You’ve probably heard that time is the fourth dimension.\u00a0 What does it mean?\u00a0 It should seem rather fishy that time should be the same sort of thing as a spatial dimension.\u00a0 We all know that you can only go in one direction in time\u2014towards the future!\u00a0 Time is measured in seconds, space in meters, etc.\u00a0 It turns out though, that time can be thought of as the same sort of thing as space, but not exactly: there’s just one tiny change that makes everything turn out different.<\/p>\n
Let’s start with two dimensional boring old Euclidean geometry of the sort one learns about in high school. It’s called two dimensional because you can specify the location of a\u00a0 point using two numbers $$(x,\\,y)$$.\u00a0 Now the most important equation\u2014from which everything else about geometry follows\u2014is the Pythagorean theorem.\u00a0 This allows you to compute the distance $$r$$ between any two points $$(x_1,\\,y_1)$$ and $$(x_2,\\,y_2)$$.\u00a0 If we define $$\\Delta x = x_2 – x_1$$ and $$\\Delta y = y_2 – y_1,$$ then we can think of $$\\Delta x$$ and $$\\Delta y$$ as the sides of a right-angled triangle.\u00a0 Then the Pythagorean theorem says that then the distance is $$!r = \\sqrt{(\\Delta x)^2 + (\\Delta y)^2}.$$Now it turns out that the exact same formula works in three dimensional Euclidean space as well; you just have to add in the $$z$$ coordinate: $$!r = \\sqrt{(\\Delta x)^2 + (\\Delta y)^2 + (\\Delta z)^2}.$$Now our world has only three dimensions of space, so far as we know (though that hasn’t stopped physicists from exploring ideas such as string theory where there are more dimensions of space).\u00a0 Nevertheless, nothing stops us from imagining that there are more spatial directions, say four.\u00a0 (If you haven’t read the classic Flatland<\/a>, go do it now.\u00a0 Geometry as social satire!)\u00a0 In that case, one would simply give the extra dimension a new name (say $$w$$ since we’re out of letters at the end of the alphabet) and then write down $$!r = \\sqrt{(\\Delta w)^2 + (\\Delta x)^2 + (\\Delta y)^2 + (\\Delta z)^2}.$$Voil\u00e1!\u00a0 Four dimensional space has arrived!<\/p>\n What has not yet arrived is spacetime.\u00a0 Each of these dimensions are all exactly the same (it doesn’t matter which one we call “first”, “second”, “third”, or “fourth” because of rotational symmetry).\u00a0 There is no notion of past or future.\u00a0 To get a time coordinate, we have to do something to make it special.\u00a0 And what we do is very simple, we just put in a minus sign: $$!s = \\sqrt{(\\Delta x)^2 + (\\Delta y)^2 + (\\Delta z)^2 – (\\Delta t)^2}.$$In deference to custom, I’ve renamed $$r$$ as the separation $$s$$, but that’s just a change of notation.\u00a0 What you should be noticing instead is how similar this equation looks to the last one.\u00a0 In fact, just as all of Euclidean geometry essentially follows from the Pythagorean theorem, so all of what’s called Special Relativity, or the geometry of Minkowski space $$(x,\\,y,\\,z,\\,t)$$ essentially follows from this one equation.\u00a0 (Minkowski was the name of Einstein’s math teacher, who first pointed out this way of understanding relativity.)<\/p>\n Now, what is the effect of this minus sign?\u00a0 It turns out that it changes things rather a lot.\u00a0 In Euclidean geometry, the thing inside of the square-root is always positive.\u00a0 But because of the minus sign, the $$s^2$$ of two different points can be either positive, negative, or zero:<\/p>\n I should say that I’m using units here where the speed of light (normally called $$c$$) is equal to one.\u00a0 That means that e.g. if we measure time in seconds, we have to measure space in light-seconds (the distance light travels in one second).\u00a0 Otherwise, the beautiful equations would get all cluttered up with $$c$$’s flying all over the place.<\/p>\n We’re used to dividing up time into three parts relative to ourselves: past<\/em>, present<\/em>, and future<\/em>. The present is just an infinitesimal sliver, so in a sense this division is into two parts: points to the past have $$\\Delta t < 0$$ compared to you, while points to the future have $$\\Delta t > 0$$ compared to you.<\/p>\n However, special relativity tells us you have to chop up spacetime in a more complicated way.\u00a0 Bearing in mind that you each live in a particular place as well as a particular time, you can chop up spacetime into three different regions.\u00a0 The future <\/em>is points that are timelike separated to you and have <\/em>$$\\Delta t > 0$$; these are the points of spacetime that you can affect.\u00a0 The past <\/em>is points that are timelike but have $$\\Delta t < 0$$; these are the points that can affect you.\u00a0 Then there is elsewhere<\/em>, the points that are spacelike separated.\u00a0 These points can neither affect, nor be affected, by each other.\u00a0 The three regions are separated by the “light cone”, which consists of the points that you could send a lightray to (or from).\u00a0 I’m too lazy to draw a picture right now, but you can see a pretty good explanation here<\/a>.<\/p>\n Next up, we’ll talk about rotations.\u00a0 As always, readers are free to ask me questions in the comments box.<\/p>\n","protected":false},"excerpt":{"rendered":" You’ve probably heard that time is the fourth dimension.\u00a0 What does it mean?\u00a0 It should seem rather fishy that time should be the same sort of thing as a spatial dimension.\u00a0 We all know that you can only go in … Continue reading \n