Why Time is a Dimension

 This post is dedicated to KP and our relentless debate on this matter.

I'd like to convince you that time is a dimension on a completely equal footing to the spatial dimensions we're used to. Along the way I hope to instill an idea of what a dimesion actually is. 

'Dimension' is one of those words that's thrown around in everyday use as a science word but in reality the definition is far more banal. We can thing of it loosely as 'the number of pieces of information necessary to fully specify something'. In 2d space (ironically I'm specifying the dimension there) i.e. a flat piece of paper which extends in all directions then I could specify the location of any point using normal (cartesian) coordinates x and y. This isn't the only way to do this though: I could also use polar coordinates to specify the distance from the origin and a certain angle. I could even use hyperbolic coordinates. The point is, there's no holy choice of coordinates. I'm given the space and I choose the coordinate system but it's an inherent feature of the space how many coordinates I need.

I'd like to show the generality of dimension. If I wanted to specify a triangle, how many numbers would I need? First instinct might be that I triangle has 3 vertices, each with 2 coordinates so I need 6 numbers. But if I have a triangle and I rotate it a little or move it around then I'm going to consider it to be the same triangle. After all, space has no choice of origin and orientation so why should it matter where it is. The first time I thought about this I actually chose to specify 2 angles (as the angles must sum to 180 so the third is fixed by the first 2) giving the space of similar triangles dimension 2 and then I could specify the area, giving a third dimension. Alternatively, I could state an angle and the 2 lengths either side of the angle. Or I could specify 2 angles and the length of the line joining them. Point being: however I choose to do it, I need 3 numbers. I certainly wouldn't say that area is the third dimension.

Now imagine you want to "specify an event in spacetime". When physicists do this, they're always talking about photons and light rays but a more parochial event would be meeting a friend for a coffee. You specify a latitude, longitude and altitude for meeting but your friend replies "you've only restricted our event to a hyperplane in spacetime, when are we meeting!" and you apologise and give the meeting time: the fourth number needed to specify a point in spacetime. Notice as well that they specified altitude but if you think that's redundant then you would be right in thinking that the surface of the Earth is 2 dimensional.

Now I agree, our relationship with time is completely different to our relationship with the spatial dimensions, primarily because we can change out spatial coordinates at will but not our time coordinate. But picture an ant which is born midair and spends its entire existence falling to Earth (say with terminal velocity) and dies before it hits the ground. The ant would consider it's progression along the radial axis to be identical to its progression through time and it has control over neither, but us outsiders would tell it that space is a dimension, regardless of whether the ant has control over it. 

And so we are ants, unable to control our time dimension but that doesn't make it any less of one.