- #1

sweetser

Gold Member

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Hello:

I will try to meet the terms of the 8 guidelines.

1. The behavior of light is explained with a rank 1 field theory, the

Maxwell equations. Gravity is explained with a rank 2 field theory,

general relativity. The two can be combined in one Lagrange density,

but they are not in any sense unified.

For my unified field proposal, gravity and EM arise from the same

4-potential and form a rank 1 field. Here is the Lagrange density

for my gravity and EM (GEM) unified field proposal:

[tex]

\mathcal{L}_{GEM}=-\frac{1}{c}(J_{q}^{\mu}-J_{m}^{\mu})A_{\mu}

-\frac{1}{2c^{2}}\nabla_{\mu}A^{\nu}\nabla^{\mu}A_{\nu}

[/tex]

where:

[itex]J_{q}^{\mu}[/itex] is the electric charge 4-current density

[itex]J_{m}^{\mu}[/itex] is the mass charge 4-current density, the standard mass 4-density times [itex]\sqrt{G}[/itex]

[itex]A_{\mu}[/itex] is a 4-potential for both gravity and EM

[itex]\nabla_{\mu}[/itex] is a covariant derivative

[itex]\nabla_{\mu}A^{\nu}[/itex] is the reducible unified field strength tensor

which is the sum of a symmetric irreducible tensor [itex](\nabla_{\mu}A^{\nu}+\nabla_{\nu}A^{\mu})[/itex] for gravity

and an antisymmetric irreducible tensor [itex](\partial_{\mu}A^{\nu}-\partial_{\nu}A^{\mu})[/itex] for EM which uses an exterior derivative

The core variance is how one gets a dynamic metric which solves the

field equations for gravity. With general relativity, one starts with

the Hilbert action, varies the metric field, and generates the second

rank field equations. Here, I work with a symmetry of the Lagrange

density, working directly from the standard definition of a covariant

derivative:

[tex]

\bigtriangledown_{\mu}A^{\nu}=\partial_{\mu}A^{\nu}+\Gamma_{\sigma\mu}{}^{\nu}A^{\sigma}

[/tex]

Any value contained in the unified field strength tensor could be due

any combination of the change in the potential or due to a change in

the metric. One is free to alter the change in the metric so long as

the change in potential compensates, and likewise the reverse. I

believe this is called a diffeomorphism symmetry (but my training is

spotty). Any symmetry in the Lagrange density must be related to a

conserved charge. For this symmetry, mass is the conserved charge.

The field equations are generated in the standard way, by varying the

action with respect to the potential. One ends up with a 4D wave

equation:

[tex]

J_{q}^{\mu}-J_{m}^{\mu}=(\frac{1}{c}\partial^{2}/\partial t^{2}-c\nabla^{2})A^{\mu}

[/tex]

For the physical situation where the mass density equation is

effectively zero, one gets the Maxwell equations in the Lorentz gauge.

If the equations describe a static, neutral system, then the first

field equation, [itex]\rho_{m}=\nabla^{2}\phi[/itex], is Newton's

field equation for gravity. If the neutral system is dynamic, then

the equation transforms like a 4-vector under a Lorentz boost.

Because this equation is consistent with special relativity, that

removes a major motivation for general relativity (consistency with

SR).

If the system is neutral, static, and one chooses a gauge such that the

potential is constant, then the first field equation is the divergence

of the Christoffel symbol:

[tex]

\rho_{m}=2\partial_{\mu}\Gamma_{\nu}{}^{\:0\mu}A^{\nu}

[/tex]

This contains second order derivatives of the metric, a requirement

for constraining a dynamic metric. The exponential metric solves the

field equation:

[tex]

g_{\mu\nu}=\left(\begin{array}{cccc}

exp(-2\frac{GM}{c^{2}R}) & 0 & 0 & 0\\

0 & -exp(2\frac{GM}{c^{2}R}) & 0 & 0\\

0 & 0 & -exp(2\frac{GM}{c^{2}R}) & 0\\

0 & 0 & 0 & -exp(2\frac{GM}{c^{2}R})\end{array}\right).

[/tex]

The easiest way to realize this is that for the definition of a

Christoffel of the second kind for a static, diagonal metric will only

involve [itex]g_{00}[/itex] and [itex]g^{0}{}_{0}^{u}[/itex].

The exponentials will cancel each other, leaving only the divergence

of the derivative of the exponent, or

[tex]

\rho_{m}=\nabla^{2}(GM/c^2 R)

[/tex]

The 1/R solution should be familiar. This metric gives a point

singular solution to the field equations.

One could have chosen a gauge where the metric was flat. With that

gauge choice, the potential (GM/c^2 R, 0, 0,0) solves the first field

equation with a point singularity, a good check for logical

self-consistency.

2. The exponential metric solution to the GEM field equations for a

static, neutral system is consistent with first-order parameterized

post-Newtonian predictions of weak field theories. The relevant terms

of the Taylor series expansion are:

[tex]

(\partial\tau)^{2}\cong(1-2GM/c^{2}R+2(GM/c^{2}R)^{2})dt^{2}-(1+2GM/c^{2}R)dR^\{2}/c^{2}

[/tex]

These are identical to those for the Schwarzschild metric of general

relativity. Therefore all the weak field tests of the metric, and all

tests of the equivalence principle will be passed. To second-order

PPN accuracy the metrics are different:

GEM

[tex]

(\partial\tau)^{2}\cong(1-2GM/c^{2}R+2(GM/c^{2}R)^{2}-4/3(GM/c^{2}R)^{3})dt^{2}

[/tex]

[tex]

-(1+2GM/c^{2}R+2(GM/c^{2}R)^{2})dR^{2}/c^{2}

[/tex]

GR

[tex]

(\partial\tau)^{2}\cong(1-2GM/c^{2}R+2(GM/c^{2}R)^{2}-3/2(GM/c^{2}R)^{3})dt^{2}

[/tex]

[tex]

-(1+2GM/c^{2}R+3/2(GM/c^{2}R)^{2})dR^{2}/c^{2}

[/tex]

This will translate into 0.7 microarcseconds more bending of light

around the Sun according to a paper by Epstein and Shapiro,

Phys. Rev. D, 22:2947, 1980. We currently can measure bending to 100

microarcseconds. Clifford Will responding to a question I posed said

there are _no_ plans in development to get to the 1 microarcsecond

level of accuracy. Darn!

The antisymmetric field strength tensor will be represented by the

spin 1 photon, where like charges repel. These are the transverse

modes of emission. The symmetric field strength tensor will be

represented by the spin 2 graviton, where like charges attract. These

will be the scalar and longitudinal modes of emission. Should we ever

measure a gravity wave, and then determine its polarization, general

relativity and the GEM proposal differ on the polarization. If the

polarization is transverse, GEM is wrong. If the polarization is not

transverse, general relativity is wrong (Will also made this point in

his living review article).

3. Once the Lagrange density is stated, everything else flows from

that. I have discussed this work as it developed and took misteps on

sci.physics.research and my own web site, but that should not be

needed here.

4. To back up the derivations, I have cranked through all this and a

bit more in a Mathematica notebook. It is available here:

http://www.theworld.com/~sweetser/quaternions/gravity/Lagrangian_to_tests/Lagrangian_to_tests.html

http://www.theworld.com/~sweetser/quaternions/ps/Lagrangian_to_tests.nb.pdf

http://www.theworld.com/~sweetser/quaternions/notebooks/Lagrangian_to_tests.nb

[Despite the URL, no quaternions are used in this body of work,

although they continue to be the wizard behind the curtain.]

5. This theory is consistent with strong field tests of gravity, such

as energy loss by binary pulsars. For an isolated mass, the lowest

mode of emission is a quadrapole moment. This proposal does not have

extra fields that can store energy or momentum, which is what is

needed to form a dipole if there is only one sign to the mass charge,

which the proposal claims.

6. I know of no physical experiments that contradict this work. There

are _thought_ experiments that claim that gravity must be non-linear

(there was a primer on GR by Price I recall as an example). These

thought experiments appear to always use electrically neutral sources.

For a unified field theory, one must consider what happens if charge

is included. What Price did was imagine a pair of boxes with 6

particles in each. Then the energy of one of the particles in one box

gets completely converted to kinetic energy of the other 5. Price

argues that the box with 6 particles should not be able to tell the

difference between the two boxes, the one with 6 still particles and

the one with 5 buzzing about. If this is the case, then the field

equations for gravity must be nonlinear. I argue that if the 6

particles were charged, there would be no way to destroy an electric

charge, so the experiment cannot be done in theory. No conclusions

can be drawn. EM puts new constraints on gravity thought experiments.

7. It has been my observation that no one is impressed by the

Mathematica notebook, even people at Wolfram Research. The notebook

is my best unbiased source that no obvious mathematical errors have

been made. Earlier versions of this body of work did have errors that

Mathematica pointed out.

8. I understand how general relativity works well enough to appreciate

that a linear, rank 1 field theory is in fundamental conflict with GR.

That is an observation, nothing more or less. GR works to first order

PPN accuracy. It is an open question if it will work to second order.

My money is riding on the exponential metric, because exponentials

appear to be Nature's favorite function (simple harmonics around the

identity for small exponents).

Sorry to be this l o n g, but the guidelines appeared to require it.

doug sweetser

I will try to meet the terms of the 8 guidelines.

1. The behavior of light is explained with a rank 1 field theory, the

Maxwell equations. Gravity is explained with a rank 2 field theory,

general relativity. The two can be combined in one Lagrange density,

but they are not in any sense unified.

For my unified field proposal, gravity and EM arise from the same

4-potential and form a rank 1 field. Here is the Lagrange density

for my gravity and EM (GEM) unified field proposal:

[tex]

\mathcal{L}_{GEM}=-\frac{1}{c}(J_{q}^{\mu}-J_{m}^{\mu})A_{\mu}

-\frac{1}{2c^{2}}\nabla_{\mu}A^{\nu}\nabla^{\mu}A_{\nu}

[/tex]

where:

[itex]J_{q}^{\mu}[/itex] is the electric charge 4-current density

[itex]J_{m}^{\mu}[/itex] is the mass charge 4-current density, the standard mass 4-density times [itex]\sqrt{G}[/itex]

[itex]A_{\mu}[/itex] is a 4-potential for both gravity and EM

[itex]\nabla_{\mu}[/itex] is a covariant derivative

[itex]\nabla_{\mu}A^{\nu}[/itex] is the reducible unified field strength tensor

which is the sum of a symmetric irreducible tensor [itex](\nabla_{\mu}A^{\nu}+\nabla_{\nu}A^{\mu})[/itex] for gravity

and an antisymmetric irreducible tensor [itex](\partial_{\mu}A^{\nu}-\partial_{\nu}A^{\mu})[/itex] for EM which uses an exterior derivative

The core variance is how one gets a dynamic metric which solves the

field equations for gravity. With general relativity, one starts with

the Hilbert action, varies the metric field, and generates the second

rank field equations. Here, I work with a symmetry of the Lagrange

density, working directly from the standard definition of a covariant

derivative:

[tex]

\bigtriangledown_{\mu}A^{\nu}=\partial_{\mu}A^{\nu}+\Gamma_{\sigma\mu}{}^{\nu}A^{\sigma}

[/tex]

Any value contained in the unified field strength tensor could be due

any combination of the change in the potential or due to a change in

the metric. One is free to alter the change in the metric so long as

the change in potential compensates, and likewise the reverse. I

believe this is called a diffeomorphism symmetry (but my training is

spotty). Any symmetry in the Lagrange density must be related to a

conserved charge. For this symmetry, mass is the conserved charge.

The field equations are generated in the standard way, by varying the

action with respect to the potential. One ends up with a 4D wave

equation:

[tex]

J_{q}^{\mu}-J_{m}^{\mu}=(\frac{1}{c}\partial^{2}/\partial t^{2}-c\nabla^{2})A^{\mu}

[/tex]

For the physical situation where the mass density equation is

effectively zero, one gets the Maxwell equations in the Lorentz gauge.

If the equations describe a static, neutral system, then the first

field equation, [itex]\rho_{m}=\nabla^{2}\phi[/itex], is Newton's

field equation for gravity. If the neutral system is dynamic, then

the equation transforms like a 4-vector under a Lorentz boost.

Because this equation is consistent with special relativity, that

removes a major motivation for general relativity (consistency with

SR).

If the system is neutral, static, and one chooses a gauge such that the

potential is constant, then the first field equation is the divergence

of the Christoffel symbol:

[tex]

\rho_{m}=2\partial_{\mu}\Gamma_{\nu}{}^{\:0\mu}A^{\nu}

[/tex]

This contains second order derivatives of the metric, a requirement

for constraining a dynamic metric. The exponential metric solves the

field equation:

[tex]

g_{\mu\nu}=\left(\begin{array}{cccc}

exp(-2\frac{GM}{c^{2}R}) & 0 & 0 & 0\\

0 & -exp(2\frac{GM}{c^{2}R}) & 0 & 0\\

0 & 0 & -exp(2\frac{GM}{c^{2}R}) & 0\\

0 & 0 & 0 & -exp(2\frac{GM}{c^{2}R})\end{array}\right).

[/tex]

The easiest way to realize this is that for the definition of a

Christoffel of the second kind for a static, diagonal metric will only

involve [itex]g_{00}[/itex] and [itex]g^{0}{}_{0}^{u}[/itex].

The exponentials will cancel each other, leaving only the divergence

of the derivative of the exponent, or

[tex]

\rho_{m}=\nabla^{2}(GM/c^2 R)

[/tex]

The 1/R solution should be familiar. This metric gives a point

singular solution to the field equations.

One could have chosen a gauge where the metric was flat. With that

gauge choice, the potential (GM/c^2 R, 0, 0,0) solves the first field

equation with a point singularity, a good check for logical

self-consistency.

2. The exponential metric solution to the GEM field equations for a

static, neutral system is consistent with first-order parameterized

post-Newtonian predictions of weak field theories. The relevant terms

of the Taylor series expansion are:

[tex]

(\partial\tau)^{2}\cong(1-2GM/c^{2}R+2(GM/c^{2}R)^{2})dt^{2}-(1+2GM/c^{2}R)dR^\{2}/c^{2}

[/tex]

These are identical to those for the Schwarzschild metric of general

relativity. Therefore all the weak field tests of the metric, and all

tests of the equivalence principle will be passed. To second-order

PPN accuracy the metrics are different:

GEM

[tex]

(\partial\tau)^{2}\cong(1-2GM/c^{2}R+2(GM/c^{2}R)^{2}-4/3(GM/c^{2}R)^{3})dt^{2}

[/tex]

[tex]

-(1+2GM/c^{2}R+2(GM/c^{2}R)^{2})dR^{2}/c^{2}

[/tex]

GR

[tex]

(\partial\tau)^{2}\cong(1-2GM/c^{2}R+2(GM/c^{2}R)^{2}-3/2(GM/c^{2}R)^{3})dt^{2}

[/tex]

[tex]

-(1+2GM/c^{2}R+3/2(GM/c^{2}R)^{2})dR^{2}/c^{2}

[/tex]

This will translate into 0.7 microarcseconds more bending of light

around the Sun according to a paper by Epstein and Shapiro,

Phys. Rev. D, 22:2947, 1980. We currently can measure bending to 100

microarcseconds. Clifford Will responding to a question I posed said

there are _no_ plans in development to get to the 1 microarcsecond

level of accuracy. Darn!

The antisymmetric field strength tensor will be represented by the

spin 1 photon, where like charges repel. These are the transverse

modes of emission. The symmetric field strength tensor will be

represented by the spin 2 graviton, where like charges attract. These

will be the scalar and longitudinal modes of emission. Should we ever

measure a gravity wave, and then determine its polarization, general

relativity and the GEM proposal differ on the polarization. If the

polarization is transverse, GEM is wrong. If the polarization is not

transverse, general relativity is wrong (Will also made this point in

his living review article).

3. Once the Lagrange density is stated, everything else flows from

that. I have discussed this work as it developed and took misteps on

sci.physics.research and my own web site, but that should not be

needed here.

4. To back up the derivations, I have cranked through all this and a

bit more in a Mathematica notebook. It is available here:

http://www.theworld.com/~sweetser/quaternions/gravity/Lagrangian_to_tests/Lagrangian_to_tests.html

http://www.theworld.com/~sweetser/quaternions/ps/Lagrangian_to_tests.nb.pdf

http://www.theworld.com/~sweetser/quaternions/notebooks/Lagrangian_to_tests.nb

[Despite the URL, no quaternions are used in this body of work,

although they continue to be the wizard behind the curtain.]

5. This theory is consistent with strong field tests of gravity, such

as energy loss by binary pulsars. For an isolated mass, the lowest

mode of emission is a quadrapole moment. This proposal does not have

extra fields that can store energy or momentum, which is what is

needed to form a dipole if there is only one sign to the mass charge,

which the proposal claims.

6. I know of no physical experiments that contradict this work. There

are _thought_ experiments that claim that gravity must be non-linear

(there was a primer on GR by Price I recall as an example). These

thought experiments appear to always use electrically neutral sources.

For a unified field theory, one must consider what happens if charge

is included. What Price did was imagine a pair of boxes with 6

particles in each. Then the energy of one of the particles in one box

gets completely converted to kinetic energy of the other 5. Price

argues that the box with 6 particles should not be able to tell the

difference between the two boxes, the one with 6 still particles and

the one with 5 buzzing about. If this is the case, then the field

equations for gravity must be nonlinear. I argue that if the 6

particles were charged, there would be no way to destroy an electric

charge, so the experiment cannot be done in theory. No conclusions

can be drawn. EM puts new constraints on gravity thought experiments.

7. It has been my observation that no one is impressed by the

Mathematica notebook, even people at Wolfram Research. The notebook

is my best unbiased source that no obvious mathematical errors have

been made. Earlier versions of this body of work did have errors that

Mathematica pointed out.

8. I understand how general relativity works well enough to appreciate

that a linear, rank 1 field theory is in fundamental conflict with GR.

That is an observation, nothing more or less. GR works to first order

PPN accuracy. It is an open question if it will work to second order.

My money is riding on the exponential metric, because exponentials

appear to be Nature's favorite function (simple harmonics around the

identity for small exponents).

Sorry to be this l o n g, but the guidelines appeared to require it.

doug sweetser

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