length of this pen, we both will agree what the length of the pen is. As soon as Bob is traveling at a constant velocity relative to me, then the time Bob is measuring between two events and the time I'm measuring between two events will be different, as well as the distance that Bob is measuring for an object, it will be different than the one I'm measuring. Sounds weird, but is like that. It was shown by multiple experiments that this is really happening. Now, luckily, we have the model of special relativity, which allows us to convert from Bob's time and length to my time and length and vice versa. The problem, however, with these equations is that I have to figure out who is actually measuring the proper time and who is measuring the proper length. Now, this really depends on what we are measuring. Let's start with the proper time. The proper time is measured in the frame of reference in which the events that we are measuring happen at the same location. For example, if we're talking about the time it takes for Bob to pass in front of my hand, we're measuring the time from when Bob's head is here to when Bob's tail is here. So in this case, the events that I'm measuring happen here and here. In my coordinate system, this happens at the same place, right in front of my hand, that is not moving. So I'm measuring the proper time in this case and Bob, for him, the events happen once at his head and once at his tail. So for Bob here, he will measure a delayed time, which is greater than the time I measure it. If, however, we measure the time that it takes Bob to go with his head from the beginning of my pen to the end of the pen. Now in this case, in my frame of reference, the events happen here and here, so not at the same location. So I'm measuring delayed time. For Bob, however, the events happen right at his, in front of his eyes here and here. So in his frame of reference, the events happen the same. Location. So in this case, Bob has the proper time and I'm measuring the delayed time, the time with the bigger numbers. As for proper length, again, it depends what we are measuring. So again, Bob is moving in front of me. The theory says the proper length is the distance between two locations that are not moving relative to the frame of reference. So for example, if we measure the length of Bob, while he's moving, you're measuring from here to here. These two points are not moving relative to Bob, so he's measuring the proper length. They're moving relative to me, so I'm gonna be measuring the contracted length, the number that is a bit smaller than what Bob is measuring. Now if, however, I wanna measure the pen that's here on the table, in this case, I'm gonna be measuring from here to here. So these two points are not moving relative to me, so I'm measuring the proper length and for Bob, the two end points are traveling relative to him, so he's gonna be measuring the contracted length. So as a little summary, proper time is measured for the frame of reference that the event happens at the same location, so you don't have to look at two different places in order to start and stop your timer, while the proper length is measured in the frame of reference where both ends of the distance that you're measuring are not moving relative to you, so basically you can lie down and measuring tape and no matter how long you leave the measuring tape there, you will always measure the same distance for the two points and it will be no difficulty because they're not moving relative to your measuring tape.