April 10, 2021

The science magazine of Imperial College

Time: Going Places

What’s the truth about the relationship between space and time? Isobel Nicholson tells us…

“the faster you travel through space, the slower you experience time… It is simply the way our universe works”

Space is a very familiar concept. We intuitively understand distance – when we go for a walk, we cut the corner without ever needing Pythagoras’s theorem. If you want to get to the shop across the square, you could take three steps forwards and then four to the left, but you will probably just walk five steps in a straight line. It’s easy to prove, even if you include the third dimension (up and down): the square of the distance travelled by the direct route is the sum of the squares of what we call the three orthogonal directions (x, y and z).

The fourth dimension is far less intuitive. We know that time is a dimension because it fulfils all the criteria: we have been measuring ‘distances’ in time for thousands of years and you only need to look at the recent cinema releases to see that we are comfortable with the concept of going ‘forwards’ and ‘backwards’ in time! We also instinctively understand that time acts in a fourth, very different ‘direction’ which is independent of the spatial directions.

However, time is far more beautiful and complicated than simply an addition to our spatial comfort zone. The key difference is not the physical difference between being able to see for ourselves that space exists in front of us while simply having to assume that the future will be there. This difference is strange, but it is not a deep question of philosophy related to the way we experience reality: after all, we do not ‘see’ anything but simply interpret a complex series of mid-range electromagenetic signals. The key difference is that if time was like one of our three spatial dimensions, we would be able to have a straight line through space, where the value of time didn’t change, just as we can have a straight line through time where your position doesn’t change. But you can’t stand still in time, or move backwards, you can only go forwards.

But do we all move forwards in time at the same pace? It turns out that the faster you travel through space, the slower you experience time. This is not a question of biology or consciousness – a clock that has been left on an aircraft will show a fractionally earlier time than one on the ground. It is simply the way our universe works, and the only reason that we don’t understand it intuitively is because we don’t travel fast enough. As you travel faster and faster, time slows down until at some point it is very, very close to stopped. This speed has been experimentally found to be 670,616,629 miles per hour, the speed of light. Only particles with no mass, like photons, can travel at this speed and nothing can break this barrier.

According to special relativity, we can quantify the relationship between time and space by trying to think of some kind of Pythagoras’s theorem that will measure the total ‘distance’ travelled through both space and time at once. We know that the square of the distance between two events in three-dimensional space just adds up the squares of orthogonal directions. But as we have seen, when you travel quickly, though you travel further in space, you travel slower in time, which we need to take into account in our equation. So the square of the distance between two events in spacetime adds up to the square of the orthogonal spatial directions minus a quantity proportional to the square of the duration in time. This ‘distance’ in spacetime is called the ‘spacetime interval’.

The spacetime interval has a special property. Take the following scenario. I am on a train moving at a constant speed, sitting in a seat, watching the ticket officer walk down the aisle. I see him move a few metres. However, if I am instead on the side of the train track, and watch the ticket officer walk down the aisle, I see him fly by, moving many metres. This because the train, as well as the ticket officer, is moving relative to me. You can see that the distance someone else moves in three-dimensional space changes according to how fast you yourself are travelling. However, in four-dimensional space, the spacetime interval of two events always stays the same, no matter how fast you yourself are travelling. So if I measured the spacetime interval of the ticket officer moving when sitting on the train, it would be the same as if I measured the spacetime interval of the ticket officer moving when sitting on the side of the train track. The fact that spacetime intervals don’t change is called ‘Lorentz invariance’ and the spacetime interval itself is called an ‘invariant’.

Things that don’t change, like invariants, are often called symmetries, and they have an important mathematical property, discovered by Emmy Noether. This is that every symmetry has a conserved quantity associated with it. For example, if distance doesn’t change in a particular physical system, then momentum is conserved; if time doesn’t change, then energy is conserved. The symmetry of Lorentz invariance means that in spacetime, the energy and momentum together are conserved: Time invariance is therefore intricately tied up with the conservation of energy.

Unfortunately, spacetime is all a bit more complicated that than. The mass of an object actually bends spacetime, making other masses travelling towards it appear to accelerate. We call this phenomenon gravity. This idea is often modelled by a ball on a rubber sheet – the ball’s mass produces a dip in the rubber sheet, so that any mass on the edge of the rubber sheet would accelerate towards to the middle. Taking these sorts of accelerations into account moves us away from the theory of special relativity to the theory of general relativity.

But we’ve seen at least one thing: time’s role as a dimension is essential. So many factors, from the finite speed limit of the universe to the conservation of energy itself, depend on time and its relationship with space. It is the interplay between space and time that makes it possible for the universe as we know it to exist

Isobel Nicholson is studying for an MSc in Quantum Fields and Fundamental Forces

Images: Cranbourn Street by Xynn Tii; Relativity by Lamerie; Wikipedia, Spacetime curvature