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Gravitation fundamental fields


computer
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All the most known theories of gravity are built on the principle of long-range action. When approximately "point" body or mass density (also electromagnetic energy) distributed in space creates gravitational potential. This article attempts to substantiate gravitational phenomena on the principle of proximity (locality). The cause of interactions is some spatial state of fundamental fields, the consequence is change of these fields over time (first derivative in time at local point of continuum).

Presumably, following fundamental gravitational fields exist:
(SI units in parentheses are m-metre, s-second, k-kilogram, A-Ampere)
scalar potential g (m2/s2)
vector potential G (m/s)
scalar strain f (m2/s3)
vector strain F (m/s2)

The gravitational constant g0 = 6.6742^-11 (m3/s2/k) is also used,
local energy density u (k/m/s2), for example electromagnetic = ε0/2 • E2 + μ0/2 • H2
and Poynting vector S (k/s3) = [E × H]

Time derivatives are expressed as follows:
g' = - f - c2 • div G
G' = - F - grad g
f' = - c2 • div grad g + fu • u
F' = c2 • rot rot G - fs • S

The constants fu (m3/s2/k) and fs (m/k) are positive, signs are selected so that scalar potential g becomes negative in presence of positive density u in vicinity of point.

The equations are similar to electromagnetic equations expressed in potentials:
a' = - c2 • div A
A' = - E - grad a
E' = c2 • rot rot A

In stationary state, for example, during formation of gravitational fields by stable elementary particle or single celestial body:
S = 0, G = 0, f = 0
F = - grad g
div grad g = fu • u / c2 = 4 • π • g0 • ρ, according to Newton's potential

Hence we get at ρ = u / c2: fu = 4 • π • g0

The effect of gravitational fields on other fundamental ones can manifest itself as a curvature of space, and direct effect on velocity vector V, mentioned in this topic:

With zero u and S, following types of "pure" gravitational waves can exist:
1. Longitudinal potential-potential: g' = - c2 • div G, G' = - grad g
2. Longitudinal with phase shift of 90 degrees: g' = - f, f' = - c2 • div grad g
3. Transverse: g' = - c2 • div G, G' = - F - grad g, F' = c2 • rot rot G
Transverse ones are probably easier to detect in experiments.

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22 minutes ago, computer said:

All the most known theories of gravity are built on the principle of long-range action.

That’s not true. General Relativity - which is the best model of gravity we currently have - is a purely local constraint on the metric of spacetime. The influence of distant sources enters only via boundary conditions.

26 minutes ago, computer said:

Presumably, following fundamental gravitational fields exist:

In order to capture all real-world degrees of freedom of gravity, you need at least a rank-2 tensor field. Scalar and vector fields aren’t enough.

30 minutes ago, computer said:

a' = - c2 • div A
A' = - E - grad a
E' = c2 • rot rot A

The div, grad and curl operators are only defined in three dimensions, but our universe is manifestly 4-dimensional.

These equations are also not covariant, so you need to specify what frame you are working in.

It is possible to formulate gravity in the way you suggest (this is called gravitoelectromagnetism), but this only works as an approximation in the weak field limit. A full description of gravity requires GR.

 

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5 hours ago, Markus Hanke said:

The div, grad and curl operators are only defined in three dimensions, but our universe is manifestly 4-dimensional.

It is possible to formulate gravity in the way you suggest (this is called gravitoelectromagnetism), but this only works as an approximation in the weak field limit. A full description of gravity requires GR.

Forth dimension is the time. Such approach is convenient to write some laws briefly, including electromagnetic, but curls anyway are existing and used. Gradient and divergency defined in a world with any dimensions amount (except zero-dimensional).

Both gravitoelectromagnetism and relativity (modified Poisson's equation) use mass or energy density in long-action manner. It is always laplacian or divergency of some operator, not cause of first derivative change.

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On 9/30/2022 at 10:26 AM, Markus Hanke said:

That’s not true. General Relativity - which is the best model of gravity we currently have - is a purely local constraint on the metric of spacetime. The influence of distant sources enters only via boundary conditions.

Indeed.

On 9/30/2022 at 10:26 AM, Markus Hanke said:
On 9/30/2022 at 9:48 AM, computer said:

 

In order to capture all real-world degrees of freedom of gravity, you need at least a rank-2 tensor field. Scalar and vector fields aren’t enough.

Indeed. You can build a workaround by using the formalism of differential forms and introducing the diff and the co-diff operators (the analogues of grad and rot.)

The co-diff operator is a contraction of the epsilon tensor and metric coefficients with the diff operator.

You're venturing into territory well-charted by others and without the proper tools.

See: 

Differential Geometry, Gauge Theories, and Gravity

By M. Göckeler, T. Schücker

Cambridge University Press 

p. 40

https://books.google.es/books?id=Lr8zo2caBWYC&pg=PA40&#v=onepage&q&f=false

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13 minutes ago, joigus said:

Indeed.

Indeed. You can build a workaround by using the formalism of differential forms and introducing the diff and the co-diff operators (the analogues of grad and rot.)

The co-diff operator is a contraction of the epsilon tensor and metric coefficients with the diff operator.

You're venturing into territory well-charted by others and without the proper tools.

See: 

Differential Geometry, Gauge Theories, and Gravity

By M. Göckeler, T. Schücker

Cambridge University Press 

p. 40

https://books.google.es/books?id=Lr8zo2caBWYC&pg=PA40&#v=onepage&q&f=false

+1

 

See also

the Wiki articles on four-vectors

https://en.wikipedia.org/wiki/Four-gradient

 

https://en.wikipedia.org/wiki/Four-vector

 

http://hyperphysics.phy-astr.gsu.edu/hbase/Relativ/vec4.html

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I had not "abandoned" Hypothesis about the formation of particles from fields, but at first it was about velocity fundamental field, then somehow switched to the equations of quantum physics, and I was afraid the administration of the forum would be dissatisfied.

Four-vectors have too unclear notation. I try to write equations in the most direct and comprehensible way.

But the biggest problem is other: long-action. Gravitoelectromagnetism (in the Wikipedia article at least) is described as follows:

div E = - 4 • π • g0 • ρ

div B = 0

E' = c2 • rot B - 4 • π • g0 • J

B' = rot E

First two equations represent the pure long-action. Arbitrary assumption, that some "charge" or even density generates field over the whole Universe. In electromagnetism charge density is really proportional to div E and changes with E'. But in gravitation theories mass or enerdy density is "external" value to these equations.

Edited by computer
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I read your hypothesis about the formation of particles from fields post. I wondered if you were going to treat gravity and time. I think I have seen a formalization using differential geometry to develop what I think is a type of gravitoelectromagnetism.

In fact, I think credit is due to @eytan_il for this, and I wonder if this can get his attention. He seems to be well versed in differential geometry as an alternate way to build up gravity. 

Electro-gravity via geometric chrononfield by Eytan H. Suchard

Quote

Abstract.
In De Sitter / Anti De Sitter space-time and in other geometries, reference submanifolds from which proper time is measured along integral curves, are described as events. We introduce here a foliation with the help of a scalar field. The scalar field need not be unique but from the gradient of the scalar field, an intrinsic Reeb vector of the foliations perpendicular to the gradient vector is calculated. The Reeb vector describes the acceleration of a physical particle that moves along the integral curves that are formed by the gradient of the scalar field. The Reeb vector appears as a component of an anti-symmetric matrix which is a part of a rank2, 2-Form. The 2-form is extended into a non-degenerate 4-form and into rank-4 matrix of a 2- form, which when multiplied by a velocity of a particle, becomes the acceleration of the particle. The matrix has one U(1) degree of freedom and an additional SU(2) degrees of freedom in two vectors that span the plane perpendicular to the gradient of the scalar field and to the Reeb vector. In total, there are U(1) x SU(2) degrees of freedom. SU(3) degrees of freedom arise from three dimensional foliations but require an additional symmetry to exist in order to have a valid covariant meaning.
Matter in the Einstein Grossmann equation is replaced by the action of the acceleration field, i.e. by a geometric action which is not anticipated by the metric alone. This idea leads to a new formalism that replaces the conventional stress-energy-momentum-tensor. The formalism will be mainly developed for classical physics but will also be discussed for quantized physics based on events instead of particles. The result is that a positive charge manifests small attracting gravity and a stronger but small repelling acceleration field that repels even uncharged particles that have a rest mass. Negative charge manifests a repelling anti-gravity but also a stronger acceleration field that attracts even uncharged particles that have rest mass.

[...]

Physical meaning: svg.image?A_{ij} transforms the vector svg.image?\frac{P_{i}}{\sqrt{Z}} to svg.image?\frac{U_{i}}{2} as a rotation and scaling transformation and is therefore, of rank 2. It can be extended to a non-degenerate matrix of rank 4, svg.image?\widetilde{A}_{ij} which defines a field of acceleration, i.e. svg.image?\widetilde{A}_{ij}\frac{V_{j}}{c}=\frac{a_{i}}{c^{2}}=g_{ij}\frac{1}{c^{2}}\frac{dV^{i}}{d\tau} where svg.image?{a_{i}} is the covariant acceleration of the mass that interacts with the field, svg.image?c is the speed of light, svg.image?\tau is proper time and svg.image?g_{ij} is the metric tensor.

[...]

If you go to pg. 32 of the .pdf, you'll find speculations on elementary particles being composed of opposite charges, which is what I noticed in your other post, although you were positing the idea for leptons, and then photons...

Quote

[...]
The problem is that there is no stable charged particle without spin and therefore our discussion could mean a temporary decomposition of electrically neutral Bosons into two energy states, one temporarily behaving like a negative charge and one like a positive one.
[...]

 

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16 hours ago, NTuft said:

what I think is a type of gravitoelectromagnetism

Classical gravitoelectromagnetism is based on long-action. I try to avoid this and use only short-action differential equations. Without "sources" of potentials and other fields, like charges or masses in their mechanical representation as "points", or some density, also referring to "cloud" of points.

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