7 - 11 - Week 6 summary (15-26, high-def) (1)

8/10/2019 7 - 11 - Week 6 summary (15-26, high-def) (1)

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Let's summarize now some of thehighlights from week six.we started off doing a number of videoson space time diagrams revisited.And this is, [UNKNOWN] where we ended up,er, one of our final diagrams.In other words the idea was we want to

combine both frames of reference in theone diagram.So both Bob and Alice, where as usual wehad Bob traveling [INAUDIBLE] I guesssometimes we had Alice doing that aswell.But Bob's traveling at velocity b to theright, Alice's observing.Using the Lorentz transformation if weare given coordinates in Bob's frame ofreference.We can transform of course, into Alice's

frame of reference into her coordinatesystem.And we want to figure out how to plotthat and, and see them on the same plot[INAUDIBLE] only two plots in, in one.And by using the Lorentz transformationand starting with Alice's normal plot,right angle plot, really get Bob's axison there as well.So of course we have Alice's X axis and Taxis.And with respect to that then Bob's Xaxis is an angle, and his T axis is also

at an angle there.And the angle depends actually on therelative velocity between the frames,because that gets us the slope of thoselines.And we also noted that of course we havelines of simultaneity.And lines of same location, so for Alice,lines of same location are just thestraight vertical lines.In other words, one, two, three, fourright here for Alice.

Everything that occurs on this verticalline is at this same location goingthrough time.So if something is just sitting there.We talked about this before, even beforethis week.If something is sitting in Alice's frameat position 4.It's world line is just going to be astraight vertical line there.And for a horizontal line for Alice, theworld line there just indicatessomething, or a series of things that are

all simultaneous.In line with each other there and if yougoing to step through time that way.


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For Bob on this spot however, because hisaxis are skewed.Remember we have to draw the linesimultaneity parallel to the time axis inthe lines of actually so that backwardsthere lines of same location parallel.Now which way is that now, so I'll say it

correctly this time.Lines of same location parallel to thetime axis line of simultaneity paralleledto the X axis.There, so you' get this spewing effect.But for a given space time eventrepresented by this say a flash or lightor whatever, this red dot here.we can it's the same event but Bob andAlice will just have differentcoordinates for that.So in this example here, we can say that

flash of light for Bob occurs, say at, xsub b equals 1 and t sub b equals 1, 2,3.Those are the coordinates for Bob.x equals 1, t equals 3.For Alice those coordinates though, samespace time event, roughly speaking, it'snot an exact diagram here.I have about two and a half for x thereand 1, 2, 3, maybe three and a half,maybe a little bit more for the tcoordinate there.So Alice and Bob same space time event,

but they read it in differentcoordinates, or observe it with differentcoordinates.We also noted that you can see, timedilation on here, because, note that forthis event, Alice is meeting it, again,time about three and a half, somethinglike that.For Bob the time is three.Alice would be observing Bob's clock,would be observing it running slow, timedilation.And, same thing for length contractionhere, that, if you look at the distancefor Bob here, it's at x sub equals 1.Alice is reading it at, x sub A equalsyou know 2 , or something like that.So as she observes Bob going by, rememberthat's the basic picture here.She's observing Bob going by at somevelocity v.She sees objects in his frame ofreference contract their length, versustheir rest length in Bob's frame ofreference.

So you can see length contraction andtime dilation if you look at the diagramcarefully here.


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You can also see remember the relativityof simultaneity.Because we did an example where Alice hada whole series of flashes that went offsimultaneously, so that'd be on one ofher lines of simultaneity here.So let's just do it on, on this one here.

So a flash there, there, there, andthere, and a few more in here roughlyspeaking there, off but.Those are all on a horizontal line inAlice's diagram here.Meaning that to her they are allsimultaneous.They all occurred at, at t equals 1, likea, having a longer flash bulbs that forAlice they all go off with the same time.They are all synchronized with hersynchronized clocks.

And so, if you were to take flashphotographs of every, at every flashthere, compare it to the clocks at thelocation.All Alice's clock would read the same,same time for those flashes.But look what happens in terms of Bobhere that, that we can see.Now, of course we know from the Lorentztransformation there's going to be adifference, but we can see this right offthe diagram.Because Bob's line of simultaneity are

like this, okay?So he sees, yeah, this is kind of draw itin here a little bit.So here's line of simultaneity.One of them look like that.another one right here.And then a sort of clicker there, buthopefully you get the idea.So, all these are sync synchronized andsimultaneous for Alice.But clearly for Bob, if you look at let'sjust pick some in the middle here.This one right here occurs between time Tequals 0 and, and 1.This one over here though is occurringafter time T equals 1.This one in simultaneity.And this one over here occurs after timeT equals 2.So in other words, to Bob, these flashesare occurring before these flashes.In fact he will see it go off as.Flash, and then flash, and then flash,and then flash; there will be a, a seriesof flashes, not simultaneous, but in line

with each other.And again, also, you may remember, we cansee leading clocks lag on this as well.


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Because to Bob, these clocks or flashesare moving to the left, because to Alicehe's moving to the right.And it's this flash over here, theleading flash.The leading clock, as it moves by, thatis behind these, this other flash if you

want to pick that one.Or at least behind any one that is behindit as it moves this way.So we can see time dilation on here, wecan see length contraction on here, wecan see the relativity of simultaneity onhere as well.Which really shouldn't be thatsurprising, because the the whole thingis based on the Lorentz transformation.And out of which we get things like timedilation, length contraction, and

relativity of simultaneity.So we spend a fair amount of timedeveloping this tool really as it is.A visual tool for seeing some thingsabout the implications of the specialtheory of relativity.Then we did another version of it, wellwe didn't do two on the same problem, wejust said okay let's do a space timediagram here.And let's note that when we put a lightbeam on the space time diagram.If we can choose the units of c to be

light years per year, or light secondsper second, then c, the speed of light,has a value of 1.It's one light year per year, or onelight second per second.Which is nice for us, because if we usesort of regular units of time anddistance, like meters per second.Then a light beam along here would be,you know Very close to the X axis, almostindistinguishable from the X axis.Because it travels a very far distance ina very short amount of one.And so choosing C equal 1, one light yearper year, one light second per second,one light day per day, one light monthper month, whatever.Then C becomes a world line on ourdiagram here that has a slope of 1, or ifit's going the opposite direction has aslope of -1 V equals -C.And once again, you'll get tired of mesaying this, this is a reminder.This does not mean the light beam isgoing at an angle, everything we're doing

is going along the X axis, just shows itsprogression in time.So time is flowing upward in this case


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it's moving along the x axis at avelocity of c, which means it goes.You know, maybe one light year here inone year of time, or one light day in oneday of time and so on and so forth.So they call this the light cone becauseas we analyzed it further we said, you

know, if, if this is an event right hereor we're standing right here.Can we influence events later in time?And we analyze this and learn that aslong the event is within this cone here.This cone of light.Then we can influence it.Because we can get a signal to that.Or actually travel to that space timeevent.In time to have some influence over it.Or down here if we want to influence an

event occurring here.Anything in this cone here would work.But if we are outside the light cone,like say we are over here in thissection.What that means is I'd have to travelfaster than the speed of light to getthere in the amount of time I have.The question we asked was given the timeI have, can I make that distance?It's like we often ask in real life ifwe're, if we're taking a trip.You know, how long is it going to take?

Do I have enough time to get there, giventhe speed I can, can go?So here, the ultimate speed is the speedof light.And therefore, we actually analyze itinto three intervals based on ourinvariant interval.To give some more insight into it, and wehave the name time like interval forreally those things that occur in thecone here or the cone here.And what that meant, we don't try toprove this but we just stated it.It means that c squared t squared, thevalue of the invariant interval between,you know, two points here, say the pointhere and a point up here.If that's greater than 0, then it'spossible to get there and have aninfluence on it.And what that means is that CT is greatthan X.If you think about it, C times T is thedistance light can travel, if that is theultimate speed limit, we can't go faster

than that.Then if that the distance light cantravel is greater than the distance we


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actually want to go.Then yes we can get there in principal atleast assuming we can travel a up to somespeed less than speed of light.that so those will be a locations inspace and time that would be within thelight going here or here.

a light-like interval then is when thisequals zero or ct equals x, in otherwords the distance light can travelexactly equals the distance we need togo.And those are the values right on thelight cone here.So if I'm here, and there's a point outhere, then I have a light-like intervalbetween me.And if I can go at the speed of light, Ican't go, but if I can shoot a laser

beam, I could, the laser beam could getthere in time to get to that space-timepoint.If however, these, the inter-variantinterval between two points in space andtime, is less than 0.Then that means CT is less than X meansthe distance is farther than a light beamcould get there even.And therefore, I can't have any cause andeffect relationship between them.And it'd be like, again, if I'm here,it'd be like I'm out over here or out

over here.Or if I'm over here and I want toinfluence this point, I have to travelfaster than the speed of light.And remember again, that world lines withslopes less than 45 degrees in ourdiagram would indicate a speed fasterthan the speed of light.So, we analyze that a little bit.That was an interesting way to thinkabout cause and effect relationships withthis light cone diagram.And then we considered a couple ofexamples at the end.Where we had a question of, isfaster-than-light travel possible?And we did it in a thought experiment asit were, of a spaceship traveling fromSan Francisco to St Louis to New YorkCity.And, and shooting off some flashes alongthe way so the flashes the images wouldcontinue on in front of them.Go in front of them to an observer in NewYork City.

And to the observer in New York City, itseemed like that they saw the image ofthe San Francisco photograph appear.


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And then the image of the St Louisphotograph appear pretty quickly afterthat such that it seemed that the ship.The spaceship went from San Francisco toSt Louis at maybe four times the speed oflight, or even faster.And, but clearly this ship was not going

fast as the speed of light.It was just traveling, it was going veryfast in our example, but clearly lessthan the speed of light.So it shows there can be situations whereyou can get observations that seem toshow a violation of the fact that thespeed of light is an ultimate speedlimit.But when you analyze it further itactually is, is not true.And finally, we did an with [INAUDIBLE]

true faster than light travel.So we said, okay what if we actually hada way, a spaceship where we could travelfaster than the speed of light?What would be the implications of that?And we did a, a simple example with thatand the result was we saw a violation ofcause and effect.That the effect of something actually didnot happen after the cause but happenedbefore the cause.And the example we gave was an inventionof a spaceship that could travel faster

than light and then catch up with anotherone and do a sneak attack on it.And what we showed in the frame ofreference of the good guys who are beingattacked.That the invention of the spaceship andtheir frame of reference actuallyhappened after the sneak attack, which isclearly a violation of cause and effect.You can't have the sneak attack if thespaceship doesn't even exist yet.So it it works backwards.Another way you could, you could say itis that the spaceship actually attackedthem first.And then moved In their frame ofreference back to its invention state.So, it'd be like, a little bit like timerunning backwards is, is one way to, tolook at that.So, anyway, true faster-than-light travelleads to, true being, it's not possible.But we do an example, where we say, whatif we could do it.Leads to violation of cause and effect.

So next week, some of the things we havedeveloped here will be useful in thinkingabout things like the twin paradox.


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Which we'll get to and also things likethe pole so called pole in the barnparadox.And unlike this example of faster thenlight travel which is its almost anoptical illusion that you get that.These sort of strange paradoxes are

actually true, and they come out of thespecial theory of relativity.So, that's one of the things we've beenworking toward them, we'll be getting tothose next week.

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7 - 11 - Week 6 summary (15-26, high-def) (1)