# Equivalence

Well, Tuesday was a big day in class. We reached a landmark – the introduction of one of my very favourite thoughts of the 20th Century: the Equivalence Principle. This is the realization of Einstein’s that there’s something profoundly odd about Newtonian mechanics and Newtonian gravity that hints at something deeper.

Quick reminder from high school (just two formulae, please bear with me): You know the mass that appears in $$F=ma$$ (Newton’s second Law) telling us how much acceleration we get for a given applied force? …and you know the mass that appears in Newton’s formula for the strength of the gravitational force in the presence of some gravitational field?: $$F=m\,\,\times\,\, \left(GM/r^2\right)$$. Well, there’s absolutely no logical reason that these two masses have anything to do with each other! The fact that they are experimentally measured to be equal to an accuracy of one part in $$10^{13}$$ is a total fluke from the point of view of Newtonian physics. Newtonian physics tells you to use the same mass in these two separate arenas, and this does indeed yield the famous result that (barring any obstructions like air resistance) all objects fall to the earth with the same acceleration, regardless of their mass. Newtonian physics does not tell you why, though.

So as I said to my students, watch out: If you see two things – such as these two types of mass that we can dub “inertial” and “gravitational” – that have no reason to be equal turn out to be equal to such accuracy, then there’s definitely something about this aspect of Nature that you don’t yet understand, and it is possibly something really huge and important.

Einstein turned it into the Equivalence Principle. Basically, the equality of the two things means that gravity is not some special phenomenon at all… it is just acceleration. You are unable to distinguish between gravity on the one hand and acceleration on the other. See the little figure in the box above. Imagine the box was closed so that they could not look outside. Then if they woke up in the box one morning and started doing physics experiments in the box – any experiments (that don’t look outside the box) – they would not be able to tell whether they are on the surface of the earth with the usual downward acceleration “due to the force of gravity” of magnitude $$g\simeq 9.8$$, (measured in metres per second per second) or whether they are in fact out in empty space being towed behind a spaceship that is accelerating with acceleration $$g$$. This is just lovely1 .

The point is that the fact that those two types of masses are equal has been elevated to a Principle. They are equal because gravity is not what we thought it was (some special force acting on things in the usual arena of space and time), but something familiar looked at in not the best way – acceleration.

The rest of the lecture then follows: What are the consequences of this statement. From this, you can arrive at several wonderful things in only a few steps – gravity bends light, and gravity can slow down and speed up time. Those are the ones we focused on, and then we did the next big step: Abandoning gravity altogether. Using the geometry I’ve been teaching them over the last few weeks we can quickly come to the point that we realize that the entire idea of a “gravitational force” is clumsy, limiting, and inelegant for making progress. (There’s a bit more needed here to make this leap smoothly, but I won’t go into it here…this isn’t the lecture.)

There simply is no gravity. It is just geometry, plain and simple. Objects move as they always move in spacetime – on the shortest (in general, the “extremal”) paths. There’s no need to invoke some special force (gravity), but instead to realize that spacetime itself can change its shape. Objects only appear to be bending this way and that under the “force of gravity” because the spacetime itself is curved (by masses and energy and so forth). There is no gravity, there’s just geometry2 .

Beautiful! I do so love giving that lecture.

-cvj

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1. Technical point not to worry about: I’ve assumed a box small enough that we can deal with the case of uniform fields here. We deal with non-uniformity later. [return]
2. I’m talking about classical (non-quantum) gravity. Quantum gravity is another story for later. Much later. [return]

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