Hello, and welcome back to MPC! Last week, we discussed an interesting implication of *general relativity*: the *bending of light*. Today, we will analyze this phenomenon in much greater detail to see if we can learn anything from it.

Before we begin, we have to ensure that we are all on the same page when it comes to basic/*classical physics*. First, what is *light*? For instance, when you look at a red *light*, what are you actually* seeing*?

**Figure 1: **Looking at a red light

Believe it or not, this fundamental question stumped physicists for a long time. Today, a convenient way of describing *light* is with ** photons**. A

*photon*can be thought of as a particle of

*light*.

**Figure 2: **A particle of *light*, or *photon*

***Note: Some of you may be familiar with the *wave-particle duality* of *light*. We will be discussing this when we start *quantum mechanics*. For now, we will think of *light *as a particle***

As a matter of fact, when you look at a red *light* from a traffic light, you are just seeing the *photons* from the traffic light as they enter your eyes:

**Figure 3:** Stream of *photons* from a red *light*

*Photons* have many interesting properties, but the one that we are most interested in is their *mass*. *Mass* is essentially how much matter (“stuff”) is “in” something. We have a basic intuition of what *mass *is: we consider the Sun *mass*ive, but we do not consider an atom *mass*ive. *Photons* are very special because their *mass *is *zero*. Honestly…that’s pretty awesome!

The next concept we have to understand is *gravity*. In the 17th century, Sir Isaac Newton spent a lot of time studying the motion of planets (making use of data from other scientists). What Newton found is that the *force* of *gravity*, *F*, can be explained by a simple law:

Let’s break this equation into pieces. The *G* in the equation is just a constant — no matter where in the world you are, *G* is *6.67 * 10^-11 N*m^2/kg^2* (do not worry about its crazy units!). How about *M*, *m*, and *r*? These are all dependent on the situation we are talking about. Let’s say that we have a tennis ball in the Earth’s atmosphere:

**Figure 4:** A tennis ball in Earth’s *gravitational field*

We can analyze the *force* of *gravity* acting on this tennis ball. *M* is the *mass* of the object that is “creating” the *force* of *gravity*. In this scenario, we are looking at the *force *of *gravity* on a tennis ball. What is the source of this *gravity*? The Earth! So, *M* is the *mass* of the Earth (a very big number: *5.98 * 10^24 kg*).

How about *m*? *m* is also a *mass*, but it is the *mass* of the object that the *gravity* is acting on. In our scenario, *m* is the *mass* of the tennis ball (we are looking at the *gravity* acting on the tennis ball).

Finally, *r*. *r* is simply the *distance *between the **centers** of the two objects that we are analyzing (i.e. the source and recipient of the *gravitational force*). In our case, this is just the *distance *between the (center) of the Earth and the (center) of the tennis ball.

**Figure 5:** Important symbols

***Note: Although *M* is just drawn on the ground, it represents the *mass* of the entire Earth. Remember, *r *represents the *distance *between the **center** of the tennis ball and the **center** of the Earth*.****

So, if we multiply *G*, *M*, and *m*, then divide by *r^2*, we will have our *force* of *gravity*.

We can do a sample problem: say our tennis ball has a *mass* of *2 kg* and is *100 meters* above the Earth’s surface (making the center of the tennis ball and the center of the Earth about *6.4 * 10^6 meters* apart):

**Figure 6:** Important quantities

Using the fact that the Earth’s *mass* (*M*) is *5.98 * 10 ^24* *kg* and *G* is always *6.67 * 10^-11 N*m^2/kg^2*, we can calculate the *force *of *gravity* acting on the tennis ball:

The *force *of *gravity* on the ball is *19.5* *newtons* (similarly to how we measure *length* in *meters* and *time* in *seconds*, we measure *force* in* newtons*). That was simple!

It is important that we understand what this *force* (that we happen to call *gravity*) is doing. A *force* is, essentially, a *push *or a *pull *on an object. In our previous scenario, *gravity* is *pulling* the tennis ball towards the Earth (specifically, towards the center of the Earth; it should be noted that the tennis ball will not actually reach the center of the Earth — it will hit the ground first):

**Figure 7:** *Gravity* acting on a tennis ball

The *gravity *that is pulling the tennis ball to the (center of the) Earth is the same *gravity* that can bend the path of objects. For instance, last week, we spoke about the Earth bending the path of an asteroid:

**Figure 8:** The Earth bending the path of an asteroid

The reason why the path of the asteroid bends is that the Earth’s *gravity *is pulling the asteroid towards the Earth (the Earth, however, is not pulling hard enough for the asteroid to crash into its surface like the tennis ball does):

**Figure 9:** The Earth’s *gravity* acting on an asteroid

***Note: The orange lines represent the Earth’s *gravity acting/pulling* on the asteroid at two instances when the asteroid is close to the Earth.***

So, when we talk about the Earth’s *gravity* bending *light*, it makes a lot of sense: just like it does to the asteroid, *gravity *is pulling the light towards the Earth, bending its path.

**Figure 10: **The Earth’s *gravity* acting on *light*

***Note: The orange lines represent the Earth’s *gravity acting/pulling* on the light at two instances when it is close to the Earth.***

Let’s have a little bit of fun: why don’t we calculate how hard the Earth pulls on *light* that is, say, *100 meters* from its surface (making the “center” of the *light* and the center of the Earth about *6.4 * 10^6 meters* apart). Remember that we can think about *light* as a particle, a *photon*, so we essentially have the following:

**Figure 11: **A *photon* passing by the Earth

***Note: This image is not drawn to scale — *photons* are actually very, very small. The blue arrow shows that the *photon* is moving down as a result of *gravity*. The red arrow shows that the *photon *is also traveling to the right (it has to move to the right to get the same effect as is seen in ***Figure 10**)*. We can ignore this rightward motion when analyzing the effects of *gravity*.***

What do we know? *G *is still *6.67 * 10^-11 N*m^2/kg^2 *(it always is). *M* is the *mass* of the Earth (like it was in ** Figure 5** and

**), or**

*Figure 6**5.98 * 10 ^24*

*kg*.

*m*is the mass of the

*photon*, or

*0 kg*.

*r*is the distance between the center of the

*photon*and the center of the Earth, or

*6.4 * 10^6 meters*.

**Figure 12:** Important quantities

Now we can use our formula to find that the *force* of *gravity* acting on the *photon* is…

…*0 newtons*. That’s strange! *0* *newtons* means that no *force* is acting on the *photon*, which means that the Earth is not pulling on the *photon* at all. In other words, if *light* were passing by the Earth (like it is in ** Figure 10**), it would just travel in a straight line; there would be nothing (specifically no

*gravity*) pulling it towards the Earth!:

**Figure 13: ***Light* without the Earth’s *gravity *acting on it

***Note: Do not forget: the ray of *light* is really a stream of *photons*.***

Yet, at the same time, we have used the *equivalence principle* to show that the *light* **has** to bend like it does in ** Figure 10** (see this post). Not only that, experiments show that

*gravity*does indeed bend

*light*. Something is not adding up!

In order to solve this predicament, we will have to completely **reshape** our understanding of *gravity*. This is exactly what we will be doing next week. See you then!

*For more information on some of the calculations we performed, be sure to check out this resource: **http://nova.stanford.edu/projects/mod-x/ad-surfgrav.html*

(featured image: http://light-radiant.com/wp-content/uploads/8589130489556-cool-light-wallpaper-hd.jpg)