https://en.wikipedia.org/wiki/Interior_Schwarzschild_metric
We have suspected for several years that the rigidity of the spatial metric under pressure cannot be true. On November 20, 2020 we wrote about a perpetuum mobile which would exploit the infinite rigidity of the spatial metric.
It turns out that Karl Schwarzschild did calculate correctly back in 1916.
Let us have a test mass inside a spherical pressurized vessel. Below we estimate that moving the test mass radially seems to do the same amount of energy shipping as moving it tangentially.
The inertia of a test mass is the same for tangential and radial motion. Light moves at the same speed radially and tangentially, measured by an outside observer. That implies that time has slowed down, but there is no stretching of the radial or the tangential spatial distances.
A spherical pressurized vessel
On November 8, 2021 we calculated exactly that one can explain the potential of pressure at the center of the Schwarzschild metric simply by considering the spatial expansion which a test mass m causes in a spherical vessel.
vessel, radius R
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/ \
| <--- • | m test mass
\ /
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Suppose that we have a pressurized spherical vessel, and we move a test mass from its side to the center. The volume of the vessel expands. We can harvest energy from the movement of m.
Thus, m has a "potential" in the vessel. The potential is the lowest at the center.
Where is the (negative) potential energy of m located?
The spatial metric is stretched close to m. If m moves at almost the speed of light, the negative energy is presumably close to m. We assume that the fluid does not have time to adjust to the change in the metric.
The negative potential energy then is very clearly localized: it is in those parts of the vessel whose volume m has increased. Pressure did work when space expanded.
Estimating the energy displacement
Let us model the stretching of the spatial metric with concentric, overlapping, spheres around m. Each sphere may represent, say, 0.1% of stretching. How much energy is shipped if we move such a sphere A a short distance s radially or tangentially? Let us denote the radius of A by R'.
vessel, radius R
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/ \
| O | sphere A, radius R'.
\ /
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If the sphere A fits entirely inside the vessel, then, of course, the energy shipping is the same if we move A radially or tangentially. In the diagram, O denotes the small sphere A.
If A is so large that the whole vessel is inside A, then there is no energy shipping inside the vessel. We ignore the effect on the walls of the vessel and only calculate energy shipping inside the vessel.
X
X
• <-- r --> × <----- R -----> X
s | X
v X
test mass m center energy E
If
R' = R,
and m is at a small distance r from the center × of the vessel, then moving m tangentially rotates the energy E (marked with XXXX) at the edge of the vessel on the opposite side. If we move m tangentially a distance s, then we move E a distance
s R / r.
The energy displacement is
E s R / r.
If we move m the distance s < r right, we ship the energy E s / r to the left to the test mass m, over a distance
R + r,
which is approximately R since r is small. The energy displacement is
E s R / r.
The displacements agree for a radial and a tangential motion.
If
R' = R + k, 0 < k,
and k is not large, then the case is very roughly symmetric to the case
R' = R - k.
A radial movement does do more energy displacement in the case R + k, but does about the same amount less in the case R - k.
We presented extremely crude arguments that suggest that the energy displacement is the same in a tangential and a radial movement. A precise calculation can be done with a computer, or it might be possible even analytically.
General relativity has an exact analytical result in the Schwarzschild internal metric. Our Minkowski & newtonian model agrees with general relativity about the potential caused by pressure. We believe that general relativity and our model agree about the spatial metric, too.
Why the November 20, 2020 perpetuum mobile does not work? The problem of conservation laws
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/ \
| ● M ----------------->
\ /
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Let us have an extremely rigid vessel which we fill with almost incompressible fluid.
We put a mass M at the center of the vessel. The volume of the vessel grows and we pour in more fluid.
The pressure in the vessel is zero, and M does not feel any pull of gravity.
We then shoot M out at almost the speed of light. Since M moves so fast, it does "know" what is happening behind it. The volume of the vessel shrinks and we can harvest huge energy from the pressure.
Why the process does not work? How does M know that it is not allowed to fly out?
We may further assume that M is a pulse of light. Is the light reflected back?
There are probably many similat settings where a light-speed object seems to break conservation of energy or momentum. We have contemplated the question of conservation laws, but have not yet found a good model which would enforce them.
We suggested that in quantum mechanics, processes are "transactional". An unknown central authority checks that conservation laws are obeyed before approving a transaction.
If we believe in the existence of fields, then the field of M must linger inside the vessel, even though M itself is speeding away. How can the field "tell" M that it should stop?
The question is clearly about the self-force of a field on the particle itself. Can the field communicate superluminally with the particle?
This question is arguably the greatest open problem in theoretical physics. We will keep trying to solve it.
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