Modification of Nordstroem's theory of gravity as a unified field theory candidate

[JMessenger]

Back again after discovering the wonderful text The Genesis of General Relativity.

 

Nordstroem's theory of gravity was the last scalar field theory of gravity that was a challenge to GR. Once the bending of light was verified it was mainly forgotten except as a teaching tool for General Relativity. What is fascinating about it is that the scalar fields have a tensor equivalent of [math]R=\kappa T[/math].

 

According to http://www.pitt.edu/~jdnorton/papers/einstein-nordstroem-HGR3.pdf

the regular Newtonian scalar function [math]\Phi[/math] was used originally within a Laplacian and then with the D'Alembertian as it was a simple logical choice (Einstein had also attempted the first Nordstroem theory previously).

 

What neither of them may have considered is determining whether or not [math]\Phi[/math] is an integrable function (and can provide an integral for gravitational energy). Assuming that it is, for a static three dimensional "field", even if the integral is "flipped" it does not change any subsequent derivatives. Meaning [math]\frac{\partial\Phi}{\partial x_u}=\frac{\partial(C-\Phi_{alt})}{\partial x_u}[/math]. Note that unimodular gravity considers the cosmological constant also as a constant of integration, but when Einstein introduced a physical use for the cosmological constant, he did so using the Poisson equation but didn't include it within the differential operator (2nd equation of Cosmological Considerations on the General Theory of Relativity).

 

In the tensor equivalence though for the alternate scalar field, it would seem that we should arrive at something like [math]\eta R g_{\mu\nu}-R_{\mu\nu}=\Gamma g_{\mu\nu}-T^{alt}_{\mu\nu}[/math] (instead of [math]R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=G_{\mu\nu}-\Lambda g_{\mu\nu}=T_{\mu\nu}-\Lambda g_{\mu\nu}[/math] which doesn't meld with QFT). I am attempting to see if using this field equation I can take the difference of [math]\Gamma g_{\mu\nu}-\kappa T_{\mu\nu}[/math] as a change in the vacuum expectation value of QFT where this is equivalent to baryonic energy density. In simple terms I am using a region of baryonic energy density as a decrease in vacuum energy density, or equivalently inverting the energy scale of baryonic energy relative to the classical zero energy of the vacuum. This would also seem to be able to mesh with Noether's theroems using the flux of a decrease in density rather than the flux of a positive or negative density.

 

Anybody want to help derive the field equations? Probably not going to be easy.

Edited February 27, 2013 by JMessenger

[elfmotat]

What neither of them may have considered is determining whether or not [math]\Phi[/math] is an integrable function (and can provide an integral for gravitational energy). Assuming that it is, for a static three dimensional "field", even if the integral is "flipped" it does not change any subsequent derivatives. Meaning [math]\frac{\partial\Phi}{\partial x_u}=\frac{\partial(C-\Phi_{alt})}{\partial x_u}[/math].

 

But those aren't equal. They're opposite in sign! Unless you've defined [math]\Phi_{alt}=-\Phi [/math]. Regardless, it seems pointless.

 

Note that unimodular gravity considers the cosmological constant also as a constant of integration, but when Einstein introduced a physical use for the cosmological constant, he did so using the Poisson equation but didn't include it within the differential operator (2nd equation of Cosmological Considerations on the General Theory of Relativity).

 

In the tensor equivalence though for the alternate scalar field, it would seem that we should arrive at something like [math]\eta R g_{\mu\nu}-R_{\mu\nu}=\Gamma g_{\mu\nu}-T^{alt}_{\mu\nu}[/math] (instead of [math]R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=G_{\mu\nu}-\Lambda g_{\mu\nu}=T_{\mu\nu}-\Lambda g_{\mu\nu}[/math] which doesn't meld with QFT). I am attempting to see if using this field equation I can take the difference of [math]\Gamma g_{\mu\nu}-\kappa T_{\mu\nu}[/math] as a change in the vacuum expectation value of QFT where this is equivalent to baryonic energy density. In simple terms I am using a region of baryonic energy density as a decrease in vacuum energy density, or equivalently inverting the energy scale of baryonic energy relative to the classical zero energy of the vacuum. This would also seem to be able to mesh with Noether's theroems using the flux of a decrease in density rather than the flux of a positive or negative density.

 

Anybody want to help derive the field equations? Probably not going to be easy.

 

You've written down a bunch of symbols but haven't bothered to define any of them. What are [math]\eta[/math], [math]\Gamma[/math], and [math]T_{\mu \nu}^{alt}[/math]? Also you've written down the EFE's wrong. They should be:

 

[math]R_{\mu \nu}-\frac{1}{2}g_{\mu \nu}R+\Lambda g_{\mu \nu}=G_{\mu \nu}+\Lambda g_{\mu \nu}=\kappa T_{\mu \nu}[/math]

 

Also, from what I can see, you're defining that equation to be the field equations. What is there to derive? Are you trying to quantize it?

Edited February 27, 2013 by elfmotat

[JMessenger]

 

But those aren't equal. They're opposite in sign! Unless you've defined [math]\Phi_{alt}=-\Phi [/math]. Regardless, it seems pointless.

 

 

You've written down a bunch of symbols but haven't bothered to define any of them. What are [math]\eta[/math], [math]\Gamma[/math], and [math]T_{\mu \nu}^{alt}[/math]? Also you've written down the EFE's wrong. They should be:

 

[math]R_{\mu \nu}-\frac{1}{2}g_{\mu \nu}R+\Lambda g_{\mu \nu}=G_{\mu \nu}+\Lambda g_{\mu \nu}=\kappa T_{\mu \nu}[/math]

 

Also, from what I can see, you're defining that equation to be the field equations. What is there to derive? Are you trying to quantize it?

It does seem pointless at the beginning but I would maintain that this is due to our mental understanding of directional derivatives of static scalar fields. Looking at the below picture, it cannot be determined from the directional derivative which integral was the source function:

 

since for instance we can get

or equivalently, if modeling the constant scalar field as orthogonal vectors with a sum of zero,

(ignore the notation).

 

The symbols don't make sense since I would have to define a physical model that would be the counterpart to the stress-energy tensor in this model. That isn't immediately necessary to make a definition akin to [math]G_{\mu\nu}=\kappa T_{\mu\nu}[/math] just yet as the first question at hand is whether or not a constant of integration of a scalar field becomes a scalar multiple of the metric. If the Poisson equation doesn't differentiate between [math]\nabla^2\Phi[/math] or [math]\nabla^2(C-\Phi_{alt})[/math] and if there is no known way to solve the magnitude of the cosmological constant with respect to the magnitude of the Newtonian phi, I would first ask on what a priori basis do we choose one scalar form over the other?

[elfmotat]

I don't understand what any of your pictures mean at all. And you still haven't defined [math]\Phi_{alt}[/math]. And I also don't see what any of this has to do with Nordstrom's theory. And I certainly don't see what any of this has to do with unification.

[JMessenger]

I don't understand what any of your pictures mean at all. And you still haven't defined [math]\Phi_{alt}[/math]. And I also don't see what any of this has to do with Nordstrom's theory. And I certainly don't see what any of this has to do with unification.

Let's stick with the explaining [math]\Phi_{alt}[/math]. Changing notation to [math]\Phi^{alt}[/math]

A more technical explanation of the Poisson equation can be found here:http://eprints.ma.man.ac.uk/894/02/0-19-852868-X.pdf In this case u is equivalent [math]\Phi[/math].

 

In a practical setting, u could represent the temperature field in Ω subject to the external heat source f. Other important physical models include gravitation, electromagnetism, elasticity and inviscid fluid mechanics

 

[math]\Phi[/math] is shown in and as a function of a single cartesian dimension [math]x_0[/math] (listed as r with boundary conditions for [math]\Phi\rightarrow\infty[/math] as [math]r\rightarrow 0[/math] and [math]\Phi\rightarrow 0[/math] as [math]r\rightarrow\infty[/math])

and [math]\Phi^{alt}=f_3[/math] as an arbitrary translation along the vertical axis and is defined such that [math]\frac{\partial \Phi}{\partial x_0}=\frac{\partial(C-\Phi_{alt})}{\partial x_0}=\frac{-\partial\Phi_{alt}}{\partial x_0}[/math] and that [math]|\Phi(x_0)|=|C-\Phi^{alt}(x_0)|[/math] (boundary conditions for [math]\Phi^{alt}\rightarrow -\infty[/math] as [math]r\rightarrow 0[/math] and [math]\Phi^{alt}\rightarrow C[/math] as [math]r\rightarrow\infty[/math])

 

The first point being made (that doesn't seem related yet) is that one cannot tell the magnitude of the function within the Poisson in relation to a constant of integration (the first derivative of each function is identical). Nor can one speak definitively of whether a vector equated to a directional derivative points away or towards a load function. It is subjective based on whether or not a constant of integration is present. In other words, you can't technically state that a "field" is attractive (nor repulsive) if the field is based on the Poisson equation.

 

The point I will be making is that the Poisson is only an approximation for a more accurate model of field theory, but I will argue that one of these identical functions should present a better approximation to another model.

Edited February 28, 2013 by JMessenger

[JMessenger]

Had someone ask by email something, so thought I would put the explanation here also.

 

Take the scalar potential definition of Wikipedia http://en.wikipedia.org/wiki/Scalar_potential

 

The potential P is assumed to be a function of the coordinate system, for example P(x,y,z). For a 3 dimensional scalar field there is a scalar value assigned to each coordinate point. If a directional derivative (gradient) is take [math]\frac{\partial P(x,y,z)}{\partial x}+\frac{\partial P(x,y,z)}{\partial y}+\frac{\partial P(x,y,z)}{\partial z}[/math] we obtain F. This definition of F however is technically incomplete. The question must be asked on whether or not P(x,y,z) is integrable. The reason is that the definition I just described of a scalar field is based on a understanding of the same type of plot as [math]\Phi[/math] above. There are always two scalar values assigned to each point for an integrable function. In the case of [math]\Phi[/math] there is [math]|\Phi|[/math] and the magnitude of a constant [math]|C|[/math] which is assigned a magnitude of 0 in this particular case. A more general case is that of the flipped integral, where the two assigned scalar values are [math]|\Phi^{alt}|[/math] and the magnitude of a constant with a non-zero value. While the directional derivatives are identical, the two scalar values must be retained for each point in order for the integrals to stay the same.

Stated another way, the scalar field as shown on Wikipedia is actually composed of a minimum of two scalar fields in order to be integrable. We normally think of one of those scalar fields as a zero scalar field and being unimportant, but this is not the most general case.