In the 18th century newtonian mechanics, if the force between m and M is canceled, then gravity has no effect whatsoever.
General relativity claims that the large mass M determines the "geometry of spacetime", and that electric repulsion cannot cancel its effects.
For example, the speed of light close to M is slower, as seen by a faraway observer. The mass m cannot move faster than a photon, claims general relativity.
Shipping of energy between different force fields
flow of energy
M -------------------------------> m
s
● <--- •
Q <------------------------------- q
flow of energy
We have two test bodies: a large mass/charge M/Q, and a test mass/charge m/q.
In the diagram, there is attraction of gravity between M and m, and repulsion of the Coulomb force between Q and q. The forces cancel each other.
If we move m a short distance s closer then energy seems to flow to both directions. Is the total energy shipping distance zero? That is, does the energy from the large body ● take a shortcut and go directly back to ●, without passing through the test body • ?
• e- electron
● proton • q test charge
In the case of positive and negative electric charges, the answer is clear: energy flows do take any shortcuts available.
Earth contains some 2 * 10⁵¹ protons whose combined charge is Q = 3 * 10³² coulombs. The combined Coulomb force on a single electron on the surface of Earth is
F = k Q e / r²
= 1.4 * 10¹⁰ N.
If we move the electron one meter closer to Earth, we ship
1.4 * 10¹⁰ J
of energy over 6,000 kilometers. That is equivalent to shipping 10¹⁷ J, or one kilogram of matter, over 1 meter! The inertial mass of the electron would be two kilograms, instead of 10⁻³⁰ kg.
The 10¹⁷ joules of energy cannot travel through our test electron. The energy must go directly from the fields of protons to the fields of electrons inside Earth.
Hypothesis. Shipping of energy always takes the shortest route between fields, even if the fields are different force fields, like gravity and the Coulomb field. Also, if an object appears electrically neutral (because of quantum mechanics, for example) then the shipping distance of electric energy is zero. A neutral object does not contribute to the inertial mass of a test charge.
Our hypothesis does not differentiate between different force fields. If we have a hamiltonian where different force fields have complex interactions, we cannot in a reasonable way tell what part of an effect is due to which field. Our hypothesis is compatible with complex hamiltonians.
Shipping of energy: the concrete view
We might be able to define the energy shipping distance in a very concrete way. It simply is distance over which we ship recoverable energy.
W energy W
pulley pulley
O O
| |
| |
| |
¿ ¿
• e- ----------------->
● Q positive charge
A positive charge pulls on an electron. We lower the electron with a pulley and harvest the energy W. Then we move the electron horizontally and lift it with another pulley. The second pulley uses its quota of energy, W.
The process shipped the electron to the right and the energy W to the left. The electron possessed this energy W as extra inertia when we were moving the electron to the right.
Now imagine that we cancel the charge Q by putting an opposite charge -Q close to it. We can no longer ship the energy W between the pulleys. There is no extra inertia on the electron.
If we cancel the pull of gravity with electric repulsion, we cannot concretely move energy, like in the pulley example. This supports our Hypothesis that energy does not differentiate between force fields.
The gravity of galaxies: how much energy we ship when we move a test mass? Mach's principle
Could it be that the entire inertia of a kilogram of mass comes from its attraction with masses far away in the universe?
Let us do a quick calculation. The mass density of the close visible universe is assumed to be
D = 10⁻²⁶ kg/m³.
The "current" radius of the visible universe is 60 billion light-years, or
R = 6 * 10²⁶ m.
The total mass is
M = 2 * 10⁵⁴ kg.
Let us calculate the inertia of a test mass m in a horizontal movement at a distance R from a mass M.
The Schwarzschild radius of the mass M is
r_s = 2 G M / c²
= 2.6 * 10²⁷ m
= 4.3 R.
At the distance of the Schwarzschild radius, the potential of gravity is -m c². The extra inertia from the force field is then m.
Since R < 4.3 R, our mass m has 4.3 m as the extra inertia?
The whole universe seems to be inside the Schwarzschild radius of its own mass, but quite narrowly.
It looks like that the entire inertial mass of our test mass m might come from its gravity with the rest of the universe. This would satisfy Mach's principle.
Also, the inertial coordinate system would be determined by other masses in the universe. The other masses would do frame dragging which totally determines what coordinates are inertial. This is Mach's principle.
Close to the horizon of a black hole Mach's principle holds in the sense that the black hole almost totally determines what is the inertial coordinate system. An observer is like in "syrup" and follows movements of the black hole, even if we accelerate the black hole with some mechanism.
Let us look at the Big Bang. Suppose that the gravity charge of elementary particles stays constant. In the early universe, the inertial mass of a particle could have been much larger than now. What would that imply?
If inertia is born from gravity, that would explain why the inertial mass and the gravity charge have a constant ratio for all known objects.
There may be a vicious circle in our definition of inertia. If we move a test mass m, then energy is shipped in the global gravity field. But the shipped energy itself holds mass-energy, and causes more energy to be shipped in the global gravity field. How do we stop the vicious circle?
The flatness problem of cosmology
Our Hypothesis solves the flatness problem of cosmology.
Let us assume that the energy of mass is
E = m c²,
like it is in special relativity. In special relativity m is the inertial mass, since there is no gravity in special relativity.
According to our Hypothesis, the inertial mass of a body is its negative potential in the gravity field of the universe.
However, we have to differentiate between the negative potential energy of a test mass, and the average binding energy of a test mass. The negative potential usually has a larger absolute value.
The extra inertia for a test mass inside a neutron star is the negative potential. It is not the average binding energy of the neutron star. The potential usually is double the binding energy.
Suppose that the universe is empty, except for the neutron star. The energy in the inertial mass inside the star will appear to be double the binding energy. That is close to what we are currently observing in the universe.
Does this bring problems in the early universe? Or do laws of nature then, at the small scale, appear to be identical to the ones now?
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