Free Falling

My recent post on the black hole information loss problem drifted off into the basics of General Relativity, and I thought it would be worthwhile to repeat some essentials, most notably what "freely falling" means. To start with, we need the Equivalence Principle which we previously discussed here:

The Equivalence Principle: Locally, the effects of gravitation (motion in a curved space) are the same as that of an accelerated observer in flat space.

The commonly used example is if you were standing in an elevator, you couldn't distinguish between the elevator being pulled up in constant acceleration (by, say, a flying pig) far away from any gravitating masses, or standing on the surface of a planet, being pulled down by gravity. Similarly, you couldn't distinguish between your elevator being not accelerated in empty space, and being freely falling in a gravitational field. That indeed was, so the story goes, what sparked Einstein's idea of the Equivalence principle:

“For if one considers an observer in free fall, e.g. from the roof of a house, there exists for him during his fall no gravitational field---at least in his immediate vicinity.”~A.Einstein [via]

The easiest way to understand how the non-accelerated observer in Special Relativity becomes a freely falling observer in a curved background is to generalize Newton's first law: “In the absence of a force, a body either is at rest or moves in a straight line with constant speed.” (:-)) Now in a curved background, speaking of a “straight line” is not particularly meaningful. Instead, one uses curves of minimal length, so called “geodesics” which is the straight-forward generalization of the flat space's straight lines. The first law then turns into: In the absence of a force, a body moves on a geodesic. And that's what we mean by freely falling.

Doesn't sound too complicated, but there's one thing people tend to get confused about. If space-time is not flat, the motion on a geodesic describes the motion in a gravitational field already. It thus seems like there's a force acting on the body, isn't that what we've learned in high school, things falling from towers etc? No! That's exactly what we mean with “gravity is not a force.” Gravity is a property of space-time. You are “freely falling” as long as there's no force acting on you that pushes you off the geodesic.

Right now, sitting in a chair trying to figure out what I'm telling you, you are not freely falling. There's a force acting on you, which is a combination of the electromagnetic interaction and the Pauli exclusion principle that prohibits you from falling through the Earth. You are not moving on a geodesic. The guy who fell off the roof was moving on a geodesic - no force acting, no acceleration - until he hit the pavement. That's where he was accelerated, which corresponds to a force acting. That's what makes the confrontation with the pavement so unpleasant.

[If you know a little bit about the mathematics, the Equivalence Principle says one can always locally chose coordinates in which space-time is flat and you can get the first derivatives of the metric to vanish too. This means in particular that in this “freely falling” coordinate system, the Christoffel-symbols vanish. That is only possible because the Christoffel-symbols are not tensors. In this freely falling coordinate system, the equation of motion is just that the derivative of the momentum with respect to the proper time vanishes, i.e. it's the same as Special Relativity - the Equivalence Principle at work. You can actually start from there and, using covariance, obtain the equation of motion in all other coordinate systems.]

If you look outside the “local” surrounding mentioned in the Equivalence Principle you can however distinguish flat from curved space. You can do that for example by measuring the distance to nearby geodesics over time. The change of this distance gives you something called “geodesic deviation” which is related to the curvature tensor. Whether or not this geodesic deviation vanishes is a coordinate independent statement: A tensor that vanishes in one coordinate system does so in all coordinate systems. In your elevator this means if you measure very precisely you could figure out that in a gravitational field there's a slight difference between particles moving at the top and at the bottom of your elevator.

Okay, if you've made it till here, there's one question remaining: how do we get back the “gravitational force” that we're used to talk about? Well, you can go and define something like a force in a particular coordinate system, for example the coordinate system labeling distances to the Earth's surface (or, more conveniently, distance to its center). Using such a particular coordinate system allows to separate the terms in the geodesic equation between the time derivative of the momentum (dot p) and “everything else,” which you can then go and interpret as a “force.” Also known as “the Newtonian limit” for obvious reasons, this allows to identify a derivative of the metric tensor in Einstein's field equations with the Newtonian potential.

[You can find this derivation in any textbook on the topic, in Sean Carroll's lecture notes page 105/106 or in my master's thesis, page 28 (in German).]

Zero G flights make active use of the Equivalence Principle. Basically, when you're inside the plane you get the floor pulled away below your feet and are freely falling: there is no force acting on you. You don't need the plane for the fall. You need the plane to slowly push you off the geodesic again and prevent you from colliding with planet Earth. Not to mention that said planet has an atmosphere which, at some hundred miles per hour, causes quite some friction that would prevent actual geodesic motion. But if being a theoretical physicist has any advantage then it's to disregard friction on cue ;-)