You probably know that objects experience a gravitational force (Fgrav) between them, and that the amount of attraction depends on their masses (m1 and m2) and the distance (d) between them. This is easy to represent symbolically (and is frequently referred to as Newton’s Law of Universal Gravitation (though he never wrote is as such):
Fgrav = G m1 m2 / d^2
Having more than 2 masses makes things a bit more complicated, but the idea and approach is similar. The other term, G, is a physical constant (6.67 x 10^-11 Nm^2/kg^2) related to just how strong the attraction is. You might be able to guess that, given the smallness of this constant, gravitational forces are only significant when the masses (or at least one of them) are great – things like stars and planets. Take two cupcakes out in space, away from strong gravitational forces, and they don’t really attract each other a whole lot. And what a waste of perfectly good cupcakes!
Now this mass, the mass that responds to gravitational forces, we refer to as gravitational mass.
But let’s talk more generally – if there is an unbalanced force, any force, acting on a mass, the mass will accelerate. In other words, if YOU push on a mass with a force greater than any resistive forces (friction, etc.), the mass will accelerate. The acceleration will be greater with greater forces, but it will be smaller with greater masses. There is a convenient way to represent this, too, with symbols. Enter Newton’s 2nd Law:
F = m a
Now this mass, the one that responds to forces in general, is called the inertial mass.
Maybe I’m not making a big enough deal out of this – these are two completely different ways to view the concept of mass: one idea of mass is gravitational (how the mass responds to gravitational forces) and the other is inertial (how the mass responds to any force at all).
Now here’s the punchline: the two masses, while conceptually very different, are numerically the same!* This idea is known as the Equivalence Principle, and it is the cornerstone of Einstein’s theory of general relativity. Let’s explore this.
We’ll use the classic physics example of the sealed elevator, so that you cannot see what’s happening outside of your box. And let’s imagine that you are conveniently carrying a tennis ball. You hold the tennis ball out at arms reach and release it - you notice (to your relief) that it falls and accelerates toward the floor. Quick, give 2 possible interpretations of what you see.
So, the ball could simply be accelerating normally – down, toward the ground. But, couldn’t the elevator be accelerating upward toward the ball? Is there a way to tell the difference?
The equivalence principle says NO.
On the other hand, if you want to change the rules a little, would it help if you could look out a (newly inserted) window? What would be evidence as to whether your box was accelerating or the ball was?
Let’s look at a more peculiar example. Get back in your box, but this time you have a flashlight. You direct the beam of light straight to the opposite wall. The spot lands exactly where you expect (directly across from the flashlight). What are the possible interpretation(s) of this?
Now imagine that your box is accelerating upwards. Draw the path that the light would take from your flashlight to the opposite wall.
Behold the genius of Einstein! The path is not straight, but rather, curved. And in the correct interpretation of the equivalence principle:
There is no difference between the light curving due to the accelerated box, and the light curving down as though it is affected by gravity.
In other words, light is affected by gravity – or better still, mass causes space to curve. Let’s look at this idea in more detail. You are no doubt familiar with the idea that the shortest distance between two points is a straight line. This is true in classic (Euclidean) geometry – the kind of geometry that you learn in high school. Another view of this is called Fermat’s Principle – light will take a path that causes the time of travel to be shortest.
But the case we’ve been discussing, the flashlight in the accelerated box, is clearly different. The path of light is NOT straight. If it is indeed affected by gravity – as “caused” by mass – then the space itself is not Euclidean. Rather, space is curved. You can even re-interpret the tennis ball problem – there is no force acting on it; it is simply following the shortest space-time path (a geodesic). Of course, for you to notice these effects, the gravitational force has to be substantial – due to stars, planets, galaxies, etc.
You might not be surprised to learn that there is a whole truckload of mathematics that could be attached to these ideas. We’ll leave that for you when you get to graduate school. It’s interesting to note that these were a great challenge to Einstein himself!
How about that – crazy stuff, eh? That’s all for today – see ya soon, see ya on the Moon!
ALL TEXT AND IMAGES COPYRIGHT SEAN LALLY 2009
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