spacetime

It all starts with coordinates.

Time is different from space: we cannot see time but measure its passing with a clock. Still, time serves the same purpose as space - namely, to locate objects. If we want to communicate the location of things in a room, we first agree on an origin from which we will measure distances. We then choose three “axis” attached to it alongside as reference for the distances.

We then quote three numbers: x, y and z - the coordinates - to find any point in the room. Let’s say the corner of the room is the origin, and the edges on the room attached to it are the coordinate system: I am sitting at the desk with my mobile next to me. I can find its location by x=2m, y=1m, z =1m. It means I follow one edge on the floor by 2m, the other by 1m and then the mobile is at 1m height.

Well, at least my mobile was there at 00:12. Ten minutes later, at 00:22, it dropped to the floor: it is now at x=1.8m, y=1m, z=0m. Clearly, for uniquely specifying the position of my phone, I need FOUR numbers: x,y,z and time. When it comes to locating things, time is on par with space. Otherwise, time cannot be more different from space: we can see the space around us, we cannot see time. We measure distances in space in metres and time in seconds. Well, this is just how we like our units. We already saw that the vacuum speed of light c is a fundamental constant in nature: we can use it to convert “seconds” to “metre” by multiplying time by c. If two events are 1 second apart, we could also say they have a distance of about 300,000 km. So, how different is time really? Less than you might think.

Finding my mobile.

We want more!

The amount of numbers you need to find an object unambiguously is called dimensions, and the numbers themselves are called coordinates. For finding a point that is not moving - say Mount Everest -, on the surface of the Earth, one needs two numbers, as the surface of the Earth is a 2-dimensional space. There is an awful lot of choice for those numbers, but common practice is to use longitude and latitude. The interior of your room with unmovable furniture items is a 3-dimensional space. In everyday life, we have not encountered a situation where we need more than four numbers: x, y, z, and time. Let us carefully call this 4-dimensional space “spacetime”.

Yet, we want more than putting four numbers together to find an object. We want to measure the distance between two points, and we would like to quantify the angle with which the two lines intersect. If you bolt these features on to your numbers, i.e., coordinates, your space becomes a “normed vector space”. This is just the maths name for it; we take from here that it is not granted that every space allows those measurements. If you find this difficult to take in, then this is because, in everyday life, we do not easily find a space that is not a normed vector space: our rooms have more or less right-angled corners, and we speak about distances between objects all the time. I drive 2 miles from home to the supermarket for shopping, to name just one example.

There is something special about our space!

The distance between two points in space is always positive, and if the distance is zero, it is the same point. Mathematically, everything is possible, meaning this is a feature of the space we are living in. In maths, we call this a “metric space”.

We use those features all the time in our lives. If my SatNav tells me that the distance to the supermarket is zero - “Arrived.”- I am banking on the fact that I am at the coordinates of the supermarket and not at any other point that would also have zero distance. There is only one point with the distance zero to the supermarket in our space: the supermarket! Isn’t our space beautiful? There could be more, different points, all with the distance zero without the extra feature above. Our SatNav would not work: even if your SatNav steered you to a location with distance zero, there would be more possible locations where you were then, and chances are, it would not be the supermarket.

Now think like Einstein!

We already said that three space coordinates are not enough to specify the location of an object since the object could move to another place. We could just add time as an additional number and keep the 3-dimensional space as it is with its ability to measure distances. This looks a bit clumsy. What if (the best mathematical discoveries start like this) we do more and consider space & time as 4-dimensional space? An immediate question arises: how should we measure distances in this bigger 4-dimensional world? Let’s now make its name official: spacetime.

What metric should we choose for spacetime?

This is when physicists talk to mathematicians. Mathematicians say “everything is possible”, but physicists want to pick the option which describes the universe. The answer for the “distance” is shown in the figure. The distance squared is not guaranteed to be positive anymore. Such a space is called Minkowski space.

Astronauts beware: SatNavs don’t work anymore for spacetime!

Assume a light pulse is moving with the speed of light along the x-axis. The centre of the pulse is described by x = c t. If we calculate the “distance” of the pulse to the origin of our coordinate system, we always find zero no matter what time it is (or equivalently where the pulse is). Even if the distance is zero, it no longer uniquely specifies the location.

What a bold move

….combining space and time to a new vector space with a norm: spacetime! Since we can measure distances now, we can start doing things: Let us consider rotations. What is rotation? A rotation changes the coordinates of a point but in such a way that the distance to the origin does not change. For instance, consider a plane with (xy) coordinates and a point located at x=2 y=0. If we rotate by 90 degrees, the point ends up at x=0 and y=2. Clearly, the distance to the origin has not changed. The interesting observation is this: what was x before ended up as y after the rotation.

What are rotations in spacetime? They are defined as always, as (linearly) mixing coordinates in such a way that the Minkowski distance to the origin does not change. Does this mean that we can rotate bits of space into time and vice versa? Yes, that is exactly what it means! To work out the transformation given the distance in the above figure is -admittedly- a little bit of numeracy.

We have encountered those rotations before: when you are moving in a spaceship with speed v, the time axis of your spaceship coordinate system is rotated out of line with that of the Earth’s system. The angle of this rotation is related to v/c. The transformations are called Lorentz transformation, and they describe how time slows down in a spaceship (see Flow of Time).