We define the mass equivalent m of an electric charge q with the following formula:
1 / (4 π ε₀) * q² = G m²,
where ε₀ is vacuum permittivity and G is the gravity constant. That is, the electric Coulomb force between the two charges q should be equal in absolute numbers to the gravity force between the two masses m.
1 coulomb is equivalent to 1.16 * 10¹⁰ kilograms.
Since gravity is attractive, it is not equivalent to the Coulomb force
The "equivalence" which we defined in above is dubious. If we move two opposite charges closer to each other, then their combined electric field grows weaker.
We can harvest the entire energy which is freed from the weakening electric field, where we define the field energy density with the familiar formula
1/2 ε₀ E²,
where E is the strength of the electric field.
However, if we move two masses closer to each other, their gravity field grows stronger. This is a big difference from the electric case. Should we define that the energy of the gravity field is negative, and it becomes even more negative when we move the masses close to each other?
Gravitational waves are kind of a gravity field "broken free" from their original source, a massive object. If the energy of the field is negative, how can the waves carry positive energy?
Robert C. Hilborn in his paper mentions that James Clerk Maxwell considered building an analogy between newtonian gravity and electromagnetics, but Maxwell thought that the apparent negative energy of the gravity field prevents us from forming of an analogy.
The famous 1865 paper by James Clerk Maxwell is freely readable on the internet. He writes that energy is "essentially positive". He suggests that the negative energy density of the gravity field should be added to some, ad hoc, constant positive energy density, to make the total energy density positive. However, he does not like the idea and does not explore gravity further.
The model of metal spheres on a rubber membrane explains things?
Let us have a tense rubber membrane stretched horizontally. If we put two heavy metal spheres on it, the spheres tend to roll together. Potential energy is freed when the spheres can sink deeper on the membrane.
There is an apparent "attractive force" between the spheres. We can harvest a part of the released potential energy by letting the spheres pull on a spring when the spheres roll together. But another part goes to the deformation of the rubber membrane.
If we identify the deformation of the rubber membrane with the gravity field of a sphere, or a set of spheres, then the gravity field has positive energy. Now we no longer have the problem of gravitational waves carrying negative energy.
Furthermore, the energy of the gravity field is not dictated by the attractive "force" between the spheres.
The energy of the gravity field is positive, and 16-fold compared to the naive analogue?
Robert C. Hilborn observed that the if the field in his electromagnetic model of gravity would be 4 times as strong, then the model would describe well gravitational waves. The field energy would be 16-fold.
In the previous section we argued that the gravity field can have arbitrary energy which is not determined by the strength of the newtonian attractive gravity force.
Conjecture. The energy of the gravity field is positive and 16-fold compared to the naive electromagnetic model.
In general relativity, there has been debate about what is meant by the energy of gravitational waves. The energy is measured with a pseudotensor, and the energy does not appear in the stress-energy tensor. Thus, is it "real" energy?
In the rubber membrane model, it certainly is real, positive energy.
Our Minkowski & newtonian model needs a major overhaul
Gravitational waves revealed a major flaw in our Minkowski & newtonian model. The structure of the gravity field is more complex than we thought.
It may be that one cannot build a model of an attractive force from the simple model of electromagnetism. Negative energy of the field cannot work.
How does the new, more complex, model affect the inertia of a test mass or a photon in the Schwarzschild solution? Does it affect the rate of clocks?
How do gravitational waves in the new model affect the apparent metric of time and space?
No comments:
Post a Comment