# Worries about the equivalence principle

I have a serious worry about the equivalence principle in general relativity. I have been learning general relativity from different sources (Quantum Gravity by Carlo Rovelli, A First Course in General Relativity by Bernard Schutz, Gravitation by Misner et al., etc.). With the help of all these sources, I have been able to make my own construction of the general theory of relativity. Using the equivalence principle in the form given by Steven Weinberg in his book "Gravitation and Cosmology" I could motivate the use of a differentiable manifold to represent the spacetime and to justify that freely falling particles follow geodesics in spacetime.

The equivalence principle, in the form given by Steven Weinberg, says that:

At every spacetime point in an arbitrary gravitational field it is possible to choose a "locally inertial coordinate system" such that, within a sufficiently small region around the point in question, the laws of nature take the same form as in unaccelerated Cartesian coordinate systems in the absence of gravitation.

I didn't have any problem with this statement. As I said, I used it as the basis to construct my own understanding of general relativity. What worries me is that I recently found a statement in the book by Ohanian and Ruffini, "Gravitation and Spacetime" (2013), where they say the following:

Einstein’s statement has often been generalized to sweeping assertions about all laws of physics being the same in a laboratory freely falling in a gravitational field and in another laboratory far away from any field. Such generalizations are unwarranted because, as we have seen, even quite simple devices signal the presence of a true gravitational field by their sensitivity to tidal forces and therefore permit us to discriminate between a gravitational field and the pseudo-force field of acceleration... If the rotational degrees of freedom of the motion of masses are taken into consideration, then the equivalence fails.

Actually, thinking carefully, even in a freely falling coordinate system, the Christoffel symbols can vanish, and they represent the first derivatives of the metric. However, the second derivatives of the metric can not vanish, and they leave a "remainder" that makes it so that the Riemann tensor is different from zero. This causes the tidal forces to be different from zero in a freely falling coordinate system, even in a small spacetime region.

Steven Weinberg insists, however, that:

Although a freely falling particle appears to be at rest in a coordinate frame falling with the particle, a pair of freely falling particles will exhibit a relative motion that can reveal the presence of a gravitational field to an observer that falls with them. This is of course not a violation of the principle of equivalence because the effects of the right-hand side (of the Jacobi equation, or the geodesic-deviation equation) become negligible when the separation between particles is much less than the characteristic dimensions of the field.

However, in section $$1.8$$, Ohanian and Ruffini say the following:

If astronauts in orbit wish to detect the gravitational field of the Earth by measuring the tide produced by the Earth on a drop of water, they will find it desirable to use a very large drop of water... The height of the tide increases with the size of the drop. This suggests that if the astronauts have been ordered to confine their experiments to the interior of a sufficiently small spacecraft, then they will not be able to detect the tide or the gravitational field... However, even in the limit $$R\to 0$$, the tidal deformation remains. We see that there exist several methods for measuring the tidal field locally, in a small neighborhood of a given point...The limitations on the minimum size of the neighborhood needed to perform measurements of a given precision do not arise from any intrinsic properties of the gravitational field; rather, these limitations arise from the quantum nature of matter, which prevents us from constructing an apparatus of arbitrarily small size... Local experiments can distinguish between a reference frame in free fall in a gravitational field and a truly inertial reference frame placed far away from all gravitational fields.

On the other hand, in chapter $$16$$ of "Gravitation" by Misner, Thorne, and Wheeler, they say that:

There is no way, by experiments confined to an infinitesimally small region of spacetime to distinguish one local Lorentz frame in one region of spacetime from any other local Lorentz frame in the same or any other region.

This is something very similar to Weinberg's statement.

Weinberg says one thing, Ohanian arguments something completely opposite, and then Misner et. al. reinforce Weinberg, but my own deduction agrees with Ohanian. I am quite confused. How can I, then, construct my understanding of general relativity? Are all my motivations and deductions incorrect? Who's right? According to Ohanian, Schutz, Misner et. al., Weinberg, and even Carlo Rovelli are all wrong. I am confused and overwhelmed.

• The principle holds for an infinite massive plane. Standing on that plate is like being accelerated in empty space. Feb 13, 2022 at 2:15
• @Felicia Please post answers as answers, not as comments Feb 13, 2022 at 2:41

The equivalence principle is valid in the limit where the spacetime region goes to zero. As any real lab has a finite size, and any experiment takes a finite time, it is always possible to imagine accurated enough sensors to detect tidal forces.

• Well, it is a matter of opinion. It is like to say that, as the Earth is spherical, we can not rely in a plumb bob to have the vertical walls parallel. Feb 13, 2022 at 0:32
• @CarlosBeltran It's not trivial for two reasons. One is mathematical: A smooth manifold has the property that it becomes more and more like Euclidean space as you zoom in. This is not a trivial property, because you could in principle imagine something with a corner or crease. The second reason is physical: Hypothetically there could be two particles with same inertial masses that fall down at different rates. So in a free-falling, small, enclosed box , you could tell you are falling easily if you had the two particles in your pocket. Put them side-by-side in front of you and ... Feb 13, 2022 at 2:36
• ... watch as one starts moving relative to the other from your point of view (imagine you are inside the falling box). This sort of situation is completely prohibited by the equivalence principle, and therefore the principle is not at all trivial. Feb 13, 2022 at 2:37
• Another consequence of the equivalence principle is that you can't have gravitational shielding. The two notions (equiv principle and grav shielding) contradict each other, so either one or the other is true (or both are false). This is also very non-trivial. Feb 13, 2022 at 2:42
• @CarlosBeltran I think the problem is the idea of "infinitesimal regions," which can be misleading at times. It's better to think about it in terms of limits. If you take that point of view, then the statement says telling gravity becomes harder and harder in the limit of smaller and smaller regions. Feb 13, 2022 at 2:47

I think the history was as follows (and this probably explains the confusion, mind you I am not an expert on gravity):

• It started with the equivalence principle, then you need to know how to move a vector to a (infinitely small distance) near position. This is done with the Christoffel-Symbols.
• Then you can show that you can always find for each point a coordinate transformation that makes the Christoffel-Symbols vanish at that point. This is interpreted as the 'elevator' Koordinate system, meaning locally your space looks like a Lorentzian-flat space.
• Then you can find (by some assumptions) that the Christoffelsymbols are related to the metric (after calculating, the metric is sort of a potential of the symbols)

This procedure is the approach Einstein took and Landau Lifshitz Vol 2 uses. Now the question Einstein faced was: What equations does the metric/Christoffelsymbols obey. Originally he dismissed coordinate independent proposals with the hole argument: https://en.wikipedia.org/wiki/Hole_argument (or see here: https://plato.stanford.edu/entries/spacetime-holearg/)

But later reconciled it in a way that only events are important and not coordinates, leading him immediately to the correct field equations and able to derive the tests of GR which were successful. So now the original concept at the beginning is not really viable anymore. If you take now coordinate invariance as the top principle, you can only talk about tidal gravity (which the field equations basically are). MTW puts the tidal part in front of the book, also shying away from single partcle EOM (only stating the geodesic deviation, if I didnt miss it).

For me the upside down approach now used in pedagogy doesnt work, and lead to the same difficulties you have. I think one has to travel the historical path to really understand it. (maybe in the sense of Wittgenstein's ladder; we know from todays perspective it is not right, but we need it until we are above and can throw it away).

TLDR: I strongly recommend studying LL Vol 2 on GR. And Einstein didnt start with tidal gravity but eventually 'had to' use it.

One should state the strong equivalence principle carefully. It does not assert that tidal effects disappear completely; it asserts that in the limit as the region size tends to zero the only difference between observed physical effects and those happening in zero curvature are the tidal effects (i.e. anything depending directly on the curvature). The point is that there isn't any other influence such as an influence of gravitational potential on the relative placement of energy levels in hydrogen or something like that. Put more mathematically: any physical effect which goes to zero when Christoffel symbols (connection coefficients) go to zero will not be observable in a local inertial frame.