제1부. 공간과 시간 (Space and Time)
1.1 균일한 우주(The Homogeneous Universe) 동영상/영문자막/한글자막
1.2 팽창하는 우주, 허블의 법칙(Hubble's Law) 동영상/영문자막/한글자막
1.3 관성 질량과 중력 질량(Inertial and Gravitational Mass) 동영상/영문자막/한글자막
1.4 공간 (Space) 동영상/영문자막/한글자막
1.5 측량(The Metric) 동영상/영문자막/한글자막
* 이번 강좌는 두 물리학자가 '측량(Metric)'의 개념을 여러가지 예를 동원하여 이해시키려고 노력한다. '측량'은 특수 상대론에서 빛이 휘어 보이는 이유를 수학적으로 설명하기 위한 기초 개념이다. 이를 이해하기 쉽지 않다. 그래서 몇가지 예를 드는데 장황 하다. 이 강좌의 요점은 마지막 두문장이다.
폴: 이제껏, 힘이 가해지지 않고도 가속되는 상황을 본거죠.
PAUL: So we seem to be getting acceleration without force.
만일 우리가 측량을 흐트러 놓음으로써 사물들이 아주 이상하게 행동 하게 만듭니다.
If we muck-up the metric, it makes things behave rather weirdly.
* 십분여에 걸친 긴 강좌다. 중간에 일부 장황한 예는 생략하겠다.
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브라이언: 좋습니다. 폴.
BRIAN: All right, Paul.
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당신이 공간에 대해 설명해 주셨는데 확실하지 않군요.
So I'm not sure if you've illuminated us on what space is,
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제가 이해한 바로는 공간이란
but as near as I can understand, the idea
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숫자로 표현될 수 있다는 거죠.
is that space is described by numbers.
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공간은 모두 숫자를 가지고 있다는 거죠. 이런 숫자들을 사용할 수 있다는 건데...
Every part of space has got a number, and we can use those numbers--
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폴: 세개의 숫자지요.
PAUL: Three numbers.
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브라이언: 네. 세개의 숫자.
BRIAN: Well, three numbers, sorry.
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맞아요. 숫자 세개.
Yes, three numbers.
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세개의 숫자라는 것은 기본적으로
And that those numbers can be used to describe, essentially,
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수학으로 사물들이 서로 얼마나 멀리 떨어져 있는지 표현할 수 있다는 거네요.
how far things are apart by using mathematics.
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그러니까 중력이라던가 그런 물리 법칙들이
And we care about that because physical laws, like gravity and things,
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숫자공간에서 거리에 달려 있다는 거죠.
depend on that distance in number space.
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숫자로 공간을 기술하는 고도의 방법이라는 거네요.
And so that's sort of an abstract way of describing space in numbers.
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폴: 여기서 중요한 것은
PAUL: So the crucial thing here is the concept
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'측량(metric)'이라고 부르는 개념 입니다.
of what we call the metric, which is, if you've
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세개의 숫자로 표현된 두 지점의
got two things with their sets of three numbers,
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거리를 재는 방법이 바로 '측량' 입니다.
to work out how close they are.
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여기 지구에서 두 지점의 좌표를 X-Y-Z 로 알고 있다고 합시다.
So let's say I've got X-Y-Z coordinates of two things on Earth.
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그 두 지점이 얼마나 떨어져 있는지 계산하는 방법이 뭘까요?
How will you work out how far apart they are?
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브라이언: 네. 그 방법은 고대 그리스 수학자가 이미 알려 줬었죠.
BRIAN: Well, that's easy, because that's just what the Greeks told us.
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피타고라스 정리라고 하는 거요.
Pythagorean theorem.
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폴: 넵.
PAUL: Yep.
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브라이언: 오.
BRIAN: Whoops.
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바로 이겁니다.
PAUL: Here it is.
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미소 거리를 계산해 볼께요.
So we're talking about small distances.
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여기 델타()라고 한 의미는 x축으로 작은 차이, y축으로 작은 차이,
So a delta means a small difference in x, a small difference in y,
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z 축으로 작은 차이를 의미 합니다.
a small difference in z.
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만일 전체 거리를 계산 할 거라면 이 미소거리를 합치면 되겠죠.
If you had to go a big distance, you just add up all these small differences
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as you go.
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이것이 바로 피타고라스 정리 입니다. 거리란
And so this is just Pythagoras' theorem, that the distance is
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x의 제곱, y의 제곱 그리고 z의 제곱을 더해
equal to the square root of the difference in x-coordinates squared,
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제곱근을 씌운 겁니다.
the difference in y-coordinates squared, the difference in z-coordinates
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squared.
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그게 바로 피타고라스 정리죠.
That's just Pythagoras' theorem.
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브라이언: 만일 삼차원 이라면, X-Y-X는
BRIAN: If you have a three-dimensional triangle, that's X-Y-Z,
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경사(hypotenuse)의 길이가 되겠네요.
that's how long the hypotenuse is.
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그리고 우리가 계산한 이 거리는 아주 작고
And we're going to make sure that our distances are really small here,
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그 미소 길이를 모두 직선으로 취급 할 수 있어요.
so we can treat everything as a straight line.
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폴: 넵. 그럼 당신이 신의 슈퍼컴퓨터를 가졌다면
PAUL: Yeah, so if you have god's supercomputer
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수많은 것들을 이 숫자들을 이용해 나타낼 수 있어요.
and we have lots of things with different sets of numbers,
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이 숫자를 이용해 그 두점이 가까워 질 수 있는지 알아 볼 수 있습니다.
we can use this to work out if they're going to be close.
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그리하여 만일 그중 하나가 폭발 하면 어디까지 영향을 줄지 알아볼 수 있다는 거죠,
And therefore, if one explodes, which things are going to be affected,
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예를 들어서 말이죠.
for example.
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브라이언: 좋습니다.
BRIAN: OK.
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문제 없어요.
No worries.
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쉽군요.
This is easy.
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폴: 네.
PAUL: Yeah.
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이런 식으로 우주를 시뮬레이션 해보죠.
So we can simulate the universe this way.
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이제 아인슈타인의 가속과 질량에 관한 문제로 돌아가볼까요.
But now, let's go back to Einstein's problem, acceleration and mass.
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브라이언: 뭐라구요?
BRIAN: Yes?
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폴: 어떻게 가속 시킬 수 있을지 그리고 서로 어떤 힘이 관계되어 있을까요?
PAUL: How can we make acceleration and a force relate to each other?
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마구 뒤섞어 놓아야 할겁니다.
Maybe we need to muck-up something.
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우리가 앞선 '격렬한 우주'강의 에서 아인슈타인의 특수 상대론에 대해 다룰 때
When we talked about Einstein's theory of special relativity
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'변환'에 관한 개념을 생각해 냈었습니다.
in the Violent Universe course, we came up with idea of transforms.
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기준 좌표계를 잡고 다른 좌표계로 변환 하는 방법을 다뤘는데
How you convert from one frame of reference to another,
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끔찍히 어렵긴 했지만 그것은 기초였습니다.
and we did horrible things to it, and that was the basis there.
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아인슈타인은 거기에서 다시 시작할 셈이었죠.(특수 상대론에 이어 일반 상대론으로 확장할 셈이다.)
So Einstein was going to be up to it again.
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우리도 서툴나마 시작해 보시죠?
What if we start tinkering with this?
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브라이언: 아.
BRIAN: Ah.
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폴: 도움이 되겠죠?
PAUL: Could that help us?
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개략적으로나마 해볼까봐요...
Could tinkering with this--
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브라이언: 우리를 괴롭히실 작정이군요.
BRIAN: Sounds to me like it's going to hurt us
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피타고라스 정리 정도에서도 헤메는데.
if we mess around with Pythagoras' theorem.
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폴: 글쎄요. 지난세기 수학
PAUL: Well, there was some mathematics from the previous century that
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had played with this purely in an abstract way,
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아인슈타인은 이 문제를 파내려고 어렵게 애썼습니다.
and Einstein managed to dig this stuff up.
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브라이언: 넵.
BRIAN: Yep.
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폴: 그리고 그가 생각하길
PAUL: And he thought, well, maybe if we muck around with this,
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it will give us something that can behave like a force.
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브라이언: 좋아요.
BRIAN: OK.
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예를 들어 주세요.
So give me an example.
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폴: 그럴께요.
PAUL: OK.
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아까 거리 공식을 먼저 제곱합시다(제곱근이 포함된 수식은 불편하다).
So let's say we take the equation we had before, square it,
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그리고 약간 변형을 시켜봐요.
but now, we're going to change it a little bit.
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평범한 것은 그대로 두고요.
Since we have the normal thing normally.
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브라이언: 옙.
BRIAN: Yep.
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폴: 정수 눈금에 경계가 쳐진 특별한 x축이 있고 이에 대한 미소간격을 델타 x라 합시다.
PAUL: But let's say that delta x crosses an integer boundary.
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자 이제 델타x는 1.3에서 1.4까지라면 그대로 적용 합니다.
So delta x is going from 1.3 to 1.4-- just use this--
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그런데 미소범위가 정수경계를 넘어 갈 수도 있죠.
but then it goes over an integer boundary.
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그러니까, 1.95에서 2.05 사이라고 합시다. 정수경계를 넘었죠.
So let's say 1.95 to 2.05-- that's across an integer boundary--
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이런 경우 별도로 취급 하기로 하죠.
and let's use that instead.
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나머지(modulo) 값을 취하기로 합시다.
So we're going to take the modulus.
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그리하여 양수로 만든 델타x 에서 1을 뺀 후 제곱하는 것으로 하죠.
So this is a positive value, that, take off 1 and square it.
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브라이언: 오.
BRIAN: Oh.
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좋아요.
OK.
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폴: 그럼 우리의 우주는 어떻게 되겠습니까?
PAUL: So what's that going to do to our universe?
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브라이언: 제생각에는 한칸씩 튈 것 같은데요.
BRIAN: I think it's going to make us jump around a fair bit
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이 근처로 가면,
when you get to these little edges.
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한번 생각해보죠.
Let's think.
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폴: 네.
PAUL: Yeah.
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여기 좌표, X, Y 그리고 Z가 있죠.
So here we have coordinates, X, Y, and Z,
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그리고 각 축에 정수 경계가 있다고 합시다.
and these are the integer boundaries.
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브라이언: 좋습니다.
BRIAN: OK
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폴: 우리는 두가지를 상상해볼 수 있겠네요.
PAUL: So let's imagine we have two things here.
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저곳에서는 무슨일 이 생길까요?
What's going to happen there?
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브라이언: 좋아요.
BRIAN: OK.
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여기 두가지 경우가 있죠. 여기서 조금 저리 움직이고
So we have two things here, and let's say one's moving a little bit that way,
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조금 저리로 이동핳 겁니다.
and we're going to move a little bit that way.
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그러니까 dx 와 dx 만큼 움직인 것인데
Well, you move by dx and dx, and so that distance
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그랬더니 완벽하게 보통(정상)이 되었군요.
is going to be perfectly normal.
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그저 당신이 예상 했던 대로죠.
It's going to be just what you had expected.
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그러니까 경계가 없던 거나 마찬가지로요.
So not on the boundary.
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따라서 내가만든 측량법은 내가 움직인 거리가
And so my metric tells me that the distance I
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내가 움직였어야할 거리랑 같다는 것을 보여줍니다.
move is just the distance I should move.
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폴: 예.
PAUL: Yeah.
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그러니까 여기있는 입자나 저기있는 입자
So a particle there and a particle there,
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신의 컴퓨터내에, 당신이 만든 측량법에 따르면
in god's supercomputer, according to this metric,
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아주 귽ㅂ해 있다는 거죠. 그러므로 서로 영향권안에 있는 거구요.
are going to be close to each other, and so they're going to affect each other.
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그러니가 제 귀에 있는 두 원자가 인접 해 있다는 겁니다. 예를 들면요.
So it could be, for example, two atoms adjacent on my ear.
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브라이언: 좋습니다.
BRIAN: OK.
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폴: 그래서 그중 하낙 산책이라도 나가면
PAUL: And so if one of them goes for a walk,
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다른 하나도 꼭 함께하죠. 왜냐면 그 둘이 서로 영향관 안에 있으니까요.
so does the other one, because they're bound together.
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브라이언: 좋아요.
BRIAN: OK.
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폴: 그렇죠.
PAUL: OK.
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그런데 여기있는 입자와 저기있는 두 입자를 생각해 보시죠.
But now, let's say, an imaginary two particles, one there and one there.
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BRIAN: And we're going to be right on side, just one side of 3 or 4.
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4 in this case.
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So just one, like 3.9999 and 4.0001.
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And in this case, if we try to cross that boundary, we jump, right?
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PAUL: Yeah.
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So if we go back to here, we see delta x is very small, minus 1 square.
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This is going to be about 1 squared, about 1, so not very small.
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It's got pretty big.
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BRIAN: So now, a long ways apart.
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And so I'm going to go think, at some level,
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you would normally think they'd be right next to each other.
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But in this case, their saying, they're a long ways apart.
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It's like this one's almost like being over there.
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PAUL: Yeah.
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So it means if that one explodes, well, it won't affect that one,
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because they're not next to each other.
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They got coordinates that are similar, but the metric is changed.
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They're not going to affect each other.
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BRIAN: But interestingly enough, they do affect something else a long ways away.
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PAUL: Yeah, so let's imagine we take these two.
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What's going to happen now?
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BRIAN: All right.
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So in this case, these things are an integer apart.
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So if I do boom here, the explosion is going to map, not here, but over there.
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PAUL: Yeah.
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So this one's at, say, 1.99 and this is 3.01,
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so they're crossing an integer boundary.
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BRIAN: Yep.
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PAUL: And so we'd have to use this one.
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So the difference is like 1.01 minus 1, small numbers.
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Squared makes it even smaller.
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So it means, go back a bit, these two actually are close to each other
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and are going to affect each other.
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BRIAN: Right.
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PAUL: So what is this giving us?
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BRIAN: Well, it's going to give us kind of a checkered universe,
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I think, where you literally have these boundaries where suddenly everything
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changes, your universe changes, and you affect things differently.
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So?
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PAUL: Yeah, so it could look something like this.
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BRIAN: Oh, OK.
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So yes.
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So this is, ah, yes.
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It looks to me like a place far, far away, long, long time ago.
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And so you literally got to go through, and here, you affect
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and you map on to here.
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And this is lost, except for it maps onto here and it maps onto there.
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So everything maps on to things that are sort of away
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from itself, but just a bit.
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PAUL: So we could actually live in a universe with a metric like this.
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We wouldn't know.
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BRIAN: Well, unless you cross the boundary, and--
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PAUL: You can't cross the boundary, because if you go here,
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you're moving that way.
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You're going to the place which is a small metric away,
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which means you'll jump to there.
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BRIAN: OK.
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So we wouldn't know.
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It'd just look like it's next door to us.
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PAUL: So this could be our universe.
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BRIAN: OK.
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PAUL: Well, that's actually a rather trivial case,
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because we wouldn't be able to tell the difference in that one.
----------------------------------------------------------------
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이번에는 좀더 진지하게 이상한 상황을 상상해 보시죠.
But let's imagine something a bit more seriously weird.
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좌표를 x, y, z로 측정하는 대신
Instead of measuring our coordinates in X, Y, and Z,
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우리는 그것을 직교좌표계라고 하죠. 직교좌표 대신 원통 좌표를 써봐요.
so-called Cartesian coordinates, we can measure them in cylindrical polars.
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그 좌표계에서도 우리가 하던대로 측정 할 수 있죠.
We can measure them anyway we like.
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원통 좌표계는 X,Y Z 대신
Cylindrical polar coordinates, instead of measuring X, Y, and Z,
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반경 r, 그 반경까지 뻗은 직선이 이루는 각도, 그리고 높이 입니다.
you measure r, out, angle, and height.
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브라이언: 이 런 좌표계에서도 원하는 곳 어디든지 표현 할 수 있는 거죠? (측량법이 바뀌어도 좌표계사이의 변환에 문제 없음을 시사함)
BRIAN: And then you can get to any place you want with that coordinate?
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펄: 맞아요.
PAUL: Yes.
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브라이언: 좋습니다.
BRIAN: Good.
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폴: 총구 조준경 처럼
PAUL: Just like a gun sight.
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위와 아래, 돌고 돌죠.
So you're up and down, round and round.
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브라이언: 예.
BRIAN: Yeah.
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폴: 그리고 우리가 r 과 쎄타 그리고 z를 사용합니다.
PAUL: And so we've got r, theta and z.
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이 경우 만일 반지름을 아주 약간 움직였다면 측량된
Now, in this case, the metric, if you move the r by a bit,
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거리는 그 반지름이 움직인 거리와 동일 합니다.
the distance moves by that same amount.
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그러니까 s의 제곱은 그저 r의 제곱과 같죠.
So the s square is just the r squared.
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브라이언: 넵.
BRIAN: Yep.
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폴: z 도 마찬가지예요.
PAUL: Likewise, z.
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위로 약간 움직이면 그것이 곧 전체 움직인 거리 입니다.
You go up a bit, the distance is also going by that.
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브라이언: 넵.
BRIAN: Yep.
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폴: 하지만 쎄타는 다르죠. 왜냐면
PAUL: But theta's a bit different, because when
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당신이 중심에 아주 가깝게 있을 때 각도의 변화에 대해
you're very close to the center, an angle change
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움직인 거리는 크지 않아요. (각도와 원호의 길이)
doesn't move you very much.
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당신의 팔뚝이 아주 짧다면 각도를 변경해도 움직인 거리(원호의 길이)는 그리 길지 않죠.
So you got a very small arm, you change the angle, it doesn't move very much.
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하지만 아주 길 팔뚝을 가졌다면 각도의 변화에 움직인 길이는 커질 겁니다.
But a very long arm, you change the angle, it moves rather more.
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그러니까 움직인 거리(원호의 길이)를 (호도법으로)계산하면 r 제곱 곱하기 쎄타의 제곱이 들어갑니다.
So you need an r square d theta squared for that.
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브라이언: 그러니까 작은 각도 근사(호도법) 군요.
BRIAN: So that's just the small angle approximation,
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원호(arc)는 반지름(r) 곱하기 각도(쎄타)죠. 그 원호가 바로 움직인 거리구요.
where the arc is r theta, and that's how much you move.
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물론 이경우에도 피타고라스 정리를 적용하기 위해 제곱을 해야 합니다.
And of course, you got to square it here for Pythagorean's theorem.
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폴: 넵.
PAUL: Yeah.
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우리가 이 강좌의 첫번째 강의에서 나눈 얘기로 돌아가 보면
The same thing we used back in the first course in the series
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공간 상에서 사물의 크기를 게산하기 위해 같은 일을 했었습니다.
to work out the size of things out in space.
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이것이 우리의 우주이며 상식이죠.
So this is our universe, common sense.
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하지만 r의 제곱을 없앴다고 상상해봐요.
But let's imagine we get rid of that r squared.
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브라이언: 좋아요.
BRIAN: OK.
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폴: 그러면 우리는 이런 상황을 격게 됩니다.
PAUL: So we've now got this.
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이런 우주에서는 세상이 어떻게 돌아갈까요?
What's going to happen in this universe?
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브라이언: 좋습니다.
BRIAN: OK.
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폴: 여길 보세요.
PAUL: So we can look here.
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여기 각도 쎄타가 있죠.
We've got angle theta again.
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저기서 저기로 움직였다고 해보죠. (반경 r이 없다면)무슨일이 벌어질까요.
Now what's going to happen is, let's say you move from there to there.
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각도만 변화 한 것일 뿐이죠.
It's going to be a given change in angle.
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브라이언: 넵.
BRIAN: Yep.
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폴: 그리고 저기서 저기로 움직임이 각도만으로 본다면 같은게 되요.
PAUL: And moving from there to there is the same given change in angle.
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브랑언: 그렇군요.
BRIAN: All right.
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폴: 생각해보세요. 이런식의 (r이 사라진) 측량법(metric)에서는 움직인 거리 ds가
PAUL: Suppose they're actually the same ds, the same distance,
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(사실 원호의 길이가 다름에도) 여기나 저기나 같다는 거죠.
according to this metric.
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브라이언: 네 그렇군요.
BRIAN: All right.
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당신이 움직이는 방식을 달리하게 할 것 같네요.
So that sounds to me like it's going to change the way you move around
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그런 측량법이 적용된 우주에서는요.
in a universe that had this law associated with it.
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폴: 네.
PAUL: Yeah.
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빛의 파동이 이렇게 들어온다고 가정해 봅시다.
So let's imagine there's kind of light wave coming like this.
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당신도 기억 하겠지만 아마 두번째 강좌에서 '하위헌스 법칙'을 다뤘었죠.
And if you remember, we talked about in the second course, Huygens' principle,
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빛의 파장은 (굴절되어) 굴절면 양쪽으로
which says that the light wave, both sides of it
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이동한 거리는 같습니다.
https://ko.wikipedia.org/wiki/%ED%95%98%EC%9C%84%ED%97%8C%EC%8A%A4_%EC%9B%90%EB%A6%AC
are going to move the same distance.
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브라이언: 그랬죠.
BRIAN: Right.
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폴: 이 특정 ds는 저기에 도달 할 겁니다.
PAUL: So this particular ds would get to there,
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그리고 저쪽 것은 같은 ds 를 움직이겠죠. 여기를 돌아 저기에 도달 할 겁니다.
and that one would move the same ds, which would get it round to here.
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브라이언: 그렇죠.
BRIAN: All right.
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폴: 거기서부터 이렇게 움직이죠.
PAUL: And from there, then move like this.
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그럼 빛이 퍼져나가면ㅅ 무슨일이 벌어질지
So what's going to happen is, you get waves spreading out
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파형의 각지점에서, 그리고 위상이 모두 합쳐지는 곳,
from each point on the waveform, and where they all add up in phase,
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그렇게 움직일 겁니다.
it's going to move.
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브라이언: 그런가요?
BRIAN: Yes?
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폴: 그리고 모두 같은 ds죠.
PAUL: And that'll be the same ds which is
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각도만 같으면 동일한 것이니까요.
going to be moving at a constant angle around.
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그말은 빛의 파동이 직선으로 가지 않고
So that means the light wave, instead of having a straight line,
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회전 한다는 뜻이죠.
would go in a circle.
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브라이언: 음...
BRIAN: Mm.
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폴: 이렇게 빙빙 돌아요.
PAUL: Just go round and round.
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같은 현상을 물질에도 적용할 수 있죠.
And the same thing would apply to matter,
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왜냐면 물질도 양자역학 관점에서 파동이잖아요(드브로이 물질파).
https://ko.wikipedia.org/wiki/%EB%AC%BC%EC%A7%88%ED%8C%8C
because matter is quantum mechanical waves.
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브라이언: 맞아요.
BRIAN: Right.
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폴: 따라서 직선으로 걷길 원한다면 둥글게 돌아야 할 겁니다.
PAUL: So if you try to walk in a straight line, you'd go in a circle.
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브라이언: 좋습니다.
BRIAN: OK, so it is possible that the distance that you would travel,
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you'd think as being a straight line actually
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ends up being curved if you have that type of metric.
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폴: 이것은 그저 시작에 불과해요.
PAUL: And this is beginning to sound like what
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아인슈타인의 개념이 옳다는 것을 설명할 수 있어야 해요.
we need to make Einstein's idea come true.
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우리가 해야할 일은 어떤 물체가 가속될 수 있다는 겁니다.
What we need is something that could accelerate
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힘이 가해지지 않고도 가속될 수 있다는 겁니다.
things that doesn't involve a force.
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브라이언: 흠...
BRIAN: Hm.
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폴: 그리고 이는 원안에 들어갈 물질과 관련있죠. 그러니까 가속인데
PAUL: And this is getting things to go in a circle, which means acceleration,
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하지만 힘이 가해지진 않았습니다.
but there's no force involved.
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이는 단지 측량(metric)만을 바꾼 것이죠.
It's just changing the metric.
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시작은 이렇게 보는 것이 우리에게 도움이 되리라는 겁니다.
So it's beginning to look like this might be helpful to us.
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브라이언: 그것은 아주 팜신한 해법인데요. 근데, 네, 어떻게 작동 하는지 알 것 같아요.
BRIAN: It's a very tricky solution, but yeah, I could see how it might work.
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폴: 측량의 이해에 도움이 될 비유를 하나 더 들죠.
PAUL: There are other analogies to this which may help.
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이쯤 되면 대부분 사람들의 뇌는 그들의 귀를 흘리죠.(흘려듣게 되죠)
At this point, normally, most people's brains are dribbling out of their ears.
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브라이언: 맞아요.
BRIAN: Yes.
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폴: 그럼 좀더 괴롭혀 보지요.
PAUL: So let's try and make it even worse.
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여기에 다른 비유가 있어요.
We'll have another analogy.
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비유중 하나는 뜨거운 철판이 있습니다.
One analogy is a hot plate.
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벌래 한마리가 있는데 뜨거운 철판은 2차원의 우주라고 칩시다.
Let's say you've got a bug and you've got a two-dimensional universe, which
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is a hot plate.
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그리고 이 철판의 일부분이 열을 받았다고 하죠.
And let's imagine some parts of this hot plate are hot,
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벌래가 그위를 기어가면서 벌래의 크기가 커진다고 하죠.
and when the bug goes over them, it gets bigger.
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확장 되는 거죠.
It expands.
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그것은 철로된 벌래인데 뜨거운 부분을 지날때 마다 점점 커져요.
It's a metal bug, and so it gets larger when it gets over it.
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브라이언: 좋아요.
BRIAN: OK.
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폴: 이제 측량(metric)을 조금 바꿔볼께요.
PAUL: So that's a bit like changing the metric.
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거리 ds가 여기(차가운 곳)에서 저기(뜨거운 곳)보다 더 커져요.
That means the distances, ds's, are bigger here than over there.
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브라이언: 넵.
BRIAN: Yep.
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폴: 이제 벌레가 차가운 지역을 지나 갑니다.
PAUL: Now the bug's just been through a cold area.
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벌레 양쪽에 다리가 달렸고 같은 한번 걸음 할 대마다 양을 기어가죠.
It's got both legs, and they both advance the same amount
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마치 탱크의 양쪽 바퀴처럼, 그럼 앞으로 똑바로 갈겁니다.
like tank tracks or something, and it will go in a straight line.
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브라이언: 좋아요.
BRIAN: OK.
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폴: 하지만 벌래가 저 위(뜨거운 곳 옆)를 지나 여기까지 올때
PAUL: But now if we move it over there, then it comes over to here.
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오른쪽 다리는 확장되겠죠. 왜냐하면 그곳의
It's right-hand legs have expanded because they're
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뜨거운 철판의 측량이 다르니까요.
in this region with a different metric, a hot plate.
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왼쪽 다리는 여전히 차가우니까, 기어간 경로는 이렇게 옆으로 굽어집니다.
Whereas, the left ones are still cold, and so it's going to turn.
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음.
Mm.
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자 보세요. 단지 측량을 바꾼 것 만으로도 가속되는게 보이죠. (기어가는 속도 벡터의 방향이 바뀜)
So once again, we get acceleration without force.
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우리는 그냥 측량만 바꿨을 뿐인데 확실히 가속이 일어나네요.
We're just changing the metric and making things apparently accelerate.
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브라이언: 네. 그렇네요.
BRIAN: Yeah.
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폴: 좀 다르게 생각해보죠.
PAUL: Another way to see it is to imagine
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삼차원 우주에 이차원 생명체가 존재한다고 가정해 보는 겁니다.
two-dimensional creatures embedded in the three-dimensional universe.
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당신이 이차원 세상에 사는 벌레라고 상상해봐요.
So let's imagine you're a bug that lives in a two-dimensional space.
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삼차원 우주에서 공간이 휘었다고 생각 할 수 있죠.
We can imagine curving that space in the third dimension.
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예를 들어 광선이 이곳을 지난다고 합시다.
So let's imagine, for example, we had a light ray coming along here
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이렇게 휜 공간을 지나죠.
in this curved space.
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이런 두개의 빛은 그저 직선으로 간다고 생각 할 수 있죠.
Two light rays like this, you'd think can just go in a straight line,
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이쪽 빛은 더 많이 가는 겁니다. 왜냐면 밑으로 떨어졌다가 가니까요.
but this one's actually going to go further, because it's got to dip down.
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브라이언: 오. 그러니까 하나는 쭉 뻗는 한편 하나는 푹패인 곳을 지나네요.
BRIAN: Oh, so it's going to rurp, like that, while this one's going straight.
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폴: 그러니까 여기에서 두 빛이 보조(위상)를 맞춰야 하니까 (하위헌스 원리)
PAUL: So they won't add-up in phase here,
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하나가 더 많이 움직여야 합니다.
because that one's had to go further.
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만일 빛이 휜다면 위상을 늘리는 수 밖에 없죠. (빛의 파동성. 위상을 늘리면 파장이 길어짐)
They'll only add-up in phase if they curve around,
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결국 같은 거리를 움직인 셈이 되는 거죠.
so they've gone the same distance.
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BRIAN: Right.
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폴: 그리하여 안쪽으로 지나는 하나는 아래로 내려갔다 올라오죠.
PAUL: So the one inside, it's gone down and up.
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바깥쪽 빛줄기도 같은 거리를 이동하는데
The outer one's gone a the same distance because it's
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회전 반경의 바깥쪽을 멀리 돌아가기 때문입니다.
gone further around the outside of a circle.
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이쯤에서 상기하면 빛은 입자입니다.
So once again, light rays are a particle,
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이런 상황에서 빛이 휘어 움직일 겁니다.
will move in a curve in a situation like this.
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브라이언: 흠. 좋아요.
BRIAN: Hm, OK.
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폴: 이제 힘이 가해지지 않고도 가속되는 상황을 본거죠.
PAUL: So we seem to be getting acceleration without force.
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만일 우리가 측량을 흐트러 놓음으로써 사물들이 아주 이상하게 행동 하게 만듭니다.
If we muck-up the metric, it makes things behave rather weirdly.
1.2 팽창하는 우주, 허블의 법칙(Hubble's Law) 동영상/영문자막/한글자막
1.3 관성 질량과 중력 질량(Inertial and Gravitational Mass) 동영상/영문자막/한글자막
1.4 공간 (Space) 동영상/영문자막/한글자막
1.5 측량(The Metric) 동영상/영문자막/한글자막
* 이번 강좌는 두 물리학자가 '측량(Metric)'의 개념을 여러가지 예를 동원하여 이해시키려고 노력한다. '측량'은 특수 상대론에서 빛이 휘어 보이는 이유를 수학적으로 설명하기 위한 기초 개념이다. 이를 이해하기 쉽지 않다. 그래서 몇가지 예를 드는데 장황 하다. 이 강좌의 요점은 마지막 두문장이다.
폴: 이제껏, 힘이 가해지지 않고도 가속되는 상황을 본거죠.
PAUL: So we seem to be getting acceleration without force.
만일 우리가 측량을 흐트러 놓음으로써 사물들이 아주 이상하게 행동 하게 만듭니다.
If we muck-up the metric, it makes things behave rather weirdly.
* 십분여에 걸친 긴 강좌다. 중간에 일부 장황한 예는 생략하겠다.
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브라이언: 좋습니다. 폴.
BRIAN: All right, Paul.
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당신이 공간에 대해 설명해 주셨는데 확실하지 않군요.
So I'm not sure if you've illuminated us on what space is,
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제가 이해한 바로는 공간이란
but as near as I can understand, the idea
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숫자로 표현될 수 있다는 거죠.
is that space is described by numbers.
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공간은 모두 숫자를 가지고 있다는 거죠. 이런 숫자들을 사용할 수 있다는 건데...
Every part of space has got a number, and we can use those numbers--
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폴: 세개의 숫자지요.
PAUL: Three numbers.
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브라이언: 네. 세개의 숫자.
BRIAN: Well, three numbers, sorry.
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맞아요. 숫자 세개.
Yes, three numbers.
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세개의 숫자라는 것은 기본적으로
And that those numbers can be used to describe, essentially,
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수학으로 사물들이 서로 얼마나 멀리 떨어져 있는지 표현할 수 있다는 거네요.
how far things are apart by using mathematics.
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그러니까 중력이라던가 그런 물리 법칙들이
And we care about that because physical laws, like gravity and things,
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숫자공간에서 거리에 달려 있다는 거죠.
depend on that distance in number space.
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숫자로 공간을 기술하는 고도의 방법이라는 거네요.
And so that's sort of an abstract way of describing space in numbers.
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폴: 여기서 중요한 것은
PAUL: So the crucial thing here is the concept
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'측량(metric)'이라고 부르는 개념 입니다.
of what we call the metric, which is, if you've
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세개의 숫자로 표현된 두 지점의
got two things with their sets of three numbers,
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거리를 재는 방법이 바로 '측량' 입니다.
to work out how close they are.
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여기 지구에서 두 지점의 좌표를 X-Y-Z 로 알고 있다고 합시다.
So let's say I've got X-Y-Z coordinates of two things on Earth.
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그 두 지점이 얼마나 떨어져 있는지 계산하는 방법이 뭘까요?
How will you work out how far apart they are?
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브라이언: 네. 그 방법은 고대 그리스 수학자가 이미 알려 줬었죠.
BRIAN: Well, that's easy, because that's just what the Greeks told us.
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피타고라스 정리라고 하는 거요.
Pythagorean theorem.
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폴: 넵.
PAUL: Yep.
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브라이언: 오.
BRIAN: Whoops.
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바로 이겁니다.
PAUL: Here it is.
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미소 거리를 계산해 볼께요.
So we're talking about small distances.
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여기 델타()라고 한 의미는 x축으로 작은 차이, y축으로 작은 차이,
So a delta means a small difference in x, a small difference in y,
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z 축으로 작은 차이를 의미 합니다.
a small difference in z.
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만일 전체 거리를 계산 할 거라면 이 미소거리를 합치면 되겠죠.
If you had to go a big distance, you just add up all these small differences
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as you go.
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이것이 바로 피타고라스 정리 입니다. 거리란
And so this is just Pythagoras' theorem, that the distance is
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x의 제곱, y의 제곱 그리고 z의 제곱을 더해
equal to the square root of the difference in x-coordinates squared,
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제곱근을 씌운 겁니다.
the difference in y-coordinates squared, the difference in z-coordinates
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squared.
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그게 바로 피타고라스 정리죠.
That's just Pythagoras' theorem.
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브라이언: 만일 삼차원 이라면, X-Y-X는
BRIAN: If you have a three-dimensional triangle, that's X-Y-Z,
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경사(hypotenuse)의 길이가 되겠네요.
that's how long the hypotenuse is.
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그리고 우리가 계산한 이 거리는 아주 작고
And we're going to make sure that our distances are really small here,
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그 미소 길이를 모두 직선으로 취급 할 수 있어요.
so we can treat everything as a straight line.
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폴: 넵. 그럼 당신이 신의 슈퍼컴퓨터를 가졌다면
PAUL: Yeah, so if you have god's supercomputer
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수많은 것들을 이 숫자들을 이용해 나타낼 수 있어요.
and we have lots of things with different sets of numbers,
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이 숫자를 이용해 그 두점이 가까워 질 수 있는지 알아 볼 수 있습니다.
we can use this to work out if they're going to be close.
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그리하여 만일 그중 하나가 폭발 하면 어디까지 영향을 줄지 알아볼 수 있다는 거죠,
And therefore, if one explodes, which things are going to be affected,
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예를 들어서 말이죠.
for example.
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브라이언: 좋습니다.
BRIAN: OK.
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문제 없어요.
No worries.
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쉽군요.
This is easy.
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폴: 네.
PAUL: Yeah.
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이런 식으로 우주를 시뮬레이션 해보죠.
So we can simulate the universe this way.
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이제 아인슈타인의 가속과 질량에 관한 문제로 돌아가볼까요.
But now, let's go back to Einstein's problem, acceleration and mass.
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브라이언: 뭐라구요?
BRIAN: Yes?
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폴: 어떻게 가속 시킬 수 있을지 그리고 서로 어떤 힘이 관계되어 있을까요?
PAUL: How can we make acceleration and a force relate to each other?
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마구 뒤섞어 놓아야 할겁니다.
Maybe we need to muck-up something.
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우리가 앞선 '격렬한 우주'강의 에서 아인슈타인의 특수 상대론에 대해 다룰 때
When we talked about Einstein's theory of special relativity
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'변환'에 관한 개념을 생각해 냈었습니다.
in the Violent Universe course, we came up with idea of transforms.
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기준 좌표계를 잡고 다른 좌표계로 변환 하는 방법을 다뤘는데
How you convert from one frame of reference to another,
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끔찍히 어렵긴 했지만 그것은 기초였습니다.
and we did horrible things to it, and that was the basis there.
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아인슈타인은 거기에서 다시 시작할 셈이었죠.(특수 상대론에 이어 일반 상대론으로 확장할 셈이다.)
So Einstein was going to be up to it again.
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우리도 서툴나마 시작해 보시죠?
What if we start tinkering with this?
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브라이언: 아.
BRIAN: Ah.
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폴: 도움이 되겠죠?
PAUL: Could that help us?
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개략적으로나마 해볼까봐요...
Could tinkering with this--
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브라이언: 우리를 괴롭히실 작정이군요.
BRIAN: Sounds to me like it's going to hurt us
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피타고라스 정리 정도에서도 헤메는데.
if we mess around with Pythagoras' theorem.
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폴: 글쎄요. 지난세기 수학
PAUL: Well, there was some mathematics from the previous century that
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had played with this purely in an abstract way,
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아인슈타인은 이 문제를 파내려고 어렵게 애썼습니다.
and Einstein managed to dig this stuff up.
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브라이언: 넵.
BRIAN: Yep.
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폴: 그리고 그가 생각하길
PAUL: And he thought, well, maybe if we muck around with this,
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it will give us something that can behave like a force.
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브라이언: 좋아요.
BRIAN: OK.
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예를 들어 주세요.
So give me an example.
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폴: 그럴께요.
PAUL: OK.
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아까 거리 공식을 먼저 제곱합시다(제곱근이 포함된 수식은 불편하다).
So let's say we take the equation we had before, square it,
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그리고 약간 변형을 시켜봐요.
but now, we're going to change it a little bit.
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평범한 것은 그대로 두고요.
Since we have the normal thing normally.
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브라이언: 옙.
BRIAN: Yep.
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폴: 정수 눈금에 경계가 쳐진 특별한 x축이 있고 이에 대한 미소간격을 델타 x라 합시다.
PAUL: But let's say that delta x crosses an integer boundary.
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자 이제 델타x는 1.3에서 1.4까지라면 그대로 적용 합니다.
So delta x is going from 1.3 to 1.4-- just use this--
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그런데 미소범위가 정수경계를 넘어 갈 수도 있죠.
but then it goes over an integer boundary.
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그러니까, 1.95에서 2.05 사이라고 합시다. 정수경계를 넘었죠.
So let's say 1.95 to 2.05-- that's across an integer boundary--
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이런 경우 별도로 취급 하기로 하죠.
and let's use that instead.
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나머지(modulo) 값을 취하기로 합시다.
So we're going to take the modulus.
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그리하여 양수로 만든 델타x 에서 1을 뺀 후 제곱하는 것으로 하죠.
So this is a positive value, that, take off 1 and square it.
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브라이언: 오.
BRIAN: Oh.
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좋아요.
OK.
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폴: 그럼 우리의 우주는 어떻게 되겠습니까?
PAUL: So what's that going to do to our universe?
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브라이언: 제생각에는 한칸씩 튈 것 같은데요.
BRIAN: I think it's going to make us jump around a fair bit
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이 근처로 가면,
when you get to these little edges.
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한번 생각해보죠.
Let's think.
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폴: 네.
PAUL: Yeah.
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여기 좌표, X, Y 그리고 Z가 있죠.
So here we have coordinates, X, Y, and Z,
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그리고 각 축에 정수 경계가 있다고 합시다.
and these are the integer boundaries.
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브라이언: 좋습니다.
BRIAN: OK
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폴: 우리는 두가지를 상상해볼 수 있겠네요.
PAUL: So let's imagine we have two things here.
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저곳에서는 무슨일 이 생길까요?
What's going to happen there?
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브라이언: 좋아요.
BRIAN: OK.
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여기 두가지 경우가 있죠. 여기서 조금 저리 움직이고
So we have two things here, and let's say one's moving a little bit that way,
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조금 저리로 이동핳 겁니다.
and we're going to move a little bit that way.
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그러니까 dx 와 dx 만큼 움직인 것인데
Well, you move by dx and dx, and so that distance
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그랬더니 완벽하게 보통(정상)이 되었군요.
is going to be perfectly normal.
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그저 당신이 예상 했던 대로죠.
It's going to be just what you had expected.
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그러니까 경계가 없던 거나 마찬가지로요.
So not on the boundary.
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따라서 내가만든 측량법은 내가 움직인 거리가
And so my metric tells me that the distance I
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내가 움직였어야할 거리랑 같다는 것을 보여줍니다.
move is just the distance I should move.
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폴: 예.
PAUL: Yeah.
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그러니까 여기있는 입자나 저기있는 입자
So a particle there and a particle there,
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신의 컴퓨터내에, 당신이 만든 측량법에 따르면
in god's supercomputer, according to this metric,
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아주 귽ㅂ해 있다는 거죠. 그러므로 서로 영향권안에 있는 거구요.
are going to be close to each other, and so they're going to affect each other.
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그러니가 제 귀에 있는 두 원자가 인접 해 있다는 겁니다. 예를 들면요.
So it could be, for example, two atoms adjacent on my ear.
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브라이언: 좋습니다.
BRIAN: OK.
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폴: 그래서 그중 하낙 산책이라도 나가면
PAUL: And so if one of them goes for a walk,
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다른 하나도 꼭 함께하죠. 왜냐면 그 둘이 서로 영향관 안에 있으니까요.
so does the other one, because they're bound together.
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브라이언: 좋아요.
BRIAN: OK.
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폴: 그렇죠.
PAUL: OK.
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그런데 여기있는 입자와 저기있는 두 입자를 생각해 보시죠.
But now, let's say, an imaginary two particles, one there and one there.
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BRIAN: And we're going to be right on side, just one side of 3 or 4.
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4 in this case.
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So just one, like 3.9999 and 4.0001.
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And in this case, if we try to cross that boundary, we jump, right?
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PAUL: Yeah.
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So if we go back to here, we see delta x is very small, minus 1 square.
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This is going to be about 1 squared, about 1, so not very small.
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It's got pretty big.
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BRIAN: So now, a long ways apart.
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And so I'm going to go think, at some level,
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you would normally think they'd be right next to each other.
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But in this case, their saying, they're a long ways apart.
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It's like this one's almost like being over there.
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PAUL: Yeah.
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So it means if that one explodes, well, it won't affect that one,
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because they're not next to each other.
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They got coordinates that are similar, but the metric is changed.
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They're not going to affect each other.
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BRIAN: But interestingly enough, they do affect something else a long ways away.
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PAUL: Yeah, so let's imagine we take these two.
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What's going to happen now?
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BRIAN: All right.
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So in this case, these things are an integer apart.
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So if I do boom here, the explosion is going to map, not here, but over there.
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PAUL: Yeah.
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So this one's at, say, 1.99 and this is 3.01,
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so they're crossing an integer boundary.
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BRIAN: Yep.
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PAUL: And so we'd have to use this one.
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So the difference is like 1.01 minus 1, small numbers.
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Squared makes it even smaller.
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So it means, go back a bit, these two actually are close to each other
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and are going to affect each other.
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BRIAN: Right.
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PAUL: So what is this giving us?
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BRIAN: Well, it's going to give us kind of a checkered universe,
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I think, where you literally have these boundaries where suddenly everything
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changes, your universe changes, and you affect things differently.
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So?
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PAUL: Yeah, so it could look something like this.
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BRIAN: Oh, OK.
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So yes.
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So this is, ah, yes.
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It looks to me like a place far, far away, long, long time ago.
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And so you literally got to go through, and here, you affect
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and you map on to here.
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And this is lost, except for it maps onto here and it maps onto there.
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So everything maps on to things that are sort of away
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from itself, but just a bit.
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PAUL: So we could actually live in a universe with a metric like this.
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We wouldn't know.
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BRIAN: Well, unless you cross the boundary, and--
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PAUL: You can't cross the boundary, because if you go here,
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you're moving that way.
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You're going to the place which is a small metric away,
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which means you'll jump to there.
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BRIAN: OK.
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So we wouldn't know.
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It'd just look like it's next door to us.
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PAUL: So this could be our universe.
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BRIAN: OK.
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PAUL: Well, that's actually a rather trivial case,
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because we wouldn't be able to tell the difference in that one.
----------------------------------------------------------------
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이번에는 좀더 진지하게 이상한 상황을 상상해 보시죠.
But let's imagine something a bit more seriously weird.
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좌표를 x, y, z로 측정하는 대신
Instead of measuring our coordinates in X, Y, and Z,
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우리는 그것을 직교좌표계라고 하죠. 직교좌표 대신 원통 좌표를 써봐요.
so-called Cartesian coordinates, we can measure them in cylindrical polars.
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그 좌표계에서도 우리가 하던대로 측정 할 수 있죠.
We can measure them anyway we like.
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원통 좌표계는 X,Y Z 대신
Cylindrical polar coordinates, instead of measuring X, Y, and Z,
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반경 r, 그 반경까지 뻗은 직선이 이루는 각도, 그리고 높이 입니다.
you measure r, out, angle, and height.
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브라이언: 이 런 좌표계에서도 원하는 곳 어디든지 표현 할 수 있는 거죠? (측량법이 바뀌어도 좌표계사이의 변환에 문제 없음을 시사함)
BRIAN: And then you can get to any place you want with that coordinate?
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펄: 맞아요.
PAUL: Yes.
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브라이언: 좋습니다.
BRIAN: Good.
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폴: 총구 조준경 처럼
PAUL: Just like a gun sight.
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위와 아래, 돌고 돌죠.
So you're up and down, round and round.
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브라이언: 예.
BRIAN: Yeah.
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폴: 그리고 우리가 r 과 쎄타 그리고 z를 사용합니다.
PAUL: And so we've got r, theta and z.
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이 경우 만일 반지름을 아주 약간 움직였다면 측량된
Now, in this case, the metric, if you move the r by a bit,
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거리는 그 반지름이 움직인 거리와 동일 합니다.
the distance moves by that same amount.
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그러니까 s의 제곱은 그저 r의 제곱과 같죠.
So the s square is just the r squared.
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브라이언: 넵.
BRIAN: Yep.
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폴: z 도 마찬가지예요.
PAUL: Likewise, z.
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위로 약간 움직이면 그것이 곧 전체 움직인 거리 입니다.
You go up a bit, the distance is also going by that.
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브라이언: 넵.
BRIAN: Yep.
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폴: 하지만 쎄타는 다르죠. 왜냐면
PAUL: But theta's a bit different, because when
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당신이 중심에 아주 가깝게 있을 때 각도의 변화에 대해
you're very close to the center, an angle change
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움직인 거리는 크지 않아요. (각도와 원호의 길이)
doesn't move you very much.
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당신의 팔뚝이 아주 짧다면 각도를 변경해도 움직인 거리(원호의 길이)는 그리 길지 않죠.
So you got a very small arm, you change the angle, it doesn't move very much.
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하지만 아주 길 팔뚝을 가졌다면 각도의 변화에 움직인 길이는 커질 겁니다.
But a very long arm, you change the angle, it moves rather more.
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그러니까 움직인 거리(원호의 길이)를 (호도법으로)계산하면 r 제곱 곱하기 쎄타의 제곱이 들어갑니다.
So you need an r square d theta squared for that.
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브라이언: 그러니까 작은 각도 근사(호도법) 군요.
BRIAN: So that's just the small angle approximation,
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원호(arc)는 반지름(r) 곱하기 각도(쎄타)죠. 그 원호가 바로 움직인 거리구요.
where the arc is r theta, and that's how much you move.
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물론 이경우에도 피타고라스 정리를 적용하기 위해 제곱을 해야 합니다.
And of course, you got to square it here for Pythagorean's theorem.
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폴: 넵.
PAUL: Yeah.
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우리가 이 강좌의 첫번째 강의에서 나눈 얘기로 돌아가 보면
The same thing we used back in the first course in the series
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공간 상에서 사물의 크기를 게산하기 위해 같은 일을 했었습니다.
to work out the size of things out in space.
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이것이 우리의 우주이며 상식이죠.
So this is our universe, common sense.
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하지만 r의 제곱을 없앴다고 상상해봐요.
But let's imagine we get rid of that r squared.
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브라이언: 좋아요.
BRIAN: OK.
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폴: 그러면 우리는 이런 상황을 격게 됩니다.
PAUL: So we've now got this.
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이런 우주에서는 세상이 어떻게 돌아갈까요?
What's going to happen in this universe?
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브라이언: 좋습니다.
BRIAN: OK.
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폴: 여길 보세요.
PAUL: So we can look here.
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여기 각도 쎄타가 있죠.
We've got angle theta again.
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저기서 저기로 움직였다고 해보죠. (반경 r이 없다면)무슨일이 벌어질까요.
Now what's going to happen is, let's say you move from there to there.
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각도만 변화 한 것일 뿐이죠.
It's going to be a given change in angle.
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브라이언: 넵.
BRIAN: Yep.
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폴: 그리고 저기서 저기로 움직임이 각도만으로 본다면 같은게 되요.
PAUL: And moving from there to there is the same given change in angle.
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브랑언: 그렇군요.
BRIAN: All right.
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폴: 생각해보세요. 이런식의 (r이 사라진) 측량법(metric)에서는 움직인 거리 ds가
PAUL: Suppose they're actually the same ds, the same distance,
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(사실 원호의 길이가 다름에도) 여기나 저기나 같다는 거죠.
according to this metric.
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브라이언: 네 그렇군요.
BRIAN: All right.
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당신이 움직이는 방식을 달리하게 할 것 같네요.
So that sounds to me like it's going to change the way you move around
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그런 측량법이 적용된 우주에서는요.
in a universe that had this law associated with it.
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폴: 네.
PAUL: Yeah.
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빛의 파동이 이렇게 들어온다고 가정해 봅시다.
So let's imagine there's kind of light wave coming like this.
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당신도 기억 하겠지만 아마 두번째 강좌에서 '하위헌스 법칙'을 다뤘었죠.
And if you remember, we talked about in the second course, Huygens' principle,
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빛의 파장은 (굴절되어) 굴절면 양쪽으로
which says that the light wave, both sides of it
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이동한 거리는 같습니다.
https://ko.wikipedia.org/wiki/%ED%95%98%EC%9C%84%ED%97%8C%EC%8A%A4_%EC%9B%90%EB%A6%AC
are going to move the same distance.
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브라이언: 그랬죠.
BRIAN: Right.
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폴: 이 특정 ds는 저기에 도달 할 겁니다.
PAUL: So this particular ds would get to there,
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그리고 저쪽 것은 같은 ds 를 움직이겠죠. 여기를 돌아 저기에 도달 할 겁니다.
and that one would move the same ds, which would get it round to here.
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브라이언: 그렇죠.
BRIAN: All right.
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폴: 거기서부터 이렇게 움직이죠.
PAUL: And from there, then move like this.
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그럼 빛이 퍼져나가면ㅅ 무슨일이 벌어질지
So what's going to happen is, you get waves spreading out
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파형의 각지점에서, 그리고 위상이 모두 합쳐지는 곳,
from each point on the waveform, and where they all add up in phase,
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그렇게 움직일 겁니다.
it's going to move.
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브라이언: 그런가요?
BRIAN: Yes?
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폴: 그리고 모두 같은 ds죠.
PAUL: And that'll be the same ds which is
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각도만 같으면 동일한 것이니까요.
going to be moving at a constant angle around.
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그말은 빛의 파동이 직선으로 가지 않고
So that means the light wave, instead of having a straight line,
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회전 한다는 뜻이죠.
would go in a circle.
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브라이언: 음...
BRIAN: Mm.
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폴: 이렇게 빙빙 돌아요.
PAUL: Just go round and round.
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같은 현상을 물질에도 적용할 수 있죠.
And the same thing would apply to matter,
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왜냐면 물질도 양자역학 관점에서 파동이잖아요(드브로이 물질파).
https://ko.wikipedia.org/wiki/%EB%AC%BC%EC%A7%88%ED%8C%8C
because matter is quantum mechanical waves.
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브라이언: 맞아요.
BRIAN: Right.
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폴: 따라서 직선으로 걷길 원한다면 둥글게 돌아야 할 겁니다.
PAUL: So if you try to walk in a straight line, you'd go in a circle.
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브라이언: 좋습니다.
BRIAN: OK, so it is possible that the distance that you would travel,
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you'd think as being a straight line actually
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ends up being curved if you have that type of metric.
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폴: 이것은 그저 시작에 불과해요.
PAUL: And this is beginning to sound like what
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아인슈타인의 개념이 옳다는 것을 설명할 수 있어야 해요.
we need to make Einstein's idea come true.
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우리가 해야할 일은 어떤 물체가 가속될 수 있다는 겁니다.
What we need is something that could accelerate
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힘이 가해지지 않고도 가속될 수 있다는 겁니다.
things that doesn't involve a force.
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브라이언: 흠...
BRIAN: Hm.
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폴: 그리고 이는 원안에 들어갈 물질과 관련있죠. 그러니까 가속인데
PAUL: And this is getting things to go in a circle, which means acceleration,
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하지만 힘이 가해지진 않았습니다.
but there's no force involved.
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이는 단지 측량(metric)만을 바꾼 것이죠.
It's just changing the metric.
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시작은 이렇게 보는 것이 우리에게 도움이 되리라는 겁니다.
So it's beginning to look like this might be helpful to us.
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브라이언: 그것은 아주 팜신한 해법인데요. 근데, 네, 어떻게 작동 하는지 알 것 같아요.
BRIAN: It's a very tricky solution, but yeah, I could see how it might work.
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폴: 측량의 이해에 도움이 될 비유를 하나 더 들죠.
PAUL: There are other analogies to this which may help.
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이쯤 되면 대부분 사람들의 뇌는 그들의 귀를 흘리죠.(흘려듣게 되죠)
At this point, normally, most people's brains are dribbling out of their ears.
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브라이언: 맞아요.
BRIAN: Yes.
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폴: 그럼 좀더 괴롭혀 보지요.
PAUL: So let's try and make it even worse.
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여기에 다른 비유가 있어요.
We'll have another analogy.
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비유중 하나는 뜨거운 철판이 있습니다.
One analogy is a hot plate.
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벌래 한마리가 있는데 뜨거운 철판은 2차원의 우주라고 칩시다.
Let's say you've got a bug and you've got a two-dimensional universe, which
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is a hot plate.
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그리고 이 철판의 일부분이 열을 받았다고 하죠.
And let's imagine some parts of this hot plate are hot,
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벌래가 그위를 기어가면서 벌래의 크기가 커진다고 하죠.
and when the bug goes over them, it gets bigger.
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확장 되는 거죠.
It expands.
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그것은 철로된 벌래인데 뜨거운 부분을 지날때 마다 점점 커져요.
It's a metal bug, and so it gets larger when it gets over it.
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브라이언: 좋아요.
BRIAN: OK.
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폴: 이제 측량(metric)을 조금 바꿔볼께요.
PAUL: So that's a bit like changing the metric.
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거리 ds가 여기(차가운 곳)에서 저기(뜨거운 곳)보다 더 커져요.
That means the distances, ds's, are bigger here than over there.
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브라이언: 넵.
BRIAN: Yep.
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폴: 이제 벌레가 차가운 지역을 지나 갑니다.
PAUL: Now the bug's just been through a cold area.
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벌레 양쪽에 다리가 달렸고 같은 한번 걸음 할 대마다 양을 기어가죠.
It's got both legs, and they both advance the same amount
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마치 탱크의 양쪽 바퀴처럼, 그럼 앞으로 똑바로 갈겁니다.
like tank tracks or something, and it will go in a straight line.
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브라이언: 좋아요.
BRIAN: OK.
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폴: 하지만 벌래가 저 위(뜨거운 곳 옆)를 지나 여기까지 올때
PAUL: But now if we move it over there, then it comes over to here.
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오른쪽 다리는 확장되겠죠. 왜냐하면 그곳의
It's right-hand legs have expanded because they're
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뜨거운 철판의 측량이 다르니까요.
in this region with a different metric, a hot plate.
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왼쪽 다리는 여전히 차가우니까, 기어간 경로는 이렇게 옆으로 굽어집니다.
Whereas, the left ones are still cold, and so it's going to turn.
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음.
Mm.
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자 보세요. 단지 측량을 바꾼 것 만으로도 가속되는게 보이죠. (기어가는 속도 벡터의 방향이 바뀜)
So once again, we get acceleration without force.
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우리는 그냥 측량만 바꿨을 뿐인데 확실히 가속이 일어나네요.
We're just changing the metric and making things apparently accelerate.
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브라이언: 네. 그렇네요.
BRIAN: Yeah.
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폴: 좀 다르게 생각해보죠.
PAUL: Another way to see it is to imagine
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삼차원 우주에 이차원 생명체가 존재한다고 가정해 보는 겁니다.
two-dimensional creatures embedded in the three-dimensional universe.
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당신이 이차원 세상에 사는 벌레라고 상상해봐요.
So let's imagine you're a bug that lives in a two-dimensional space.
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삼차원 우주에서 공간이 휘었다고 생각 할 수 있죠.
We can imagine curving that space in the third dimension.
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예를 들어 광선이 이곳을 지난다고 합시다.
So let's imagine, for example, we had a light ray coming along here
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이렇게 휜 공간을 지나죠.
in this curved space.
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이런 두개의 빛은 그저 직선으로 간다고 생각 할 수 있죠.
Two light rays like this, you'd think can just go in a straight line,
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이쪽 빛은 더 많이 가는 겁니다. 왜냐면 밑으로 떨어졌다가 가니까요.
but this one's actually going to go further, because it's got to dip down.
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브라이언: 오. 그러니까 하나는 쭉 뻗는 한편 하나는 푹패인 곳을 지나네요.
BRIAN: Oh, so it's going to rurp, like that, while this one's going straight.
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폴: 그러니까 여기에서 두 빛이 보조(위상)를 맞춰야 하니까 (하위헌스 원리)
PAUL: So they won't add-up in phase here,
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하나가 더 많이 움직여야 합니다.
because that one's had to go further.
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만일 빛이 휜다면 위상을 늘리는 수 밖에 없죠. (빛의 파동성. 위상을 늘리면 파장이 길어짐)
They'll only add-up in phase if they curve around,
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결국 같은 거리를 움직인 셈이 되는 거죠.
so they've gone the same distance.
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BRIAN: Right.
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폴: 그리하여 안쪽으로 지나는 하나는 아래로 내려갔다 올라오죠.
PAUL: So the one inside, it's gone down and up.
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바깥쪽 빛줄기도 같은 거리를 이동하는데
The outer one's gone a the same distance because it's
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회전 반경의 바깥쪽을 멀리 돌아가기 때문입니다.
gone further around the outside of a circle.
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이쯤에서 상기하면 빛은 입자입니다.
So once again, light rays are a particle,
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이런 상황에서 빛이 휘어 움직일 겁니다.
will move in a curve in a situation like this.
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브라이언: 흠. 좋아요.
BRIAN: Hm, OK.
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폴: 이제 힘이 가해지지 않고도 가속되는 상황을 본거죠.
PAUL: So we seem to be getting acceleration without force.
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만일 우리가 측량을 흐트러 놓음으로써 사물들이 아주 이상하게 행동 하게 만듭니다.
If we muck-up the metric, it makes things behave rather weirdly.
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