6 - 8 - Exploring the Lorentz Transformation, Part 2 (11-56, High-Def)

8/10/2019 6 - 8 - Exploring the Lorentz Transformation, Part 2 (11-56, High-Def)

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Moving onto part two of the Lorenztransformation, we've brought back one ofour props here, taped it up on the board.We'll see what we're going to do withthat in a minute.But, in this part we want to focus on thetime equation, the time transformationequation.In the last video clip we were focusingon the, the x location, the xtransformation equation.So we have t rest equals gamma, remember,rest frame, and moving frame here.And we're measuring Alice's in the restframe, Bob is in the moving frame, movingto the right at velocity v in thepositive x direction.So gamma times a time value in the movingframe plus gamma v over c squared timesan, an x value in the moving frame.In other words, x and t, the moving framevalues, are measurements made by Bob.We wanted to transform that into a timevalue for Alice in this case.

So just as we did with the last equation,let's explore this a little bit.Let's just say, what you know?Let's just let x moving here.In other words, Bob's measurement of, ofsome event in the moving frame.Let that equal 0.'Kay?Very simple.And then of course this term becomes 0.And we're left with t in the rest stringequals gamma t in the moving frame, whichshould remind ourselves of the time

dilation equation.Because that's exactly what it is.Does, does this make sense?Well remember, when we derive the timedilation equation we use the light clock.And the light clock in the frame here.And Bob's frame, okay, is just, you know,a beam of light bouncing up and down.Like that.In Bob's frame, you know, he's carryingalong with him.And Alice, of course, she's at adiagonal.

And that's where we get the time dilationeffect.But as far as Bob is concerned, if he'scarrying the light clock with him as hegoes along.That bouncing up and down is happening inhis moving frame, what we're labeling themoving frame from Alice's prospective.But in Bob's frame at x equals zero.'Kay, because it's in his cockpit with


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him and that's his definition of theorigin, right.As far as he's concerned, he's notmoving.And so essentially for that light clockexample, we had the x value for Bob inthe moving frame b equal to 0.And we see it falls right out of theLorentz Transformation equation, as it,as it should.We just get t and the rest frame, timemeasured in the rest frame is gamma,times the time measured in the, themoving frame.Gamma is greater than than one, greaterthan or equal to one, of course.And so, the way we wrote it previouslywas something like this, we said time inthe moving frame equals 1 over gamma,times time of rest.And obviously you can write it, write iteither way there, but this emphasizes forus at least, the time dilation effect,gamma being greater than or equal to 1.

So time, the elaspsed time in the movingframe Is slower than in the rest frame.moving clocks run, run slow is how we,how we term that.So, Lorentz transformation equation forthat, checks out.It gives us our basic time dilationresult, that we derived earlier.Now though, let's say, what if we justlet x moving be anything, here.So, this is where we come back to Alicehere.So, can't simplify our equation at all.

We're assuming that V is some value, Iguess we could say certainly, if V iszero here, as we did before, then gamma'sone gamma's one.And you just get time in the rest frameequals time in the moving frame.In other words, both Alice and Bob aresitting there with their clocks, they'renot moving with respect to each other,and they measure the same, same time.So, that's a trivial case there.But what about the general case where Bobis measuring some time in his moving

frame.And of course, he's not consideringhimself to be moving, this is fromAlice's perspective in his frame, and anx value in the, the frame.Well let's imagine the situation thatAlice is here, we'll, we'll put her in aspaceship but we're, we'll assume she'sat rest.Let's bring back Bob in his spaceship.


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With their, each with their lattice ofclocks.Each of then synchronized in their ownframe of reference.And as we go out here of course Bob ismoving, moving like this.So let's imagine that right about, let meget this lined up here a little bit.Right about here, obviously probablywould have traveled much farther thanthat.But that Bob observes a certain event.Actually, let's, let's not even go thatway.Let's say that Alice at an instant intime on her clocks sets off photographsall along her lattice of clocks.And she photographs not only her clocks,but the corresponding clocks of Bobwherever he happens to be at that instantin time.Okay?And let's just say it's 10 seconds, okay?So on Alice's clock.

So at 10 seconds, you know, Bob is flyingby here.When they're right next to each other.Both are times T equals zero, TA equalszero, TB equals zero.And then later at some point way down theline, of course, but we want to show thatone, so about right here, 10 seconds havegone by.Alice takes photographs at all of herclocks at that instant in time.The question is, what are the readingson, on Bob's clocks?

We don't want to put numbers in here.We just want to get the qualitative.sense of this at, at this point.So, let's imagine it's something,something like that there.And note one thing about our equationhere is that let's say, you know, we'vegot a time right here.These are all 10 seconds reading onAlice's clock.So we've got 10 seconds on that one.What is the value of the correspondingclock right here going to be on, on Bob's

clock at that instant in time?Well, according to the Lorentztransformation equation here, if this is10 seconds over here, according to Alice.I've got something plus something overhere and assuming both those terms areboth positive terms here.We're assuming nothing is zero because Vis not zero and gamma is not zero,obviously.


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X is not zero.In other words, x, the x values may be 1,2, you know, the third clock out there,the second clock or something like that.So what that essentially means is if thisis 10, then this term has to be less than10.And in particular, T moving has to beless than 10.In other words, the time on Bob's clock,whichever one you point to out there, hasto be less than 10.And that again, is an indication of thetime dillation effect.That the moving clocks are going to runslow.There are, they are perfectly, at leastthe clocks right here at t equals zerosaid fine.Everything was set up, not all the clocksbut just the clocks at t equals zero.But down here, as it went on, we findthat when Alice took the photo that thetime on say I guess, been pointing the

second one there.Whichever one had been pointing to.Time has to be less than 10, becausegamma's greater than 1.This term also is positive and therefore,the value of time on Bob's clock in themoving frame has to be less than 10.Okay, so that's sort of an interestingresult.It just reminds us of of, time dilation.But now, let's, let's, also think aboutsomething else.Because remember, Alice took pictures

along the whole line here of her clocks.And so, 10 seconds on this clock.And she, we can read off the value onBob's clock at that instant in time.And then what about the next block over?Also 10 seconds for Alice, so again thisis 10.But know what happens as we move over oneclock, the x value here in Bob's framealso gets larger by whatever amount.It is, we could use the, our otherequation to get that.We're interested in more of the

qualitative argument here.So, as we move this way, so 10 seconds onthis clock, some value on Bob's clockthat's less than ten.As we move this way, let's just say it'seight, to give us a number here.So, say it was eight seconds on Bob'sclock for that one.As we move this way, this value isgetting bigger.


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As we move from that clock to that clock.Therefore for the next clock over, thishas to be even less.If this was 8 for this clock, then I'mgoing to move it one over, this term isgetting a little bit bigger, this termhas to get a little smaller.So this has to be a little smaller than 8now.If we move over one more, still 10seconds on all of Alice's clocks.So this is still 10 but this termsgetting a little bit bigger even andtherefore this has to get a littlesmaller.So as you move in this direction, againall of Alice's clocks are 10 seconds butBob's clocks the time as Alice observesthem here in her photos that she's taken.The time on these things is gettingsmaller.It, It's really prior in time.Again, if this was eight seconds, thismaybe has to be 7.6, and the next one

over is 7.2 or something like that as itgoes on.And what that means is, this is anotherverification of something we talked aboutbefore and that is leading clocks lag.In other words, leading clocks lag isbuilt in to Lorenz transformationequation through this term right here.Because otherwise, put this down for aminute, it looks strange because theGallian transformation, we only just hadTA equals TV.T rest equals T moving.

There's not change.So we'd expect okay special theory ofrelativity.Gamma's going to be involved so we mightexpect something like T rest equals gammaT sub moving without this term here.And we sort of get the handle out.But then This term, it's just weird.We say how, how does that work in termsof not only an affect on time here, butyou're adding some distance in here, asit were, in a sense.A distance factor that's affecting the

time observed in Alice's frame of, ofreference compared to Bob's frame ofreference.And what this term essentially is doingthat another way to put it is leadingclock's lag.As we we're just trying to argue here isthat, again, Alice takes all thephotographs up at t equals 10 seconds.So all of hers, all her clocks say 10.


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But, but maybe Bob's clock here becauseI've got two terms here.They're both positive.If this is 10, that has to be less than10.We said maybe it was 8.But then as you move over one, this isgetting bigger, and moving in morepositive X direction there.And therefore if this is 8 that's got tobe 7.6 or something like that.Move over again, that'll be down to 7.2now because this term keeps gettingbigger, this term has to keep gettingsmaller.Again, we don't see these effects.At least not in everyday life, becauselook at the factor here.Not only the gamma, of course, but V overC squared.This is a very, very small number.So it does show that leading clocks lag,and in fact, in a later video clip, we'regoing to look at that in more

quantitative fashion.Figure out exactly how much do Do leadingclocks lag by.But this isn't showing, therefore, thatthe time equation, as part of the Lorentztransformation equations, has this builtin.Not only the time dilation effect whichwe got from our light clock but also theleading clocks lag affect, as well.So that's part two of explaining Lorentztransformation.In part three, we're going to look at the

case where, what about from Bob'sperspective, looking at Alice.Or in general, what if we have some frameof reference moving to the left insteadof to the right?[BLANK_AUDIO]

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6 - 8 - Exploring the Lorentz Transformation, Part 2 (11-56, High-Def)