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 massive, but we do not consider an atom massive. 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 Figure 6), or 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)