In this blog we have never studied degrees of freedom, and our understanding of the matter is not very good yet.
Suppose that we have a spherically symmetric static electric charge. The electric Coulomb field around it has a "freedom" of only one real number: the charge.
Similarly, in general relativity, the field around a spherical mass is the Schwarzschild metric, and the "freedom" is the mass.
In our 2-dimensional surface model of March 8, 2024, the solution is a cone made of paper around a circular "mass". The freedom is the angle of the cone.
In all these cases, all "freedom" is eventually lost because of a symmetry, and because the solution must match the "charge" at the center.
Is there a paradox in this? A gravity field, or a wave, seems to be much more "complex" than an electromagnetic field. How can it have the same two degrees of freedom?
Controlling the metric around matter
Suppose that we have a configuration of light masses sitting in space. We can make the metric of time to match the newtonian gravity potential. That fixes one degree of freedom. After that, we can try to find a spatial metric which makes the Ricci tensor zero outside matter. In the Schwarzschild case, we can find such a spatial metric. Is there any guarantee that we can find it in the general case?
We can use the mass distribution to control the temporal metric g₀₀. If we can dynamically create pressure inside matter, we can presumably control also some spatial components of g. Does this remove too much freedom from the field, so that we cannot make Ricci curvature zero there?
On November 5, 2023 we tentatively proved that a change in pressure "breaks" the Einstein field equations. We cannot make the Ricci tensor zero in such a case.
Degrees of freedom may clarify our result. If the Ricci tensor R is zero around a spherically symmetric system, then the only possible freedom is the amount of mass at the center. But if we manipulate the pressure at the center, the system cannot be described by a single parameter.
In electromagnetism, the only relevant parameter is the amount of charge. Pressure plays no role.
How about a general system of masses? Can we "fix" the metric of time and some components of the spatial metric around it, so that the remaining freedom makes it impossible to make the Ricci tensor R zero?
Static masses: is there enough freedom to make R = 0 outside matter?
If we have two masses floating freely in space, it is not a static system. Gravity accelerates those masses.
Let us consider a subcase where the matter sits static, because of pressures which counteract gravity. Schwarzschild was able to handle a spherically symmetric ball of incompressible liquid. What about more complex shapes?
----------------------------------
| |
| distribution of |
| mass and pressure |
| |
-----------------------------------
block of matter
The block contains both mass and stress forces: pressure and shear forces.
The metric in the vacuum around the block must be able to match the mass distribution and the stresses inside the block. Is it plausible that such a metric can be found?
In the case of electromagnetism, the Coulomb field in the vacuum can easily match the charge distribution. Simply linearly sum the fields of elementary charges.
In the case of gravity, we can control g₀₀ around the block by tuning the mass distribution. We can probably control several other elements in the metric tensor by tuning the pressure and the shear stresses. How many degrees of freedom are there in elastic stresses of a material? The sum of forces on a cube of matter must be zero, to each direction x, y, z.
The pressure in the cube can have independent values to each direction x, y, z. The stresses, apparently, form a 3 × 3 stress tensor.
Such a tensor has 6 degrees of freedom? Is there a gauge invariance which would reduce them?
At first sight, the two degrees of freedom in general relativity are too few to match 6 degrees in the stress tensor and one degree in the mass distribution. This would imply that the Einstein field equations have no solutions in almost all cases which involve pressure or shear stresses.
Pressure at the surface of the block, obviously, cannot have a component which is normal to the surface. Also, shear stresses have to be 2-dimensional at the surface.
We are dealing with a 2 × 2 stress tensor, which has three independent components?
If the vacuum metric around the block seriously restricts the allowed mass density / stress distributions in the block, then we obtain strange forces which guide us to make the distributions into the allowed ones! This would be very strange, and an indication that general relativity is an incorrect theory.
Question. Did we heuristically prove that the Einstein field equations only have solutions for very special pressure distributions?
The question if the Einstein equations have a solution for an arbitrary mass distribution, is an open one. Above we employ the freedom in stresses. Do these definitely show that general relativity lacks the required degrees of freedom?
In our 2D surface model on March 8, 2024, we struggled to define the metric for "mass" distributions. We obviously in that model do not have any means of handling pressure distributions. The paper which we use is rigid, and cannot communicate the pressure in the "matter" which we embed.
A spherical mass with a tense membrane around it
General relativity says that the metric around a spherically symmetric system must be Schwarzschild, and the metric cannot change.
Let us put a rubber membrane around the sphere, and vary the tension in the rubber. The metric inside the membrane must change because the "source" of the gravity field changes. Could it be that the metric at the surface of the membrane stays constant?
Jürgen Ehlers et al. (2005) study this configuration and conclude that the metric stays the same outside the sphere, regardless of the pressure.
However, if the body is not spherical, what happens?
Conclusions
If we have a block of rubber, it can be strained with a stress tensor. The stretching of the block can be described with a "metric tensor". There intuitively exists a solution. Counting degrees of freedom does not bring anything more to this analysis.
For now, we will not analyze the degrees of freedom further.
We will look at zero Ricci curvature as a problem in optics. A bundle of parallel rays of light must not be focused or defocused if Ricci curvature is zero. How to design such an optical device? The Schwarzschild metric is one. But if we perturb that metric, does it necessarily follow that the resulting optics focus or defocus certain beams of light? If that is the case, there is no solution for the Einstein field equations, except for artificial, symmetric configurations.
Our analogy with 2D surfaces and bending a paper into a cone suggests that the Schwarzschild metric might be the only solution where the Ricci tensor R = 0, just like a cone (or a cylinder) is the only way to bend a paper if we want to keep its metric straight.
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