Calculus of variations and a carpet

(Photo Wikipedia)

Let us present a simple example where charge conservation keeps the action such that it can be optimized strictly locally, but a change in the charge makes it impossible to optimize the action locally.

                   

                         ___

            ______/ ■■ \_____________    carpet

                      board  -->

We have a long corridor and a carpet which is almost as long as the corridor. Someone forgot a board under the carpet. The board is normal to the corridor and extends from the wall to wall. The volume of the board is a "charge".

The action is the volume V between the floor and the carpet minus the volume of the board under the carpet. Initially the volume V is 0.

If we slide the board along the corridor (i.e., the charge is conserved), we can find the minimum of the action V strictly locally. Just press the carpet tightly to the floor or to the board. This configuration corresponds a typical field theory with charges, like electromagnetism. We can keep the far parts of the carpet as is: the carpet only needs to be adjusted locally.

But if we magically remove the board, the carpet becomes loose. There is a wrinkle, and the action suddenly is V > 0. The charge was not conserved.

We assume that one cannot move the carpet infinitely fast. The action V must stay > 0 for some time.

Now it is obvious that the carpet cannot be optimized strictly locally. We can make V = 0 again by moving the wrinkle to the end of the carpet, but that takes time.

Note that we could have a spare board B which we insert inside the wrinkle. Then we could again optimize the action strictly locally and slide the spare board B out at the end of the carpet. This shows that we can find a solution to the global optimization problem by restoring charge conservation and sliding the spare charge "far away".

                        _

              ____/   \____    carpet

                 wrinkle

Let us study how we can vary the form of the wrinkle in the carpet if there is no board under the carpet. We might have a very large potential which prevents the carpet from curving into a circle of less than, say, 1 cm of radius. We can minimize the volume V under the wrinkle. The optimum is not unique. The wrinkle can be put at many locations. We have a global optimization problem which has many solutions. The solutions do not determine the elevation of the carpet at a specific location.

General relativity

In general relativity, the analogue of the board is a mass M, and the analogue of the carpet is the metric. Suppose that M is a spherically symmetric. Then the field outside M is the Schwarzschild metric.

Suppose that M is suddenly reduced by magic to a mass M' < M. The optimization of the Einstein-Hilbert action suddenly becomes a global problem.

A new optimum, which would satisfy the Einstein field equations, would be the Schwarzschild metric for M'. But we can never get to that metric since the speed of light is finite.

We could use the spare board trick above to switch to the metric of M'. Make a shell of matter whose mass is M - M'. Let that shell recede from M' at the speed of light. However, we can never complete this operation since Minkowski space is infinite.

Conjecture. It is impossible to satisfy the Einstein equations and switch (locally) from the metric of M to the metric of M', if we require the Ricci tensor to be zero in the vacuum around M'. We have to eject some kind of matter to transition to the metric of M'.

In our previous blog post we sketched a proof for this conjecture. The proof uses the "focusing" of a cube of test masses.

Conclusions

The carpet is a "field" and the board is a "charge". We illustrated what happens if we magically reduce the charge. Nice, local optimization conditions are replaced with a much harder global optimization problem.