UPDATE January 19, 2022: We estimated today the radial speed. Karl Schwarzschild was right.
----
In this blog we have been wondering why pressure appears in the energy-momentum tensor in general relativity, while from the interior Schwarzschild solution we see that pressure does not change the spatial metric at all. Pressure does slow down time, though.
It is as if the spatial geometry of spacetime would have infinite rigidity against forces which try to stretch it.
_____________
| |
| ● M |
|_____________|
pressurized vessel
We have been trying to devise a perpetuum mobile which would take advantage of the infinite rigidity of the spatial metric.
Let us have an infinitely rigid vessel filled with incompressible fluid. By moving a mass M in and out of the vessel we can change the volume of the vessel, and potentially create an infinite pressure. That might make a perpetuum mobile possible.
Now we realize that the "gravity" produced by pressure is not gravity at all, but the combined effect of the gravity of the mass M and the force field which creates the pressure.
The mass M has an approximate Schwarzschild metric around it. Radial lengthts have grown a little, but horizontal lengths are as in the surrounding space. The spatial volume around M is a little larger. Moving M into the vessel frees energy from the force field which creates the pressure, because the volume of the vessel increases slightly.
Why does light slow down in the vessel? It is because a photon entering the vessel acquires some extra energy from the pressure force field, and carries it to the other side of the vessel before leaving the vessel. The photon carried some extra baggage, which caused extra inertia and slowed down its progress.
Let us assume that the extra baggage on a photon is 1%.
We can explain the slowdown of clocks inside the vessel just as in the case of the Schwarzschild metric: clock parts have 1% larger inertia than in outer space because moving them causes energy to be displaced in the pressure force field. The electromagnetic field, which moves the parts of the clock, has grown 1% weaker inside the vessel because the gravity of the electromagnetic field did some work when we lowered the clock into the vessel.
We still have to explain why the radial speed of light in the vessel is the same as the horizontal speed. In the Schwarzschild metric, the radial speed is slower. The potential inside the vessel is the lowest at the center, just like in the Schwarzschild metric.
The reason might be that the field energy in the Schwarzschild case is spread over a large volume of space. Moving a photon radially in the field can cause a displacement of field energy over large distances.
But in the vessel, the extra volume generated by the mass M is (?) close to the mass. Maybe that is the reason why the energy in the pressure field has to be displaced the same distance s that we move the mass M.
The Einstein-Hilbert action
The Einstein field equations are derived from the Einstein-Hilbert action through varying the metric. The action contains a lagrangian for the matter fields, as well as a term for gravity.
It is no wonder that the "gravity" which is generated by pressure really is interplay of the pressure force field with the Schwarzschild geometry which is around a test mass M.
The stress-energy tensor contains terms for "simple" interactions of the metric with matter fields. We can imagine a more complicated interaction which cannot be reduced to pressure or shear stresses.
If we have "pure" gravity between point masses, with no other interactions, then the stress-energy tensor has the pressure and shear stress elements zero. We could say that this is the "real" gravity.
A photon inside glass versus a photon in the vessel
photon ---- electromagnetism ---- charges in atoms in glass
photon ---- gravity ---- charges in atoms in liquid
A photon slows down in any interaction. Inside a glass pane, the electromagnetic field of the photon interacts with the charges of the atoms in glass.
In the pressurized vessel experiment, the photon interacts through its gravity with the charges in the atoms of the liquid which fills the vessel.
General relativity claims that in the vessel case, it is the geometry of spacetime which guides the photon, while inside glass it is an ordinary force. Why is gravity special?
Philosophical aspects - is spacetime really curved?
In newtonian gravity, only mass-energy produces gravity - pressure or shear stresses do not produce gravity. It looks like the same holds for general relativity, once we recognize that pressure and shear stress elements in the stress-energy tensor only describe indirect effects which are caused by gravity along with other fields.
Does pressure alter the metric of spacetime?Pressure certainly attracts matter, but our analysis did not require us to assume that pressure itself causes any change in the spacetime metric. We did assume that the Schwarzschild metric surrounds our test mass M, but we showed in our October 7, 2021 blog post that the Schwarzschild geometry may be just an illusion caused by the interplay of a newtonian gravitational field with matter and other fields.
Let us again look at the philosophical question if gravity changes the spacetime metric at all. If we work in special relativity and newtonian gravity, we may be able to derive all the effects of general relativity. If that happens to be the case, why should we not do newtonian mechanics, this time correctly, taking into account all the effects of various fields?
General relativity is supposed to be a better description of natural phenomena than newtonian mechanics. That is the reason why people in the 20th century adopted the claim that matter changes the Minkowski metric and creates a curved spacetime. If newtonian mechanics, done correctly, produces the same phenomena, why should we accept the superfluous claim that the metric of spacetime has changed?
There are some weaknesses in general relativity. The most prominent weakness is the appearance of singularities. In newtonian mechanics we seem to be able to avoid the singularity at the center of a black hole.
There is still one good reason to claim that the geometry of spacetime can be changed from the Minkowski metric. The reason is the equivalence principle. If we claim that in a freely falling laboratory, the "real physics" is what would happen in an inertial laboratory far away in space, then we are tempted to claim that the geometry of spacetime itself has changed. The claim that the "real physics" in the freely falling case is the same as far away in space is strange, though. The laboratory is not floating freely. It is close to a huge mass. Moreover, there are tidal forces which are not present far away.
There is an equivalence principle in the old gravity model introduced by Isaac Newton in 1687. Physics in a freely falling laboratory looks the same as in a laboratory which floats freely in space. The simple explanation is that gravity attracts everything in the laboratory in the same way. There is no need to claim that the metric of spacetime has changed.
If we do newtonian mechanics more precisely, and observe that light moves slower close to a large mass, why should we then start claiming that the metric of spacetime has changed?
Is general relativity completely equivalent with special relativity & newtonian mechanics?
We probably can always claim that the curved geometry of spacetime in general relativity is just an illusion which is caused by the interplay between the fields and matter. It might be that it is only a philosophical question which interpretation we adopt.
A brave physicist could do an experiment and jump into a black hole. If he freezes for an eternity at the event horizon, then the newtonian model might be the right. If he continues unharmed through the horizon, then general relativity might be correct.
But we need to think carefully. Is there a more practical way to test the different interpretations?
No comments:
Post a Comment