Having a test mass m with "negative energy" in a "potential"
We can use a rope to lower a test mass m to a location in the system, and measure how much energy we can harvest. Let us assume that we were able to harvest more energy than m c². Is that a problem? Do we have now an object with negative mass-energy?
We may imagine that the forces between particles are complex. Maybe we have mechanical robots in the system, and they pull certain test masses so vigorously that we can harvest more than m c² of energy when lowering the test mass down. Having such robots does not introduce any fundamental problem. Having "negative energy" for a test mass does not lead to any paradox.
On the other hand, the energy of the whole system cannot become negative, since then we would have all the paradoxes of negative energy objects.
An old problem is how to localize the energy of a system of particles. If we harvest the binding energy when we build the system, from where is that energy missing?
If we have a very rigid shell of mass, the potential of each of its parts may be close to -m c², but the binding energy of the shell is only 1/2 of the negative potential energy. Thus, the shell has large mass-energy. Where is that energy located?
It may be a wrong approach to try to assign the energy of a complex system to its parts.
The Schwarzschild metric within the event horizon is strange: it breaks a time symmetry of laws of nature, as well as thermodynamics
In the Schwarzschild metric formula,
r_s = 2 G M / c²
is the Schwarzschild radius. It is the radius of the event horizon in the standard Schwarzschild coordinates. M is the mass of the spherically symmetric object, as measured by a faraway observer.
In general relativity, the event horizon is a one-way membrane. The metric inside the event horizon is strange. If we in the formula above have r < r_s, the signs of the time metric and the radial metric flip. When the proper time τ of an observer increases, his radial coordinate r must decrease. The metric claims that all objects must approach the center?
Actually, the metric says that the radial coordinate r must change: the formula just says that dr² must be nonzero. People usually think that r must decrease, but is it really so?
Just above the event horizon the metric says that the force of gravity, as seen by a static local observer, tends to infinity. If the photon does not move outward exactly radially, gravity will pull it to the event horizon. Since no photon moves exactly radially, the event horizon is a one-way membrane.
In traditional classical physics it would require an infinite force to ensure that all objects will approach the center.
In our Minkowski & newtonian model, gravity is about inertia and Newton's gravity force, which is analogous to the Coulomb force. We cannot make a one-way membrane in electromagnetism. Thus, it should not be possible in gravity either.
The metric for r ≤ r_s is not compatible with the Minkowski & newtonian model. On the other hand, the metric for
r > r_s + ε
is just what the Minkowski & newtonian model predicts, where ε is some very short distance which depends on M.
One can simulate the metric for r > r_s with a finite force, but for r = r_s it would require an infinite force.
After Albert Einstein introduced general relativity in 1915, some people criticized the apparent singularity at r = r_s. The critics were right.
Having infinite forces is unnatural. They break a time symmetry of laws of nature: a certain process can only happen to one direction.
A one-way membrane breaks thermodynamics, too. Jacob Bekenstein and Stephen Hawking realized this, and tried to fix general relativity. They did not realize that the error is in general relativity itself, and one cannot remove the problem with Hawking radiation.
Where is the flaw in general relativity?
For thermodynamics to work, physical processes cannot be just one-way. Having a one-way membrane in the event horizon is a major flaw in general relativity.
What is the root cause for the one-way nature of general relativity?
One of the reasons is that people think that the "geometry of spacetime" is something which has "infinite strength". Even a small mass M can make an infinitely strong geometry around itself.
General relativity, as understood by most people, has a similar infinite strength problem in gravitational waves. People think that the waves fundamentally change the "geometry of spacetime" in their zone. However, as we have argued, the waves only have finite energy. They cannot significantly change the geometry as seen by very large masses and very rigid rods. The true Minkowski space geometry is hiding below the wave which only has finite energy.
Is the flaw in the Einstein-Hilbert action, or is it just in its interpretation in the study of black holes?
There is some kind of a lagrangian or a hamiltonian behind the Einstein-Hilbert action. One would expect that a process in a typical hamiltonian cannot be just one-way, since that would require an infinite force.
We conjecture that the flaw is in an incorrect interpretation of the Einstein-Hilbert action.
A correct interpretation would not involve a one-way event horizon. The fact that general relativity treats gravity in a way different from other forces is a contributing factor in the wrong interpretation. For an ordinary force, one would immediately suspect a calculation error if one would get infinite forces.
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