Sharp Blue: VBHT1: Newtonian Spacetime


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Italicised articles are old and unrevised.


In this first instalment, I’ll talk about Newtonian spacetime, which is how we thought space and time were until Einstein invented Special Relativity in 1905.

Let’s start by getting rid of one dimension, so that everything becomes easier to visualise. Make lots of thin glass sheets and etch each one with a regular grid. Each sheet will represent the state of the universe at some particular time. Give each one a time label - this will be Newton’s absolute coordinate time. Draw on each sheet the state of all the particles and stuff at the time represented by the absolute time label. You can also mark little arrows that show how fast the stuff is moving and in which direction. Stack them up one on top of the other - this will make a model of Newtonian spacetime. Each “event” is labelled by the place it takes place (the coordinates in the sheet) and the time when it happens (the sheet number it is on). Now imagine that there is a particle moving around in space. In each of the time slices it is in a slightly different position, and if you look at the whole stack then it appears to trace a line in spacetime - the particle’s worldline.

Anybody moving at constant velocity (that is, at a constant speed and in a constant direction) will have a straight worldline (remember that this is a line in spacetime, made up of points in each of the timeslice sheets). Anybody moving in such a way is an observer in an inertial frame (it’s possible to tell if you are accelerating, so you can rule yourself out as an inertial observer if you find that you are). Now, first look at it from my point of view as I sit in my spaceship and watch you race by in yours. In my frame, I am not going anywhere, so my wordline is both straight and vertical (I seem to myself to be moving with zero speed, so I am at the same place in each successive slice). On the other hand, when I draw your worldline it appears straight, but tipped over at some angle: in each successive time slice you appear to have moved along by the same distance. The faster you’re moving, the more tipped-over your worldline will be. This means that you will label a given event somewhere in spacetime with different position coordinates, as the point from which your spatial coordinate grid is measured is moving with respect to mine in each successive slice. The change in coordinates on changing from one viewpoint to another in Newtonian physics is called a Galilean transformation; it changes the spatial coordinates of an event, but not the time coordinate. The Galilean transformation will turn out to be wrong in special and general relativity.

Now, here’s a way to visualise the Galilean transformation. Take each of the sheets and drill a hole through both of our positions on each one. Reassemble the model Newtonian spacetime and stick straight rods through each of the sets of holes, so that one rod passes along each wordline. Now in my frame, my worldline rod is vertical. To do the Galilean transformation, grab your rod worldline and pull it across until it is vertical (taking care not to rotate the sheets around it!). My worldline will then be slanted. Thus from your viewpoint, you appear static, whereas I am racing by in my spaceship. The whole thing is nicely symmetric. This symmetry means that you can do your calculations in any inertial frame and the laws of physics will look the same. This is the Principle of Relativity (remember, though, that we aren’t even talking about the theory of relativity yet). Not surprisingly, the Principle of Relativity remains true in special relativity and is actually extended in general relativity.

If we were both moving through space at the same speed and in the same direction, then our worldlines would be parallel: we would share the same inertial frame. Even looked at from the point of view of some third observer (Earth say) we would still appear to be moving along parallel worldlines - this is good as the parts of you are moving along parallel worldlines, and if from somebody else’s inertial frame they weren’t, then that person would see you falling apart! (Of course, we might still give different spatial coordinates to events, but that is just because our zero coordinates are in different places; it’s much easier to fix that up.)

Again I should stress that this only works if the two observers don’t accelerate. The Principle of Relativity is deeper than the Galilean transformation - it says that you can look at things in any inertial frame, and all your laws of physics will work in it. Importantly, it means that any inertial frame can be considered to be “at rest”. This is why when you’re flying in an aeroplane in straight, level, constant speed flight you can almost forget that from the point of view of people on the ground you are hurtling by at hundreds of kilometres an hour. Of course, if the aeroplane accelerates, then it’s no longer an inertial frame - you can tell as you feel like you’re being pushed back in your seat, and no longer have a sensation of being “at rest”.

The laws of classical physics fall into two classes: constraint equations and evolution equations. The constraint equations tell you the restrictions on how you are allowed to draw stuff on each sheet. For example, in classical electromagnetism there are charges, electric fields and magnetic fields. Say that the charges are represented by black spots (marked with pluses or minuses) on the sheets, with little arrows to show their current velocities. The fields will be lines with arrows on them - say green for magnetic and blue for electric. Some of the rules for the fields are that the green magnetic lines have no ends, and the blue electric lines flow out of positive charges and into negative charges. You simply aren’t allowed patterns of field lines for which this isn’t true. The evolution equations then tell you how the positions of charges, the arrows representing their velocities and the patterns of field lines change from one sheet to the next. (Of course, things are much more complicated than that, but that’s the gist of the thing).

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