Sometimes such radio telescopes are connected with telescopes on the other side of the Earth, forming a baseline comparable to the Earth’s diameter – in a certain sense, a telescope as large as the planet. In the future we may have telescopes in the Earth’s orbit, around toward the other side of the Sun, in effect a radio telescope as large as the inner solar system. Such telescopes may reveal the internal structure and nature of quasars. Perhaps a quasar standard candle will be found, and the distances to the quasars determined independent of their red shifts. By understanding the structure and the red shift of the most distant quasars it may be possible to see whether the expansion of the universe was faster billions of years ago, whether the expansion is slowing down, whether the universe will one day collapse.
Modern radio telescopes are exquisitely sensitive; a distant quasar is so faint that its detected radiation amounts perhaps to a quadrillionth of a watt. The total amount of energy from outside the solar system ever received by all the radio telescopes on the planet Earth is less than the energy of a single snowflake striking the ground. In detecting the cosmic background radiation, in counting quasars, in searching for intelligent signals from space, radio astronomers are dealing with amounts of energy that are barely there at all.
Some matter, particularly the matter in the stars, glows in visible light and is easy to see. Other matter, gas and dust in the outskirts of galaxies, for example, is not so readily detected. It does not give off visible light, although it seems to give off radio waves. This is one reason that the unlocking of the cosmological mysteries requires us to use exotic instruments and frequencies different from the visible light to which our eyes are sensitive. Observatories in Earth orbit have found an intense X-ray glow between the galaxies. It was first thought to be hot intergalactic hydrogen, an immense amount of it never before seen, perhaps enough to close the Cosmos and to guarantee that we are trapped in an oscillating universe. But more recent observations by Ricardo Giacconi may have resolved the X-ray glow into individual points, perhaps an immense horde of distant quasars. They contribute previously unknown mass to the universe as well. When the cosmic inventory is completed, and the mass of all the galaxies, quasars, black holes, intergalactic hydrogen, gravitational waves and still more exotic denizens of space is summed up, we will know what kind of universe we inhabit.
In discussing the large-scale structure of the Cosmos, astronomers are fond of saying that space is curved, or that there is no center to the Cosmos, or that the universe is finite but unbounded. Whatever are they talking about? Let us imagine we inhabit a strange country where everyone is perfectly flat. Following Edwin Abbott, a Shakespearean scholar who lived in Victorian England, we call it Flatland. Some of us are squares; some are triangles; some have more complex shapes. We scurry about, in and out of our flat buildings, occupied with our flat businesses and dalliances. Everyone in Flatland has width and length, but no height whatever. We know about left-right and forward-back, but have no hint, not a trace of comprehension, about up-down – except for flat mathematicians. They say, ‘Listen, it’s really very easy. Imagine left-right. Imagine forward-back. Okay, so far? Now imagine another dimension, at right angles to the other two.’ And we say, ‘What are you talking about? “At right angles to the other two!” There are only two dimensions. Point to that third dimension. Where is it?’ So the mathematicians, disheartened, amble off. Nobody listens to mathematicians.
Every square creature in Flatland sees another square as merely a short line segment, the side of the square nearest to him. He can see the other side of the square only by taking a short walk. But the inside of a square is forever mysterious, unless some terrible accident or autopsy breaches the sides and exposes the interior parts.
One day a three-dimensional creature – shaped like an apple, say – comes upon Flatland, hovering above it. Observing a particularly attractive and congenial-looking square entering its flat house, the apple decides, in a gesture of interdimensional amity, to say hello. ‘How are you?’ asks the visitor from the third dimension. ‘I am a visitor from the third dimension.’ The wretched square looks about his closed house and sees no one. What is worse, to him it appears that the greeting, entering from above, is emanating from his own flat body, a voice from within. A little insanity, he perhaps reminds himself gamely, runs in the family.
Exasperated at being judged a psychological aberration, the apple descends into Flatland. Now a three-dimensional creature can exist, in Flatland, only partially; only a cross section can be seen, only the points of contact with the plane surface of Flatland. An apple slithering through Flatland would appear first as a point and then as progressively larger, roughly circular slices. The square sees a point appearing in a closed room in his two-dimensional world and slowly growing into a near circle. A creature of strange and changing shape has appeared from nowhere.
Rebuffed, unhappy at the obtuseness of the very flat, the apple bumps the square and sends him aloft, fluttering and spinning into that mysterious third dimension. At first the square can make no sense of what is happening; it is utterly outside his experience. But eventually he realizes that he is viewing Flatland from a peculiar vantage point: ‘above’. He can see into closed rooms. He can see into his flat fellows. He is viewing his universe from a unique and devastating perspective. Traveling through another dimension provides, as an incidental benefit, a kind of X-ray vision. Eventually, like a falling leaf, our square slowly descends to the surface. From the point of view of his fellow Flatlanders, he has unaccountably disappeared from a closed room and then distressingly materialized from nowhere. ‘For heaven’s sake,’ they say, ‘what’s happened to you?’ ‘I think,’ he finds himself replying, ‘I was “up.” ’ They pat him on his sides and comfort him. Delusions always ran in his family.
In such interdimensional contemplations, we need not be restricted to two dimensions. We can, following Abbott, imagine a world of one dimension, where everyone is a line segment, or even the magical world of zero-dimensional beasts, the points. But perhaps more interesting is the question of higher dimensions. Could there be a fourth physical dimension?*
* If a fourth-dimensional creature existed it could, in our three-dimensional universe, appear and dematerialize at will, change shape remarkably, pluck us out of locked rooms and make us appear from nowhere. It could also turn us inside out. There are several ways in which we can be turned inside out: the least pleasant would result in our viscera and internal organs being on the outside and the entire Cosmos – glowing intergalactic gas, galaxies, planets, everything – on the inside. I am not sure I like the idea.
We can imagine generating a cube in the following way: Take a line segment of a certain length and move it an equal length at right angles to itself. That makes a square. Move the square an equal length at right angles to itself, and we have a cube. We understand this cube to cast a shadow, which we usually draw as two squares with their vertices connected. If we examine the shadow of a cube in two dimensions, we notice that not all the lines appear equal, and not all the angles are right angles. The three-dimensional object has not been perfectly represented in its transfiguration into two dimensions. This is the cost of losing a dimension in the geometrical projection. Now let us take our three-dimensional cube and carry it, at right angles to itself, through a fourth physical dimension: not left-right, not forward-back, not up-down, but simultaneously at right angles to all those directions. I cannot show you what direction that is, but I can imagine it to exist. In such a case, we would have generated a four-dimensional hypercube, also called a tesseract. I cannot show you a tesseract, because we are trapped in three dimensions. But what I can show you is the shadow in three dimensions of a tesseract. It resembles two nested cubes, all the vertices connected by lines. But for a real tesseract, in four dimensions, all the lines would be of equal length and all the angles would be right angles.
Imagine a universe just like Flatland, except that unbeknownst to the inhabitants, their two-dimensional universe is curved through a third physical dimension. When the Flatlanders take short excursions, their universe looks flat enough. But if one of them takes a long enough walk along what seems to be a perfectly straight line, he uncovers a great mystery: although he has not reached a barrier and has never turned around, he has somehow come back to the place from which he started. His two-dimensional universe must have been warped, bent or curved through a mysterious third dimension. He cannot imagine that third dimension, but he can deduce it. Increase all dimensions in this story by one, and you have a situation that may apply to us.