The Physical Universe – Part 2
Space, and time.
Continuing this essay series on the laws of our physical universe, in this essay we’ll explore more implications of Einstein’s discovery that space, time, and gravity are inter-related.
Jim Al-Khalili, in The World According to Physics, tries his best to use common language (rather than mathematics) to describe why time runs more slowly in stronger gravity:
Just as two people moving relative to each other will, according to special relativity, measure each other’s clock ticking at a slower rate, a similar situation arises between the two observers if they are a fixed distance apart, but one of them is feeling a stronger gravitational pull—say, on the surface of the Earth, while the other is hovering far out in space. Again, the two of them will disagree on the time interval between events. As before [see Part 1 of this essay series], their clocks will tick at different rates: being deeper within the Earth’s gravitational well, where there is more spacetime curvature, the Earth observer’s clock will tick more slowly. However, unlike in special relativity, the situation here is no longer symmetrical, as she would see the clock out in space ticking more quickly. In a very real sense, gravity slows the flow of time. We can say that the reason that a body ‘falls’ to Earth is because it always moves to where time runs the slowest—it is trying to age more slowly. Isn’t that beautiful?
And Einstein showed that, without the matter and energy that creates gravity, there would be no space or time either:
According to his general theory, matter and energy create a gravitational field, and spacetime is nothing more than the ‘structural quality’ of this field. Without the ‘stuff’ contained within spacetime, there is no gravitational field and hence no space or time! This may sound somewhat philosophical, and I suspect even some physicists will be uncomfortable with it. The problem is, in part, down to the way we teach physics. We tend to start with special relativity and ‘flat’ spacetime (because it is easier to teach and because Einstein hit upon it first), then we progress on to the more difficult general relativity, in which this flat spacetime is filled with matter and energy, causing it to curve. In fact, conceptually we should think of it the other way around, beginning with matter and energy within spacetime. This way, special relativity is just an idealised approximation that only works when gravity is so weak that spacetime can be regarded as ‘flat’ … To understand Einstein’s thinking, we must understand the concept of a ‘field’ in physics. The simplest definition of a field is that it is a region of space containing some form of energy or influence, in which every point can be assigned a value that describes the nature of the field at that point. Think of the magnetic field surrounding a bar magnet. The field is strongest close to the poles of the magnet and becomes progressively weaker the further away in space from the magnet we get. The pattern of iron filings that arrange themselves along the magnetic field lines is simply their way of reacting to the field they are immersed in. But the point I wish to make sounds too obvious to warrant stating: the magnetic field needs space to exist in. In stark contrast, the gravitational field, as described by Einstein and created by the mere existence of matter, is more than just a region of influence within space and time. It is spacetime. Einstein went to great lengths in appendix 5 of his ’booklet’ to clarify his thinking on this. In a new preface to the 1954 edition, he says: [S]pacetime is not necessarily something to which one can ascribe a separate existence, independently of the actual objects of physical reality. Physical objects are not in space, but these objects are spatially extended. In this way the concept of ‘empty space’ loses its meaning. Then, in appendix 5, he clarifies this further: ‘If we imagine the gravitational field … to be removed, there does not remain a space of the type (1) [i.e., flat spacetime], but absolutely nothing.’ Flat spacetime, ‘judged from the standpoint of the general theory of relativity, is not a space without field, but a special case … which in itself has no objective significance.… There is no such thing as an empty space, i.e., a space without field.’ He concludes, ‘Spacetime does not claim existence on its own, but only as a structural quality of the field.’ Building on the ideas of Aristotle and Descartes, Einstein generalised the notion that there is no space without material bodies and showed that there is no spacetime without a gravitational field. Just like our magnetic field, the gravitational field is a real physical thing—it can bend, stretch, and undulate. But it is also more fundamental than the electromagnetic field: the electromagnetic field needs the gravitational field to exist, since without a gravitational field there is no spacetime.
As a side note, Al-Khalili writes:
We speak of the Standard Model as describing all the elementary building blocks of matter and energy, but we are now pretty sure that everything we have found only makes up 5 percent of the universe. The other 95 percent, known as dark matter and dark energy, is still to some extent mysterious. We are confident it’s out there, but we don’t know what it is made of or how it fits into our current theories … The rotational speed of galaxies, the motion of entire galaxies within galaxy clusters, as well as the large-scale structure of the entire universe, all point towards a significant component of the universe consisting of a near-invisible matter component. We call it ‘dark,’ not because it is hidden behind other, visible matter, or even because it is actually dark, but because, as far as we can tell, it doesn’t feel the electromagnetic force and so does not give off light or interact with normal matter, other than gravitationally,1 and so a better name for it would have been invisible matter. Think for a moment about why, if you slam your hand down on a solid table, it doesn’t pass straight through. You might regard this as trivial: surely it is because both your hand and the table are made of solid stuff. But don’t forget that down at the level of atoms, matter is mostly empty space—diffuse clouds of electrons surrounding a tiny nucleus—and so there should be plenty of room for the atoms that make up your hand to easily pass through the atoms of the table without any physical matter coming into contact. The reason they don’t is because of the electromagnetic force between the electrons in the atoms of your hand and the electrons in the atoms of the table, repelling each other and providing the resistance we experience as solidity. However, if your hand were made of dark matter, then it would pass straight through as though the table weren’t there—the gravitational force between them being too weak to have much effect. [That’s because dark matter interacts extremely weakly with ordinary matter, primarily through gravity, and does not engage in electromagnetic interactions that typically prevent objects from passing through one another.] It has long been known that galaxies have much more mass than can be accounted for if one measures all the normal matter they contain in the form of stars, planets, and interstellar dust and gas. At one point it was thought that dark matter might be made up of long-dead stars and black holes—objects made of normal matter, but which do not emit light. However, overwhelming evidence now suggests that this invisible stuff must be made up of a new form of matter, most likely a new type of particle yet to be discovered. Originally, dark matter was proposed to explain the large-scale dynamics of entire clusters of galaxies. Further evidence then came from the way stars moved within spiral galaxies, circulating like undissolved coffee granules on the surface of a mug of instant coffee after it has been stirred. Most of the stars—and hence, you would think, most of the mass—in a galaxy are concentrated at its core, which would require those in the outer rim to be moving around the centre more slowly. The observed higher-than-expected orbital speeds of these outer stars suggest that there must be some additional invisible stuff present, extending out beyond the visible matter we can see and providing the extra gravitational glue to stop the outer stars from flying off. Dark matter can also be seen from the way it curves space around it. This phenomenon manifests itself in the way light bends while on its path from very distant objects to our telescopes. The amount of bending can only be explained by the extra gravitational curvature of space provided by the dark matter of galaxies that the light passes on its way to us.
Al-Khalili then describes a way of understanding the expansion of the universe, and what it means to say that space itself is expanding between galaxies, which are the spaces beyond the strong gravitational forces within galaxies:
There is one final point I wish to make before I move on. A common confusion many have with the idea of spacetime curvature becomes apparent when physicists describe the expansion of the universe. If spacetime is one big static four-dimensional block, what does it mean when physicists talk about it expanding? How can something that includes time embedded within it expand? After all, the word ‘expand’ suggests something changing with time, but that something contains time! The answer is that the expansion of space that we observe through our telescopes does not involve any stretching of the time coordinate too. It isn’t spacetime that is stretching, but rather only the three dimensions of space expanding as time moves forward. Although spacetime is in some sense democratic, with time as just one of the four dimensions, we can algebraically manipulate the equations of general relativity (by which I mean recast them in a slightly different form) so that all distances will now be multiplied by a ‘scale factor’ that increases as time moves forward and only space expands. Remember also that this expansion only happens in the vast expanses in between the galaxies, because within the galaxies themselves the gravitational field that holds them together is strong enough to withstand the overall cosmic expansion. Galaxies are like the raisins embedded within a loaf of rising bread in the oven. The loaf expands, but the raisins themselves remain the same size—they just become more separated from each other. In terms of the block universe, imagine that our local spacetime sits within a ‘bread universe’ in which successive slices of the loaf, as we move along the time axis from past to future, get bigger. Floating outside of spacetime, you’d just see the static loaf with its increasing slice sizes. But from our vantage point trapped within the loaf (or within a figurative raisin within the loaf), all we can experience is successively larger slices, and so we see a point (a distant galaxy, say) moving further away from us as we move through the slices.
As Al-Khalili summarizes, and elaborates:
The lesson from Einstein is that matter, energy, space, and time are all intimate companions … Einstein’s equation expresses how a gravitational field, or rather, the shape of spacetime, is determined by matter and energy. It is often said that his field equation shows how spacetime is curved by matter and energy and, at the same time, how matter and energy behave in curved spacetime. The point is that, just as matter and energy cannot exist without somewhere to exist in, there would, equally, be no spacetime without matter and energy. So, let us explore what we know about the ‘stuff’ of the universe … Whenever we talk about the nature of matter, we also need to understand the concept of mass. At the most basic level, the mass of a body is a measure of the amount of ‘stuff’ it contains. However, even mass does not always remain constant. The faster a body moves, the more its mass increases. This is not something you will be taught at school, and Isaac Newton would have found it astonishing, because it is yet another consequence of the nature of spacetime as elucidated by Einstein’s Special Theory of Relativity. If you are wondering why we don’t see this in everyday life, it is because we do not typically encounter things moving close to the speed of light, where the effect becomes noticeable. For example, a body moving at 87 percent of the speed of light, relative to some observer, will be measured by that observer to have double the mass it has when it is not moving, and a body moving at 99.5 percent of the speed of light will have ten times the mass it had when it was ‘at rest’. But even the fastest bullet only travels at 0.0004 percent of the speed of light, which means we generally do not experience relativistic effects or changes in moving bodies’ masses. The increase in the mass of a body as it reaches a significant fraction of the speed of light does not mean that it grows larger in size, or that the number of atoms it contains increases, but rather that it gains more momentum (making it harder to stop) than you might expect based simply on its ‘at rest’ mass.
Regarding momentum and mass, Al-Khalili writes:
According to Newtonian mechanics, a body’s momentum is the product of its mass and its speed, meaning that its momentum increases in proportion with its speed—you double its speed and its momentum doubles. But Newtonian mechanics says nothing about masses increasing when a body is moving. Special relativity gives us a different (and more correct) ‘relativistic’ formula for momentum, which is no longer proportional to a body’s speed. In fact, momentum becomes infinite when a body reaches light speed. This is a useful way of understanding why nothing can travel faster than light (another of the predictions of Special Relativity). Think of the energy needed to make a body move faster. At low speeds, this energy gets transferred into kinetic energy (energy of motion) as the body speeds up. But the closer the body gets to the speed of light, the harder it gets to make it go even faster, and the more of the energy being put into it gets used to increase its mass instead. This notion leads to the most famous equation in physics: E = mc2, which links mass (m) and energy (E) together (along with the square of the speed of light, c) and suggests that the two quantities are transformable into each other. In a sense, mass can be thought of as frozen energy. And because the speed of light squared is such a large number, a small amount of mass can be converted into a large amount of energy, or conversely, a large amount of energy freezes into very little mass. Therefore, we see that the law of conservation of energy is more accurately generalised to the law of conservation of energy and mass: the total amount of energy plus mass in the universe is constant over time. Nowhere is this notion clearer or more important than in the subatomic world, where E = mc2 led to an understanding of nuclear fission and the unlocking of the energy of the atomic nucleus. And it is E = mc2 that lies behind half a century of accelerator laboratories in which beams of subatomic particles are smashed together at ever higher energies to create new matter—new particles—out of the energy of the collision.
The force that binds atoms together is the electromagnetic force (which my kids and I explored in more detail in this video here):
At school, we learn about the electromagnetic force in the form of electrical or magnetic attraction or repulsion, but it plays an even more crucial role down at the atomic scale. Atoms bond together in all sorts of combinations, to make simple molecules and complex compounds and ultimately the huge variety of different materials we see around us. But how the atoms bind together comes down to the way their electrons arrange themselves around the nuclei, which is of course the very essence of chemistry, and this binding together of atoms to make up the stuff of our world is almost entirely due to the electromagnetic force between the electrons. In fact, together with gravity, the electromagnetic force is responsible, either directly or indirectly, for nearly all the phenomena we experience in nature. On the microscopic scale, materials are held together by the electromagnetic forces between atoms. On the cosmic scale, it is gravity that holds matter together.
As Al-Khalili reminds:
Despite all these deeply profound concepts, everything about spacetime that I have described in this chapter comes from just one of the three pillars of modern physics. But space, according to relativity theory, is smooth and continuous. If we zoom in, smaller and smaller, we will ultimately reach the domain of the second pillar of modern physics, quantum mechanics, where everything is fuzzy and subject to chance and uncertainty … [T]he third pillar of physics, thermodynamics, tells us that the idea of time as ‘just another dimension’ is inadequate. Thermodynamics describes the way systems change with time; more than that, it gives a directionality to time that is missing from the three dimensions in space. Independently of our own perception of time flowing in one direction only—born from the fact that we remember the past, live in the present, and anticipate the future—there exists an arrow of time that points from the past to the future, ruining the neat symmetry of the block universe.
In the next essay in this series, we’ll explore thermodynamics, which describes why time can only move in one direction, before proceeding to discuss the much stranger findings of quantum mechanics.
Links to other essays in this series: Part 1; Part 2; Part 3; Part 4; Part 5; Part 6.