Castles in the Air

John D. Norton
Department of History and Philosophy of Science
University of Pittsburgh
This page at http://www.pitt.edu/~jdnorton/Goodies

https://commons.wikimedia.org/wiki/File:Caricature_Vernon_1740.png
Caption: "English Satirical Print - Spain builds castles in the air, Britain makes commerce her care in the War of Jenkins' Ear."

1. Introduction

It is a term of derision that dates back to the late 16th century. To accuse someone of building a castle in the air is to accuse them of folly and fantasy, a wondrous construction that is sustained only by imagination, but without any proper foundation.

This rebuke derives its power from a commonplace of physics. Nothing can evade the downward pull of gravity unless somehow there is a counteracting force that pulls in the opposite direction.

This commonsense of ordinary experience is supported by a standard result in classical physics. The net effect of gravitational forces on an ordinary body in a gravitational field is to accelerate its center of mass downward.

Is it possible to conceive systems that evade these results? If the system consists of finitely many component parts, interacting by the usual physical laws, then these results cannot be evaded. If, however, we consider bodies consisting of infinitely many components, then matters are otherwise.

The structure to be described here consists of a castle that rests upon a course of stones. That course rests upon another smaller course of smaller mass; and so on for an infinity of courses. The structure can be so contrived that its total mass is finite; it occupies a finite extent of space; and its matter density remains everywhere finite below some fixed bound.

There are many forces acting on and in the structure. There are the external gravitational forces that pull the castle and each course of stones downward. Then there are the forces that each course of stones exerts upon the one below it because of the weight each course carries. Finally there are the reaction forces each course of stones exerts back on the course of stones above it.

One might expect that on summation all these forces would cancel out and that we are left with just the net external gravitational forces that must then accelerate the structure downward.

Because of the infinity of the courses of stones, it is possible to set up the forces between the stones in such a way that they do not mutually cancel. Rather there is a residual force. That residual force can be set to cancel exactly the net gravitational force.

Then the castle and its courses of supporting stones floats in the air.

This castle in the air example arises as a special case of the analysis of the "turtles all the way down" problem in cosmology.

The literature in supertasks exploits the unexpected behavior of infinitely many interacting systems  to enable a classical system of masses to spring into life spontaneously. The construction of the castle in the air depends upon an analogous behavior of infinitely many components. Rather than producing a dynamical effect where none is expected, it produces no dynamical effect where one is expected. It is a static supertask.

2. Distributing the forces so the castle stands

The structure consists of a massive castle in a gravitation field that imparts a constant downward acceleration to free bodies. The castle rests on the first course of stones. This course of stones rest on a second course of stones, and so on for infinitely many courses.

Such a structure may fall freely; or it may remain at rest. If we assume the case of a structure that remains at rest, we can determine the forces that must be in place for this to be so and affirm that they conform with Newtonian physics.

In this static case, each component object of the structure is pulled downward by a gravitational force and the weight of the objects above. However the component does not fall. It passes all these forces as a loading to the next lowest course of stones upon which it rests. This next course of stones exerts a reaction force back onto it. The reaction force is equal in magnitude but opposite in direction to the loading forces.

As an aid to visualization, we will imagine the courses of stones to be connected by very stiff springs. In the configuration described, they are compressed by just the amount needed to sustain the balanced loading and reaction forces.

This account is already sufficient to display qualitatively the mechanism that allows the castle to float in the air. It is:

The castle is supported at rest by reaction forces from the the first course of stones.
The first course of stones is supported at rest by reaction forces from the the second course of stones.
The second course of stones is supported at rest by reaction forces from the third course of stones.
The third course of stones is supported at rest by reaction forces from the fourth course of stones.
...

This list continues for each course of stones. Since each course of stones in the infinite structure is supported at rest, it follows that the entirety of the structure remains at rest.

This same analysis cannot be applied to the case of a castle with a finite number of courses of stones. We might try to set up a similar chain of inferences:

The castle is supported (??) at rest by reaction forces from the the first course of stones.
The first course of stones is supported (??) at rest by reaction forces from the the second course of stones.
The second course of stones is supported (??) at rest by reaction forces from the third course of stones.
The third course of stones is supported (??) at rest by reaction forces from the fourth course of stones.
...
The last course of stones is supported (??) at rest by reaction forces from ... What? There are no further courses of stones beneath it to provide support.

The failure in the finite structure of the last course of stones to be supported undoes the analysis. This last course will fall. It follows that the course above it is unsupported and falls; and so on up the structure until the castle itself falls.

The essential element in the support of the infinite structure is its infinity: there is no last course of stones. The mode of failure just seen for the finite case is eluded.

Nothing in these relations of support requires the courses of stones to be all the same size. We can require each successive course (including its springs) to be half the height of the one before it. If we assume the density of the materials used remains the same, then each successive course will be half the mass of the one above it.

It follows that the total mass of the structure is finite. If we set the mass of the first course to unity, that mass is:

        mass castle + 1 + 1/2 + 1/4 + 1/8 + ... = mass castle + 2

Correspondingly, setting the height of the first course of stones to unity, the total height of structure is finite:

        height cast + 1 + 1/2 + 1/4 + 1/8 + ... = height castle + 2

The structure can hover in the gravitational field with nothing but empty space underneath it.

3. Applying Newton's laws

The analysis of the last section is sufficient to display the mechanism through which the castle is able to stand in the air. Here we see a more formal analysis. It adds nothing other than a display of formulae that express the ideas of the last section and a resulting assurance that Newton's laws are respected.

Once again, the structure consists of a massive castle of mass m₀ at vertical position x₀ in a gravitation field that imparts a constant downward acceleration g to free bodies. The castle rests on the first course of stones of mass m₁ at vertical position x₁. This course of stones rest on a second course of mass m₂ at vertical position x₂, and so on for infinitely many courses.

The net force  f_i acting on the ith object (castle or course of stones) is

(1)      f₀ = m₀a₀ = m₀g + f₀₁
           f_i = m_ia_i = m_ig + f_i,i-1 + f_i,i+1 for i>0

where f_ik is the force exerted on the ith object by the kth object. (The only cases we will consider are those in which i and k are one number separated, so that k=i-1 or k=i+1.) The acceleration of each object is a_i.

The weight of all the objects above the ith object is transmitted to it as the loading  f_i,i-1 which is the force with which the i-1th object bears down on the ith object. The ith object exerts a reaction force back on the i-1th object of f_i-1,i

Newton's third law requires that the force with which the ith object acts on the kth object is equal but opposite in sign to the force with which the kth object acts on the ith object. That is:

(2)     f_ik = -f_ki

All these forces must vanish in case the structure is to remain at rest, hovering in the air. Then the accelerations must also vanish:

(3)     0 = f₀ = f₁ = f₂ = f3 = ...      0 = a₀ = a₁ = a₂ = a₃ = ...

We can find the forces f_ik that realize this case by solving equations (1), (2) and (3) iteratively:

(4)     -f₀₁ = f₁₀ = m₀g
         -f₁₂ = f₂₁ = m₁g + f₁₀ = m₁g + m₀g
         -f₂₃ = f₃₂ = m₂g + f₂₁ = m₂g + m₁g + m₀g
         ...
         -f_i-1,i, = f_i,i-1 = m_i-1g + ... + m₂g + m₁g + m₀g
         ...

If the structure is initialized with these inter-object forces, the springs will all be compressed by  just the amount needed to sustain these forces. Since the net force on each object is zero, the objects do not accelerate. They remain separated by the same distances, so that the springs retain their compression and the forces of (1) remain unchanged from zero through time.

The structure stands, supported by the forces displayed in (1).

Were there only finitely many courses of stones, N say, then this distribution (4) of forces would not realize a static system. For then the forces acting on the component objects would not be given by (1) but by

(5)      f₀ = m₀a₀ = m₀g + f₀₁
          f_i = m_ia_i = m_ig + f_i,i-1 + f_i,i+1     for 0 < i < N
          f_N = m_Na_N = m_Ng + f_N,_N-1

The inter-object force distribution (4) does yield an acceleration free solution (3) when substituted into (5). For both m_Ng > 0 and f_N,_N-1 > 0, so that f_N > 0.

4. A center of mass theorem

The analysis of the last section is sufficient to show conformity of the castle in the air with Newtonian mechanics. However the structure appears to conflict with a familiar result in mechanics. If we have a body in a gravitational field, its overall motion is determined by the total gravitational force acting on it and its total mass. Internal forces between its components need not be considered. The effect of the gravitational force is an acceleration of the center of mass, where we have

     total gravitational force = total mass x acceleration of center of mass

This familiar result is not an independent law, but is a result deduced from the compounded behavior of the components of the system in a gravitational field. This section will show that, if we carry out the corresponding deduction for this infinite structure, the castle can remain in the air.

The calculation that follows computes the net force on the first N components. Since these first N components are always a subsystem of the infinite structure, there are always reaction forces acting on it from further components and they can be set to cancel out the gravitational forces acting on these first N components. This cancellation persists when we take the limit over infinitely many components, yielding an unaccelerated center of mass for the castle in the air.

To proceed, we define some intermediate quantities. The mass M_N of the first N components is

 (6)         M_N = m₀ + m₁ + m₂ +... + m_N

The total mass is the limit of this partial mass

            M = Lim _{(N → ∞)} M_N  

The net force f_i acting on the ith component object is

        f_i = m_ia_i = m_i (d²/dt²) x_i

The sum of these net forces for the first N objects is

        F_N = f₀ + f₁ + f₂ +... + f_N
                  = (d²/dt²) (m₀x₀ + m₁x₁ + m₂x₂ +... + m_Nx_N)
                        = M (d²/dt²) (m₀x₀ + m₁x₁ + m₂x₂ +... + m_Nx_N)/M
                                = M (d²/dt²) X_N

where the center of mass of the first N components is

       X_N = (m₀x₀ + m₁x₁ + m₂x₂ +... + m_Nx_N)/M

and the center of mass of the infinite structure is

            X = Lim _{(N → ∞)} X_N

This limit is well-defined since we have assumed that the mass M is finite that the positions x_i are confined within a finite magnitude.

Using (1) we compute the net force on the first N objects as

F_N =  M (d²/dt²) X_N
            = (m₀ + m₁ + m₂ +... + m_N)g
                        + f₀₁ + (f₁₀ + f₁₂) + (f₂₁ + f₂₃) + (f₃₂ + f₃₄) +
                                               ... + (f_N-1,N-2 + f_N-1,N) + (f_N,N-1 + f_N,N+1)

This expression can be simplified using (6) and applying (2) to the sequence of loading and reaction forces so that

(f₀₁ + f₁₀) = (f₁₂ + f₂₁) = (f₂₃ + f₃₂) = ... = (f_N-1,N + f_N,N-1) = 0

We arrive at

(7)           F_N =  M (d²/dt²) X_N = M_N g + f_N,N+1

This result (7) marks the point at which the derivations for a finite structure and an infinite structure separate.

If the structure is finite and has a total of N components, then there is no N+1th component and thus no reaction force f_N,N+1. We have:

Center of mass theorem for finite structures
F_N =  M (d²/dt²) X_N = M_Ng

If the structure is infinite, then there is always an N+1th component and we can arrive at the theorem for the infinite case by computing the total net force acting on the infinite structure as

            F = Lim _{(N → ∞)} F_N  

One possibility is that the castle is in free fall. In that case, all the forces f_ik vanish, including  f_N,N+1, and the limit gives us:

Center of mass theorem for infinite structures in free fall
F =  M (d²/dt²) X = Mg

These last two cases conform with our expectations. The net forces acting on the finite or infinite structures are just the gravitational forces: M_Ng and Mg in the two cases respectively. This net force manifests as an acceleration of the centers of mass X_N and X respectively.

However, in the infinite static case of a castle in the air, the inter-object forces are given by (4). From it we read that

      f_N,N+1 = -M_Ng

Combining this value with (7) we find that

          F_N =  M (d²/dt²) X_N = M_Ng - M_Ng = 0

Taking the limit we recover:

Center of mass theorem for static infinite structures
F =  M (d²/dt²) X = 0

That is, there is no net force F acting on the infinite structure as a whole. The external gravitational forces have been cancelled exactly by the inter-object forces. The center of mass X is unaccelerated. Since it is by supposition initially at rest, it remains so.

John D. Norton, April 18, 2018