Phil. Space and Time

Free Fall, Inertial Frame, and Equivalence Principle

When, in the year 1907, I was working on a summary essay concerning the special theory of relativity for the Jahrbuch fuer Radioaktivitaet und Elektronik, I had to try to modify Newton's theory of gravitation in such a way that it would fit into the theory [of relativity]. Attempts in this direction showed the possibility of carrying out this enterprise, but they did not satisfy me because they had to be supported by hypotheses without physical basis. At that point, there came to me the happiest thought of my life, in the following form:

Just as is the case with the electric field produced by electromagnetic induction, the gravitational field has similarly only a relative existence. 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, manuscript written in 1919, quoted from G. Holton, Thematic Origins of Scientific Thought, Harvard Univ. Press, 1973, 364. The original manuscript in Pierpont Morgan Library, New York, and in Einstein Archives, Hebrew University of Jerusalem.)

In special relativity, we did not treat gravity, and in general relativity gravity is essential. The equivalence principle (equivalence between gravity and acceleration) is introduced in order to bridge a gap between the two theories. Some students are confused at this stage, as regards the notion of inertial frames. "Inertial frames are such that they move with a uniform velocity with each other. But gravity is equivalent with acceleration, and why an accelerated frame can be an inertial frame? A frame moving with the gravitational acceleration cannot be an inertial frame in the sense of special relativity!"

The trick is that, in general relativity, inertial frames (in other words, Local Lorentzian frames) are only local, for a small region for a short period, or for a local observer;whereas in special relativity, inertial frames are grobal. Aside from these limitations, "inertial frames" can make a perfect sense in general relativity too, and Einstein's equivalence principle (also with limitation of local character) is closely related to them. How can you tell your frame is inertial? You cannot presuppose this and that are inertial, without any means of telling it. If you know in advance that this frame is inertial, then another in uniform motion relative to it is known to be inertial; but how do you know, in the first place, that this frame is inertial? Thus we are led to the question: what is a good test for an inertial frame within itself? You've got to ascertain that there are no forces acting on matter, no acceleration, no bending of a trajectory are detectable in your frame; and you have to ascertain the law of inertia holds in your frame. A simplest way, on the earth, is to undergo a free fall! This way, you can see, at least, that two different frames can be equivalent in the sense Einstein intended.

Recall that, according to Newton's second law, force and acceleration are interchangeable; and inertial motions are those without force acting on them. If you wish to keep this relationship, then "inertial" may be equivalent with "no force acting". In this sense, an inertial frame is a frame where forces, or fields generating such forces are eliminated (uniform motion then becomes secondary).

As many writers proposed already, consider a frame with gravity, and another frame without gravity but with the corresponding acceleration: that's free fall. It is easy to imagine such frames. One is a system with a shooter and an arrow in the gravitational field (as is the case on the earth), and another is a system in which the shooter begins a free fall at the moment of shooting. Suppose the shooter, the arrow, and their reference frame fall freely, as in the following figure (but don't imitate this experiment on a high building!). Then this frame is a perfect inertial system (for a short period, where we can neglect the change of gravity) in which the Newtonian law of inertia holds, and gravitational field disappears! The two descriptions, in the following figure, of the same motion, are equivalent; but which is simpler? However, we have to consider also unification of many local frames in general relativity, and that's a harder part, as you can see with Schwarzschild Geometry.

(You should also see Einstein's treatment of uniform acceleration in his 1907 paper, since we may have to be more careful, depending on the problem we wish to address; see Genesis of General Relativity (1). But this was only the beginning, and a genius of Einstein's stature had to spend 8 more years to reach the correct form of the new law of gravitation.)

All right. This imaginary experiment is dangerous enough. If your want to see a far more dangerous experiment of free fall (across the horizon of a black hole; yes, you can cross the horizon alive, if the black hole is large enough!) , see Project B of Taylor & Wheeler (2000).

References

Variations on the theme of free fall:

Misner, Thorne, Wheeler (1973), Gravitation, Freeman, 13-19.

Taylor, E. F. and Wheeler, J. A. (2000), Exploring Black Holes, Addison Wesley Longman, 2-4, 2-31, Project B.

Thorne, Kip S. (1994), Black Holes and Time Warps, Papermac, 97-99, 449-453.

Wheeler, J. A. (1999) A Journey into Gravity and Spacetime, Scientific American Library, 18-33.

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