Hello, and welcome back to MPC! Last week, we continued our discussion of *spacetime diagrams* and introduced the concept of a *light cone*. Today, we will be relating our discussion of *light cones* to a topic we discussed a few weeks ago: the *spacetime interval*.

Let’s start off with a brief review of the *spacetime interval*. Recall that in spacetime, time and space are *relative quantities*. In a previous post, we were searching for a quantity that was *absolute* in spacetime, and we came across the following:

We called this quantity the *spacetime interval*. The *spacetime interval* is a very special value. Imagine that you were throwing a ball across a field:

**Figure 1:** A ball flying across a field

If someone were just standing and watching the ball, he/she would measure a certain distance that the ball traveled (*L*) and a certain amount of time that it took the ball to travel that distance (*t*). Another observer who is traveling close the speed of light, though, may measure completely different values of *t* and *L* due to time dilation and length contraction. However, if both observers were to take their *t*’s, multiply them by the speed of light (*c*), square that quantity, and subtract the square of their *L*’s (in other words, calculate the *spacetime interval* for the event), they would get the same exact value!

There is no doubt that the *spacetime interval* is amazing. Believe it or not, there is actually more that we could do with it! Let’s return back to the concept of the *light cone*:

**Figure 2:** The *light cone*

***Note: We are making the *y*-axis into the *L-axis* in order to stay consistent with our notation***

Recall from last week’s post that everything outside of the green region cannot occur in *spacetime*. *Why not?* Any event starting from the origin of the graph and reaching a point outside of the light cone is traveling faster than the speed of light, which is physically impossible.

Let’s imagine that we were given information from someone who saw a ball fly past him. This person says the ball flew 1.5 * 10^8 meters in 1 second (it was thrown by someone with a really good throwing arm). Let’s ask ourselves: *can this event have happened (i.e. is it within the light cone)?*

For those of you who are good at mental math, you may be able to answer this question immediately. Others of you may choose to draw a *spacetime diagram *in order to check if the event is within the *light cone*. However, perhaps we can think of a better, more efficient way of solving this problem that will work for more complicated numbers/scenarios. Let’s try calculating the *spacetime interval*:

We see that the *spacetime interval *is *6.75 * 10^16 m*. More importantly, though, the *spacetime interval* is greater than zero. This means that this event is possible (this event is within the l*ight cone*)!

*What? How were you able to tell so quickly? *Any event whose *spacetime interval* is positive is within the light cone!

Perhaps you do not believe me — this may seem like magic. However, it is completely logical. Just imagine any line within the light cone:

**Figure 3:** An event that occurs within the *light cone*

***Note: The circle just represents the end of the event***

Look at the blue line. Its final position on the *ct*-axis is clearly greater than its final position on the *L-*axis. Remember our formula for the *spacetime interval *is:

So, for the *spacetime interval *of the blue line in *Figure 3*, we are subtracting a small number ( *L*^2 ) from a large number (* (ct)^2 *). This results in a positive number. As a matter of fact, you can look at any line within the *light cone* and you will find that its position on the *ct-*axis is greater than its position on the *L*-axis. This means that any event occurring within the *light cone* will have a *spacetime interval* greater than zero.

*But what about someone else who saw the ball travel a different distance for a different amount of time (because of time dilation and length contraction)? How can you be sure that they also think that the event happened (i.e. they also get a positive number)? What happens if they get a negative number with their values? Did the event just not happen for them?* Remember: the *spacetime interval* is *absolute*! This means that even if someone did measure different distances/times, they would still get the same *spacetime interval* (a positive number).

Let’s now look at lines outside of the *light cone*:

**Figure 4: **An event that occurs outside of the light cone

Look at the red line. Notice that the red line’s final position on the *ct*-axis is **less** than its final position on the *L*-axis. This means that any events outside of the light cone will have a *spacetime interval *that is negative (a small number minus a big number is negative).

Finally, think about an event that is happening at the speed of light (i.e. on the edge of the *light cone* itself):

**Figure 5:** An event that occurs at the edge of the light cone

For reasons explained in last week’s post, all points on this line will have the same position on the *ct*-axis as on the *L*-axis. This means that any event occurring at the speed of light will have a *spacetime interval *of 0.

Let’s summarize what we discussed about *spacetime intervals *today:

*spacetime interval > 0 → event is within the light cone*

- spacetime interval = 0 → event is on the edge of the light cone

- spacetime interval < 0 → event is outside of the light cone

That is all for this week. Some of you (I hope not many of you…) may have found this post to be a little dull. *Who cares if something is inside or outside of the *light cone*? Events that are outside of the *light cone *cannot happen, so why should we even care about them* Well, initially, when I said that events outside of the *light cone* cannot possibly happen, I left out a few details. There is actually an interesting and important way of thinking about events outside of the *light cone*. We will discuss this next week when we talk about the concept of *causality*. See you then!

For more information on spacetime diagrams, be sure to check out: http://www.farmingdale.edu/faculty/peter-nolan/pdf/UPCh34.pdf

(featured image: https://www.ukworkstore.co.uk/media/catalog/product/cache/1/image/9df78eab33525d08d6e5fb8d27136e95/t/r/traffic-road-cones.jpg)