The Equivalence Principle


June 2016
Jan Onderco

Abstract

The central building block of the general relativity is known as the equivalence principle and it states that accelerations and gravitational fields are equivalent. There is no experiment to distinguish one from the other.
A logical consequence is that no force is acting on a free falling body that follows a geodesic trajectory.
Having said that the geodesic trajectory being described by a reference system that is at the center of the gravity source is a function of the direction and the velocity of the falling body. The end result is that it is possible to detect the falling from within a falling reference frame and distinguish between “no force acting” on a body in a free fall and “no force acting” on a body in an intergalactic space where gravitational field would be close to 0 in a thought experiment. The slightly modified thought experiment can distinguish between gravitational field and acceleration.
This invalidates the equivalence principal.

Introduction

The following paragraphs are a definition of a Lorentz frame (free falling frame of reference) from Benjamin Crowell’s book General Relativity. The definition is followed by the short thought experiment that can detect whether the Lorentz frame is falling within a gravitational field or it is in the intergalactic space where the gravitational field is close to 0.

Operational definition of a Lorentz frame


We can define a Lorentz frame in operational terms using an idealized variation (Figure i) on a device actually built by Harold Waage at Princeton as a lecture demonstration to be used by his partner in crime John Wheeler. Build a sealed chamber whose contents are isolated from all nongravitational forces.

Lorentz Frame experiment
Figure i: The spherical chamber, shown in a cutaway view, has layers of shielding to exclude all known nongravitational forces. Once the chamber has been calibrated by marking the three dashed-line trajectories under free-fall conditions, an observer inside the chamber can always tell whether she is in a Lorentz frame.

Of the four known forces of nature, the three we need to exclude are the strong nuclear force, the weak nuclear force, and the electromagnetic force. The strong nuclear force has a range of only about 1 fm ($10^{-15}$ m), so to exclude it we merely need to make the chamber thicker than that, and also surround it with enough paraffin wax to keep out any neutrons that happen to be flying by. The weak nuclear force also has a short range, and although shielding against neutrinos is a practical impossibility, their influence on the apparatus inside will be negligible. To shield against electromagnetic forces, we surround the chamber with a Faraday cage and a solid sheet of mu-metal. Finally, we make sure that the chamber is not being touched by any surrounding matter, so that short-range residual electrical forces (sticky forces, chemical bonds, etc.) are excluded. That is, the chamber cannot be supported; it is free-falling.
Crucially, the shielding does not exclude gravitational forces. There is in fact no known way of shielding against gravitational effects such as the attraction of other masses (example 2, p. 251) or the propagation of gravitational waves (ch. 9). Because the shielding is spherical, it exerts no gravitational force of its own on the apparatus inside.
Inside, an observer carries out an initial calibration by firing bullets along three Cartesian axes and tracing their paths, which she defines to be linear.
We’ve gone to elaborate lengths to show that we can really determine, without reference to any external reference frame, that the chamber is not being acted on by any nongravitational forces, so that we know it is free-falling. In addition, we also want the observer to be able to tell whether the chamber is rotating. She could look out through a porthole at the stars, but that would be missing the whole point, which is to show that without reference to any other object, we can determine whether a particular frame is a Lorentz frame. One way to do this would be to watch for precession of a gyroscope. Or, without having to resort to additional apparatus, the observer can check whether the paths traced by the bullets change when she changes the muzzle velocity. If they do, then she infers that there are velocity-dependent Coriolis forces, so she must be rotating. She can then use flywheels to get rid of the rotation, and redo the calibration.
After the initial calibration, she can always tell whether or not she is in a Lorentz frame. She simply has to fire the bullets, and see whether or not they follow the precalibrated paths. For example, she can detect that the frame has become non-Lorentzian if the chamber is rotated, allowed to rest on the ground, or accelerated by a rocket engine.
It may seem that the detailed construction of this elaborate thought-experiment does nothing more than confirm something obvious. It is worth pointing out, then, that we don’t really know whether it works or not. It works in general relativity, but there are other theories of gravity, such as Brans-Dicke gravity (p. 283), that are also consistent with all known observations, but in which the apparatus in figure i doesn’t work. Two of the assumptions made above fail in this theory: gravitational shielding effects exist, and Coriolis effects become undetectable if there is not enough other matter nearby.

Locality of Lorentz frames

It would be convenient if we could define a single Lorentz frame that would cover the entire universe, but we can’t. In Figure j, two girls simultaneously drop down from tree branches — one in Los Angeles and one in Mumbai.

Lorentz Frame experiment
Figure j: Two local Lorentz frames.

The girl free-falling in Los Angeles defines a Lorentz frame, and in that frame, other objects falling nearby will also have straight world-lines. But in the LA girl’s frame of reference, the girl falling in Mumbai does not have a straight world-line: she is accelerating up toward the LA girl with an acceleration of about 2g.
A second way of stating the equivalence principle is that it is always possible to define a local Lorentz frame in a particular neighborhood of spacetime.$^{16}$ It is not possible to do so on a universal basis.
The locality of Lorentz frames can be understood in the analogy of the string stretched across the globe. We don’t notice the curvature of the Earth’s surface in everyday life because the radius of curvature is thousands of kilometers. On a map of LA, we don’t notice any curvature, nor do we detect it on a map of Mumbai, but it is not possible to make a flat map that includes both LA and Mumbai without seeing severe distortions.

Terminology

The meanings of words evolve over time, and since relativity is now a century old, there has been some confusing semantic drift in its nomenclature. This applies both to “inertial frame” and to “special relativity.”
Early formulations of general relativity never refer to “inertial frames,” “Lorentz frames,” or anything else of that flavor. The very first topic in Einstein’s original systematic presentation of the theory$^{17}$ is an example (Figure k) involving two planets, the purpose of which is to convince the reader that all frames of reference are created equal, and that any attempt to make some of them into second-class citizens is invidious.

Lorentz Frame experiment
Figure k: One planet rotates about its axis and the other does not. As discussed in more detail on p. 104, Einstein believed that general relativity was even more radically egalitarian about frames of reference than it really is. He thought that if the planets were alone in an otherwise empty universe, there would be no way to tell which planet was really rotating and which was not, so that B’s tidal bulge would have to disappear. There would be no way to tell which planet’s surface was a Lorentz frame.

Other treatments of general relativity from the same era follow Einstein’s lead.$^{18}$ The trouble is that this example is more a statement of Einstein’s aspirations for his theory than an accurate depiction of the physics that it actually implies. General relativity really does allow an unambiguous distinction to be made between Lorentz frames and non-Lorentz frames, as described on p. 26. Einstein’s statement should have been weaker: the laws of physics (such as the Einstein field equation, p. 240) are the same in all frames (Lorentz or non-Lorentz). This is different from the situation in Newtonian mechanics and special relativity, where the laws of physics take on their simplest form only in Newton-inertial frames.
Because Einstein didn’t want to make distinctions between frames, we ended up being saddled with inconvenient terminology for them. The least verbally awkward choice is to hijack the term “inertial,” redefining it from its Newtonian meaning. We then say that the Earth’s surface is not an inertial frame, in the context of general relativity, whereas in the Newtonian context it is an inertial frame to a very good approximation. This usage is fairly standard,$^{19}$ but would have made Newton confused and Einstein unhappy. If we follow this usage, then we may sometimes have to say “Newtonian-inertial” or “Einstein-inertial.” A more awkward, but also more precise, term is “Lorentz frame,” as used in this book; this seems to be widely understood.$^{20}$
The distinction between special and general relativity has undergone a similar shift over the decades. Einstein originally defined the distinction in terms of the admissibility of accelerated frames of reference. This, however, puts us in the absurd position of saying that special relativity, which is supposed to be a generalization of Newtonian mechanics, cannot handle accelerated frames of reference in the same way that Newtonian mechanics can. In fact both Newtonian mechanics and special relativity treat Newtonian-noninertial frames of reference in the same way: by modifying the laws of physics so that they do not take on their most simple form (e.g., violating Newton’s third law), while retaining the ability to change coordinates back to a preferred frame in which the simpler laws apply. It was realized fairly early on$^{21}$ that the important distinction was between special relativity as a theory of flat spacetime, and general relativity as a theory that described gravity in terms of curved spacetime. All relativists writing since about 1950 seem to be in agreement on this more modern redefinition of the terms.$^{22}$
In an accelerating frame, the equivalence principle tells us that measurements will come out the same as if there were a gravitational field. But if the spacetime is flat, describing it in an accelerating frame doesn’t make it curved. (Curvature is a physical property of spacetime, and cannot be changed from zero to nonzero simply by a choice of coordinates.) Thus relativity allows us to have gravitational fields in flat space — but only for certain special configurations like this one. Special relativity is capable of operating just fine in this context. For example, Chung et al.$^{23}$ did a high-precision test of special relativity in 2009 using a matter interferometer in a vertical plane, specifically in order to test whether there was any violation of special relativity in a uniform gravitational field. Their experiment is interpreted purely as a test of special relativity, not general relativity.

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$^{16}$This statement of the equivalence principle is summarized, along with some other forms of it, in the back of the book on page 347.
$^{17}$Einstein, “The Foundation of the General Theory of Relativity,” 1916. An excerpt is given on p. 321.
$^{18}$Two that I believe were relatively influential are Born’s 1920 Einstein’s Theory of Relativity and Eddington’s 1924 The Mathematical Theory of Relativity. Born follows Einstein’s “Foundation” paper slavishly. Eddington seems only to mention inertial frames in a few places where the context is Newtonian.
$^{19}$Misner, Thorne, and Wheeler, Gravitation, 1973, p. 18
$^{20}$ibid, p. 19
$^{21}$Eddington, op. cit.
$^{22}$Misner, Thorne, and Wheeler, op. cit., pp.163-164. Penrose, The Road to Reality, 2004, p. 422. Taylor and Wheeler, Spacetime Physics, 1992, p. 132. Schutz, A First Course in General Relativity, 2009, pp. 3, 141. Hobson, General Relativity: An Introduction for Physicists, 2005, sec. 1.14.
$^{23}$arxiv.org/abs/0905.1929

The Thought Experiment

Let us assume there is an Earth like non-rotating planet in the intergalactic space.
The $g$ is approximately $6.2m/s^{2}$ at $1700km$ above the surface of the planet.
The Newton’s cannon ball velocity at this height is approximately 7km/s. That’s a velocity when “falling” around the planet follows geodesic that is a circle.
Our thought experiment is going to use the same apparatus but we will replace the rifles with a cyclotron. The outer diameter is $1m$. There is vacuum in the cyclotron. The Waage’s apparatus with the cyclotron is in the space at $1700km$ height above the planet surface held by a spaceship. The cyclotron axis points to the center of the planet. The spaceship does not move in the inertial reference frame attached to the center of the planet, using thrusters to maintain its position.

Lorentz Frame experiment
Figure 1: The cyclotron.

Positive charges (atom nuclei stripped of electrons) fly clockwise in the cyclotron at $7km/s$. This is the cutaway profile view of the cyclotron left side. The sides are $20cm$.

Lorentz Frame experiment
Figure 3: The cyclotron cutaway profile view.

Let us assume the cyclotron requires $+100kV$ voltage on the outside wall in order to generate the centripetal force to keep charges flying on a trajectory that is a perfect circle and it is centered ($10cm$ from each wall) in the cyclotron. No voltage is required at the bottom wall or top wall of the cyclotron in order to maintain center position of the charges because their ‘falling’ velocity is $7km/s$.

At time $T0$ the Waage’s apparatus is released and it is in a free fall.
At time $T1$ the apparatus fell/dropped $5cm$.

The charges in the cyclotron ‘keep falling’ in their ‘geodesic’ – $7km/s$ and they will not fall with the Waage’s apparatus and the cyclotron towards the planet.
Lorentz Frame experiment
Figure 3: The cyclotron cutaway profile view.

The charges will be $5cm$ from the top of the cyclotron at $T1$ because the apparatus and the cyclotron dropped $5cm$. In order to maintain the center position inside of the cyclotron a voltage needs to be applied to the top wall of the cyclotron. The voltage would be proportional to the velocity of cyclotron falling towards the gravity source and the gravitational acceleration $g$.

Discussion

The ‘falling’ in the intergalactic space where no planet is present is not going to require any voltage at the top wall of the cyclotron to center the charges.
If the Waage’s apparatus with the cyclotron was accelerated by a spaceship in a straight line with $a=6.2m/s^{2}$ then a voltage at the bottom wall of the cyclotron is required even though charges would be moving at $7km/s$; on the contrary to time prior to $T0$ when the apparatus was held by the spaceship and the gravitational acceleration was $g=6.2m/s^{2}$ and no voltage was required to maintain the center position. The gravitational acceleration can be distinguished from the straight line acceleration.

It appears the equivalence principle is not universal.

Conclusion

The reference frame at the center of the gravity is preferred reference frame.
The reference frame at the center of the Lorentz frame is not valid for all the velocities.