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## Physics library

### Course: Physics library>Unit 16

Lesson 3: Lorentz transformation

# Evaluating a Lorentz transformation

We'll consider an example of the Lorentz transformation with actual numbers, and analyze the results we get.

## Want to join the conversation?

• I'm finding it hard to word this, but I still have to try. What does it mean to say we observe anything? Is this a nonsense question or does physics have a definite answer to such a question? [I got thinking I guess a little too hard about the extreme cases here of beta = 0 and beta = 1.]
(3 votes)
• Observation is data in any situation being recorded by an object foreign to the system. As jaketudan suggested, observation always changes the system.
(3 votes)
• Using special reletivity, I was thinking about the time distortions when traveling at the speed of light, and since time is relative like light, I started thinking of how it would affect what you and others observe. For exgample, suppose you are traveling at the speed of light for 5 years in a round trip, and during that time, the uinverse to you seems to not move in time. Now your friend, who is not moving relative to you, sees your time stopped while you travel that distance. When you reach your friend, you stop, and meet your friend. What would happen, and would both realities exist, or would there be a minor paradox?
(2 votes)
• This is called the twin paradox, but it is not really a paradox.
It seems like the situations of the twins are symmetrical, but they are not.
One twin experiences acceleration when he turns around to come back so that they can compare ages. Special relativity is for inertial reference frames, but only one twin stayed in the same, inertial frame the whole time, so this problem cannot be solved by simple application of special relativity.

We need to use a "more advanced" version of special relativity that accommodates acceleration, or use general relativity. When we do that, it will turn out that the one who was on the rocket will be younger when they are back together.
(4 votes)
• Don't quite understand something, if ct is the value of how long it takes for light to go one meter, shouldn't we cal the points
(1m , 1 l*m) or light meter?
(0 votes)
• c*t is measured in meters, but Sal misspeaks by calling the units "light meters." They're just meters.

I think what he meant to say is that 3x10^8 meters is a "light second" - the distance light goes in one second (in the same way that 9.46x10^15 meters is a "light year," because that's how far light goes in one year).
(5 votes)
• when I try some values I get a negative number. what does that mean?
(2 votes)
• As you said that for every 1 s in your frame of reference ,your friend travelling at a speed of C/2 ,has 0.58 s.
But , for her ,you too are travelling at a speed of C/2 ,so shouldn't for each 1 s of hers ,you would have 0.58 s too?
How does this play out?
(2 votes)
• I want is Lorentz transformation is related to wormhole? As in Lorentz transformation the distance is contracted for fast moving observer,the same as in wormhole in which the space is contracted
(1 vote)
• Did anyone try calculating the Lorentz transformations for the velocity of bullet (at ) i did and my answer was that x' and ct' are 1.99 when x and ct are equal to 2

Could someone confirm this?
(1 vote)
• Why did Sal take the reciprocal of the square root of 0.75?
(1 vote)
• What if V goes above the speed of light C.
Will the lorentz factor be an infinite number?
(1 vote)
• IF (and it won't, but if) v were to exceed c, then you would have a negative number underneath the square root sign. This makes no sense and shows an algebraic reason v cannot exceed c. But, as v approaches c without exceeding it (which can happen), 1 - (v/c)^2 approaches zero and the lorentz factor approaches infinity.
(1 vote)
• So, as I understand it, the final values for x' and ct' should be a point on the graph with the tilted axes. Where exactly would this point be? It's hard for me to visualize it without more lines for this graph.
(1 vote)

## Video transcript

- [Voiceover] Let's now dig a little bit deeper into the Lorentz Transformation. In particular, let's put some numbers here, so that we're, we get a little bit more familiar manipulating and then we'll start to get a little bit more intuition on how this transformation or sometimes it's spoken of in the plural, the transformations behave. So let's pick the scenario in which our friend passes us by, and this is the same scenario that we've been doing in previous videos, with a relative velocity, from my frame of reference, at half the speed of light. So the magnitude of her velocity is half the speed of light. She is moving in the positive x direction and our space-time diagrams, they coincide at the origin. And so let's pick an event in space-time. And so let's say in my coordinate system, in my frame of reference, this event that we focused on in the last video, let's say that is at x is equal to one meter... Let's use the same color... X is equal to one meter and let's say that time or ct is also equal to one meter. And like we said in, I think it was several videos ago, we could do this as a light meter, the time it takes for light to go one meter. So we will also say this is one meter. So in my coordinate system, in my frame of reference, this would be the point one comma one. One meter in the x direction, one meter in the ct direction. Now, based on that, think about what would be the prime coordinates. What would be the coordinates in her frame of reference? And I encourage you to pause the video, evaluate the Lorentz Factor using v and c, and then evaluate what x prime and ct prime would be. All right, I'm assuming you've had a go at it. Now let's work through this together. So first let's figure out what the Lorentz Factor... Actually, let's first figure out what beta would be. That will simplify everything. So, beta, we'll do it in the blue color, beta in this case is going to be equal to zero point five c. That's her relative velocity in my frame of reference. The ratio between that and the speed of light, so that's just going to be equal to zero point five. You could just view beta as what fraction of the speed of light is that person traveling in my frame of reference since we're using that as kind of the non-primed frame of reference. And so let's now think about what gamma is going to be, the Lorentz Factor. The Lorentz Factor is going to be... We'll do it in that reddish color not the magenta... it is going to be, so gamma is going to be one over the square root of one minus beta squared. Beta squared is, zero point five squared is going to be zero point two five. Actually let me just write zero point five squared just so you can see what I'm doing. So this is zero point five squared and if we were to evaluate that... Let's see this is going to be one minus zero point two five. So that's going to be point seven five. This is going to be equal to one over the square root of point seven five, one over the square root of zero point seven five. Let me get my calculator out. We can at least approximate it. So point seven five, let's take the square root. And I'll just take the reciprocal of that, so approximately one point one five. So our Lorentz Factor is approximately one point one five. And now using that we can figure what x prime and ct prime are going to be. X prime is going to be equal to my Lorentz Factor which is approximately one point one five. So maybe I'll write it as approximately going to be equal to one, we'll do that same color, it's going to be one point, I'm having trouble switching colors today, one point one five. One point one five times, now we're saying x is one meter, so x is one meter minus beta, beta is zero point five, so zero point five. And ct we're also saying is one meter, so times one. And then t prime or I should say ct prime, ct prime is going to be approximately the Lorentz Factor one point one, I always have trouble switching colors for the Lorentz Factor, it's going to be approximate to one point one five times... Well, ct is one. I think you see a little bit of symmetry here. This one in particular because it had the same x and ct coordinates. So one minus beta, one minus beta, so zero point five times x, which is once again one. So times one. So in this particular case, it simplifies to half of the Lorentz Factor because this one minus zero point five times one that's just going to be zero point five. And same thing over here, zero point five. So these things are going to be approximately equal to zero point five times the Lorentz Factor. We already had the Lorentz Factor in my calculator, so let me just multiply by zero point five. And I get, it's approximately point five, I'll just say point five eight. So, zero point five eight, zero point five eight. And once again the units are in meters. So even though this is one meter and one meter, this over here x prime is zero point five eight, zero point five eight meters and ct prime is also zero point five eight meters. So this is also equal to zero point five eight meters. So one way to think about it, if right when our two, right as she was passing me at x equals zero, time equals zero in my frame of reference, If I were to shoot my lazer gun or I were to turn my flashlight on and that very first photon starts traveling and so I could think about it's path through space-time, it would look, that very first photon would look something like that. When I think that a, when I think that that photon has traveled one meter in the positive x direction and one light meter of time has passed, from my friend's frame of reference, she would say, "No, no, no, no." At exactly that moment, let's say it hits an asteroid at that moment, it lights up an asteroid. She would say, "No, no, no, no, no." That happened zero point five eight light meters after she passed me up. And it happened zero point five eight meters in the positive x direction. So something very, very, very interesting is going on. And I encourage you to think about what's actually going on with these different parts of the Lorentz Transformations. The most interesting is what's going on, well, actually it's all interesting. In fact, the symmetry's interesting. But the Lorentz Factor, think about what's happening here. Think about what's happening here for low velocities when v is a very, very, very small fraction of the speed of light. Well then beta is going to be pretty close to zero, and then the Lorentz Factor is going to be pretty close to one. And think about what happens when v approaches the speed of light. Well then this thing just booms. This thing gets larger and larger and larger as we see this denominator getting smaller and smaller and smaller. If v were actually equal to the speed of light, well then you're going to be dividing by zeros. Well, you know, that's when all sorts of silliness starts to happen. So I really encourage you to try out different numbers. We tried very high relative velocity, half the speed of light, incredibly, incredibly high velocity. Try it out for something more mundane like the speed of a bullet or something like that. But definitely get very familiar with this. And also manipulate it algebraically. In fact, maybe in the next video I'll manipulate this a little bit algebraically so that you can reconcile the way I've written the Lorentz Transformation or the Lorentz Transformations with the way that you might see it in your textbook or other resources.