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Numerade Educator

University of Washington

Simon Fraser University

Hope College

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Welcome to the next section in our unit on relativity. This section we're going to be discussing how this issue with simultaneity and the speed of light effects are measurements of time and distance. So thinking about this in kind of some simple scenarios, if we consider a little object that can produce some light and then also receive it back, we call the production of light event one and the reception of the light event, too. Now, if everything's holding still, it reflects off a mirror and then comes back to the exact same point at which it was admitted. This means we have to events at the same location, which means we can measure the time that it takes with a single clock that stays there at that location. On the other hand, if we have our little block move, then that means is going to admit some light. The light bounces off a mirror and then bounces back where it is received. So we have event won an event to they happen at different locations and will require two different clocks. Remember, we don't want to move clocks. What we're picturing here is kind of an array of many clocks, and whichever clock is the closest at a particular position. That is the time that that we read when we want to give a time to a particular event. Remember, events require space time coordinates. So we have our two different scenarios here. Let's see how they line up mathematically looking at these, we can see that the time it's gonna that will be measured by the first scenario is going to be two times D. That's the distance travel that goes up D and then it comes down d divided by the speed of light seat. Meanwhile, over here we have a different distance. You know, this is gonna be the high pot news of this triangle. If we say that it travels a distance v Delta T, then that means it will go one half v. Delta T. And it'll have a side D, which means that the high partners will have a length of d square square root of D squared plus one half V Delta T quantity squared so we can plug that in and say Delta T is going to be equal to two because it does this twice multiplied by the high pot noose length divided by C Well, over here we can solve for D. That D is equal to one half. See Delta t not and plug it in and then solve for Delta T in terms of tea not giving us the equation, Delta T is equal to Delta t not divided by the square root of one minus v oversee Square. Now you might ask, why the dance here? Well, because we just found a way to express the difference in time for our second frame of reference with respect to the difference in time for our first frame of reference, which was at rest with respect to the two events that occurred. So looking at this, then what we're going to define is a term called proper time, proper time. The Delta Tina is the time between two events that occur in the same location in an inertial reference frame as measured by a clock in that reference frame, meaning if we have to. Events that occur say an object moves from position A to position B, and we have a clock that is in the same inertial reference frame as thes two events, which in this case would mean that it's traveling with the exact same speed as this reference frame is traveling. That amount of time that the clock measures here is our proper time. And then the amount of time measured by any other reference frame, say, a reference frame that's holding steel or one that's moving the opposite direction or one that's moving the same direction but just faster is going to be measured as delta T Z equal to one over the square root of one minus B oversee squared, multiplied by the proper time. This is a really powerful idea. In fact, a lot of people will take this factor. Gamma is equal to one over the route of one minus. We oversee, squared and call it the Lorentz factor. And by taking the learns factor multiplied by the proper time, we have the idea of time dilation. Notice that if we really dig into this route of one minus v oversee squared. We've already talked about how we have this speed of light, and the speed of light is essentially the fastest that you're going to be able to go because every time that you go faster and faster and light hits you, you're still going to measure the same speed of light. So speed of light axes this sort of cosmic speed them in. People have even tried to accelerate electrons to the fastest possible speed, and they have come awful darn close. I think it's something like 0.99999 I can't remember how many nines there, but there's a lot of nines here, 5% of the speed of light. But they can never get to the speed of light. And that is because it is what we call the the cosmic speed limit for any object that has mass. So photons, which don't have mass that is to say, light waves that don't have masks, can travel, etc. Things that have mass can't. And so because this will always be less than one, we square it and we're going to end up with a positive number down here. That is less than one, which means the ratio the Lawrence factor here, one over the rue of one minus we oversee square is always going to be greater than one, so time dilation means that the amount of time measured by any other reference frame will be greater than the proper time now. This also has an effect on distances. For example, if we have reference frame A, that's a person standing on a platform and a cart goes by. That is a reference frame B and has a velocity V relative to the person holding. Still, the person holding still says, Well, I measure a length l not of the platform that I'm standing on, and Elna is going to be equal to the speed of this thing, multiplied by the amount of time it takes for it to travel from the beginning of the platform to the end of the platform. Meanwhile, someone who's writing in reference frame be will say I measure a length l because they are moving relative to this distance, and that is going to be equal to this vast city that I know I have, or that is moving by relative to me at multiplied by the proper time, which is what I am majoring. So this guy measures the time interval with a clock that is at rest relative to him, and this guy is measuring a delta T because V here is applied to reference frame be and therefore he is measuring a delta T for it and not a delta t. Not. This is a little confusing, but when you put it together, when you come up with something amazingly simple that we have l is equal to one over gamma multiplied by l not. So we're going to call l not then the proper or rest length, and it is a measurement of the oven. Objects length in the same reference frame as the object itself. So if there is a wooden beam and you are at rest with respect to the wooden beam, you could measure its proper length. On the other hand, if there is a wooden beam that is going by you with some velocity V, the best you can do is measure ah length L, which will be really related to the proper length by a factor of one over gamma. This is known as length contraction again because gamma is always going to be greater than one. This will always be less than one, which means the length measured by someone in a different restaurants frame will always be less than the proper length. Now you might ask, Is this all really true? Absolutely. And measured. It is one of the most major defects in all of science to many, many decimal places, in particular proper time, which has been measured using all sorts of quantum relationships. And without it, you wouldn't be able to use your GPS in your phone. It would get within a square mile or so, but it certainly wouldn't get as close as it actually does, so these are important factors.

University of North Carolina at Chapel Hill