This basic principle has as the consequence that superluminal communication in the Minkowski background metric cannot happen.
Gravitational waves in the Minkowski & newtonian model are analogous to electromagnetic waves. Gravitational waves impose forces on charges of gravity, that is, on photons and other elementary particles. Those forces imitate a change in the spatial metric, but the true metric does not change.
Let us study gravitational wave phenomena in more detail in our Minkowski & newtonian model.
Why clocks slows down in the Minkowski & newtonian model?
The slowing down of mechanical clocks close to a mass is due to two effects:
1. Any packet of energy, when lowered down in the gravitational potential, does work when it is lowered. There is less energy available in a low potential. Forces are weaker.
2. The inertia of any particle increases when lowered down in a gravitational potential. Besides the mass-energy of the particle, we also have to move some (negative) energy in the combined gravity field.
Time itself does not slow down close to a mass. It is clocks which slow down. Light slows down because a photon has to carry besides its own inertia, also some inertia of the gravity field. These effects create the illusion that time itself would have slowed down.
Why the radial Schwarzschild metric is stretched in the Minkowski & newtonian model?
The stretching of the radial metric in the Schwarzschild solution is in our model explained by the fact that the inertia in a radial movement is larger than in a horizontal movement. If we move a test mass deeper in the potential, it receives energy from distant parts of the gravity field. Moving this energy around causes extra inertia. Light propagates slower to the radial direction than to a horizontal direction.
We have to assume that the local geometry of all physical phenomena between elementary particles is controlled by the light speed. Then all things and phenomena are squeezed in the radial direction according to the ratio
radial light speed / horizontal light speed.
The fact that inertia is, say, 1% larger to the radial direction, cannot alone explain the squeezing by 1%. Consider a particle which is moving horizontally and collides elastically with a wall at a 45 degree angle. The particle after the collision moves radially, 1% slower. The absolute momentum |p| of the particle is preserved, but its kinetic energy is now lower:
E = p² / (2 m),
where m is the inertia of the particle. Where did some energy go? It had to go to a deformation of the force fields. The deformation energy is freed if the particle starts to move horizontally again.
Consider an almost light-speed particle which starts to interact with other particles. For example, we may have a photon propagating in air: it moves at almost the light speed. Then the photon enters a pane of glass. The interaction is much stronger inside glass. The photon gains more inertia and moves considerably slower. But again we have the problem: where did the extra kinetic energy go? Also in this case the extra energy had to go to a deformation of the system. The extra energy is given back to the photon when it exits the glass pane.
Any interaction increases the inertia of a light-speed particle and slows it down?
We can slow down light easily: let it go through a glass pane, or fly past Earth at a close distance.
We are not aware of any process which could speed up the travel of light.
Conjecture. If a light-speed particle interacts with some other particle or field, the particle always slows down. The speed of the particle is the largest in otherwise empty space.
Gravitational waves in general relativity seem to break the conjecture because they allow time to run faster in some areas.
The weak energy condition is a hypothesis which bans negative mass in general relativity. Negative mass would make time to run faster in its vicinity. The purpose of the weak energy condition is to prevent paradoxes which would come from too fast a speed of light.
What is the inertia of a test charge or a test mass inside an electromagnetic or gravitational wave?
In the Minkowski & newtonian model, a gravitational wave is similar to an electromagnetic wave. There is a force field which is analogous to the electric field, and another field which is analogous to the magnetic field.
The movement of test masses (= test charges) is due to the "electric field". If we could have dipole gravitational waves, they would make masses to oscillate up and down in the plane of polarization.
Quadrupole waves squeeze a ring of test particles. The squeezing has two components: + and ×. The component + squeezes horizontally, and the component × at a 45 degree angle to horizontal.
Let us analyze a dipole wave.
We have the field of a test mass, and the "electric" field of the wave. We want to find out what is the inertia of the test mass. This is just like finding the inertia of an electric test charge inside an electromagnetic dipole wave.
^ E electric field of the wave
|
| ●
q test charge
Suppose that the wave has an almost zero frequency. Then E is almost constant. How much does the ambient field E increase the inertia of the charge q?
If the charge q is positive, it makes the total field stronger than E above q and weaker than E below q. The field of q is
E' ~ 1 / r²,
where r is the distance. The increase in energy density is
~ E E'
and the volume element is
~ r² dr
This is strange. The inertia could increase without bounds inside a large plane wave. The energy of such a plane wave is infinite, which is not realistic, though.
The quadrupole wave from a black hole merger carries huge energy. If our test mass would move significant energy around the field of the wave, then the inertia of a small test mass could be astronomical? Let us calculate an example.
The mass-energy of a wave from a merger of black holes can be 10³⁰ kg. If our test mass is 1 kg, it might be able to increase the field by a factor 10⁻³⁰. The inertia of the test mass might double. Moreover, the wave does not need to be close to us. A distance of a few billion light years is no problem because the gravity field of our 1 kg test mass extends that far. Everything would happen much slower because of gravitational waves which exist in the visible universe. That does not seem plausible.
With an electric charge we face a similar problem in its coupling to all electromagnetic radiation in the universe. If moving the charge alters the energy distribution of every electromagnetic wave in the whole universe, the inertia of the charge might be huge.
Estimating the inertia of an electric charge inside an electromagnetic wave
We believe that the inertia of a test charge q increases close to macroscopic charge Q, because by moving q we can transfer energy from one place to another. Maybe inertia is not about the energy density of fields, but about the ability to transfer concrete packets of energy from one place to another?
For a static field of a charge Q, moving a test charge q for a distance s does allow us to transfer the associated potential energy. But for a light-speed wave we only have a certain time to store potential energy to q and harvest the potential energy from it.
^ ^ ^ ^
| | | | E
● q
| | s
| |
------------------
If the cycle time of the wave is t, then we, in principle, might be able to store and harvest energy during a trip whose vertical displacement in the diagram
s = 1/4 c t.
The stored energy:
W = q E * 1/4 c t.
That gives us an estimate of the increase of inertia of the charge:
W / c².
That is, only the electric field E relatively close to the test charge q can increase the inertia of q. Electromagnetic fields light-years away do not contribute.
Estimating the increase in the inertia of a test mass, caused by a gravitational wave
The first observation of a gravitational wave happened on September 14, 2015. The strain was 2 * 10⁻²¹, the frequency 35 ... 250 Hz, the distance 1.3 billion light-years, and the radiated energy 3 solar masses. The masses of the merging objects were 29 and 36 solar masses.
Let us calculate the effect on the inertia of a test mass. Both black holes had a radius of 100 km. The radius of the merging system was ~ 300 km at the final stage.
Close to the system, the extra inertia of a test mass comes from the energy deficit in the gravity field of the system, caused by the infinitesimal field of the test mass.
In the final round of the spiral, the black holes have relativistic speeds. We can guess (or calculate) that a signigicant part of their field g is "detached" and escapes as gravitational waves.
The energy density of the gravity field is
~ - (g + g')²,
where g' is the field of the test mass. The change in the total energy of the field due to g' is
~ - g g'.
The inertia of a test mass at a distance 500 km might have been elevated 10% because of its interaction with the detaching field g of the system.
The field in a gravitational wave decreases as
~ 1 / r
with the distance.
We assume that far away, the extra inertia of the test mass m still comes from its interaction with the gravitational wave field in a volume whose diameter is only
~ 500 km.
We can then apply the 1 / r rule to calculate the inertia at a great distance.
The increase in the inertia at the distance of 500 km = 5 * 10⁵ m was ~ 0.1. The distance 1.3 billion light years is 1.3 * 10²⁵ m. We get
~ 0.1 * 5 * 10⁵ / 1.3 * 10²⁵
= 4 * 10⁻²¹
as the increase of the inertia at the distance of Earth. The figure is of the same order of magnitude as the measured strain. Because of the extra inertia, a mechanical clock on Earth may tick 2 * 10⁻²¹ slower when inside the gravitational wave.
What causes the undulation of the spatial metric in a gravitational wave?
In the Schwarzschild metric, the slowing down of time has the same ratio as the stretching of the radial metric. Close to the source, the changes in the rate of a clock and measured distances have similar magnitudes. If this carries over to large distances, the strain and the slowing down of time should have similar magnitudes on Earth. Our very crude calculation agrees with this.
However, we are not sure if the apparent change in the spatial metric in a gravitational wave is due to the force of the "electric" field, or if it is caused by differences of inertia, like in the Schwarzschild metric. We need to investigate this.
It is difficult to visualize quadrupole waves. What does the electric field look like in them?
Question. If we have a very rigid rod which connects very large masses, then the strain of a gravitational wave can put very large energy to the deformation of the rod. The energy obviously cannot exceed the total energy of the wave. How do we model this? What is the maximum force that the wave can impose on the very large masses?
What is the rate of a clock inside a gravitational wave?
If the inertia of a photon increases 1%, it will fly 1% slower, and the speed of light is 1% slower.
If we have a clock which measures time by letting a photon to bounce between two walls, the rate of the clock depends on the inertia as well as the distance of the walls. The rate of the clock depends both on the inertia and the stretching of the spatial metric.
If we have a mechanical clock, how should we apply the rule 1 of the first section? Do forces grow weaker inside a gravitational wave?
In the Schwarzschild metric, the light clock and the mechanical clock tick at the same rate. However, it may be that the rates are different inside a gravitational wave. This might offer a possibility to test experimentally if general relativity is correct versus if the Minkowski & newtonian model is correct.
Another way to test is to try to detect if the speed of light is > c inside a gravitational wave. General relativity may predict superluminal speeds.
Conclusions
The Minkowski & newtonian model predicts gravitational waves where the metric of time and spatial distances undulate, and the amplitude of undulation has the same order of magnitude as in general relativity.
However, in the Minkowski model, the rate of clocks and the speed of light can only decrease when measured in the global Minkowski coordinates. A gravitational wave slows down clocks and the speed of light. In general relativity, these might grow above the values of the asymptotic Minkowski metric. This offers us an opportunity to show experimentally that general relativity is incorrect.
We do not understand well enough how electromagnetic or gravitational waves change the inertia of objects, or if they affect the strength of other forces. We do not know what causes the spatial metric to change. We need to do more research.
No comments:
Post a Comment