Frame dragging is a universal effect in interactions

In this blog we have studied the inertia of an electric test charge close to a large charge. We observed that the inertia is larger, because if we move the test charge, we have to move some energy in the electric field, too.

The same holds for a test mass close to a large mass. The effect is dramatic near the horizon of a black hole, and adds some 10% to the inertia of a test mass on the surface of a neutron star.

The speed of light is slower on a neutron star than in faraway space.

Slow speed of light is "frame-dragging"

The slow speed of light can be regarded as a kind of "frame-dragging".

The neutron star restricts the speed at which a test mass can move relative to it. Near the horizon of a black hole, the speed of light is very slow (relative to a faraway observer). The frame-dragging is extreme.

What is usually called frame-dragging is the effect that a rotating mass forces a test mass to orbit along with the surface of the mass.

The rotation effect is a special case of frame-dragging.

The syrup model of a neutron star or a black hole

Syrup is a substance where there are strong interactions between molecules. Frame-dragging is strong in syrup for small test objects, like an ant crawling in the syrup. This is the underlying reason why the syrup model is nice for neutron stars and black holes.

The speed of gravity is the speed of the photon?

In this blog we have our own Minkowski & newtonian model of gravity which claims that gravity is an ordinary force and does not affect the "true" metric of spacetime.

We have not been certain of what is the speed of gravity in the sense that how fast can a the static gravity field of a mass spread if we move the mass. Is the speed the light speed of the underlying Minkowski space, or is it the speed of the photon? Photons move slower in a low gravitational potential.

Let us analyze an electromagnetic analogue

A charge inside a polarizable ball

Let us put an electric charge at the center of a ball made of an electrically polarizable material.

                                                

                                                   field lines

                    _____                          \         /

                  /           \                   

                |      -        |              <-------   +

                  \______/

                                                       /         \

                charge in            opposite charge

       polarizable material

The polarization takes some of the internal central charge to the surface of the ball.

Let us move the external charge very quickly closer to the ball.

                    _____ 

                  /           \                   

                |       -       | -               +

                  \______/

                            induced

                             charge

As the external positive charge moves closer, it induces more polarization at the surface of the ball. It draws more negative charge at that edge of the ball which is closest to it. The induced polarization "shields" the charge at the center from seeing that the external charge has moved closer.

How quickly is the charge inside the ball aware that the external charge moved?

When electric field lines are suddenly bent, there is an associated magnetic field. The bent field line is like a half of a photon which propagates. What is the speed of the propagation?

In our example, bent field lines become denser and turn to be closer to the normal of the line between the external charge and the ball. They will induce more opposite charge to the edge of the ball. That may be enough to shield the central charge from seeing that the external charge moved closer.

Electromagnetism in a medium is determined by the permittivity ε and permeability μ of the material. It seems to be an empirical fact that field lines and their movement inside a medium obey these values.

Then it is not possible to change the field felt by the central charge faster than the speed of light in the polarizable material.

When the external charge moves close to the ball, it immediately starts pulling on the charge at the surface which the central charge induced.

But the central charge itself will not feel a stronger pull until the the field lines change, and changes in them only propagate at the speed of light in the material. We may assume that the speed of sound in the material does not exceed the speed of light. Then there is no mechanical pull either, until the field lines change.

              ------->

             |

             |            -   central charge

             |

      moving

      surface

As the surface of the ball moves, then the center of the ball would eventually bump into the surface. Then the center, at the latest, feels that something has changed and should start moving.

The speed of "local" gravity is the same as the speed of the photon

Our analogue suggests that in gravity, changes in the gravity field lines only can propagate at the speed of the photon.

If we move a mass quickly close to a neutron star, the surface of the neutron star will feel the pull earlier than the center.

A black hole is an extreme case. If we use the time of a faraway observer, does the horizon of the black hole ever feel the pull of the external mass?

But the black hole itself starts to move. Can its horizon stay at the "same" location indefinitely?

The mystery of the static location of the event horizon: frame dragging lets the horizon to move

We have our own Minkowski & newtonian model of gravity. It is counter-intuitive if the horizon stays static while the black hole seems to move. How can we solve this paradox?

The solution might be the following: the slow speed of light is caused by the collective action of all the mass-energy in the black hole. The slow speed is not relative to a fixed Minkowski coordinate system. When the external mass pulls on the outer parts of the black hole gravity field, the whole system starts to move. A kind of "frame-dragging" moves the center of the system, even though the center does not yet know that anything has changed.

The slow speed of light close to the horizon is a result of a photon "borrowing" inertia from the whole system. As most of the system starts to move, then the photon - and any field lines - must move along the whole system.

The maximum speed of a low-energy signal. Frame dragging seems to be a bad way to send data to an observer. An observer does not notice anything if his whole frame is dragged.

Local changes in the gravity field can carry information and their speed is restricted by the speed of the photon. We conclude that the speed of the photon is the maximum speed of a weak signal, a signal whose mass-energy is much smaller than the gravitating system.

On the other hand, if we use a mass similar to the gravitating system as a "signal", and let it collide with the system, the signal will drag its own frame, and may move faster than a weak signal.

Mach's principle

https://en.wikipedia.org/wiki/Mach%27s_principle

Albert Einstein wished to show that general relativity satisfies Mach's principle: inertial frames are determined by the collective movement of faraway large masses.

The frame dragging, which we described, is reminiscent of Mach's principle. Close to the horizon of a black hole, the inertial frame is almost entirely determined by the movement of the black hole.

Question. Minkowski space is empty of matter. Could we somehow simulate Minkowski space by putting a massive shell of mass far away from us?

In this blog we have argued that nearby masses increase the inertia in a linear motion of a test mass, while a radial motion of a test mass shell has no extra inertia. Is it really true, in the light of the analysis above?

In our electromagnetic analogue, the slow speed of light resists moving an electric test charge linearly. If we have a test shell of charge expanding, the electric field lines do not change outside of the sphere. The inertia should be less?

Empirically, the inertia of a test mass in a linear motion is the same as in a radial motion. That suggests that there cannot exist any large faraway large masses which would dictate the inertial frames in our universe. The answer to the Question above is negative.

Conclusions

"Local" propagation of changes in the field lines of the gravitational field happens at the speed of the photon. That is what our reasoning above strongly suggests.

But the speed caused by "frame-dragging" may exceed the local speed of the photon. If we start to move the outer layers of a massive spherical system, then the center will follow in the movement. This is because the slow local speed of the photon is a result of the collective action of the masses in the system. If we move the masses, the photon moves with them. The local speed of light does not restrict the movement.

The universal speed limit is the speed of light in the underlying Minkowski space.

We did not find a semiclassical model for the electron - a Feynman path integral is the way?

For the past six months we worked very hard to construct a semiclassical model for the electron. The goal was to build an intuitive model which would explain the magnetic moment and the gyromagnetic ratio of the electron.

We failed.

The magnetic moment and the spin of the electron

https://en.wikipedia.org/wiki/Dirac_equation

The magnetic moment follows quite simply from the nonrelativistic approximation of the Dirac equation. The ultimate reason for it is that the momentum operators

       p  =  i d / dx

act on the vector potential of the magnetic field, too, when we construct a solution for the Dirac equation in the nonrelativistic case under a magnetic field B.

If we take the axioms:

1. the Dirac equation describes the electron wave, and

2. the "correct" way to add a magnetic field B is to use the minimal coupling to the vector potential,

then we get the correct magnetic moment. Why is the minimal coupling the way to add a magnetic field? We do not know.

The spin of the electron follows from the Dirac equation if one guesses that the sum of the angular momentum J and the spin S should be conserved.

Using a Feynman path integral as a "particle model" of the electron

In our blog we hold the view that empty space is strictly empty of fields, except of the Higgs field. This solves the infinite energy problem of empty space (except for the Higgs field).

We would like the electron to be a particle. Then empty space is an intuitive concept: it contains no particles.

In a Feynman path integral, the electron is, in a sense, a particle. A single path can be viewed as a path of a particle.

In a path integral it is important that alternative paths of the particle must not interact. The only "interaction" allowed is the linear superposition of the end results. The probability amplitude of a final result is obtained by summing the amplitudes for each path. The weight of each path is a fuzzy concept, though.

The propagator, or the lagrangian action over a path is the probability amplitude of that path.

Conjecture. A wave equation must be linear for the Feynman path integral approach to work. We assume that the action of a path is calculated using the propagator of the wave equation.

Conclusions

We wanted to construct a semiclassical particle model for the electron, but failed.

We should determine if we can build a satisfactory particle model for the electron using the Dirac equation and a Feynman path integral.

For example, in high-energy collision experiments the electron behaves quite like a classical point particle with an electric charge. Can we explain this in an intuitive way using a Feynman path integral?