특수 상대론을 적용하면 어떤 일이 벌어질까? 시간지연(Time Dilation)과 거리수축(Length Contraction) 그리고 쌍둥이 모순(Twin Paradox)
If you decide to hold a party, you need to tell your guests where and when they should arrive. It's not enough to simply tell your friends where the party is without telling them when it occurs or vice-versa. Therefore, when we use the word event, we are using it to describe where and when something is happening. Events can describe things like your arrival at a party, the time you spilled your drink, or even something as simple as snapping your fingers can be an event. I could say that I snapped my fingers about five seconds ago at this location, right here. So, an event must include details of both the position and the time.
"사건(event)"은 말할때는 장소(Position)와 시간(Time)을 명시해 주어야 한다. 파티 초대장에 장소와 시간을 명시하지 않으면 손님이 어찌 찾아오겠는가?
Our universe has three spatial dimensions. So, a defined event could look something like X, Y, Z, and T where X, Y, and Z define the position and T defines the time.
우리는 3차원 공간의 우주에서 살고있다. 그리고 이 우주에서 일어나는 사건은 3차원의 공간과 시간이 포함된 4ㅏ차원으로 기술되어야 한다.
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[아래 내용 이해불가]
A good example of this occurs in the popular TV show, Big Bang Theory. In an episode entitled The Cushion Saturation, Dr. Sheldon Cooper explains the first time he sat on his favorite spot on the couch as follows,
"In an ever-changing world, it is a single point of consistency. If my life were expressed as a function on a four-dimensional Cartesian coordinate system, that spot at that moment, I first sat on it would be zero, zero, zero, zero."
Our universe has three spatial dimensions. So, a defined event could look something like X, Y, Z, and T where X, Y, and Z define the position and T defines the time.
우리는 3차원 공간의 우주에서 살고있다. 그리고 이 우주에서 일어나는 사건은 3차원의 공간과 시간이 포함된 4ㅏ차원으로 기술되어야 한다.
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[아래 내용 이해불가]
A good example of this occurs in the popular TV show, Big Bang Theory. In an episode entitled The Cushion Saturation, Dr. Sheldon Cooper explains the first time he sat on his favorite spot on the couch as follows,
"In an ever-changing world, it is a single point of consistency. If my life were expressed as a function on a four-dimensional Cartesian coordinate system, that spot at that moment, I first sat on it would be zero, zero, zero, zero."
Sheldon feels most comfortable and at home at the X, Y, Z coordinates of his spot, all zero, zero, zero. Although he can return to the spot on the couch many times, and he does, he can never return to the exact moment when he first sat on the couch to experience that event again. The reason for this is that the fourth coordinate in the event is time, T, and is always increasing as time passes.
Suppose he originally sat down to enjoy a 40-minutes episode of Star Trek. We can describe the end of the show as an event that happened at zero, zero, zero, 40 minutes. The only way to return to the original event would be to use a time machine such as the one Reto Hofstetter took a ride in, in Big Bang Theory episode "The Nerdvana Annihilation."
If we were to return to Sheldon's first time on the couch and saw Penny riding a skateboard past the scene, Sheldon would see her in his reference frame. Would Penny see the first moment he sits in that spot followed by a 40-minute episode of Star Trek in the same order Sheldon experiences it?
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Considering our previous discussion about slicing of space-time, do you think all observers see the two events in the same order or with the same time interval between the events? Let's explore this further using one of Sheldon's favorite objects, trains.
사건의 순서는 같지만 시간의 간격은 다를 수 있다. 관찰자마다 빠르게 혹은 느리게 간다.
Sheldon is preparing for a trip on the Napa Valley Wine Train in his favorite 1915 Pullman-Standard lounge car. Ever the physicist, Sheldon would like to conduct an experiment to test the consequences of a constant speed of light. To do this, he sets up a light source in the center train car and has two of his friends standing in his light detectors in the caboose and the engine at either ends of the train.
While the train is in the station, we say that it is at rest. Here in the station, Sheldon conducts his first experiment by flashing the light and recording the arrival times that his friends measure at each end of the train. Since the distance to the light source is the same from both friends, they each detect the light at the same time. We call this a simultaneous detection.
등속으로 움직이는(관성계) 기차안에서 빛을 발사하여 기차 양끝에 도달하는 시간을 재보자. 광원에서 양측의 관측자 사이의 거리는 동일하다. 실험자(observer)와 관측자 모두 기차에 타고 움직인다. 빛은 동시에 관측된다(Simultaneous Detection).
The train then leaves the station and begins its journey traveling at a constant speed through the mountains. Sheldon prepares the experiment again, this time with the train in motion. Since the train is traveling at a constant speed, once again, the flash of light arrives at each of his friends at precisely the same time. In both the stationary case and the one with the moving train, the observers are at rest with respect to Sheldon and the light source. As a result, they observe the pulses arriving simultaneously on each occasion.
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Sheldon wants to know what his experiment would look like if he was stationary with respect to a moving train, so he gets off at the next stop. He gets ahead of the train and get set up to redo his experiment as the train passes at a constant velocity. This time, the light source flashes the moment that it passes Sheldon. Since the train is moving from left to right, Sheldon observes the light pulse arrive at the caboose of the train first followed by the arrival of the pulse at the engine of the train second.
이번에는 실험자가 기차에서 내렸다. 광원과 두 관찰자는 여전히 등속의 기차에 타고 있다. 광원에서 반짝인 빛이 두 관측자에 도달한 시간을 측정하기로 한다. 지상에 고정된 실험자는 기차 뒷칸의 관측자가 먼저 인지했다는 보고를 받게된다. 고정된 관측자의 입장에서 기차 뒷칸의 관찰자에게 빛이 이동한 거리가 짧아지기 때문이다.
Sheldon is puzzled, he observed the light pulse arrive at the back of the train first, while his friends on board report that the light pulse arrives simultaneously. In Sheldon's frame of reference, the light pulses do not arrive simultaneously even though his friend's frame of reference, they do.
실험자의 좌표계가 광원과 함께 움직일 때와 고정되었을 때 동일한 사건이 이상하게도 다르게 관측된다.
This disagreement between observers is a result of light traveling at a constant speed no matter how quickly the source of light moves. This is called the relativity of simultaneity and it describes that stationary and moving observers will report the order of events differently depending on their proper motion with respect to one another.
고전 상대론에서 실험자의 속도에 관측 대상이 움직이는 속도를 더한다. 그런데 빛의 속도는 우주불면이므로 속도를 가감할 수 없다. 따라서 광원이 고정되든 움직이든 동시검출되어야 한다. 동시성의 상대성(relativity of simultaneity) 실험자의 기준 좌표계(frame of reference)가 고정되어 있을 때와 등속운동을 할때 사건의 검출이 달라진 이유를 설명한다.
As strange as it seems, both Sheldon and his friends on the train are right. In some cases, the order of events depends on the motion of the observer. Einstein explained this through the concepts of length contraction and time dilation.
아인슈타인은 이 이상한 현상을 설명했다. 만일 빛이 검출된 시간이 다르다면 두 관측자의 시간 흐름의 간격이 서로 다르던가 거리가 달라져야 한다. [광원에서 두 관측자의 거리가 동일하게 놓고 실험 한 것이므로 거리는 변하지 않아야 한다.]
So, why don't observers in different reference frames agree on the order of events?
그렇다면 실험자의 기준 좌표계가 달라짐에 따라 사건이 일치하지 않는 이유는 뭔가? [관성계에 물리법칙은 동일하게 적용되어야 함에도 불구하고......]
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Well, think back to our block of cheese space-time analogy. [치즈 썰기에 비유한 내용은 썩 유익하지 않음]
A moving observer relative to another observer actually slices up space-time differently. Not only can they potentially see events in different order than other observers, but they also measure length and time differently.
Have a look at this graph which represents space-time for a stationary observer. A stationary observer sees objects as they naturally are and clocks run as expected. However, a moving observer sees space-time through a different slice. Here's what space-time looks like to a moving observer. Both the moving observer's time axis and distance axis have shifted. Physicists say that the time and space coordinates are rotated due to the motion of the observer.
빛의 속도가 불변이라면 실험자(observer)의 시간과 거리의 좌표축이 다르다고 보자. 고정된 실험자의 좌표축에 비교하여 움직이는 실험자의 시간과 거리 축이 기울어 있다. [치즈썰기에 비유되었다.]
This is a convenient picture to paint, but what other real observable effects of this skewed reference frame? Well, earlier we mentioned that moving observers measure changes in length and time.
찌그러진 기준 좌표계가 서로다른 조건(등속운동)의 두 실험자에게 어떤 영향을 줄까? 앞서 얘기 했지만, 시간 지연과 거리 수축 현상.
Length contraction is given by the equation, L is equal to L zero times the square root of one minus the velocity of the observer divided by the speed of light squared. Well, this equation describes is how the length of the object appears to a moving observer. If I jump into a spacecraft, and I'm moving at nearly the speed of light, objects I observe will appear to shrink in my direction of motion.
앞서 말한대로 움직이는 관측자의 기준에서 길이가 줄어드는 효과(Length Contraction)를 낸다. 속도가 아주 빠른경우 길이 수축이 효과를 낸다. 아주 빠른 우주선에 올라타면 움직이는 방향으로 길이가 수죽된다.
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Something like the [inaudible] would pass by the ship appearing to be almost as flat as a pancake. [???????]
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It's not just length that changes though, time also changes.
길이뿐만 아니라 시간(측정)도 바뀐다. [길이와 시간 측정 장치의 눈금(텐서, tensor)이 다르다.]
Time dilation is given by the equation t is equals to t zero divided by the square root of one minus the velocity of the observer divided by the speed of light squared. In this case, time is modified by dividing by these terms under the square root sign. So, instead of decreasing, the time observed by the moving observer increases. So, the moving observers see all external clocks slowing down the faster they move.
움직이는 실험자의 시계가 느리게 가는 효과(Time Dilation)를 보여준다.
Both length contraction and time dilation are effects that are measured by a moving observer, the faster the observer moves with respect to any object such as a clock, the thinner it appears and the slower it ticks. Hopefully, your head isn't spinning too much yet because we have to discuss an important issue with special relativity. Since a moving observer appears to be stationary in their own frame of reference, who can we trust when we say that the lengths have been contracted and the clocks have been slowed? To illustrate this problem, we need to introduce you to the twin paradox.
움직이는 실험자는 길이 수축과 시간 지연의 효과를 경험하게 된다. 그렇다면 이동하는 실험자의 입장에서 자신의 기준 좌표계를 잡으면 (물리현상은) 서있는 것과 다를바없다. 과연 누가 길이 수축과 시간 수축을 말해줄 수 있을까? 특수 상대론의 대표적인 논쟁꺼리인 쌍둥이 모순(twin paradox)을 살펴보자.
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Like to imagine that twins named Leia and Luke are preparing for an interstellar voyager. Leia will stay behind on the earth to monitor the journey while Luke, the adventurous one, takes off in his spaceship capable of reaching nearly the speed of light.
쌍둥이의 이름은 '레아'와 '루크'다. 이란성 쌍둥이는 우주여행을 준비하고 있다. 레아는 지구에서 루크가 빛의 속도에 가까운 빠른 속도로 우주 여행을 지켜보기로 한다.
Since Leia stays on the earth as Luke speeds away, Luke's clock appears to slow down the faster he zooms off his ship.
'레아'는 지구에서 머물며 루크가 빠르게 멀어져가는 것을 관측한다. 루크의 시계가 느리게 가고 있다고 관측한다.
But to Luke, Leia appears to be speeding away, so to Luke, Leia clocks have slowed down.
'루크'의 입장에서는 레아가 멀어지고 있으므로 그녀의 시계가 느리게 간다고 관측한다.
Both observers see the other clock as being slow. While our own clocks are at regular speed, how can that be? Surely both observers can't be right.
양측의 입장에서 보면 (서로) 상대방의 시계가 느리게 간다고 말한다. 우리가 가진 시계는 정상적으로 흘러 간다. 어떻게 그럴수 있을까? 두 관측자 모두 틀린것리 분명하다.
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Welcome to the twin paradox, proposed by Einstein, not as a paradox but as a peculiarity of special relativity.
아인슈타인이 제시했던 쌍동이 모순에 대해 알아보자. 사실 모순(paradox)이라기 보다 특수 상대론의 가설(peculiarity)이다.
In Einstein's 1905 paper, he reasoned that if two clocks were synchronized and one of them where to go on a lengthy journey, the traveling clock would return to the original location with its time lagging behind the stationary clock.
아인슈타인은 1905년의 논문에서 느리게 가는 여행자의 시계가 원래 자리로 되돌아 왔을때 고정된 시계보다 여전히 늦게 간다는 모순을 설명 하였다.
However, since relativity says that either clocks could view the other as being the one in motion, that is the traveling clock could consider itself at rest and the stationary clock would therefore be the one moving away. Shouldn't the stationary clock be the one lagging behind when the other returns again?
여행을 떠났다가 다시 제자리로 돌아온 시계는 머물러 있던 시계보다 느리게 간다. 왜 그럴까?
Let's watch as Luke flies to a nearby star system six light years away while Leia stays behind on the earth. Luke can travel at a significant fraction of the speed of light, say north 0.6 c.
루크가 6광년 떨어진 별로 여행을 간다. 레아는 지구에 남아 있다. 루크의 속도는 아주 빨라서 빛의 0.6배다. 루크가 떠나기전 두 사람의 시계를 맞춰 놨다.
When we say that something is traveling at one c. It is the same as saying that it goes one light year per year. So, if Luke travels at 0.6 c he will travel north 0.6 of a light year for each year of travel. As such, Leia will say that Luke's journey takes 10 years.
이 속도로 6광년 떨어진 별로 여행을 가면 레아의 입장에서 가는데 10년이 걸리는 것으로 계산된다.
However, let's carefully assess what Luke observes on his particular journey. Luke sets his spaceship to travel at north 0.6 c. In doing so, the distance to his destination changes, at north 0.6 c, the length between the earth and the destination has shortened by 20 percent. So, instead of six light years, the star appears to be only 4.8 light years away from Luke's perspective.
루크가 탄 우주선의 속도는 매우 빠르다. 광속의 0.6배나 된다. 이 속도라면 루크가 여행할 거리는 4.8 광년으로 줄어든다.
At north 0.6 c, Luke's time of arrival is only eight years after his departure from the earth.
루크의 관점으로 보면 6광년의 거리가 4.8 광년으로 줄었다. 광속의 0.6배로 비행하면 도착하는데 8년이 걸린다.
Meanwhile, Luke has tend to shift around and begins his journey back to the earth again at 0.6 c. So, the same length contraction applies instead of flying across six light years of space, Luke only flies the length contracted, 4.8 light years, and he is back from his distance star system after another eight year journey.
별로 여행갔던 루크는 다시 같은 속도로 지구로 되돌아 오기로 한다. 루크의 입장에서 되돌아 오는 여정은 역시 4.8광년으로 여행시간은 8년이다.
For Leia, Luke has been gone for 20 years, but from Luke's perspective he has only been gone 16 years. Luke is now four years younger than Leia.
왕복 여행에 걸린 시간을 보자. 루크의 시계로는 16년, 레이의 시계로는 20년이 흘렀다.
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Hopefully, you can now see why Einstein said this was a peculiarity and not a paradox. Luke is the moving observer, sees space itself foreshortened, but in Leia's reference frame, the distances haven't changed, merely the shape of Luke's ship.
아인슈타인이 왜 쌍둥이 모순이라 하지 않고 쌍둥이 가설이라고 했는지 이해 했길 바란다. 루크는 움직이는 관측자로 길이 수축의 영향을 받았고 레아의 표준 좌표계에서는 거리 수축이 없다.
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We've seen this square root equation come up twice already.
So, what is it? Well, this is a useful little equation that represents the strength of the time dilation or length contraction based on the speed of the moving observer. It's more commonly found in this form called the Lorentz factor or the gamma factor.
The Lorentz factor is a convenient tool when discussing length contraction and time dilation because it converts these equations into these vastly simpler forms.
So, someone traveling at 10 percent of the speed of light or
속도에 따른 로렌츠 인자의 변화를 보자. 속도가 광속에 가까울 수록 급격히 증가하는 모습을 볼 수 있다.
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We should note that Einstein developed special relativity to describe physics for observers who are not experiencing a strong gravitational force. But what happens near an object with a strong gravitational field such as a black hole?
아인슈타인이 광속에 관한 특수상대론을 전개한 이후 강력한 중력의 영향을 받는 관측자에게 미치는 물리현상이 포함 되지 않았다는 것을 깨닳았다. [특수 상대론 > 광속불변 > 시간 지연과 거리 단축> 중력에 의한 시공간 왜곡 > 일반 상대론]
Einstein realized that he had to modify his theory of special relativity to make it more general and to allow for gravity, Einstein called this relativistic theory of gravity, general relativity. In order to understand the theory of general relativity, we'll begin with Einstein's first ponderings on the subject, something called the equivalence principle.
중력의 실체를 밝히려는 노력이다. 중력과 동일한 원리를 물리 법칙에서 찾아보자.
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