Think of a photon, a particle of light. It travels, of course, at the speed of light. As one year of time passes, it travels one light-year. The sum of 1 and 1 is 2, but that's not how you get the interval. The interval is the difference 1 - 1, which is 0. So the interval is related to the apparent rate of passage of time for a moving observer. The faster an object moves, the slower time on it appears to pass. This effect is called time dilation. As you travel closer and closer to the speed of light, the passage of time, as you experience it, slows down. If you could travel at the speed of light, time would be frozen. No time passes on a photon.
In Newtonian physics f particles that move when no forces are acting follow straight lines. Straight lines minimise the distance between points. In relativistic physics, freely moving particles minimise the interval, and follow geodesics. Finally, gravity is incorporated, not as an extra force, but as a distortion of the structure of spacetime, which changes the size of the interval and alters the shapes of geodesics. This variable interval between nearby events is called the metric of spacetime.
The usual image is to say that spacetime becomes `curved', though this term is easily misinterpreted. In particular, it doesn't have to be curved round anything else. The curvature is interpreted physically as the force of gravity, and it causes light cones to deform.
One result is `gravitational lensing', the bending of light by massive objects, which Einstein discovered in 1911 and published in 1915. He predicted that gravity should bend light by twice the amount that Newton's Laws imply. In 1919 this prediction was confirmed, when Sir Arthur Stanley Eddington led an expedition to observe a total eclipse of the Sun in West Africa. Andrew Crommelin of Greenwich Observatory led a second expedition to Brazil. The expeditions observed stars near the edge of the Sun during the eclipse, when their light would not be swamped by the Sun's much brighter light. They found slight displacements of the stars' apparent positions, consistent with Einstein's predictions. Overjoyed, Einstein sent his mum a postcard: `Dear Mother, joyous news today ... the English expeditions have actually demonstrated the deflection of light from the Sun.' the Times ran the headline: REVOLUTION IN SCIENCE. NEW THEORY OF THE UNIVERSE. NEWTONIAN IDEAS OVERTHROWN. Halfway down the second column was a subheading: SPACE `WARPED'. Einstein became an overnight celebrity.
It would be churlish to mention that to modern eyes the observational data are decidedly dodgy - there might be some bending, and then again, there might not. So we won't. Anyway, later, better experiments confirmed Einstein's prediction. Some distant quasars produce multiple images when an intervening galaxy acts like a lens and bends their light, to create a cosmic mirage.
The metric of spacetime is not flat.
Instead, near a star, spacetime takes the form of a curved surface that bends to create a circular `valley' in which the star sits. Light follows geodesics across the surface, and is `pulled down' into the hole, because that path provides a short cut. Particles moving in spacetime at sublight speeds behave in the same way; they no longer follow straight lines, but are deflected towards the star, whence the Newtonian picture of a gravitational force.
Far from the star, this spacetime is very close indeed to Minkowski spacetime; that is, the gravitational effect falls off rapidly and soon becomes negligible. Spacetimes that look like Minkowski spacetime at large distances are said to be `asymptotically flat'. Remember that term: it's important for making time machines. Most of our own universe is asymptotically flat, because massive bodies such as stars are scattered very thinly.
When setting up a spacetime, you can't just bend things any way you like. The metric must obey the Einstein equations, which relate the motion of freely moving particles to the degree of distortion away from flat spacetime.
We've said a lot about how space and time behave, but what are they? To be honest, we haven't a clue. The one thing we're sure of is that appearances can be deceptive.
Tick.
Some physicists take that principle to extremes. Julian Barbour, in The End of Time, argues that from a quantum-mechanical point of view, time does not exist.
In 1999, writing in New Scientist, he explained the idea roughly this way. At any instant, the state of every particle in the entire universe can be represented by a single point in a gigantic phase space, which he calls Platonia. Barbour and his colleague Bruno Bertotti found out how to make conventional physics work in Platonia. As time passes, the configuration of all particles in the universe is represented in Platonia as a moving point, so it traces out a path, just like a relativistic world-line. A Platonian deity could bring the points of that path into existence sequentially, and the particles would move, and time would seem to flow.
Quantum Platonia, however, is a much stranger place. Here, 'quantum mechanics kills time', as Barbour puts it. A quantum particle is not a point, but a fuzzy probability cloud. A quantum state of the universe is a fuzzy cloud in Platonia. The `size' of that cloud, relative to that of Platonia itself, represents the probability that the universe is in one of the states that comprise the cloud. So we have to endow Platonia with a `probability mist', whose density in any given region determines how probable it is for a cloud to occupy that region.
But, says Barbour, `there cannot be probabilities at different times, because Platonia itself is timeless. There can only be once-and-forall probabilities for each possible configuration.' There is only one probability mist, and it is always the same. In this set-up, time is an illusion. The future is not determined by the present - not because of the role of chance, but because there is no such thing as future or present.
By analogy, think of the childhood game of snakes and ladders. At each roll of the dice, players move their counters from square to square on a board; traditionally there are a hundred squares. Some are linked by ladders, and if you land at the bottom you immediately rise to the top; others are linked by snakes, and if you land at the top you immediately fall to the bottom. Whoever reaches the final square first wins.
To simplify the description, imagine someone playing solo snakes and ladders, so that there is only one counter on the board. Then at any instant, the `state' of the game is determined by a single square: whichever one is currently occupied by the counter. In this analogy, the board itself becomes the phase space, our analogue of Platonia. The counter represents the entire universe. As the counter hops around, according to the rules of the game, the state of the `universe' changes. The path that the counter follows - the list of squares that it successively occupies - is analogous to the world-line of the universe. In this interpretation, time does exist, because each successive move of the counter corresponds to one tick of the cosmic clock.
Quantum snakes and ladders is very different. The board is the same, but now all that matters is the probability with which the counter occupies any given square - not just at one stage of the game, but overall. For instance, the probability of being on the first square, at some stage in the game, is 1, because you always start there. The probability of being on the second square is 1/6, because the only way to get there is to throw a 1 with the dice on your first throw. And so on. Once we have calculated all these probabilities, we can forget about the rules of the game and the concept of a `move'. Now only the probabilities remain. This is the quantum version of the game, and it has no explicit moves, only probabilities. Since there are no moves, there is no notion of the `next' move, and no sensible concept of time.