9 - 5 - The Happiest Thought (16-53, High-Def) (1)

8/10/2019 9 - 5 - The Happiest Thought (16-53, High-Def) (1)

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[SOUND].In 1907, two years after the miracleyear, Einstein had been asked to write areview article on the special theory ofrelativity.For a a science yearbook.It was really designed not for the

general public, necessarily.But for other, other scientists.Just summarizing some of the key resultsof previous years.And he wrote later, reminisced later,that he had what he called The HappiestThought or the most fortunate thought ofhis life.In fact, here's how he, he described thesituation.he, he had been bothered as he thoughtmore about the special theory of

relativity, he was bothered by a couplethings.And one thing was there's only constantvelocity in motion and ,therefore itwasn't general enough.And also didn't take into account anygravitational effects.So he had been mulling those things overand later he reminisce that I was sittingin a chair in the patent office in Bern,Switzerland, when all of a sudden, athought occured to me.If a person falls freely, he will not

feel his own weight.And he says this, this revelationstartled him.And as he thought more about it over thenext eight years the end-result was thedevelopment of his General Theory ofRelativity.So even though in one sense it's, it'sbeyond the course, we want to get atleast a little insight into what that isall about.And see, actually in a couple of

instances, well one instance at least, isthe one we're going to talk about in thisvideo, it gives us a result that issimilar to the Special Theory ofRelativity but different in a couple ofkey respects and that is a time dilationeffect due to gravity.Oh, he talked about that if, if a person,as he, as he says.if a person falls freely he will not feelhis, his own weight.And so, the classic example of doingthat, and Einstein himself used this

would be in an elevator.So let's imagine [UNKNOWN] it's a thoughtan elevator out there in the middle of


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space.If it's falling freely or even just areal elevator if you cut the cable andthere are no other safety mechanisms, asyou fall freely, you will not feel yourweight.if you drop a ball at that point, it will

just stay there because [UNKNOWN] you'reall falling down towards the center ofthe earth or down towards the surface of,of the earth.So, it will be like gravity doesn't existanymore.And Einstein started analyzing thatsituation a little bit more and notedalso that if you, say for your elevatoragain, you, you attach a cable to it,right?And maybe there's a crane or something,

getting out in the middle of space,somewhat fanciful here.But the idea is you have something wherenow you can lift the, the elevator up.In fact, you can accelerate it in anupward direction at some acceleration a.Again, just like you, we would do in areal elevator.As the elevator goes up you feel thatmomentary push downward.You feel heavier.It's like gravity has actually increased.And so, an acceleration upwards, if

you're in the elevator is, therefore,equivalent to a gravitational field.And this was essentially Einstein'sHappiest Thought.And it came to be known as theEquivalence Principle.[SOUND] The Equivalence Principle.That, essentially, a gravitational fieldis equivalent to an accelerated frame ofreference.Right?That this person inside here if, ifthey're walled off from everything.They could be in null space for all theyknow if the crane or the elevatorwhatever the cable pulls up on them at anacceleration that's equivalent to theacceleration due to gravity, then itfeels like gravity to them.They couldn't tell whether they were inthe elevator, just on the surface of theearth, feeling gravity pulling them down,or they're actually out in the middle ofspace, and some imaginary crane isactually pulling them at an acceleration

such that they press down into the floor,as it were, and you feel that normalgravitational pull.


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So that's the Equivalence Principle.Gravitational field is equivalent to anaccelerated frame of reference.And so you see that, that opened up somethings for Einstein because he isinterested in going beyond the SpecialTheory of Relativity, which is only

constant velocity and inertial frames of,of reference.And here was something that enabled himto perhaps, if he pursued this, to, tobring in accelerated frames of referenceand also tied it into gravity as well.And so again over the next eight years ina number stops and starts and, and deadends, and so on and so forth, plus he'sworking with other things along the way,he was able to come up with the generaltheory of relativity.

And the, the key thing for analysis isthat what it enables, the EquivalencePrinciple enables you to do is, if youcould do an analysis involvingacceleration, then the conclusions fromthat analysis also apply to agravitational field.The situation where you just had agravitational field occurring.And we want to do two examples of that.In this video we're going to do it, thetime dilation example, and in the nextvideo we'll do the bending of light

example.And so, let's imagine a situation likethis.So we're going to have an elevator here.And we'll have our, our two observers[UNKNOWN] two clocks.And so we'll put a clock at the top ofthe elevator, clock U for upper.And down here, we'll have clock L forlower.And these clocks emit little pulses oflight.So, here's one [INAUDIBLE] get some nicered pulses of light coming up here inthat direction and then appear anotherlittle laser and pulses of light goingdown and we'll have a, a detector hereand there.Okay?And so the, the clock, each clock andthese are, are synchronized clocks, theidentical clocks and they, they emitthese little pulses of, of light.That maybe they emit you know, ten pulsesof light per one tick of each clock.

Okay?Now another factor here is that we'regoing to assume any velocities involved


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are much less than the speed of light.So, this example will have nothing at allto do with the special theory ofrelativity.We'll assume the velocities are lowenough that we can ignore allrelativistic effects.

They're very, very small compared toeverything else going on here.so we've got the clock.So that means the, the two clocks are insync with each other.Right?We don't have to worry about leadingclocks lag and all that because anyvelocities involved are much less than c.And we're going to now put this intoacceleration in upward direction.So we're going to accelerate it in upward

direction, but again, at very, very smallvelocities.You know, just sort of ordinary, everydayvelocities so we can ignore any specialrelativistic affects there.And so lets think about what happens thento say clock L down here, the observer atclock L.they will be observing these pulses oflight coming down to their detector here.And meanwhile, their pulses go up todetector for clock, clock U.But because, as these pulses are emitted

here, of course they're traveling alongnow.But while they're traveling, the elevatormoves up slightly.It's accelerated up a little bit here,such that, these, let's say there are youknow again, 10 pulses per 10 pulses pertick of each clock.Okay?So clock L detected it though because itsaccelerated upward a little while thesepulses are in motion.Will collect more pulses per tick of itsclock.Or collect more of these pulses comingdown, because it's, you know, sort ofmoving up a little bit and sort ofsweeping through them a bit fasterbecause of the acceleration involvedhere.And so maybe instead of receiving 10pulses, which it normally would do pertick of it's clock, it receives 12 ofthese pulses of light coming down fromthe upper clock.

So let's write that down.For, for clock L here, let's say itreceives 12 pulses during one tick.


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And you've got the acceleration goingupwards so, if you put somebody in herestanding in the elevator.[SOUND] You know, if you make this equalto the acceleration due to gravity onearth, then that person would feel agravitational field.

It'd be like as a gravitational field andEinstein would say according to theEquivalence Principle, it's the sameanalysis either way.So the conclusion here is we did theanalysis with the accelerated frame ofreference and concluded that the upperclock runs a little faster than the lowerclock.The lower clock runs a little slower.In another words, lower down in agravitational field, clocks are going to

run slower than up high.Lower altitude, at a lower altitudegravitational field like at the surfaceof the earth.One clock is at the surface of the earthand one clock is up on an airliner or upin space on a satellite.This clock on the surface is going to runslower than the clock on the upper partthe higher, higher altitude.and so this is gravitational [SOUND]gravitational time dilation.[SOUND].

That's what we're talking about here.So not only do we have time dilationeffect at high velocities.Special theory of relativity.When we, when we're just in agravitational field, there's a timedilation effect between lower clock and aclock higher up in the gravitationalfield.And they've actually done experimentswith this.[INAUDIBLE] The Physics building, justfrom the basement, you know, to the topfloor.They have very accurate clocks um,[INAUDIBLE] it's not quite as set up likethis with the pulses.But something similar and you canactually see that the lower clock runslower than the upper clock even over justthe height of just a building.Again if you have very accurate clocksinvolved.another thing we've actually mentionedthis a couple times before but the Global

Positioning System, the GPS system[SOUND] has to be designed to take thisinto account.


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And actually has to designed to take intoaccount the special theory of relativeand the general theory of relativityinvolving gravitation here.It turns out if, if you do the analysison the satellites that involved in theglobal positioning system the, they're

moving in pretty high velocity comparedto the Earth.And compared to, you know, just areference point on, on the Earth.And, therefore turns out if you do theanalysis for the Special Theory ofRelativity.So, Special Theory [SOUND] says that theclocks, okay, since, since they're goingfaster, time dilation, right?If you observe the clock of a movingobject that's going fast.

You the observer, say here on the surfaceof the Earth, and we watch the satellitegoing by overhead, or really thesatellite might be stationary but we'rerotating on the Earth.However you want to, to look at it.We're at rest, satellite is moving.Therefore, we observe the satellite clockrunning more slowly, and it's about 7microseconds slow per day [SOUND] is theeffect.This is abbreviation for microsecond.Millionth of a second.

Okay?7 millionths of a second slow per day.Gravitational time dilation, however, sothe general theory of, of relativity.[SOUND] The general theory fromgravitational time dilation, if you dothe analysis there.Remember the lower clock is going to runslower.The higher clock, in this case, asatellite up there in space is going torun faster.And so the satellite clock, from ourperspective, runs about 43 [SOUND]microseconds fast [SOUND] per day.So there's sort of competing effectsthere.And you put them together and you can seethat, you know, it's, it's roughly, youknow, why is it here, 36 microseconds,something like that, using that range.Forgot the numbers quite there right but,you know, 36, 37 microseconds when you dothe analysis fast per day.And you know, I say, well, I mean let's,

that's, that's a very big difference.We'll try millionths of a second, 38, 40millionths of a second, whatever it is.


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But at the speed of light, okay, as thesignals go from the ground to thesatellites at the speed of light, speedof light in about 30 to 40 microseconds,travels 10 or 11 km.What that means is if you, in designingthe GPS system, if you don't take that

into account the special theory ofcontribution, and then the general theoryof contribution and the other direction.Fast for the general theory.Time dilation, gravitational timedilation.And slowing down for the special theoryof relativity.If you don't take both of those intoaccount and get the numbers right there,your GPS system is going to drift by tenor 11 kilometers per day.

In other words, it's not going to be avery accurate system.So they actually had to design the systemsuch that the clocks ran slow on thesatellites.Okay, in other words, they had to runslow by, you know, 36, 37 microseconds,something like that, per day compared toa clock, an identical clock, at theground.There, and so by taking that intoaccount, they were able to get a veryaccurate, accurate system.

So again, that's Einstein's happiestthought or most fortunate thought of hislife, depending on how you do thetranslation of the German.Occurred in 1907.Developed it over the next several yearsinto the general theory of relativity orone key part of the general theory ofrelativity.And one of the consequences of thatagain, is gravitational time dilation,that a clock in a gravitational field,lower down in a gravitational field, runsslower than a clock higher up in agravitational field.[SOUND].[BLANK_AUDIO]

9 - 5 - The Happiest Thought (16-53, High-Def) (1)

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