03.08 - 요약: 생활속의 상대론(Summary: Living Relativistically) [커세라 강의 페이지]
Earlier in this module, we came to understand how event horizons work by drawing an analogy to a fish falling over a waterfall. At a certain point in its descent, the falling fish can no longer communicate with its friend above because the water is flowing downwards at a greater speed than the speed of sound. Luckily, in the story, we rescued our fishy friend using a rocket pack. This was possible because the speed of sound is not a universal limit.
소리의 속도가 우주의 한계가 아니기 때문 소리의 지평선(sonic horizon)에서 탈출할 수 있었다.
The speed of light on the other hand is. Were we to drop our fishy friend into a real black hole, a rocket pack would be of little use because once you're beyond the event horizon of a real black hole, escape is no longer possible. But don't worry, no fish were harmed in the making of this course.
[중력에 의해 가속되는 폭포수를 블랙홀이 끌어 들이는 에너지로 비유해 봤다. 강력한 블랙홀이라면 가속된 속도가 빛의 속도를 넘을 수도 있지 않을까?] 빛의 속도는 우주의 한계속도다. 실제 블랙홀 이었다면 물고기는 탈출할 수 없다.
We also learned about a revolution in physics which began in the early 1900's. Einstein, in a single year and with four extraordinary papers, turned physics on its head. One of the results of these papers was that space and time are no longer absolutes and in their place was this amalgamation called space-time.
1900년대 초에 아인슈타 인이 발표한 네개의 논문으로 물리학을 완전히 뒤집어 놓았다. 이제 시간과 공간이 절대 적이지 않다.
In this framework, relative motion is what matters and many of our traditional notions about how the universe works no longer apply. Some truly weird things start happening like length contraction, time dilation, and relative simultaneity.
이체제를 시공간(spacetime)이라고 하자. 전통적인 물리학이 더이상유효하지 않다. 빛의 속도는 우주의 상수 이며 빠른 속도로 움직이는 기준 관성 좌표계는 시간의 지연과 거리의 단축을 격게 된다. 그로 인해 '동시성'도 상대적이 된다.
When Einstein went about generalizing his theory of relativity, he realized that there ought to be an equivalence between gravitational fields and the acceleration of a reference system, meaning, if we stuff you in a windowless rocket, you can't tell the difference between a rocket sitting at rest on the surface of the earth or a rocket that is accelerating uniformly upwards at a rate equal to the acceleration due to gravity on Earth's surface.
Under these two conditions, the two are indistinguishable. In addition, general relativity comes with its own quirks, one of which being that mass deforms space-time.
가속에 의해 작용하는 힘과 질량에 의한 중력이 같다는 등가의 원리를 제시하고 이를 일반화한 일반 상대론을 펼쳤다. 질량이 시공간을 왜곡 시킨다.
This is what causes the effect known as gravitational lensing, something not exclusive to black holes but something they are nonetheless known for, that is their ability to bend light rays. In the next module, we'll explore black holes in more detail and learn how to weigh a black hole.
03.07x - 일반상대론의 증거 찾기(The history of the earliest test of general relativity.) [커세라 강의 페이지]
Interview with Dr. Robert Smith, Professor of History at the University of Alberta.
Einstein develops special relativity in 1905, the famous publication. By 1907 though, he's beginning to realize that he would like to generalize special relativity.
No longer talking about reference frames which are moving uniformly with respect to one another. But what about reference frames that are moving in an accelerated fashion with respect to one another? What about rotating reference frames?
So, by 1911, he's got his first, what's called the Prague theory of general relativity and that will change.
By 1913, he's got a version which he thinks is superior. Then, the final version of general relativity that Einstein would develop, he's got by the end of 1915 start of 1916.
But the famous light deflection that he predicts in 1911, comes out of an earlier version of general relativity. So, he calculates how big should the deflection of light be, as light, from a distant star just passes by the sun.
How much will it be affected? The calculation he makes turns out to be an answer of less than one second of arc. An astronomer at the Berlin Royal Observatory, a man called Erwin Finlay-Freundlich, is really interested in Einstein's theory, is one of the very few astronomers who's actually interested in trying to test the results of general relativity. So, Freundlich tries to look at past photographs of eclipses.
Can he look at these earlier photographs and see star images, and compare the positions of those stars with the Sun there at the eclipse? And also when the sun has moved away from the star, because then we should see the deflection. But the images were just not good enough for a range of different reasons, so he decides in 1913 that he would like to engage in an expedition. Go to the Crimea in August 1914, take photographs of the sun during eclipse in order to get the star positions during the eclipse. Take photographs at a different time when the sun is moved away, compare the star positions between the two sets of photographs, and then you've got the measurement of the deflection of light.
Freundlich leads an expedition and they go to Russia, and the eclipse will occur at the end of August 1914. But this is not a good time to be a German traveling in Russia, because World War I breaks out and so Freundlich and his companions and their instruments are taken into custody, they're thought to be spies and they're held in Odessa for several weeks before, in fact, they're released but the instruments are confiscated.
But what is very interesting about this expedition is if they had in fact gone ahead, made the measurements of the deflection of light, they would have come up with an answer which would have been roughly twice the answer that Einstein had calculated, based on one of his earlier versions of general relativity.
So, the actual shift that Einstein predicted later would be 1.75 seconds of arc. At this point, the prediction that Einstein makes is 0.85 seconds of arc. The first measurements made at an eclipse of the deflection of light that are successful anyway, the result is 1.64 seconds of arc.
So, what that means is, if Freundlich had gone ahead, made the observations in 1914, the answer would have been about double Einstein's prediction. Then Einstein would have come along later and redone his calculations, "Oh look, my answer is now double what I pretty much got the amount that the light is deflected by."
Now, would that have aided the acceptance of general relativity? I suspect not because coming in after the fact, after the measurements have been made and say, "Oh, I goofed by a factor of two" is not really a very convincing way to verify your theory I think.
One of the most important realizations Einstein made while developing special relativity is that there is no such thing as a universal time or distance. Instead, special relativity introduced the notion of an invariant spacetime interval.
아인슈타인이 특수 상대론을 발전 시키면서 알아낸 가장 중요한 것은 시간과 거리가 각각 절대적이지 않다는 점이다. 그대신 시공간(spacetime)의 간격이 불변하다는 것이다(시간과 공간의 유기적 결합).
When astronauts travel at different speeds and experience differences in the duration of time intervals, they are effectively trading distance for time or vice versa. They are experiencing a distortion or warping of space and time together. This is required in order for all observers to agree on the speed of light. Einstein realized that he could explain the effects of gravity by combining the equivalence principle with the concept of an invariant spacetime interval used in special relativity. When Einstein developed the general theory of relativity, he came to the realization that gravity is the warping of spacetime. So, stars like our Sun, which have strong gravitational field due to their large mass, actually bend and stretch the fabric of the universe itself. The warping of spacetime causes planets and light to travel on curved paths near massive objects. An early test of general relativity depended on the bent spacetime around the Sun. In 1919, the astronomer, Sir Arthur Eddington, led an expedition to an island of the West Coast of Africa in order to measure how much the gravity of the Sun warped spacetime. They did it by observing a total eclipse of the Sun. During the eclipse, a star could be seen next to the eclipse Sun. Those of you who have observed a total solar eclipse, like I have, know that the Sun looks eerily like a black hole in the sky surrounded by white hair. The locations of all the stars in our neighborhood of the galaxy were mapped more than 100 years ago. So it was known that this star should really be located behind the Sun when viewed from the Earth on the day of the eclipse. So, how did Eddington and his team see the star? The light from the star traveled on a curved path around the Sun to the astronomers' telescopes. This effect was predicted by Einstein's theory of general relativity. The measurement they made confirmed and has since been reconfirmed that the theory of general relativity is accurate and that spacetime itself is bent by matter. This changes our notion of what a straight line is, of course, because if spacetime itself is bent, how can we possibly know that we're going in a straight line? Instead of calling them straight lines, in general relativity, we call them geodesics, which represent the straightest possible path of an object in a curved spacetime. Even though geodesics represent straight lines in curved spacetimes, they wouldn't be considered straight by our standards. Just as the Sun's mass bends the spacetime around it, any light crossing bent spacetime will appear to have its path bent. Since we are talking about curving spacetime, let's consider the surface of this chalk ball as a section of a curved two-dimensional space. This works equally well if you imagine the chalk ball to be the Earth. If I asked you to draw a straight line between two points on opposite sides of the ball, the same thing as asking for the flight path between two cities on Earth, you might be tempted to draw along an equatorial line to join them together. Even though I asked you to draw a straight line, already it's curved. Instead, the smallest distance between two points on a curve surface is considered straight if it's also the shortest line joining the two points together. The smallest distance between two points on a curved surface is called a geodesic. If you look at the flight path of an airplane from Toronto to London, the airplane crosses the ocean near Greenland. The shortest route joining the two cities is a curved path. The same is true for any object traveling through curved spacetime. Now, where do you think the most convoluted curvatures of spacetime in the universe exist? That's right, black holes. Not only do black holes warp spacetime, they warp it to the point that even light will travel on highly curved paths. Photons, by definition, travel on geodesic paths in spacetime. Close to the black hole, the curvature becomes so high that light is bent into paths that all terminate at the black hole singularity. General relativity interprets gravity as the warping of spacetime. When we view a picture of the gravitational field around a massive object, it's usually represented as a depression in space. However, we need to understand that gravity also warps the passage of time. It's strangely difficult for the human mind to grasp the concept of warping spacetime. We understand what it means to bend or warp a material like plastic, but what does it mean when the actual space and time that we live in are bent and twisted? In a sense, warped spacetime means that the paths we choose to cross space and time will be shorter or longer in distance between two points and in the duration it takes to travel between them depending on what the gravitational fields are along the path. Let's focus specifically on how gravitational fields warp the time component in an effect called gravitational time dilation. Let's start with an example by considering two astronauts exploring an unstudied planet around a distant star, perhaps planet e in the nearby Trappist-1 System, which we'll shorten to trappy. One astronaut needs to stay with the ship in order to orbit around the parent star while the astronaut descends to trappy surface. Since we are talking about time, both astronauts will need to carry clocks, which they synchronize before they separate. Far from the surface of the planet, both clocks tick in perfect synchronicity. One astronaut now descends to the surface of trappy. On the surface, he is deep in the planet's gravitational well and therefore, experiences a greater gravitational force. The spacetime in the vicinity of the planet will also be warped. The effect that the warping has on the astronauts' clocks causes it to tick more slowly than the one in orbit. For every tick of the clock on the surface, the orbital clock ticks more rapidly. On the surface of trappy, the astronaut doesn't experience the change in the passage of time because all biological processes are likewise slowed down by the warping of gravity. Just like the ticks of the clock, a distant observer would see the heartbeat of an astronaut on the surface to beat more slowly. Once the surface mission is complete, the two astronauts rejoin one another in orbit around trappy. The astronaut who stayed in orbit will be dismayed. She experienced a longer time than the astronaut who was on the surface. Depending on the duration of the stay and the strength of the gravitational field, the astronaut who went down to the planet's surface will experience fewer ticks of the clock and therefore, be several seconds younger than the one who stayed in orbit. To calculate how time has worked in a strong gravitational field, the following equation is employed. Delta t planet, the elapsed time on the surface of the planet, is equal to Delta t orbit, the elapsed time on the orbiting spaceship, times the square root of one minus two times G times mass divided by radius times c squared. In this formula, the mass and radius refer to the mass and radius of the planet. But if instead of a planet, you were a distance R from a star or a black hole with mass M, you could use the same formula. The important thing in this formula is that the quantity inside the square root sign is smaller than one. So the amount of time that passes when you're in a gravitational well is smaller than if you're out in space far from the gravitating object. Note that this formula doesn't make sense if the ratio of the mass to radius gets too large. This formula only makes sense if R is larger than two times G times mass divided by c squared. You might think that your everyday life is not much affected by time dilation due to special or general relativity. However, you may be surprised to learn that almost everyone carries a piece of technology that would be useless without both theories, GPS. The Global Positioning System that you use every time you navigate with a map on your smartphone depends on Einstein's theory of relativity to function correctly. Handheld GPS works because the device inside your smartphone is capable of measuring and comparing the signals from multiple satellites in orbit around the Earth. These satellites are placed in well-known orbits and carry very precise clocks. By broadcasting a timing signal that can be picked up on a GPS receiver, the difference in timing signals from different satellites can be used to triangulate your position. Since GPS satellite travel at about 14,000 kilometers per hour, they experience a very slight time dilation due to special relativity. Each day, a satellite's clock would appear to slow down by about seven microseconds. That doesn't sound like much, but if you neglected this drift, your GPS would accumulate an error of about two kilometers every day. General relativity predicts that the clocks aboard a GPS satellite traveling at an altitude of 20,000 kilometers would appear to tick faster than clocks on Earth. Every day, a satellite clock would appear to speed up by 45 microseconds compared to clocks on Earth's surface. If this error wasn't corrected, the GPS would accumulate an error of over 13 kilometers a day. Since special relativity works to slow down the apparent rates of the clock on a GPS satellite, and general relativity speeds up their apparent rates, the combined effects add up to a 38 microseconds per day error. Without relativity, our GPS devices would drift by over 10 kilometers every day, roughly the same as 12 centimeters a second. Luckily, we know about the effects of relativity. So we can correct for this drift. GPS devices are some of the most robust tests we have for Einstein's theories of relativity. In the movie "Interstellar", the main character Cooper is sent to retrieve a fellow explorer Mann from the surface of Miller's planet orbiting the nearby black hole Gargantua. On the surface of Miller's planet, an hour of time is equivalent to seven years on Earth. This is an example of the correct use of an effect called gravitational time dilation.
Start transcript at 11 minutes 35 seconds11:35
Since gravitational time dilation slows down the passage of time in intensely strong gravitational fields, it equally effects physical processes that are time-dependent. That means that someone observing an astronaut in orbit around the black hole would see their clocks ticking slow and their hearts beating slower, and everything about them slow down. So, what happens to a beam of light when it's generated deep in the gravity well near a black hole? The beam of light experiences gravitational redshift. Recall the Doppler effect that we discussed earlier. When a moving object like a rocket ship is emitting light, the light can be blueshifted or redshifted depending on the ship's motion towards or away from the observer. If a ship were to accelerate away from you, you would see the light from its engines becoming redder and redder as it accelerated to ever increasing speeds. Light emitted from deep within gravitational well has to work against gravity in order to leave a planet, or a star, or the region near a black hole. When light travels away from a planet, the photon has to convert kinetic energy into gravitational potential energy. If we remember that red photons have less energy than blue photons, we can predict that the photons emitted from the surface of a star will appear redder to an observer far from the star. This effect is called gravitational redshift. The gravitational redshift effect is very small, but it has been measured in the light emitted by a white dwarf star and it agrees with the predictions in general relativity.
Interview with Dr. Jeremy Heyl, Professor at the University of British Columbia.
One fun thing that I did recently with this was to ask the question whether black holes, like if you think of Galileo, the first astronomer, there's this story of him dropping things off towers and whether they fall faster or slower depending on what they're made of, but you can do the same experiment with black holes, because there's black holes in galaxies, and these galaxies are falling through the universe, and you can ask, does the black hole fall any differently than the rest of the galaxy, for example?
최근에 블랙홀에 관해 한가지 재미있는 질문을 해봤어요 (받았어요?). 갈릴레오가, 그는 최초의 천문학자죠, 어느 탑위에 올라 서로 다른 물질로 만든 공을 떨어트려, 떨어지는 속도가 다른지 보려고 실험을 했다는 이야기를 들어봤을 겁니다. 만일 블랙홀을 가지고 이 실험을 하면 같은 결과를 얻을까요? 블랙홀은 은하내에 있고 이 은하들은 우주를 향해 떨어지는 중이 잖아요. 그럼 블랙홀은 은하 내의 다른 것들과 다르게 떨어지지 않겠어요?
Amazingly, I mean, maybe not surprisingly they do, but what's maybe more surprising is that you can actually pose the question that, "Hey, how does a black hole fall? Does it fall like normal stuff or not?" I mean, we're not very precise, they fall like normal stuff to within 50 percent, maybe about as precise as Galileo was, 400 years ago. But I mean, we can get better.
