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I am trying to unknot an obvious error in my understanding of the following example:

Fact 1: An observer on the moon watches a tiny ball of uranium and a tiny ball of iron fall. Both fall fall at the same speed.

Fact 2: In the gravitational field of venus, both balls will fall faster than in the gravitational field of the moon.

Rephrasing of fact 2: For an observer on the tiny ball of iron the moon falls faster in the gravitational field of the iron ball than the venus.

Problem: This rephrasing somehow clashes with fact 1 and with my (probably) wrong understanding of the weak equivalence principle.

I recall that the weak equivalence principle sometimes is formulated only for small test objects. But then: How small are they supposed to be? And then: If we assume this limitation, then it actually is not true but only a weak field approximation.

Quite obviously I am getting something very much wrong - but I currently fail to see what it is.

Add on: Quite clearly, the physical description should not depend on where we place the observer. So, how would a description of the weak equivalence principle look like which does not place the observer on the "heavier" object?

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    $\begingroup$ See this related question and links therein. $\endgroup$
    – rob
    Mar 26, 2022 at 20:38
  • $\begingroup$ Great link. The conclusion I draw is that the frequently found formulation that objects fall at the same speed lacks the necessary precision and may be regarded only as an approximation. Right? $\endgroup$ Mar 26, 2022 at 20:56

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"Everything falls at the same rate," is, ultimately, a statement about the behavior of test particles—bodies whose masses and energies are sufficiently small that they do not disturb the spacetime curvature around them to a measurable degree. Test particles fall following geodesics in a background spacetime* that they do not affect. This is an approximation, of course, but it is an approximation that is already part of the usual approximation scheme used in discussion of the equivalence principle.

The equivalence principle(s) describe what happens in a region where the gravitational field is approximately constant. Within a small spacetime patch—small enough that gradients of the gravitational field may be neglected—the gravitational field is well approximated as being identical in affect to a uniform acceleration of the observer frame. However, for examples of Venus or the moon falling in the field of a small sphere, the gravitational field of the sphere is nowhere near uniform over the size of the "falling" celestial body; nor is the field of those large bodies themselves uniform over that scale; so the approximations described in the weak equivalence principle do not apply.

*Note that while (infinitesimally light) test particles follow geodesics in the background spacetime they are passing through, it is not known whether two real bodies, falling toward one-another, actually follow geodesics, or whether that is merely a useful approximation. Even in special relativity, it is not possible to solve this kind of two-body problem exactly to find the trajectories of the two particles.

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  • $\begingroup$ Do you have any reference on "it is not known whether two real bodies ... follow geodesics". I am aware that the SRT 2-body problem cannot be solved (by closed algebraic expression) - but don't the Einstein field equations tell us how two heavy, non-test bodies move? We do not have a closed algebraic solution but we know the condition. We could solve numerically and compare with numerical solutions for geodesics? And for both we have abstract conditions, even if we have no "closed" solution. $\endgroup$ Mar 27, 2022 at 12:03

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