Main content

## Physics library

### Course: Physics library > Unit 16

Lesson 3: Lorentz transformation- Introduction to the Lorentz transformation
- Evaluating a Lorentz transformation
- Algebraically manipulating Lorentz transformation
- Lorentz transformation derivation part 1
- Deriving Lorentz transformation part 2
- Lorentz transformation derivation part 3

© 2023 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# Lorentz transformation derivation part 3

Finishing our Lorentz transformation derivation for t'.

## Want to join the conversation?

- When sal begins Lorentz transformation derivation part 2 he makes no assumptions about the chosen point in space time (X, CT). Later in the derivation, he assumes X = CT and X' = CT'. This seems to limit the set of points to those on a

Light ray,, because the speed of light is the same in both reference frames. Doesn't this choice limit the derivation to sets of points lying on the light ray i.e. isn't there some loss of generality?(7 votes)- He's not actually assuming x=ct and x'=ct', he's using the framework of the problem. He originally stated the problem was essentially "if x=ct and x'=ct' then how do x and x' relate, and how to t and t' relate." The no generatily lost in this case because it's the question itself was limited. For a more general answer, we have to turn to general relativity.(0 votes)

- I think there is an easier way to derive t' by assuming that the speed of light is the same: x=ct and x'=ct'. We take x'=ɣ(x-vt) that we derived in the previous video and just substitute x' with ct', x with ct and t with x/c. We get ct' = ɣ(ct - βx). Is it correct?(2 votes)
- Not quite. That would only prove the equation for the special case of x = ct, which means for a light ray. But the Lorentz transformation transforms the coordinates of
*any*event from one reference frame to another. When you plug in x = ct, you are choosing a special subset of all (x, t) coordinates, namely the path of a light ray moving to the right that passes through x = 0 at t = 0. But we need to make sure the transformation is applying to the entire x-t plane, not just a single path.(3 votes)

- I didnt get the step after 1-gamma^2/(gamma^2)(v). Kindly some one explain the step where 1 is replaced by c^2-v^2/c^2-v^2?(1 vote)
- he just multiplies denominator and numerator with the same number( when you divide some number with itself you will get 1) so he could simplfy eq.(2 votes)

- At 4.06 I don't understand what sal meant by "Lets factor out gamma".Everything was going well for me until this moment.How did that happen?What you mean "factor out gamma".Please someone explain(1 vote)
- by assuming 1=gamma/gamma, you can take out gamma from the numerator and just multiply it with the other gamma at denominator(1 vote)

- How is it possible that gravity has impact on light? Why near blackholes time flows slower? My best guess would be that it is because near black hole in order to maintain the orbit a massive velocity is required, but it's nowhere near the speed of light.(1 vote)
- Time slows down near a gravitational field like a black hole because of the fact that space is connected to time, remember spacetime? Anyway, gravity squeezes the time dimension in the same way it squeezes the space part of spacetime, causing the object to see the time of the surrounding universe to slow down and eventually stop in time. Also, to an observer, the light you see actually stops being emitted. Light cannot escape the event horizon, which means that once you see the object at the event horizon, it will appear to stop and then dissappear as the last photons escape just before the object goes past the event horizon.(2 votes)

- I'm from Australia and for the Time dilation and Length Contraction parts of Special Rel, we have just two equations:

t=gamma t0

and

L=(L0/gamma)

Our equations look a lot simpler, so is the American way of teaching more complex or is there another reason?(1 vote)- What's gamma?

Once you write that down you will see the equations are the same.(0 votes)

- his final answer uses x however the wikipedia page for lorentz transformations has it as x'. Can anyone clear this up for me?(0 votes)
- The Wikipedia page does not have it as x'. https://en.wikipedia.org/wiki/Lorentz_transformation#Coordinate_transformation(2 votes)

## Video transcript

- [Voiceover] We've
made some good progress in our derivation of parts of
the Lorentz transformation. We've been able to
express x prime in terms of our Lorentz factor and x and v and t. And we've been able to
switch things around and represent x in terms
of the Lorentz factor and x prime and v and t prime. And we were able to solve
for the Lorentz factor. Now, the final missing
piece in order for us to have the full transformation
is to express t prime in terms of x and t. So, how can do we do that? Well, the way I'm going to
tackle it is I'm just going to take this equation right over here, let me underline it. I'm just gonna take that
equation right over there and solve for t prime. And the parts that have an x prime in it, I'm gonna substitute with this. So let's do that, let's
solve this for t prime. The first thing that I wanna do is I wanna divide both sides of this equation by the Lorentz factor or by gamma. So if I do that, I'm going to get, I'm gonna move it over to the
left so I have more space,. It's going to be x over gamma, x over gamma is equal to x prime. x prime plus v times t prime. t prime, let me do it
in that same blue color. Now everything I'm gonna do here is pretty straightforward algebra, but it's gonna get a little hairy, so that's why I wanna take some caution with the colors and progress slowly. Now, since I wanna solve for t prime, let me subtract x prime from both sides. So the left-hand side is
going to be x over gamma. I think that looks like a v too much. x over gamma minus x prime. Minus x prime is equal to v times t prime. v times t prime. And now, to solve for t prime, let's just divide both sides by v. And so, we are going to get x, let me do that white color. We're gonna get x over, well, now it's going to be gamma v. So, gamma v, v's in that orange color. Gamma v minus x prime over v is going to be equal to, is equal to t prime. So now we'll do what I said before. We've solved for t prime
in terms of now gamma v, x, and x prime, but now
we can take that x prime and replace it with gamma
and all of this business right over here, so let's do that. If we take this and substitute it in for x prime, actually, let me swap sides too, we are going to get t prime is equal to x over gamma. Let me do the gamma in that red color. Over gamma times v minus this stuff. So we're gonna have gamma times, it's a little bit tedious,
but we'll power through it. x minus vt . And if at any point you get inspired, I encourage you to run with it. So we just replaced x prime
with this stuff over here and then we're gonna
have all of that over v. So, all of that over v. And now what we can do, let's see. Let's factor out a
gamma out of everything. So we will get t prime is equal to gamma times, and we're gonna have, it's
gonna get pretty hairy now. Gamma times x over gamma squared. x over gamma squared, you
factor out a gamma here, or another way to think
about gamma times x over gamma squared is
gonna be x over gamma. We still have that v over there. v and then minus. Minus. And actually, let me just
distribute the minus sign, the negative sign. So, minus x over v. Minus x over v. x over v. Looking forward to this
getting a little bit simpler. And then a negative times
a negative is a positive, so it's gonna be plus vt divided by v. Well, that's just going to be plus t. Plus t. So, simplify it a little bit. Plus t. We're making some progress here. And so that is going to be
equal to, actually, let me just, so I don't have to keep
rewriting everything, let me just focus on this
part right over here, try to simplify it and
actually, even better, let me just focus on that
part right over there. That part, we can factor out an x. If we factor out an x, it is going to be equal to x times one over gamma squared, v. So let me write that down. One over gamma squared, v. Get the colors right. Gamma squared, v. Gamma squared, v, and then minus one over v. W have the minus one over v. One over v. Now, if we want to
subtract these two things, and let me put the close parentheses. If we wanna subtract these
two things, it's nice to have a common denominator, so
let's multiply the numerator and denominator here by gamma squared. This is gamma squared, gamma squared. And so, now I'm going to focus on this part right over here and hopefully this will simplify nicely. This is the same thing as one minus gamma squared, over gamma squared v. Over gamma, I picked a different color. Over gam, (chuckles) I'm having
trouble switching colors. Over gamma squared, v. Now what does this simplify to? Well, it seems like it
will be useful to have a different way of writing, well, let's just think
about what gamma squared is. Gamma is this business right
over here, which we could, if we were to square it, gamma squared. Gamma, that looks like a v again. Gamma squared is going
to be equal to one over one minus v squared over c squared. I just squared the numerator,
one squared is one. Take the square of the square root, you're just gonna get that. And if I wanted to
simplify it a little bit, I could multiply the numerator
and denominator by c squared. And so then that's going to be equal to c squared over, you
multiply the denominator by c squared, you're gonna get c squared minus v squared. So how does that help us? Well, this business. One, we can write as c squared minus v squared over c squared minus v squared. So, let's do that. c squared minus v squared over c squared minus v squared. I just did that so it
has the same denominator as gamma squared. And then, we're going to
subtract gamma squared. So we're going to subtract, we're going to subtract c squared over c squared minus v squared. And it looks like this is
going to simplify nicely. Well, let me just do one step at a time. And we're gonna have all of that over, gamma squared is once again c squared over c squared minus v squared. And we're gonna multiply that times v. Times, the v in that same color, all of that times v. Now, let's see, up here in the numerator, I have c squared minus v
squared minus c squared. So, what's going to happen is this, and we have the same denominator, so this and that are going to cancel. And so this whole expression
is going to simplify to negative v, and let me do
it in that same v color. Negative v squared over c squared minus v squared. c squared minus v squared. And we're dividing by this and that's the same thing as
multiplying by the reciprocal, so let's multiply b the reciprocal. So, times c squared minus v squared over c squared times v. And we are now in the home stretch. c squared times v. And we could do some simplification now. That's going to cancel with that. And then, v squared divided by v, you're just gonna be left
with a v right over there. So, all of this crazy business has simplified to a negative v, and I'll just write it
in that orange color. It has all simplified to negative v over c squared. So all of this has simplified to
negative v over c squared and so we're in the home stretch. This expression, t prime, it's going to be equal to gamma times, let's write this t first. So, t . And then this expression has simplified to negative v over c squared times x. So we can write this as minus v over c squared times x. And we're done. We have just completed our
Lorentz transformation. We started with this, that
we've been able to show in the last few videos, and
we did a little bit of hairy, carefully, did a little
bit of hairy algebra to get this result: t
prime is equal to gamma times t minus v over c squared times x.