So imagine that you’re standing at some spot, holding a big round bulb that can shine in all directions. At a particular time, you switch on the bulb and it shines out brightly. Who has a chance to see the light from your bulb? Let’s not worry about things like buildings, cars, etc., getting in the way of the light, but just imagine it can shine free in all directions. You’ll realize that there is a whole ball of light expanding away from the bulb – light going away from you in all directions. (The edge of that ball describes what is called a sphere, a two-sphere to be precise, denoted [tex]S^2[/tex] in the trade.) The sphere expands at the speed of light. You might imagine that eventually every point in space around you would be reached by your expanding sphere (let’s stay local and not worry about the expanding universe and so forth), and you’d be right. But when? When does it get to some particular point that you might care about? As soon as it takes the light the time it takes to get there, of course. It can’t get there any sooner than that.
Let me put it another way. Your sending out the light is a particular “event”. It took place at a particular place -where you stood- and at the particular time you flicked the switch -three o’clock, saya. One second later, since light travels at about 300,000 kilometers per secondb, all points about 300,000 kilometers away (in any direction) will receive that light. A person that distance away looking your way at that time (1 second later) will see the flash of light from you. Any time less than a second before, they will not have seen that light. It could not have got there yet. No signal from you could have got to them faster. You cannot have any effect on that person sooner than one second after you emitted the light. You can “cause” nothing to take place at their position, earlier than one second later. We can call that position (300,000 kilometers away) and such a time (half a second after 3 o’clock) an “event” as well. So said more succinctly, the event of you sending out the light is not “causally connected” to that event (300,000 Km away, half a one second later), or similar events, such as 1 second later but 400,000 km away.
In physics, it is very important to appreciate the above, and people spend a lot of time drawing pictures to make clear what is going on. Let’s try to do that. Rather than three spatial dimensions, people often consider two of them, just for ease of drawing what we’re about to draw. In this case -imagine the two dimensional world is the surface of your desk or dining room table- the ball of light that is expanding away at the speed of light is a disc, and the edge is a circle (we’d call that [tex]S^1[/tex] if we were trying to impress our friends in the bar. We’ll stick to the word “circle” here). Everything I said above is still true, I’ve just replaced a sphere with a circle.
Now for a useful diagram. It will explicitly represent what is going on in time as well as space, and it actually called a “space-time diagram”. Let us call three o’clock “time [tex]t=0[/tex]”. A tiny fraction of a second after that time, the circle of light expanding away from you is quite small. A bit later, it is bigger, as the light is moving away from you. Later, a bit bigger… One second later, the radius of the circle is 300,000 km, two seconds later it is 600,000 km, and so forth. So we have a family of growing circles – the later the time, the bigger the circle. We can draw all these circles on the plane, all centered at the place from where the light was emitted. Just for fun, let’s lift the circles off the table. We raise a circle higher off the table according to the later elapsed time that it represents. What we are doing is representing the time that has elapsed in terms of how far we lift them above the table, positioning them along a line in a vertical direction in our diagram (we just chose vertical for no good reason… it does not have anything to do with vertical in space). If we are careful to accurately preserve the proportions to represent accurately the time elapsed for each circle, we’ll get a familiar shape.
You know this shape. It has a name. It is a cone. The vertex of the cone is the original event of you (or your two dimensional avatar) hitting the switch to the light bulb. Any point in this diagram is also an event, since it represents a place (in the two dimensions) and a time (how far in the vertical). Any event on or inside the cone (and you must imagine it going on forever), where the ice-cream would be, can be reached or affected by you (you’re at the apex, recall) since you don’t have to go faster than light to get to it. Any event outside the cone cannot be affected by you – it is not “causally connected” to you – since you’d have to send a signal or other effect at faster than the speed of light.
You now are acquainted with the “light cone”, the central concept in Einstein’s Special Relativity.
You can go further and extend the cone downward in the other direction…. into the past. What does that mean? Well, it is talking about the events from the past that could possibly effected you. A point on the cone at some earlier time than when you emitted the light (say, one second, 300,000km away) is causally connected to you since someone 300,000km away could have flashed a light bulb, and you’d receive the light at the original event (your original spot, at 3 o’clock – they emitted it at 2:59:59, and it took one second to get to you, you see.) An event one second earlier than [tex]t=0[/tex] but say, 400,000 km away, is outside the cone. It is again causally not connected to you. They are too far away for emitted light to have reached you one second later – it’s still 100,000 km away at that time.
So the top part is the “future light cone” – it separates points representing events that you can effect from ones that you can’t, and the bottom part is the “past light cone” – it separates events that can affect you from events that cannot. Light cones are useful for lots of reasoning. Maybe I’ll talk more about them some other time.
Well, that’s it. In a sense, you actually carry a light cone with you where ever you go, representing the borderline between the possible and the impossible, the accessible andthe inaccessible. Look after your light cone. Here are some solid ones you can buy and put on your mantelpiecec:
aA time and a place is an event. Recall that party you had in your room or apartment last night? That was an event, right? It had a place and a time, like you put in the evite, the text, or in embossed faux gold lettering on the invitations you sent out. You need both a time and a place to specify the event. Your apartment is not a party, or any other type of event. 10:00pm is not a party. Your apartment at 10:00pm, however…. Absolutely.
bIt’s actually 299,792.458 kilometres per second.
cThis is a picture of some of the lovely work of Tamsin van Essen. You can see more here. My writing more about subjects touched on by some of the her pieces, and you reading it, may well be inside all of our future lightcones.