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### Course: Physics library>Unit 16

Lesson 3: Lorentz transformation

# Lorentz transformation derivation part 3

Finishing our Lorentz transformation derivation for t'.

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• 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?
• 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.
• 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?
• 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.
• 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.
• 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)
• AT , why did you transport and calculate x' to the left side of the equation? And then continued to calculate the other side?
(1 vote)
• They are trying to solve for t' so to do so they are first isolating the term is t' on one side of the equation so that they can work on that t' term.
(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.
• 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.