《Black Holes》——55条滴墨书摘

作者: Brian Cox, Jeff Forshaw

出版: Mariner Books

出版时间: 2023-03-28

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Classical theories describe a reality that fits nicely with our intuitive mental picture of the world.

2024-02-11

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person can be both spaghettified and vaporised: Spaghettified from their own perspective and vaporised according to outsiders. This idea is known as ‘black hole complementarity’.36

2024-02-11

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From the outside, nothing is ever observed to fall into a black hole. We might say that the interior lies beyond the end of time for someone lurking outside. Stuff just falls into a hot atmosphere where it gets burnt up. From the outsiders’ point of view, it seems that a black hole is not so different to a hot, glowing coal.

2024-02-11

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We tend to think, based on our experience of the world, that large physical objects like books or astronauts can only be in one place at once and only a single fate can befall them. Quantum theory destroys this picture when we ask questions about the behaviour of sub-atomic particles, and complementarity appears to be an even more radical challenge to our intuition. It asks us to accept that there are two equally valid views of what happens to a big thing like an astronaut – you, for instance – freely falling towards a black hole. You are both spaghettified (inside) and vaporised (outside).

2024-02-11

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From the perspective of someone freely falling through the horizon, they feel nothing and are oblivious to this magical encoding. Moreover, their fate is to be both spaghettified in the singularity (from their own perspective) and burnt up on the horizon (from an outsider’s perspective). But that is no problem for the laws of Nature because no observer can be present at both events.

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entropy puts a number on ignorance; in particular, on our ignorance of the exact configuration of the component parts of a system.

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things that shuffle and jiggle will, if left alone, tend to mix and disperse, and our ignorance increases as a result.

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This ‘one way’ evolution of a system is often called the thermodynamic arrow of time because it draws a sharp distinction between the past and the future: the past is more ordered than the future. In our Universe as a whole, the arrow of time can be traced back to the mysterious highly-ordered, low-entropy state of the Big Bang.

2024-02-11

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Entropy is telling us about the amount of information that is hidden from us when we describe a system in terms of just a few numbers. Seen in this way, entropy is a measure of our lack of knowledge, our ignorance.

2024-02-11

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Now imagine instead dropping a super-high entropy object with the same mass as before into the black hole. The horizon area will increase by precisely the same amount, proportional only to the mass of the object. This means that as we add mass to the black hole, it must increase its entropy by the largest possible amount. It is as if objects thrown in get completely scrambled up, to guarantee that our ignorance is maximised.

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everything that exists in a volume of space can be completely described by information on a surface surrounding the region. This is our first encounter with the holographic principle.

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Firstly, the singularity is a ring.* The plane of the ring is aligned at a right-angle to the spin axis, which means that only trajectories in the equatorial plane will encounter it. All other trajectories will miss it. An astronaut could therefore fall into a Kerr black hole and dodge the end of time.

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Secondly, the hole has two event horizons,which we’ve labelled the inner and outer horizons.

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Thirdly, there is a region outside of the outermost horizon in which space is being dragged around so violently that it is impossible for anything to stand still.* This region is known as the ergosphere.

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For the spinning black hole, the astronaut regains her freedom to navigate once she has crossed the inner horizon. The singularity does not lie inexorably in her future, so time does not have to end.

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We already know there can be an infinite volume of space inside the event horizon of a black hole, but in the maximally extended Schwarzschild case we only had to contend with one extra infinite universe through the wormhole. Inside the eternal Kerr black hole there reside an infinity of infinite universes, nested inside each other like Russian TARDIS dolls.

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The origin of this special initial state of the Universe – a reservoir of low entropy without which we would not exist – is one of the great mysteries in modern physics

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Dramatically, Red never even sees anybody pass through the horizon because light emitted very close to the horizon heads off at 45 degrees and only reaches her in the far future. This means she continues to receive light from the other astronauts forever. She sees in-falling objects move ever slower as they approach the horizon, until they eventually freeze there. In principle, she can see everything that ever fell into the black hole.

2024-02-11

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Red sees the in-fallers in ever-increasing slow motion as they approach the horizon. As we’ve emphasised, the slowing down of the astronauts and watches is not specific to astronauts and watches. Everything slows down, from the rate of ageing of the cells in the astronauts’ bodies to the inner workings of atoms. Time is distorted, and that means every physical process is distorted too.This includes light.

2024-02-11

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A distant observer such as Red sees the in-fallers by the light they emit, and the frequency of that light reduces as time slows down. The images of the in-falling astronauts therefore become redder as they approach the horizon, and ultimately fade away as the frequency drops out of the visible range and into the microwave and radio bands beyond. This effect is known as redshift. Red sees her in-falling colleagues freeze and fade away as they approach the horizon.

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nobody sees anyone else fall through the horizon until they themselves are inside. Even more bizarrely, nobody even sees their feet cross the horizon until their eyes have crossed it.

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Green begins her journey alongside Blue and freefalls towards the horizon with him, but on crossing the horizon she panics, shouts ‘Burma’* and turns on her rocket engine in a vain attempt to escape. There are only 16 dots along her worldline once she crosses the horizon,which means that accelerating away has resulted in the end of time arriving sooner.*

2024-02-11

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she gently accelerates towards the end of time at 480g (which is five times less acceleration than Green). She presses play on Joy Division’s ‘Unknown Pleasures’ and flicks the switch. Perhaps irritatingly for her,this lengthens her stay inside the horizon and she lives longer than Green – her worldline has 17 dots. The spacetime geometry inside the black hole is certainly counter-intuitive.

心得：2024/1/4 发表想法

加速驶向奇点，却更晚抵达奇点？真的很反直觉。

2024-02-11

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she gently accelerates towards the end of time at 480g (which is five times less acceleration than Green). She presses play on Joy Division’s ‘Unknown Pleasures’ and flicks the switch. Perhaps irritatingly for her,this lengthens her stay inside the horizon and she lives longer than Green – her worldline has 17 dots. The spacetime geometry inside the black hole is certainly counter-intuitive.

2024-02-11

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The maximum time anyone can spend inside the horizon of a black hole corresponds to someone who starts out from rest on the horizon and does absolutely nothing but fall freely to the singularity. Apathy pays.

2024-02-11

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Nobody ever sees anyone else hit the singularity. The reason is the horizontal nature of the singularity on the Penrose diagram. It is a moment in time, and we can never see events that are simultaneous with a moment in time. We always see things slightly in the past, because it takes light time to travel to our eyes. This means that nobody falling into the black hole sees anyone else reach the singularity before they themselves reach it – they quite literally never see it coming.

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Imagine now that you are falling into the black hole feet first. Your head and feet will try to move apart but, since they are connected by your body, you’ll instead feel like you are being stretched.

2024-02-11

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Closer to the singularity, your constituent atoms would be ripped apart. Even more dramatically, for a typical stellar mass black hole you’d be spaghettified before you even reached the horizon.

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The Schwarzschild R and tcoordinates do happen to have a nice interpretation far away from a black hole, but inside the horizon their roles flip. The startling consequence is that an object inside the horizon moves inexorably towards the centre of attraction at R = 0, just as surely as you move inexorably towards tomorrow.

2024-02-11

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the singularity isn’t really a place in space. It is a moment in time: the end of time that lies in the future for all who dare to cross the horizon.

2024-02-11

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Green begins her journey alongside Blue and freefalls towards the horizon with him, but on crossing the horizon she panics, shouts ‘Burma’* and turns on her rocket engine in a vain attempt to escape. There are only 16 dots along her worldline once she crosses the horizon,which means that accelerating away has resulted in the end of time arriving sooner.*

心得：2024/1/4 发表想法

为什么开动引擎想逃离反而加速驶向奇点呢？

2024-02-11

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Schwarzschild’s solution then describes an infinite, eternal Universe in which the spacetime becomes more and more distorted as we head inwards towards the singularity at R = 0: a perfect eternal black hole.

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on the shell at the Schwarzschild radius: the (1 – RS/R) factors are equal to zero.Even more dramatically, as we continue further inwards, these factors become negative.What does this mean? From the perspective of someone in freefall across the shell at the Schwarzschild radius the Equivalence Principle informs us that nothing untoward happens.And yet, from a distant perspective, the shell is a place where clocks stop and space has an infinite stretch.

2024-02-11

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Far from the Schwarzschild radius, the light cones do look like those in flat spacetime, but as we approach the Schwarzschild radius the cones get narrower and narrower. At the Schwarzschild radius the light cones are infinitely narrow, which means an outgoing beam of light can only travel in the time direction and can never climb away from the hole.*

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Now we can appreciate that the Schwarzschild radius is also the event horizon: an outgoing beam of light emitted at the Schwarzschild radius stands still.

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Inside the horizon, the light cones have flipped round.

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what is time for someone inside the black hole is space for someone far away, and vice versa. As we’ve been at pains to emphasise, the coordinates we use don’t have to correspond to anyone’s idea of space and time: to quote Einstein again ‘they do not have to have an immediate metrical meaning’.

2024-02-11

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the path a freely falling object takes over curved spacetime must be the path that maximises the time on a wristwatch carried by the object. On each little flat patch, this path will be a straight line across the patch, but in curved spacetime the patches sew together to make a curve. The result is entirely analogous to the case of the little flat rulers on Earth. The straight lines must match up with each other end-to-end, but the resulting path is curved. The result in spacetime is what we see as an orbit – the paths of the planets around the Sun. Or, for that matter, the fall as someone slips unfortunately off a roof. In a way, the path the unfortunate faller takes on their way to the ground is entirely logical – they are maximising the time they have left.

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space is being stretched and time is being slowed down as we get closer to a star.

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The Sun’s Schwarzschild radius is approximately 3 kilometres and its radius is approximately 700,000 kilometres. This gives a distorting factor of 1.000002 at the surface of the Sun. This means that, for two Sun-sized shells whose radii differ by 1 kilometre in flat space, the measured distance between them would be 2 millimetres longer than 1 kilometre.Likewise, an observer far away from the Sun would see a watch at the Sun’s surface run slow by 2 microseconds every second, which is around a minute per year.

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Acceleration is locally indistinguishable from gravity, and vice versa.* That very important idea is known as the Equivalence Principle.

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By virtue of his acceleration, he has cut himself off from some regions of spacetime.The 45-degree boundary lines to his region are generically referred to as ‘horizons’because information cannot flow both ways across them.

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Previously we encountered horizons in the context of gravity and black holes. Now we see that they also appear for accelerating observers. Is there a conceptual connection between acceleration and gravity? Indeed there is, and when Einstein first realised it, he called it the happiest thought of his life.

2024-02-11

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The space station is falling towards the Earth in just the same way as you would if you jumped out of an aircraft, but it’s travelling fast enough relative to the surface of the Earth – around 8 kilometres per second – to continually miss the ground. It can continue to orbit in this way with very little intervention from rockets because there is very little air resistance at an altitude of 400 kilometres.

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the space station is in freefall around the Earth; forever falling towards the ground but never reaching it. The crucial point is that freefall is locally indistinguishable from floating freely in deep space, far away from any stars or planets; that is to say, if the astronauts had no windows and could not look outside to see the Earth below, they would be unable to do any experiment or make any observation to inform them that they are in the gravitational field of a planet.

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there is something interesting about the force of gravity, because gravity can be removed by falling.

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It’s hard to accept that the delineation between space and time is subjective, because our personal experience is that they are fundamentally different things that cannot be mixed. But this is not true. The separation between them is personal; it depends on our point of view.

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Fascinatingly, there is no limit on how far into the future one can travel with access to a fast enough spaceship. The Andromeda galaxy is 2.5 million light years away from Earth. If Pink had access to a spaceship that could travel at 99.9999999999 per cent of the speed of light, it would take her 18 years to make the round trip to Andromeda. She would, however, return to Earth 5million years in the future.

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Relativity is so-called because of this relative aspect of motion – who is at rest and who is moving is just a point of view.

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Spacetime tells matter how to move; matter tells spacetime how to curve.

2023-12-27

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The future light cone of an event therefore tells us which regions of spacetime are accessible from that event and which regions are forbidden. Likewise, the past light cone of an event tells us which events in spacetime could have possibly had any influence on that event. If you look back at the worldline in Figure 2.1, you’ll see that travelling to revisit moments in your past, the people and memories left behind, is impossible because we can never move from inside to outside the light cone at any event in our lives. To do so, we would have to travel faster than light. But the interval is invariant, so we can’t do that.

2023-12-25

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To see that this must be the case, remember that the interval between two events is invariant: if the interval is timelike from one perspective, it’s timelike from all perspectives. This means that events that can influence each other have their time-ordering preserved from all perspectives. Events that can’t influence each other do not have their time-ordering preserved, but that doesn’t matter because it does not mess with cause and effect.

2023-12-25

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Events inside each other’s light cones can have a causal relationship because it is possible that some signal or influence could have travelled between them. Events outside each other’s light cones cannot have a causal relationship. The interval there- fore contains within it the notion of cause and effect. Certain events can cause others, and the light cones at each event tell us where in spacetime the dividing lines lie.

2023-12-25

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The Exclusion Principle states that particles like electrons cannot occupy the same region of space. If lots of electrons are squashed together by gravitational collapse, they will separate themselves into their own individual tiny volumes inside the star in order to stay away from each other. Heisenberg’s Uncertainty Principle now comes into play. It states that as a particle is confined to a smaller volume, its momentum becomes larger. In other words, if you confine an electron it will jiggle around, and the more you try to confine it, the more it will jiggle. This creates a pressure in much the same way that the heat from nuclear fusion reactions earlier in the star’s life causes its atoms to jiggle and halt the collapse. Unlike the pressure from fusion reactions, however, electron degeneracy pressure requires no energy release to power it. It seemed a star could resist the inward pull of gravity indefinitely.

2023-12-25

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Georges Lemaître, a Catholic priest and one of the founders of modern cosmology, described the Big Bang singularity at the origin of our Universe as a day without a yesterday. A singularity formed by gravitational collapse is a moment with no tomorrow. What remains outside is a dark imprint of what once shone: a black hole.

2023-12-25

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