Special Relativity X: Spacetime Interval and Light Cones

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:

Eq_1.PNG

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:

STLC_1.PNG

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:

STLC_2.PNG

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:

Eq_2.PNG

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 light 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:

STLC_3.PNG

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:

Eq_1

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:

STLC_4.PNG

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):

STLC_5.PNG

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)

 

 

 

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