7 - 19 - Regions of Spacetime (29-14, Low-Def)

8/10/2019 7 - 19 - Regions of Spacetime (29-14, Low-Def)

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We've been exploring space-time diagramsover the last number of video lectures,and seeing how they can help us visualizeand gain some insight into some of thekey things about the special theory ofrelativity such as time dilation, lengthcontraction the Lorentz transformationand so on and so forth.And so in this video lecture we want toconsider this idea of regions of spacetime and the so called light cone.We'll get to that in a second but beforewe do that I want to go back to somethingwe were doing in the last video in termsof representing two sets of coordinates,two reference frames really on the sameplot.So remember how we, how we did this.We didn't do this one, we'll get to thatin a second, but we did this where we hadagain, Alice observing Bob going by in aspaceship to the right and the positive xdirection at velocity v.And clearly we could have 2 separate

coordinate systems.We could have Alice's.We could have Bob's.Just as we normally do at right angles,but we showed last time that we canactually, or couple, last couple times.So that we can actually put bothcoordinate systems on the same plot.And we did that by using the LorenzTransformation to take points from Bob'scoordinate axis and plot those,transformed onto Alice's coordinates,leaving Alice isn't the normal right

angle coordinates we have there.or the axis, the x and t axis for Alice.And we show that you end up withsomething at an angle here.We also made the point that you have ofcourse, lines of same location and linesof simultaneouty, lines of same time, andso on Alice's plot.The lines of same location are thevertical lines, represent by the tickmarks and the lines of same time, line ofsimultaneity, are the horizontal line,parallel to the, the x sub a.

Axis again represents by the tick marks,meaning of course, anything on thehorizontal line, in Alison's frame ofreference, everything on that line, anyspace time event that occurs on that lineis simultaneous to Alice.And then, any point on the vertical line,or any set of points on the verticaliline are things that occurred at the samelocation, although at different time as


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time goes on there.and then we put Bob's axis on there, and,it's important to see they're parallelagain.In other words, Bob line the samelocation, get the green in there, are.These lines here, here is his T sub Baxis, so this is the line, of samelocation for X sub B equals zero, ofcourse and then the line for X of B equalone two, three and then negative one,negative two, negative three and so onand so forth.And then the lines of same time,simultaneity, are oriented parallel tothe X of B axis 'cuz this is the line forT sub B equals zero, anything at occursat T sub B equals zero is along there andT sub B negative one and negative two areone, two, three and so on and so forth.So we can actually take it, given aspace-time event, say, represent by thisred dot here, and in Alison's frame ofreference, so I've drawn it such that it

looks to be about 2 and a half over andthen went about 4 up, maybe not quite 4up.So that would be the location 2 and ahalf, you know, some place in there.x sub a two and a half.And two sub a four.And if we're, as we started doing, we'reusing units of c being one.So light years per year, light secondsper second.So this could be maybe in terms of thelocation, might be two and a half light

years over.And time of 1, 2, 3 about 4 years or 2and a half light seconds over time about4 seconds.And that would be in Alice's frame.But in Bob's frame, according to hiscoordinate system, we'd see that, here'shis line of same location, so we go upthere from that tick mark, and we seeit's approximately 1.And then, the parallel line, parallel tothe x axis, for the line of simultaneity,looks like about 3.

Just roughly, you know, this isn't quitedrawn exactly to the scale there.But we can see that, so in Bob'sreference frame he would measure thatspace time event as x b.Equals about one, it looks like and TBequals about three, okay, again we're,these are approximations there.And then, for Alice, she would measurethat same space time point as we, as we


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noted.x of a, should have x sub a equal, we'lljust call it 2 point 5 and we'll justcall it 4, t sub a equals 4.So we see they'd measure differentcoordinates of course and we couldtransform between those differentcoordinates using Lorentz transformation.Now my numbers might not quite work outyou know, because I was just eyeballingit.There for an actual transformation butthat's the idea.A single space time event.The event's not changing.It's the same thing, it's a flash oflight that occurs someplace in space andtime.Someplace along the x axis, because we'redoing this in one dimension for the xaxis.And Bob measures it to occur at x sub zequals 1.Time equals 3 on his clock, on his

lattice of a clock, whereas Alisonmeasures it at x sub a, 2.5, and time forher lattice of a clock at 4 seconds oryear or whatever units we're happen tobe, be using there.And, and we did a little bit more that toshow how we can you know, see timedilation on there, and a little bit oflength and traction and so on and soforth, so.That's just a reminder of how we can putboth coordinate systems, both frames ofreference really on the same plot and get

some use out of it, in terms of analyzingthings with the special theory ofrelativity.You might ask what if we'd done it wherewe did it sort of from Bob's perspective.And then Alice was moving, which Alicethen would be moving to the, the left.So, if Bob, if we consider Bob to bestationary in his frame of reference andAlice moving negative v to the left, whatwould our combination plot look like?We're not going to try to prove this,we'll just say here, here is what it

looks like and that's what this is here.So you notice we draw the greencoordinate.Bob sits on here as x would be, t sub bin the normal 90 degree right angleconfiguration.And when you have the reference framemoving to the left, negative v, then itturns out, and you do the Lorenztransformation equations.


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So it'd be You know, you could do twolight years travelling in two years.So two light years per year, would be apoint, well let's just put that in hereroughly speaking.Right there.Okay.Three light years, and then those, andlet's do, one light year in one year, isgoing to be about right there.And roughly speaking three light years inthree years, it's going to be up heresomeplace, my tick marks are a littleoff.But remember what you get here, that'sway off actually, let's put it over here.You get a line at a 45 degree angle forthe speed of light.Travelling in in that direction andactually.Going down that way as well.So that is a light beam traveling in thepositive x direction, remember, it's easyto get confused with all the angles here.

Everything is happening just along the xaxis.And we're just plotting it the motionthrough time here.So this represents a light beam goingalong the x axis, traveling in, atvelocity c and this tells us how faressentially it goes in a certain amountof time at the speed of light, 45 degreeangle.so it splits not only Alice's axes herebut it splits Bob's axes as well.They're symmetrical across that 45 degree

line representing the speed of light or,or an object traveling at the speed oflight.Really not an object, a photon, a lightbeam traveling at the speed of lightthere.you can also of course have, the, thelight beam going in a negative vdirection, and look something like this,at a 45 degree angle and also you can gothat way as well.so if it's traveling from here up, thenit's going in the negative direction.

And it works the same on this plot aswell.45 degree angle, really splits the axishere so the lightening goes like that.And the same thing here.So those are the world lines for lightbeams on our plots assuming that c is 1,1 light year per year, 1 light second persecond.the world line for a light beam, or a


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photon will always be at a 45 degreeangle.Either in that direction or the oppositedirection here.So having said that, and sort of messedup our diagrams over there now, let'sturn to this diagram and, and you shouldhave already guessed.You probably already have that, reallytalking about the same thing here.So this is We're going back just asingle, plot, single diagram for ourreference frame, time up here, x axishere.We will again assume that the speed oflight is 1, we're referencing it as 1,like 1 light year per year, 1 lightsecond per second.Sometimes you'll see this diagram in sortof books with, and we mentioned thisbefore I believe, with CT here, c timest.T and that also essentially puts thingsin, in units of, of one.

But we'll just assume that, so we'llassume t is in years and then x has to bein light years or seconds, light seconds,etc., etc.So, what, what's going on with thisdiagram?Well, here's the idea, you can see allthe dots we've got.And we've, we've drawn in what's known asthe light cone.And we'll see why, we can see why it'scalled that.It sort of has a cone like feature but it

represents from the origin here where wehave dot number 1, point number 1.A light beam traveling in the positive xdirection would go like that.That's how we'd represent it.When it go like that, that's the worldline, it literally would be going alongthe x axis Axis, and this is its worldline through time.If it happened to start someplace overhere, it this is zero.If it started over here someplace, andwas going that way, then we'd have it

starting down here and maybe they'll,there's a flashlight or laser right hereshooting out the light beam in the xdirection.And therefore, the world line would begoing up like that.And if we shoot it in the otherdirection, maybe start over here.And do it this way, then world linestarts here and move up that way.


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So, the red lines represent the worldlines of a light beam.Now the key question here that we want toask, and this gets into what the themefor, for this week that we've beenworking toward, and that is, what happenswith cause and effect?And what happens with speed of light?You go faster than the speed of light.We've talked about how you can't.We did some analysis of what happens ifyou had a light clock say moving, for youto get up to the speed of light, whatwould happen.Then gamma becomes infinite and so on anddo forth, so that hints strongly that youcan't get up to the speed of light.But we want to consider some situationsnow in terms of, specific question hereis, if I have an event happening at 0.1here.Say anything at all, okay?Can that event affect any of these otherevents here, alright?

And, one way to think about it is let'sjust go from 1 to 2, okay.So, note that from 1 to 2 here, if I.Think about, think about a message orjust a spaceship traveling.Okay?Really, literally speaking too, ishappening right here on the x axis, okay,and one over here.So, could I, in a spaceship say, get from1 to 2.In time to affect something happening atthat point up here.

So it's really at, at position 2 andtime, whatever time we have over here.Well the key thing here is note that if Iwere to draw the world line here from 1to 2, something like that.That world line has a velocity less thanthe speed of light.Remember how this works, velocity isproportional to the inverse of the slopeof a world line so, faster velocities arelike this, slower velocities are up here,this is the speed of light here, so ifI'm going less than the speed of light,

I've gotta be over here, someplace interms of my slope.So from 1 to 2 here represents a worldline of something that is travelingslower than the speed of light.And we know that's possible.So therefore, in theory yes, if I have aspaceship here and I need to get toposition 2 over here in this amount oftime here, I can make it assuming my


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spaceship is fast enough here.Doesn't have to go to the speed of light,just has to go at some speed less thanthe speed of light, pretty fast probably,but it would get there, okay?And, so we say, yes, an event at one canaffect an event at two and in fact, ofyou think about it a minute, any eventswithin this cone here.Are able to be affected by an event atone because again I could, I could get ina spaceship that was fast enough and Icould travel to point 13 here, which isactually just on the x-axis.I really wouldn't have to do anytraveling.I just, I'm sorry not the x-axis, thetime axis.I wouldn't have to do any traveling.I would just sit there and wait until 13came along, then 10 second later, 10years later, whatever it happened to be.so I can get to places like that.Note where I can't get however.

Look at number 3 over here.Okay?If I say okay, I'm going to get in myspace ship, go as fast as I can, I'mgoing to try to get to point 3, which isout here at some position.Not position 12, that's just another dot.Some position out here at some latertime.To do that I'd have to have a world line,something like that.Well I missed the dot there, but I atleast hit the 3.

That would be the world line to getthere.And note that that world line has a slopeless than the speed of light slope andtherefore it's a velocity faster than thespeed of light.In other words, to get from one to threethrough space-time, which means travelingboth through space and through time, asit were, as time just ticks on.I'd have to have a spaceship that goesfaster than the speed of light.So I, I cannot get there.

and similarly for, for 12 here.Note for point 12, it's right on the xaxis.And that's a line of simultaneity ifyou're, if you're in this reference frameright here.In other words.No matter where you are in here, if it'sany distance away from you, to affectsomething you'd have to have


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Without traveling faster than the speedof light.This represents a world line that is ofan object traveling faster than the speedof light there.let's see what else we talked about.12, that's instantaneous with point 1, soclearly can't make it there.Talked about 3.mention 13.you just sort of sit there and wait for13 to occur in, in time.Cause this again is x equals zero here.11 over here, again just like 10 really.It can't get, it can't get to point onein time.It has to go faster than the speed oflight.What about nine and four?Notice, and also seven down here, notethat they are right on the light cone.Okay.What that means is that if I'm at point 1and I want to effect something going on

at point 4, if I travel at the speed oflight I can just get there in time.And the same thing over here going thereup negative extraction I'll be able toget to point 9 in time.and then down at point 7 if I want toinfluence something at point 1, I have totravel at the speed of light.So I have to send a laser beam orsomething, or a light signal with amessage that would get there in time.So, diagrams getting a little messed upthere, but again, the idea is the light

cone.And you have different regions of spacetime that it defines.Just by these 45 degree axis representingthe world lines of a light beam.And we have some other names for this aswell.And also it allows us to go back andrevisit the invariant intervals.Let's, we don't need this here.Let's introduce this again.Okay, so again, remember the questionsare: I'm at point one here and can, if

I'm at point one, can I influencesomething later on.In time, and at another point in space.Or if I'm in time before 0.1, then whereI am.Can I get a message there or travel therein time to influence something at 0.1?And so we have three names for thesesituations.Cause if you think about it.


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I can either get there in time, whetherI'm going to point one or to anotherpoint.So that's something.And we call that a time-like interval.A time-like interval.And, and really the, the key questionhere is, I'll write it as do you haveenough time, to, you could say reallyjust to make the distance.Let's put it like that.And this should be a little intuitive ifyou think about it, because In oureveryday lives we think about this a lotperhaps, if you drive a car or even ifyou're taking a bus or train or somethingyou think, I have to travel a certaindistance and I have a certain amount oftime to do it and do I have enough timeto get there given the distance and thespeed that I can go.So that's really what we're asking here,do I have enough time to make that, thatdistance, given with the parameters I'm,

I'm dealing with, whether car or walkingor however, I'm going from one space toanother point through time as well,'cause as maybe I'm walking, time isticking on, too.So I'm travelling through both space andtime that way.And a timelike interval It's such that,yes, you do have time to get there, okay?This, and, I'll write that, remember ourinterval equation?C squared, t squared minus x squaredequals a constant.

And it turns out we're not going to tryto prove this rigorously or anything,we're just going to state it.And that is if you have a timelikeinterval, what that really means is thatthis invariant interval, c squared tsquared minus x squared.Turns out to be greater than zero.And you might say, well, why?Well you can motivate this a little bitactually, because this means that, if youdo a little bit of the algebra here wedo, this means c squared t squared is

greater than x squared.Okay, just moving the x squared over.Or if we take the square root and,everything that's positive here we cansay that c t In general is greater than,than x.Now what does that mean?This is essentially, c is of course thespeed of light.C, the speed of light, times a certain


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time, this is the distance that light cantravel.Okay?And, but that, I should say, that lightdoes travel in a given time.If I say, okay.I've got 10 seconds, I've got 10 minutes.How far does light travel in that time.Just do C times T.Let will tell me how far light travels.And if that's the ultimate speed limit,if that's the fastest I can get someplace, I know if I have a certain amountof time to get some place, CT is thefastest I can go.You're like, of course you can't quite goto CT, but I can get pretty close to it.If that distance the light can travel isgreater than the distance I need to go,then yes I can get there, or at least thelight beam can get there, in that time.So this essentially is saying thedistance light can travel in that certainamount of time I have is greater than the

distance and yes it's reachable then.I'm, I have enough time to make thedistance in that case.And so that's called a time like intervaland again is represented in a diagramwhen you have things within, pointswithin the light cone that you can traveleither at the speed of light to get topoint 4 from point 1 or point 9 frompoint 1, or less than the speed of lightto get from like 1 to 2.Or down here from point 5 to 1 or 6, 6 to1 or 7 to 1 at the speed of light.

Well, and back up there.We've a special name for the speed oflight cases here.So the time-like interval, ct greaterthan x is, for example, for 5 and 6 and 1and 2, there.Now, so that's the time-like interval andnext one is actually called a like-likeinterval.A light like interval, and as you mightguess just from the name this is when youdo have things going at the speed oflight.

In that case you have the invariantinterval c squared t squared minus xsquared equals 0.And for the x squared on the other sidetake the square root and from this youget that ct equals x, and as we were justtalking about, if ct is, is the maximumdistance anything can go in that giventime, because this represents the speedof light traveling over a given time.


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That means that distance x is exactlyequal to that maximum amount of distanceI can cover.Alright, in other words, do I have enoughtime to make the distance?If that distance is reachable, justbarely by something going the speed oflight there, we call the light likeinterval.And so in that, in our diagram that wouldbe like going from 1 to 4.The only way to get from 1 to 4 in spaceand time is to travel at the speed oflight.And again really you're going out here toposition x but you only have this muchtime to do it and you have to travel thatmuch time or that much distance and toget there you have to travel at the speedof light.And that's called a lightlike interval.Or from one to nine, or from seven toone, going that way.Those are lightlike intervals.

So a given point here, if it's withinthis cone this way or down here, it's atime-like interval from this point tothat point.If it's on the light cone it's alightlike interval.And our third case is a spacelikeinterval.Third case, spacelike interval, and justfrom what we've done you may, can guesswhat this one is.It's c squared t squared minus x squaredis less than zero.

And the same analysis then tells us Thatc t, the maximum distance we can cover interms of what light can cover, is goingto be less then the distance x.The, in terms of the location of where xis.So in other words this is saying that,back to the question, do you have enoughtime to make the distance?No, because the maximum speed we can gois the speed of light and that means wecan get a distance ct away, or a lightbeam can get a distance ct away from

where we are.But if that's less than the distance itneeds to be traveled then you can't getthere.And that's called a space like interval,because space, the space coordinatethere, dominates over the timecoordinate.Light like is when space and timecoordinates are equal in a sense and


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time-like means the time coordinatedominates over the space coordinate.And so space-like interval would berepresented by like from 1 to 3 here.That is space-like interval.I can't get from 1 to 3 in my spaceshipor even a beam of light.Cannot get from 1 to 3.It's too far out here for the amount oftime I have.Or again 1 to 8 over in this direction.To get over here, in the amount of time Ihave.Don't have enough time.Same thing from 11 to 1.Starting at 11, can't get to 1 in time.Or 10 to 1 going that way.Okay, so time-like intervals, light-likeintervals, space-like intervals.This is another way to see sort of thelimitations that the special theory ofrelativity puts on us as we travel, in asense, through, through space and time.And in the next few video clips we're

going to explore some other aspects ofthis.

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7 - 19 - Regions of Spacetime (29-14, Low-Def)