Elevator Thought Experiment

Problems with the "elevator thought experiment" of General Relativity

Science Editorial 7:May 22,2010

Everyone who studies General Relativity learns about the elevator thought experiment. It is sometimes presented as the paradigmatic illustration of the Principle of Equivalence. The whole of General Relativity is basically just the mathematical working out of this foundational concept, which can be succinctly expressed as the dictum: inertial mass is identical to gravitational mass.

Picture a physicist, J (man, woman or machine (which allows us to use the neutral pronoun "it")) stationed inside a box that functions as an elevator. It has no windows, and is tough enough to be impervious to colliding objects or cosmic waves. One allows that the physicist, J, carries with it all the measuring devices it so desires: voltmeters, spectrographs, Geiger Counters, magnets, etc. In particular there are scales and weights to measure very sensitive deviations from the gravitational constant. Any detectable internal event can be quantified.

There is also another observer, O, stationed outside the box, and in the same rest frame as J. O notices that the elevator has suddenly begun to accelerate upwards, moving in the direction from floor to ceiling. The "elevator thought experiment" asks: what conclusions will J draw from this change of state?

This way of posing the question is already a mistake. The correct question to ask is: what is happening inside the box?

By the Principle of Equivalence J observes, using its weights and scales, that the measured weight of everything inside the elevator has increased. This increase can be interpreted in two ways, and they are equivalent:

(a) The elevator is accelerating in the direction from floor to ceiling, away from its previous state of rest
(b) The elevator remains at rest.What has really happened isthat a gravitational field has entered the vicinity, causing the weights of all objects to increase in proportion to the increase of the local gravitational constant, g.

Another way of looking at this is to imagine that the floor of the elevator has suddenly become very heavy through the introduction of matter of a very high density.

There are two classes of objections to this picture. The first class concerns its status as an abstraction. that is to say the "thought" part of the phrase "thought experiment".

The second class of objections has to do with the word "experiment", that is to say, what one is actually able to measure inside the box without stepping outside.

Class I: The viability of the insulated box in thought experiments

The modern frameworks of all the ramified branches of Quantum Theory, and even that of General Relativity itself, do not allow for the conceptualization of this experiment in the form in which it is presented. Closed boxes in isolation have ceased to be permissible whenever one is dealing with universal forces that make no distinction between inside and outside, gravity being the most notable: there can be no such thing (to our knowledge) as a gravity shield.Nor can there be shields against quantum fields or heat dissipation. One can summarize this state of affairs in the form of a cute theorem, one that might be considered to be an "anti-theorem": The natural topology of our universe violates the conditions for a proof of the Jordan Curve Theorem.

In its 2-dimensional form, the Jordan Curve Theorem (JCT) states that any nonself-intersecting arc in the plane, homeomorphic to the closed circle, divides the plane into two mutually disjoint regions, the "inside" and the "outside". Though intuitively obvious, it is difficult to prove directly from the basic axioms of topology.

The extension of the JCT into 3 dimensional 3-space is easily accomplished. Replace the circle by the 2D sphere : any embedded surface K in 3-Space which is topologically equivalent to the sphere divides 3-space into two regions, the inside and the outside of K.

The following phenomena of modern physical theory violate the conditions of this theorem:

  1. Quantum tunneling .

    Unless a potential well is truly infinite, there is always a constant probability that information about any phenomenon inside the elevator, or even the phenomenon itself, will leak to the outside, and vice versa. This is a universal reality which has been observed on many levels. No elevator made of real materials can prevent information from leaking back and forth through its walls. One might argue that the "thought elevator" is made of "ideal materials", but ideal materials went out with the overthrow of Aristotelianism by Galileo in the 17th century.

    The existence of quantum tunneling can be also be used to critique another classical thought experiment, Schrodinger's cat.If the cat and the apparatus are placed inside any real world box, there will be quantum tunneling in both directions. Some of the leaked information may be sufficient to determine if the cat is still alive or dead.

  2. Archibald Wheeler's notion of a "quantum foam" is based on the recognition that the enclosed boxes of classical physics and classical thought experiments, must be replaced by porous objects, with rigid bodies and solid or point particles being replaced by fields, and waves that stretch across the entire universe. The universe, in its entirely is universally present (through entanglement if nothing else) at every point in space and every moment in time.

  3. Mach's Principle.

    This principle states that the "weight" of any massive object is a function of the distribution of all massive objects in the universe. It follows that measurements of the gravitational constant inside the elevator will give information to its occupant, of the shifts in the distribution of matter everywhere.

    In fact, General Relativity claims that one should be able to determine Newton's universal gravitational constant (an invariant!) from the curvature (another invariant!) in any region of space. One need not even appeal to Mach's principle to claim this. It is enough to replicate an Eötvös experiment using delicate torsion balances.

    Once J computes this constant, it will be able to use a kind of "gravitational lensing" based on the shifting values of the local gravitational constant inside the elevator, to map out the distribution of matter across the entire universe!

    At the very least this forces a modification of the elevator thought experiment. (Mach's Principle, of course, is controversial. One is not even entirely sure of exactly what it says.)

  4. The Leibnizian character of General Relativity.

    Leibniz maintained that everything in the universe is causally connected in such a way that (in theory) its entire state may be determined from complete knowledge of the activity inside any open region. His Monadology is a system of (conscious!) mirrors within mirrors, by which knowledge of all things happening everywhere is, in theory and in potential, present at every point of space or space-time.

    General Relativity is, in many respects, a Leibnizian, as opposed to a Lagrangian, theory: "Matter tells space how to curve, space tells matter how to move". There are no systems in isolation in Leibniz's epistemology, which is why it's of little practical use in the real day to day activity of science. One might even call it a kind of scientific astrology, in the sense that far distant influences need to be taken into account inside any laboratory.

    The framework of General Relativity is Leibnizian to the extent that one cannot banish all outside influences when speaking of the universal effects of gravitation. Quantum entanglement has given this close-connectedness an additional dimension.

Class II: Experimental Objections

I will now argue that, even in quite practical terms, the "elevator thought experiment" is at most a first-order approxination, at best.

What this thought experiment is really doing is to providing a representation of the phenomenon of free fall : no observer standing on a material body X, moving under the effects of the gravitational field between itself and another massive object B, can do any experiment on A to detect its motion. It is only when A collides with B, or when A is forced out of its orbit relative to B by another force , that the observer will feel an increase in weight. For someone standing on the Earth's surface, this is the opposition of the Earth's density and bulk, which is so tightly bound (by electro-chemical bonds) that it can resist the natural tendency of persons or objects to fall (weightlessly!) right to the center.

Countless experiences with astronauts and pilots have verified the existence of the phenomenon of free fall. If there were no atmosphere, someone falling out of an airplane would experience the sensation of weightlessness all the way down to the collision with the Earth's surface. It is clear that the "elevator effect" is correct to the first order of approximation.

However, if an elevator is forced up and away from the earth, its occupants will feel a corresponding heaviness due to the accelerating force. But what is this accelerating force? In the normal narration of the story of the elevator , it is customary that one says nothing about the force that is producing the acceleration.

Now, although one can reasonably exclude bombardments by cosmic rays, or collisions with other objects as being irrelevant to the idea behind the metaphor, one cannot exclude the effects on the elevator interior of the forces that are physically pulling it out of orbit . The thought experiment thereby loses its "physicality" and turns into an abstract exercise with limited, if any, relevance to the real world.

There are only a small number of forces operative in our universe: gravity, electro-magnetism, the weak force, and the strong force. One can also consider heat as a kind of force generated by collisions of billions of molecules or minute particles, that is to say, by the accumulation of many velocities.

If the elevator is pulled away from the earth by another gravitational field, say a huge star floating into the neighborhood, its occupant will experience weightlessness. Because both the occupant and the elevator are material bodies, gravity acts equally on both of them. One might argue that the occupant's own physical body is irrelevant to the metaphor, but this cannot be so, since we have already spoken about the "sensation of increasing weight" as a key indicator of some major change in the environment. The observer cannot have an abstract body in this thought experiment.

Let us imagine on the other hand, that there is a giant magnet which is pulling up the metal container in which the observer is placed. If the elevator is rising under the effect of its magnetic field, then all of the magnets inside the elevator will have their North poles aligned in one direction and their South poles in the other. Not only can the observer record the presence of acceleration, it can identify what direction it's coming from.

Likewise for an electromotive force of some kind: all of the positive charges will migrate in one direction, and the negative charges in the other.

If the source of the acceleration is the strong force, we are speaking of a thermo-nuclear explosion, with transmutations of elements that can be detected in the substance of the container.

If the source of the acceleration is the weak force, one is talking about chemical bonds, therefore electro-chemical and radioactive interactions that may be transformed into heat. These will be detectable inside the elevator as a rise in temperature. By the Second Law of Thermodynamics, this change in heat must leak inside the elevator itself as the system seeks to attain an equilibrium state.Unless the elevator is made of some "ideal" material, no amount of shielding can prevent this. However, ideal materials are banished from our universe. Universal Entropy must increase, and this is theoretically detectable from anywhere.

Thus, even if one does not consider the universe-wide waves and fields of Quantum Theory, the local statistical physics of Thermodynamics, in any realistic scenario for the increase by force in the acceleration of an elevator, will give the occupant of the elevator a way of concluding that an accelerating force is being applied.

Conclusion: any force of nature that creates an acceleration of the elevator is detectable, even gravity itself: if the occupant of the elevator suddenly experiences weightlessness,it will have no doubt that a gravitational force is pulling him upwards!

These considerations may be a small contribution to the invention of a successful theory of Quantum Gravity.

NOTE: A thorough discussion of these issues and more, can be found in the article Does Quantum Mechanics Clash with the Equivalence Principle - and Does it Matter? Authors: Elias Okon, Craig Callender

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