UPDATE May 21, 2024: The Einstein-Hilbert action seems to work correctly if we define the matter lagrangian as "kinetic energy - potential energy" and require that a variation of the metric only alters a finite volume of space. We are investigating what happens in the case of a changing pressure in a spherical mass M.
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We have to get a lot of energy into gravitational waves. To that end, we introduce a private field for each particle, such that the field is "attached" to the particle which "owns" it. The field acts in a different way on other particles than the one which owns the field.
|
| particle
------ ● --------------- "steel wire"
|
|
The private field can be interpreted as a physical body which is attached to the particle – like steel wires attached to the particle. The wires interact with other particles, but they have a special particle which they are attached to.
In a rubber sheet model of gravity, the depression in the rubber sheet is not specifically assigned to any particle. It is a "public" field.
For a public field, the energy of a wave in it is controlled by the energy that a static field has around a mass M. We cannot arbitrarily increase the energy of a wave. But a gravitational wave contains 16 times the energy density of an analogous electromagnetic wave.
The wire model differs from a traditional field theory. In this blog we have been suggesting that gravity cannot be accurately modeled as a field, or a metric, because the interaction is too complicated.
Self-force of an electron: the rubber plate model
In this blog we have tried to find a model for the self-force of an electron when the electron is moved back and forth. The elecric field of the electron itself exerts a force on the electron.
We had the "rubber plate" model where we imagine that the electric field of the electron is a rubber disk attached to it. Electromagnetic waves are mechanical waves in the rubber disk.
See, for example, our post on December 20, 2020.
The "steel wire" model above is similar to the rubber plate model. We do not know if it is possible to write a traditional lagrangian for the rubber plate model.
Particles embedded into a block of rubber
If we fuse together the "rubber disk" of each particle, then we have a model where particles are embedded into a solid block of rubber.
But there is a problem with this model: particles moving at a constant speed must be allowed to move undisturbed. Only accelerating particles interact with the rubber, sending waves into it.
Is there any way to implement this with a public field?
Teleportation through the Huygens principle
If the particles are otherwise static, but oscillate back and forth, then the rubber block model of the preceding section works better.
| |
---- • ---- • ----
| |
---- • ---- • ----
| |
particles with
interactions
Rather than rubber, we can imagine a matrix of particles coupled to their neighbors through a gravity-like interaction.
The Huygens principle says that each point in space acts as a "source" for a new wave. This is a simple way to explain how a complex interaction can be teleported from a mass M to another mass light-years away.
We want to teleport the stretching of the spatial metric.
Can we make a public field from the "inertial frame"? A strange rubber block
<--->
● /\/\/\/\/\/\ ●
M M
oscillating quadrupole
Let us look again at the oscillating quadrupole. The acceleration of the masses M interacts with what is considered the inertial frame for a test mass m.
This is in the spirit of general relativity. We must couple an acceleration of a mass to a field.
It is like particles moving in a strange compressible liquid. A particle P can move without friction at a constant velocity. We may imagine that there is a laminar flow of the liquid past the particle P.
But if P is accelerated relative to the liquid, then P interacts with the liquid. The liquid is compressed, and sound waves are sent through it.
Since gravitational waves are transverse, we cannot really use a liquid. The particles have to be immersed into a strange rubber substance which allows the particles to move at a constant velocity without friction. Only when a particle is accelerated, does it disturb the rubber.
The tense water hose model
<---------- tension ------------>
------------------------------------------ tense hose
^ acceleration
| • P'
• P water
------------------------------------------
<~~~ waves ~~~>
Above we have a real-world example of a system where the acceleration of a particle disturbs the inertial frame far away. A particle is accelerated vertically inside a tense water hose. Waves propagate along the tense hose to both directions. A wave changes what is the "inertial frame" of a particle P' far away. The particle P' starts to oscillate up and down.
If P moves at a constant speed, then it does not disturb the hose.
Frame-dragging
m
•
● ---> a
M
We have written in this blog about the problem of how general relativity treats accelerating masses. In the diagram, if we accelerate the large mass M, then it "drags the inertial frame" and the test mass m, too, starts to accelerate.
If we try to describe the process with a metric around M, it is not clear how the metric behaves if M is accelerated.
Quantum field theory approach: treat the fields in a gravitational wave differently from static fields
In quantum field theory, we imagine that a field of sine waves can be "excited" by interactions with particles. An excitation can, for example, be a photon. It is born and absorbed through an interaction with a charge, e.g., an electron.
The field of excitations, or sine waves, is separate from static fields, which are carried by virtual particles.
Can we in this model explain why the energy density of a gravitational wave is 16 times the density of the analogous electromagnetic wave?
If we make a mass to M oscillate, it creates 16 gravitons while the analogous charge Q only creates one photon?
The 16-fold energy must, in the end, be due to a coupling which is 16 times as strong as in the electromagnetic analogue. A gravitational wave does couple with a mass in many ways.
Can we write a lagrangian where the metric of time, g₀₀, imitates Coulomb's potential, but the energy of the wave is set 16X the energy of an analogous electromagnetic wave?
Also, the effects of a gravitational wave are diverse: they replicate many of the effects of the static gravity field of an oscillating mass M. The coupling of a test mass m to the wave is complicated.
Then we would have a lagrangian which is not very precisely defined, but would work at least in some cases.
The static gravity field of a mass M has to be written separately from the field of a gravitational wave. The energy density of a static field is like in the analogous Coulomb field.
The coupling of a static gravity field to a test mass m is complicated, too, like the coupling of a gravitational wave to a mass.
Can we decompose a field into a static field and a wave propagating at the speed of light?
Can we really differentiate between waves and static fields? Does the Fourier decomposition of a field uniquely differentiate between static fields and waves propagating at the speed of light?
We can certainly simulate waves propagating at a speed less than light by letting a chain of masses M fly at a large speed past the observer.
A wave whose wavelength is very large locally looks like a static field.
This is a general question about waves. If we have a tense drum skin, can we uniquely divide its disturbances into waves, and into static depressions caused by weights lying on the skin?
Let us keep hitting the drum skin with a sharp hammer at the same point, at a large frequency. We obtain a static depression in the skin. An individual hit does produce also sine waves which would carry energy away. But there is a total destructive interference of such waves if we hit the skin at an infinite frequency, i.e., have a weight lying on the skin.
Let us guess that one can separate static fields from sine waves.
Changes in the spatial metric and a lagrangian
A complete lagrangian should, in principle, describe all the effects of a gravity field. But how do we make the lagrangian to shrink a measuring rod if it is turned radially pointing to a large mass M? The Einstein-Hilbert action is supposed to do that feat by redefining the metric at the rod.
Another option is to use the newtonian gravity field to determine the spatial metric, and mention in the matter lagrangian LM that LM has to be calculated in that metric.
A traditional lagrangian describes the movements of particles. It does not describe how a measuring rod shrinks. The shrinking is due to the force fields behaving in a different way. Thus, a lagrangian which refers to the "metric of space", describes how force fields "move" or change.
ADM formalism
The ADM formalism is a hamiltonian approach to a system sitting in an asymptotically Minkowski space. We have to check if ADM contains the same errors as the Einstein-Hilbert action.
If ADM works, then the Einstein equations should have a solution for a cylinder with shear? They should find the lowest "energy" state. We have to check if the ADM formalism is correct.
The generalized stress-energy tensor
The stress-energy tensor in general relativity measures how a specific field, a perturbation of the flat metric, changes the value of a matter lagrangian LM.
Analogously, we can define a generalized stress-energy tensor, which measures how an arbitrary field K changes the value of a lagrangian LM. In this way we can include complex interactions of the field K with matter.
For example, K could be a wave in a drum skin, where a complex mechanical system of levers and springs is embedded into the skin. Or, K can be a gravitational wave, which meets masses and rigid bodies.
Making the energy of an electromagnetic wave 16-fold
The electromagnetic tensor in the (- + + +) metric signature is:
The lagrangian is usually written:
It has a more intuitive vector form:
Let us analyze it. We set the polarization P and the magnetization M to zero. The energy density of the electric field E is the same in a static electric field and in an electromagnetic wave. The interaction with matter is assumed to be extremely simple: the inner product of the four-potential A and the the four-current field J.
Now, let us separate the field belonging to waves from static fields. We obtain a lagrangian density:
L = 8 (ε₀ E²wave - 1 / μ₀ * B²wave)
+ 1/2 (ε₀ E²static - 1 / μ₀ * B²static)
- φ ρfree
+ A • Jfree.
The energy of an electromagnetic wave is 16X the normal energy. Could this work?
The lagrangian density for gravity
The lagrangian density would be completely analogous to the electromagnetic lagrangian with a 16-fold energy density of waves.
L = 8 (E²gwave - B²gwave)
+ 1/2 (E²gstatic - B²gstatic)
+ LM.
We have set the appropriate natural constants to 1.
Eg
is the gravitoelectric field and Bg is the gravitomagnetic field, completely analogous with the electric counterparts.
The matter lagrangian LM depends in a complex way on both the static gravity field and the field of a gravitational wave. The "metric" comes to play when we calculate the value of LM.
If all the energies in the lagrangian above are counted like an observer far away sees them, then the action integral of the lagrangian L above is the simple integral over the cartesian coordinate space and time.
Can we do like in the Einstein-Hilbert action, and use "local" values for energies, and integrate over the volume element which is "discounted" by the redshift:
sqrt(-g₀₀) * dx⁴?
A problem in this is that the binding energy of a mass M is not calculated right if we discount with the full redshift.
For purely static fields with no pressure or shear, our lagrangian works like the lagrangian of newtonian gravity. It will produce reasonable results, while the Einstein-Hilbert action calculated everything wrong inside a mass M.
The reaction of the new gravity lagrangian to pressure or an oscillating mass M
How does our lagrangian react to a positive pressure? If we have a pressurized vessel, then our new lagrangian can free potential energy from the pressure by making the newtonian gravity field stronger, which, in turn, stretches the radial metric?
The stretched radial metric is an indirect consequence of a newtonian gravity field. Is it plausible that a positive pressure can create a newtonian gravity field on its own?
In a rubber sheet model of gravity, pressure stretches the rubber sheet, and makes the sheet to sag lower. Pressure does create a certain gravity field.
In general relativity, pressure is unable to stretch the radial metric.
Our new lagrangian optimizes the gravity field around a mass M in such a way that the potential of M is lowered and we pay the price of increasing the energy of the gravity field. Now, if there is a positive pressure around M, it is logical that the gravity field can become stronger. Thus, pressure does create gravity, and is able to stretch the radial metric.
An alternative model of gravity would claim that the gravity field of M is "inherent" to M, and nothing else can create such a field. That model does not allow a lagrangian, because in a lagrangian model, anything can be adjusted. The mass M cannot dictate what happens in the gravity field around it.
Is there a similar problem in electromagnetism? Let us have an oscillating charge Q. If the electric field of Q becomes weaker, then the system can save in the energy of the radiated waves. Could this weaken the apparent charge of Q?
In a rubber sheet model of gravity, a vigorous oscillation of a weight M raises M higher: it weakens the gravity field of M.
Conclusions
We now have a tentative lagrangian for gravity, such that:
1. it gets right the newtonian gravity field around static or slowly moving masses, and
2. it correctly predicts that the energy density of a gravitational wave is 16 times the analogous electromagnetic wave.
Our lagrangian is somewhat fuzzy. How exactly do we define the effect of a gravity field on the matter lagrangian LM?
The lagrangian is our first attempt as the lagrangian for the Minkowski & newtonian gravity model.
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