*I thought you’d both understand when you saw the different dimensions inside from those outside*

How can a box be bigger on the inside than on the outside?

Let’s look at it another way. Why *shouldn’t* a box be bigger on the inside than on the outside? And why is Ian Chesterton so agitated about it?

We carry around with us an intuitive model of space and time, a model so apparently obvious and reasonable that no one until the nineteenth century seriously questioned it, and which turned out to be utterly wrong.

In our inbuilt mental model, space is three dimensional. That is to say, the position of anything in space can be specified by three numbers. In your room, you could pinpoint every object by giving the shortest distance to the front wall, the left hand wall, and the floor. Describing geography, you might give latitude, longitude and height above sea level. For the positions of stars in deep space, you might state the right ascension, declination and parallax. Whatever system of coordinates you use, you always need to give three numbers to determine a position. That’s what “three dimensional” means. (If you only need two numbers – a chessboard, say, or a graph on a sheet of paper – then your space is two-dimensional, while a line has only one dimension.)

Time, in our intuitive model, is a separate thing. It ticks along at a steady rate, quite independent of where objects are in space. In principle, we could describe the entire history of the Universe as a series of three dimensional snapshots, each taken at a different instant of time.

People believed this without question, without even thinking about it, without even realising they were making assumptions that might be questioned. Then Einstein came along, and proved that this was all bollocks.

Einstein showed that our our Universe has four dimensions, not three. Time does not tick along independent of space: different people moving at different speeds will measure distances and durations differently, seeing time turning into space or space into time.

This was a pretty staggering revelation at the time, but there was more to come. During the nineteenth century, various mathematicians had been playing around with alternative forms of geometry. The standards rules of geometry had been laid down by Euclid around 300 BCE, and schoolchildren were still being taught from translations of his *Elements*, more than two thousand years after that text had been written. This ancient text collected the mathematical knowledge of classical Greece in the form of basic definitions, axioms and postulates, from which all the laws of geometry and other mathematical fields were derived by strict logical reasoning. Most of these basic postulates – the starting points for the whole business – seemed self-evident, but there was one that had been niggling at mathematicians for a while: the Parallel Postulate. This states that two parallel lines can never meet, and it seemed a bit arbitrary. Various people had tried to prove it from the other postulates, without success, and in the nineteenth century some mathematicians decided to try something new and radical. They decided to see what geometry would look like without the parallel postulate, and came up with the idea of curved space.

It’s not that hard to imagine a curved space, as long as it’s a two dimensional space, like the surface of the Earth. If you’re standing at the equator, and you draw two parallel lines some distance apart, both pointing due north, and then extend these two lines northward for thousands of miles, they will come closer and closer together and eventually, at the North Pole, they will meet. Put that way, it seems kind of obvious, but the conceptual leap required to treat curved spaces as an alternative form of geometry was profound. New terminology developed for these new ideas. A space, of however many dimensions, where parallel lines never meet is a flat, or Euclidean, space. A space where parallel lines meet is positively curved, and one where they diverge is negatively curved: in both of these non-Euclidean geometries the rate at which the parallel lines meet or diverge tells you the degree of curvature.

Now all this had been kicking around for a while before Einstein, but no one thought it had any connection to reality. The Universe was clearly described by the geometry of Euclid, and that was that. But Einstein, in his crowning intellectual achievement, showed that this was all wrong. He had already shown that we live in a four-dimensional world. Now he showed that the four dimensional geometry of spacetime is not Euclidean, but curved. The degree of curvature depends on how much mass there is in the vicinity: the greater the mass, the greater the curvature. When spacetime curves, objects moving through it follow curved paths, not straight lines. People had observed this weird phenomenon for all of history, but had misunderstood it. Even the great genius Newton, the most brilliant scientist humanity has ever produced, misidentified it as the result of some weird force that acts at a distance. He called it “gravity”. But now we knew better. There is no force of gravity. There is only the curvature of spacetime, that makes objects move together as they move through four dimensions. This is the General Theory of Relativity, and it remains the fundamental theory of space, time, gravity and cosmology in modern physics. It’s been tested, too. Indeed, the GPS system on your phone makes calculations that depend on general relativity, as it determines your position based on signals from satellites moving through the curved spacetime around Earth. If general relativity wasn’t true, your GPS wouldn’t work.

So the world is four dimensional. Time is not independent, but is just another coordinate in four dimensional spacetime, like height or latitude. This spacetime is curved by the masses that inhabit it, and there is no such thing as the force of gravity. So much for intuition.

Imagining the real curved, four-dimensional manifold in which we live is basically impossible. Our brains aren’t built for it. That’s why we need all the hard maths. But we can imagine a curved two-dimensional surface embedded in our intuitive three-dimensional space, and that can give us all sorts of insights into the real curved spacetime that we inhabit.

So, imagine a two-dimensional Ian Chesterton (all right, an even more two-dimensional Ian Chesterton). He lives in a two-dimensional world, like the surface of a sheet of paper, and can never leave it.

One day he comes across a box, guarded by an irascible two-dimensional old gentleman. This being a 2-d world, the box is just a square, one side of which can swing open or closed to allow 2-d people to move in and out. 2-d Ian assumes the interior of the square is slightly smaller than its exterior, and is astonished to discover that it is in fact much bigger!

What he doesn’t realise is that someone has carefully cut out the surface on the inside of the square and replaced it with a little tube leading through the third dimension to another sheet, much larger than the area inside the original square. To us, used to the third dimension, it is easy to see what has happened, but to poor Ian it all seems impossible. He walked all round the outside of the square and it just wasn’t this large.

You can do a similar thing in our four-dimensional spacetime. A nice paper by Arvind Borde shows how you can connect two separate surfaces in spacetime via a manifold that acts like a tunnel between them. The focus of Borde’s paper is on 3-d surfaces within 4-d spacetime, but there’s no reason why the same equations shouldn’t work for connecting up two separate 4-d spacetimes. Intriguingly, in that case the connecting manifold would be five-dimensional, perhaps explaining Susan’s odd fixation on the fifth dimension in her science class.

The fact that you can do this in general relativity perhaps isn’t so surprising. The fundamental equation of general relativity, the Einstein equation, can be written in its simplest form as

**G** = 8π**T**

where **G** describes the shape of spacetime, and **T** describes how matter and energy is distributed in spacetime. 8π is just a constant, so the equation simply says that spacetime curvature depends on mass-energy. (As is often the case, it looks simple because the complexity is hidden. **G** and **T** are both tensors, a mathematical object that is a generalisation of the concept of a vector, and actually solving this equation is a massive pain in the tits, even in simple cases.)

Now there are two ways of using this equation. One, the traditional and sensible way, is to pick some sensible mass-energy distribution, plug it into **T**, then crank the handle and calculate **G** and use that to describe the spacetime curvature. That’s how you get descriptions of black holes, gravitational lenses, the big bang and all that sort of thing. The other way, less sensible but more fun, is to come up with whatever bizarre shape you would like to contort spacetime into, plug it into **G**, then calculate from **T** the mass-energy distribution you need to create your wacky universe. This is how you get fun stuff like time machines and warp drives. The problem with this is that there is no reason for your **T** to end up being at all physically reasonable, and it usually won’t be. You generally end up needing things like negative-mass particles, which would be less of a problem if anyone had ever observed a negative-mass particle, or had any idea what one might look like, or had any theoretical reason for believing they might exist.

But this is science fiction, and we can safely assume that whatever exotic, dangerous or downright unhealthy kinds of matter and energy you need to create your Borde tunnel and transition to a new spacetime, the Doctor’s people have it by the sackful.

So that’s all fine, but there’s a deeper issue that we’ve so far only touched on implicitly. It’s all very well coming up with scientific models for a box that’s bigger on the inside, but it’s still a jumped-up parlour trick. The real point of the Tardis is that it can travel freely in spacetime. Why should a time machine have weird geometry?

We’ve already seen that there are deep links between space, time and geometry. Einstein’s theory of curved spacetime is a geometric theory of gravity, and as we shall see in more detail when we get to The Space Museum, travelling freely throughout spacetime requires manipulating the geometry of the Universe in very particular ways. So it’s no surprise that a civilisation that can build machines for spacetime travel can also make them bigger on the inside than the outside. Indeed, it would be quite odd if they couldn’t.

And this, perhaps, is what has got Mr Chesterton so worked up. He can’t imagine how anyone could have achieved this incredible feat of spacetime engineering, but he knows that, if it isn’t just a conjuring trick, the implications go far beyond revolutionising interior design. If you can make a box bigger on the inside than the outside, you have technology that gives you complete power over space and time.

Assuming, of course, it doesn’t break down.

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Lovely idea for a blog, and interesting and lucid writing. Thanks. Unscientific rumination follows:

Is a Borde manifold necessary to confound 2-d Ian? How about if the 2-d space inside the box merely bulged into a third dimension, or bulged to a greater degree that the space around the box? It would seem fitting somehow if the impression of greater 3-d space inside the Tardis was a result of it “bulging” into time.

The message I take from the Borde paper is that if you want to connect two separate n-dimensional spaces, you need a connecting tunnel of n+1 dimensions.

In the case of the 2-d Ian, this is intuitive enough. He moves on a surface that passes through the third dimension to get into the 2-d Tardis interior.

When it comes to the actual situation we see on the telly, both the junkyard in Totter’s Lane and the Tardis interior are four dimensional spaces: everything that happens in them can be pinpointed by three coordinates in space plus one measurement of time. To connect two 4-d spaces, you need to move through the fifth dimension.

As far as i can remember, this is the only episode of Doctor Who that mentions the fifth dimension at all, and it makes quite a big thing of it. Of course, it’s not as if Anthony Coburn had done all this relativistic mathematics thirty years ahead of Borde, but he clearly had some intuitive sense that higher dimensions were bound up with the remarkable spacetime properties of the Tardis, and it’s pleasing that the correspondence with the results of real physics turns out to be so close.

This post alone is more revelatory for me than “The Doctor’s Wife.” You’ve got a devoted reader for as long as you keep writing this blog!

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I now regret all my misspent hours daydreaming in math class! Your explanations however are lucid and enjoyable.