A Conceptual (philosophical) doubt on equivalence principle

Question

The reference is: Elements of Newtonian Mechanics by J.M. Knudsen; Springer; page 108 to 113.

My doubt is about the significance of the apparent distinction between Gravitation and Inertia. In Knudsen's book, he says that gravitation and inertia aren't different properties of matter, the by equivalence principle.

Can you give me an example that we can verify the apparent distinction of gravitation and inertia (which we can then see as different aspects of matter; different reference frames)? Why was gravity considered a particular reference frame, and why this "bored Einstein"? What exactly did Knudsen intend to say by:

Gravitation and inertia do not seem to be separate properties of matter, but rather two different aspects of a more fundamental and universal property of space and material particles.

I know the math behind this, but I have some difficultt imagining a good example beyond "Einstein's box" thought experiment and the insight that "gravity and acceleration are the same thing"

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If you want to help me even more:

I really want to know why this particular idea (Equivalence principle) leads Einstein to verify the necessity for a more general framework, I mean: how can we start from Einstein's box (equivalence principle that works for a constant gravitational field--a "little laboratory") and reach the framework of Riemannian Geometry with gravity as the problem?

Please, fell free to use mathematical arguments if you like.

Answer 1

I'll try my best to answer this in a qualitative but coherent fashion.

By changing reference frames, gravity can be made to look like inertia, and inertia can be made to look like gravity. The obvious example of this, as you mentioned in your original post, is the Einstein box thought experiment. Imagine a scientist confined to a closed, empty elevator far out in space, away from planets and stars and galaxies (basically anything with a gravitational field). The elevator is uniformly accelerating in some direction $\hat{n}$ at $1g$. Basically, according to the equivalence principle, there exists NO experiment that the scientist can perform in order to determine whether he is on a stationary elevator on Earth's surface (with normal $\hat{n}$) or whether he is in empty space accelerating upwards.

Actually, this isn't quite right; because the elevator and scientists are not point particles, there will be some very small, detectable effects due to the local curvature of the gravitational field. Nevertheless, the argument above holds if we approximate the Earth's gravitational field at the surface as uniform.

Another application: if someone jumps off the Eiffel Tower, during the $\sim \sqrt{\frac{2 h}{g}}$ seconds they are in free fall, they will be effectively weightless. An accelerometer would not be able to tell whether the person was freely floating in space (an inertial effect) or freely falling near the Earth's surface (a gravitational effect). If the person was confined to a box that was always magically surrounded them while they were in free fall or out in empty space, then there would be no experiment that could tell the difference. (Again, neglecting finite size effects).

In a more advanced sense, this has to do with geodesics. A popular interpretation of general relativity says that the presence of matter curves spacetime, and particles moving under the gravitational field are really "just" following the curvature of the spacetime manifold. This is succinctly expressed by the geodesic equation:

$$\frac{d^{2}x^{\beta}}{d\tau^{2}}+\Gamma^{\beta}{}_{\alpha\nu}\frac{dx^{\alpha}}{d\tau}\frac{dx^{\nu}}{d\tau}=0$$

You don't have to know exactly what this means, just know that the first term represents the effect of inertia and the second term represents the effect of gravity or spacetime curvature (the Christoffel symbols $\Gamma$ have to do with manifold curvature). Basically, the equation must always be satisfied (equal zero), but by changing reference frames, the relative contributions of each of the two terms can be modified. Thus gravity and inertia can be changed into each other just by changing reference frames. There is no "real" difference between the two (the effects are the same up to reference-frame shifts), and thus they are unified.