Hello, and welcome back to MPC! Last week, we discussed the concept of spacetime and the role it plays in our lives. Today, we will be looking at spacetime a little more closely by analyzing a quantity known as the *spacetime interval*.

In last week’s post, we discussed how *space* and *time* are *relative quantities*. In other words, the meaning of *space *(e.g. 1 meter) and *time* (e.g. 1 second) are different for different people/observers. *How so?* Using what we have learned about special relativity, we know that time is dilated for some observers and length is contracted for other observers. Therefore, *space* and *time* do not have an *absolute* meaning: if I were to tell you that it took Albert *5 hours* to drive to a city that is *300 miles* away, a reasonable question would be “*relative to whom were these measurements made?*” We typically consider *space* and *time *to be the most fundamental features of our world, but the fact that they are *relative quantities* makes them seem a little less “meaningful.”

As we also discussed last week, *spacetime* is more fundamental than *space* and *time*. We also briefly discussed that *spacetime* provides an *absolute *quantity, describing *space* and *time*, on which everyone will agree (regardless of their motion). Today, we will derive this quantity, known as the *spacetime interval*.

Let’s think back to the bouncing particle of light we described when we were discussing time dilation. Recall that we had a particle of light starting at the bottom of a rocket ship and had that particle of light move upwards towards the top of the rocket ship. To learn more about the *spacetime interval*, we will use a similar contraption. This time, however, the particle of light will not be in a moving rocket ship — rather it will be in a box (with a piece of glass as its front) of height *h* that is “stationary” in outer space:

**Figure 1:** The bouncing particle of light

***Note: The speed at which the particle moves towards the top of the box is the speed of light, *c*.***

Now, let’s imagine that we have a rocket ship zooming past the box in *Figure 1*. We’ll say that this rocket ship is traveling at a certain speed, *v**a*, and contains a person, *person a*:

**Figure 2:** Person a* traveling past the box*

As *person a* travels past the box, he will think that the box is traveling past him. To elaborate, *person a* will see the box approaching/passing him at a speed of *v**a*. This is important, for it means that, although the particle of light is only traveling vertically inside of the box, *person a* will see it traveling diagonally:

**Figure 3:** The particle of light’s motion according to *person a*

*(**Note: If the concept illustrated in *Figure 3* is confusing, be sure to check out this post**)*

So, *person a* sees the particle of light traveling along the green arrows in *Figure 3*. We can create a right triangle using the green arrows as the hypotenuse:

**Figure 4:** A very important triangle

*Figure 4* claims that the hypotenuse of the triangle is of length *c(t**a**)*. In this case, *t**a* is the amount of time it takes for the particle of light to reach the top of the box **according to*** person a* (recall that speed, *c*, times time, *t**a*, is distance). Additionally, one of the legs of the triangle is *h* and the other leg is *L**a*, or the horizontal distance traveled by the particle of light **according to** *person a*. Using the Pythagorean theorem, we can relate the aforementioned quantities as follows:

Let’s now imagine that there is another rocket ship passing the box. This rocket ship is traveling at a certain speed, *v**b*, and contains a person, *person b*. We will say that *v**b* is greater than *v**a* (meaning that *person b*’s rocket ship is traveling faster than *person a*’s rocket ship):

**Figure 5:** Person b* traveling past the box*

As was the case for *person a*, *person b* will see the particle of light traveling diagonally as he passes it:

**Figure 6:** The particle of light’s motion according to *person b*

And, finally, we can use the green arrows in *Figure 6* as the hypotenuse for a right triangle:

**Figure 7:** Another very important triangle

Although the triangle in *Figure 7* is very similar to the triangle in *Figure 4*, there are a few important things to note. First of all, the hypotenuse of the triangle in *Figure 7* is *c(t**b**)*, where *t**b* is the amount of time it takes for the particle of light to reach the top of the box according to **person b**. It is important that we establish that *t**a* and *t**b* are not necessarily the same — if *v**a* and *v**b* are different, the time dilation that is “influencing” the particle of light will be different for *person a* and *person b* (see the equation for the relativistic factor). Furthermore, one of the legs of the triangle is *L**b*, or the horizontal distance traveled by the particle of light **according to*** person b*. Because of length contraction, *L**a* and *L**b* are also not necessarily the same.

*What about the other leg of the triangle, *h? Recall when we were first discussing length contraction how we saw a ruler “shrink” based on the motion of the observer. This is illustrated by the difference between the ruler in *Figure 4* and the ruler in *Figure 5* in this post. Notice how the division between the tick marks are smaller in *Figure 5*’s ruler than they are in *Figure 4*’s ruler. Also notice that the heights of the rulers in the figures are **the same**. This is a very important concept: length only contracts in the direction of motion of the observer. *So what?* Well, in our current situation, the horizontal distance traveled by the particle of light will be contracted because the rocket ships are moving horizontally (which is why *L**a* and *L**b* are not the same). However, the rocket ships are not moving vertically, therefore the vertical distance traveled by the particle of light, *h*, will **not** be contracted at all. This means that *person a* and *person b* will measure the same value for *h*.

Let’s use the triangle in *Figure 7* to write an equation for *h* (once again making use of the Pythagorean theorem):

Let’s also rearrange *Equation 1* to be in terms of *h*:

Because both *person a*’s *h* and *person b*’s* h *are the same, we can combine *Equation 3* and *Equation 4* and write the following:

Let’s square both sides of *Equation 5*:

And there we have it! *What do we have?* Well, let’s think about what *Equation 6* is saying: if we take the speed of light and multiply it by the amount of time it takes for the particle of light to reach the top of the rocket ship according to *person a*, square that quantity, and add the horizontal distance the particle of light travels according to *person a *squared we get the **same value** as if we were to take the speed of light and multiply it by the amount of time it takes for the particle of light to reach the top of the rocket according to *person b*, square that quantity, and add the horizontal distance the particle of light travels according to *person b* squared (talk about a long sentence!). Remember that we started with *person a* moving at *v**a* and *person b* moving at *v**b*. *v**a* and *v**b* are just variables that we can set to be any value that we want (for example, *v**a** = 5 m/s *and *v**b** = 10 m/s* or *v**a* = *0.5c* and *v**b* = *0.9c*). This means that *Equation 6* holds true for **any** values of *v**a* and *v**b*. In other words, no matter what speed *person a* and *person b* are moving at, *Equation 6* still holds. It must be emphasized that *t**a* and *t**b* are not the same and *L**a* and *L**b* are not the same. Nonetheless, *Equation 6* will **always** be true.

*Always be true?* *That sounds a lot like a value that is **absolute**. *And this is exactly true, in our situation the value

will always be the same, regardless of who the values of *t* (time) and *L* (horizontal distance) are measured by. **Each and every observer of the particle of light moving in the box, regardless of his speed, will calculate the same value for ***Expression 7***.**

*So we have found an **absolute quantity** for this random box situation, but what about other situations?* We will soon see that *Expression 7* is very important and applies to many other situations. That’s for next week, though! See you then!

*If you would prefer a video of the math performed in this post, be sure to check out: **https://www.youtube.com/watch?v=SVcJnNMo2wA*

(featured image: http://wallpapercave.com/wp/u8PEixm.jpg)