The Maxwell equations for gravity

doug  <sweetser@world.com>

Introduction
Constraints on a gravity theory
The Lorentz gravitational force and the Maxwell equations of gravity
Links to Newton gravity and general relativity
Quantum gravity
Implications

Based on a post to sci.physics.research

Introduction

In this post, I will propose a new theory of gravity based on quaternion operators and potentials. Because the constraints on any theory of gravity are so tight,I feel I have no choices to make, just tight spots to slip through. Some of the consequences will be examined that make this proposal different from general relativity even in the classical region (good-bye dark matter??).

Constraints on a gravity theory

A proposal for gravity should satisfy the following three constraints:

1. For a classical point mass, reduces to Newton's law of gravity, F = - G ((m m')/(r^2)).  
2. The form should be manifestly covariant under a Lorentz transformation. 
3. The law should work with quantum mechanics.

Electromagnetism can pass through classical, relativistic, and quantum constraints. Therefore, to achieve similar goals, a theory of gravity could clone the structure of electromagnetic theory to the letter. The laws governing electromagnetism--the Lorentz force and the Maxwell equations--can be written as operators acting on the same 4-potential.

The Lorentz force\(\(\((∂\/∂t, 
              \(-▽\&⇀\))\)\+\(--
              \(--\(--\(---\)\)\)\)\) \((ϕ, \(-A\&⇀\))\) 
        \(\(\((γ, \(-γβ\))\) - \ 
              \((γ, \(-γβ\))\) 
                \((∂\/∂t, 
                  ▽\&⇀)\)\)\+\(--
              \(--\(--\(--
                    \(--\(--
                        \(--\(--
                            \(--\(--
                                \(--\(--
                                    \(--\(--
                                        \(--\(----
                                        \)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\) 
        \((ϕ, A\&⇀)\)
 = 
      \(γ 
          \((\(-β\)·\((
                  ▽\&⇀, 
                    ϕ + ∂A\&⇀\/∂t)\), 
              \(-\((▽\&⇀, 
                      ϕ + ∂A\&⇀\/∂t)
                    \)\) + \ 
                β\ X 
                  \((\(▽\&⇀\) X 
                       A\&⇀)\), )\)
 = \ 
        γ\ e \((\ β.E, E\  + β\ ×\ B)\)\)\)

The Maxwell equations 
{(Underscript[((∂/∂)t, 
                                    (Overscript[▽, ⇀])),[((∂/∂)t, 
                                      (Overscript[▽, ⇀]))
                                      , ----
                                    ----
                                        ----
                                        ----
                                        -------]),(ϕ, (Overscript[A, ⇀]))]} 
+[(Underscript[((∂/∂)t, 
                              (Overscript[▽, ⇀])),vector{((∂/∂)t, 
                                  (Overscript[▽, ⇀]))
                                  , ----
                              ----
                                  ----
                                      ----
                                        ----
                                        ------
                                        -------
                                        ]),(ϕ, -(Overscript[A, ⇀]))}]
=(-(Overscript[▽, ⇀])·(Overscript[▽, ⇀]) X (Overscript[A, ⇀]) ,-(Overscript[▽, ⇀]) X (Overscript[▽, ⇀]) ϕ)=(-(Overscript[▽, ⇀])·(Overscript[B
                        , ⇀]), 
                    (Overscript[▽, ⇀]) X 
                        (Overscript[E, ⇀]) + 
                      ∂((Overscript[B, ⇀])/∂)t)=(0,  (Overscript[0, ⇀]))

		[(Underscript[((∂/∂)t, 
                            (Overscript[▽, ⇀])),[((∂/∂)t, 
                              (Overscript[▽, ⇀])), ----
                            ----
                                ----
                                    ----
                                        ----
                                        ----]),(ϕ, (Overscript[A, ⇀]))]]
	- {(Underscript[((∂/∂)t, 
                    (Overscript[▽, ⇀])),vector{((∂/∂)t, 
                        (Overscript[▽, ⇀])), ----
                    ----
                        ----
                            ----
                                ----
                                    ----
                                        ---
                                        ------
                                        ]),(ϕ, -(Overscript[A, ⇀]))}}
=(-(Overscript[▽, ⇀])·(Overscript[▽, ⇀]) ϕ  - 
            (Overscript[▽, ⇀])·∂((Overscript[A
                    , ⇀])/∂)t, 
          (Overscript[▽, ⇀]) X 
              (Overscript[▽, ⇀]) X (Overscript[A, ⇀])  + 
            (∂^2) ((Overscript[A, ⇀])/∂)(t^2) + 
            ((∂(Overscript[▽, ⇀]), ϕ)/∂)t) = 4 π (ρ, (Overscript[J, ⇀]))

If the underlined operators act on a different potential, then equations with the same properties would be created that apply to that different potential. A potential for gravity should satisfy the following three constraints.

1. Apply to all particles, including light, reducing to mass in the classical limit. 
2. Be unidirectional.
3. Involve curvature in some way.

Every particle has a 4-momentum, composed of the scalar energy E and the 3-momentum P. The mass of a particle in flat spacetime is the scalar part of the 4-momentum squared.

(((E,  (Overscript[P,  ⇀]) ))^2)  =  
      ((E^2)  -  (((Overscript[P,  ⇀]) )^2),  
        2 E  (Overscript[P,  ⇀]) )

Define flat spacetime operationally: when the scalar of the square of a quaternion--scalar(q^2)--is the scalar squared minus the 3-vector squared--scalar(q)^2 - vector(q)^2--then  spacetime is flat. In flat spacetime, the mass of two particles is the sum.

\(\((E, P\&⇀)\)\^2 + 
        \((E\^′, P\&⇀\^′)\)\^2 = 
      \(\((E\^2 - P\&⇀\^2, 2\ E\ P\&⇀)\) + 
          \((\(E\^′\)\^2 - \(P\&⇀\^′\)\^2, 
            2\ E\^′\ P\&⇀\^′)\)\ 
 = \ \ 
        \((m\^2\  + \ \(m\^′\)\^2, \ 
          2\ \((E\ P\&⇀\  + \ 
                E\^′\ P\&⇀\^′)\))\)\)\)

If we want to be consistent in calculating the mass, add the two 4- momenta together first and then square.

\(\(\((E + E\^′, \ P\&⇀ + P\&⇀\^′)
          \)\^2\  = \ 

      \((E\^2 - P\&⇀\^2 + \(E\^′\)\^2 - 
          \(P\&⇀\^′\)\^2 + 
          \(2 \((E\ E\^′ - 
                  P\&⇀\ P\&⇀\^′)\)\)\+\(--
              \(--\(--\(--\(--\(--\(---\)\)\)\)\)\)\), 
        2 \((E\ P\&⇀ + 
              E\^′\ P\&⇀\^′ + 
              \(E\ P\&⇀\^′ + 
                  \(E\^′\) P\&⇀\)\+\(--
                  \(--\(--\(--\(---\)\)\)\)\))\))\)\ \)\)

The underlined scalar term indicates that spacetime containing two particles in it is no longer flat. The scalar term is unidirectional. For real particles,

E  > (Overscript[P,  ⇀])     and    (E^′)  >  
      (((Overscript[P,  ⇀]) )^′),  
    so    E (E^′) - 
        (Overscript[P,  ⇀])  (((Overscript[P,  ⇀]) )^′)  >  0

If E >> P and E' >> P', then

E (E^′) - 
        (Overscript[P,  ⇀])  (((Overscript[P,  ⇀]) )^′)  
      ≅  m (m^′)

The Lorentz gravitational force and the Maxwell equations of gravity

The quaternion field 2 (E E' - P P', E P' + P E' + PxP') has the three properties required of a gravitational field (reduces to m m', is unidirectional, and involves curvature). Generate the quaternion laws of gravity by having the operators act on the gravitational field.

The Lorentz gravitational force((∂/∂)t, 
              -(Overscript[▽, ⇀]))(E (E^′)  - 
                  (Overscript[P, ⇀]).((Overscript[P, ⇀])^′),-E ((Overscript[P, ⇀]))'-(E^′) (Overscript[P, ⇀]))(γ, -γβ)
			- (γ, -γβ)((∂/∂)t, 
              (Overscript[▽, ⇀]))(E (E^′)  - 
                  (Overscript[P, ⇀]).((Overscript[P, ⇀])^′),E ((Overscript[P, ⇀]))'+(E^′) (Overscript[P, ⇀]))
=γ(-β·(Overscript[▽, ⇀]),(E (E^′)  - 
                    (Overscript[P, ⇀]).((Overscript[P, ⇀])^′))-β·((∂E ((Overscript[P, ⇀]))'+(E^′) (Overscript[P, ⇀]))/(∂t))),
			-((Overscript[▽, ⇀]),(E (E^′)  - 
                      (Overscript[P, ⇀]).((Overscript[P, ⇀])^′))+((∂E ((Overscript[P, ⇀]))'+(E^′) (Overscript[P, ⇀]))/(∂t)))+ β X((Overscript[▽, ⇀])X(E ((Overscript[P, ⇀]))'+(E^′) (Overscript[P, ⇀])),)
= γ 
          ( β.(Overscript[G, ⇀]), 
            (Overscript[G, ⇀])  + β × (Overscript[F, ⇀]))

The Maxwell equations of gravitational   {((∂/∂)t, (Overscript[▽, ⇀]))
              ,[((∂/∂)t, 
                  (Overscript[▽, ⇀])),(E (E^′)  - 
                      (Overscript[P, ⇀]).((Overscript[P, ⇀])^′),E ((Overscript[P, ⇀]))'+(E^′) (Overscript[P, ⇀]))]} 
+[((∂/∂)t, (Overscript[▽, ⇀]))
              ,vector{((∂/∂)t, 
                    (Overscript[▽, ⇀])),(E (E^′)  - 
                        (Overscript[P, ⇀]).((Overscript[P, ⇀])^′),-E ((Overscript[P, ⇀]))'-(E^′) (Overscript[P, ⇀]))}]
=(-(Overscript[▽, ⇀])·(Overscript[▽, ⇀]) X (E ((Overscript[P, ⇀]))'+(E^′) (Overscript[P, ⇀])) ,-(Overscript[▽, ⇀]) X (Overscript[▽, ⇀]) (E (E^′) -(Overscript[P, ⇀]).((Overscript[P, ⇀])^′)))
=(-(Overscript[▽, ⇀])·(Overscript[F
                , ⇀]), 
            (Overscript[▽, ⇀]) X (Overscript[G, ⇀]) + 
              ∂((Overscript[F, ⇀])/∂)t)=(0,  (Overscript[0, ⇀]))

		[((∂/∂)t, 
                    (Overscript[▽, ⇀])),[((∂/∂)t, 
                        (Overscript[▽, ⇀])),(E (E^′)  - 
                            (Overscript[P, ⇀]).((Overscript[P, ⇀])^′),E ((Overscript[P, ⇀]))'+(E^′) (Overscript[P, ⇀]))]]
	- {((∂/∂)t, 
                    (Overscript[▽, ⇀])),vector{((∂/∂)t, 
                          (Overscript[▽, ⇀])),(E (E^′)  - 
                              (Overscript[P, ⇀]).((Overscript[P, ⇀])^′),-E ((Overscript[P, ⇀]))'-(E^′) (Overscript[P, ⇀]))}}
=(-(Overscript[▽, ⇀])·(Overscript[▽, ⇀]) E (E^′) -(Overscript[P, ⇀]).((Overscript[P, ⇀])^′)-(Overscript[▽, ⇀])·((∂E ((Overscript[P, ⇀]))'+(E^′) (Overscript[P, ⇀]))/(∂t)),
					(Overscript[▽, ⇀]) X  (Overscript[▽, ⇀]) X (E ((Overscript[P, ⇀]))'+(E^′) (Overscript[P, ⇀]))+(((∂^2) E ((Overscript[P, ⇀]))'+(E^′) (Overscript[P, ⇀]))/(∂(t^2)))+((∂(Overscript[▽, ⇀]), (E (E^′)  - 
                              (Overscript[P, ⇀]).((Overscript[P, ⇀])^′))
                            )/∂)t)
=( (Overscript[▽, ⇀])·(Overscript[G
                      , ⇀])  ,  
                  (Overscript[▽, ⇀]) X (Overscript[F, ⇀]) 
                     - ∂((Overscript[G, ⇀])/∂)t) = 
                8 π 
                  (mass density, (Overscript[(mass flow), ⇀]))

Looks complicated, but it was only an exercise in cutting and pasting new fields into the Lorentz force and the Maxwell equations.

The key question is how well this proposal fits the data.

Links to Newton gravity and general relativity

Newton's law of gravity works well, but where exactly is it hiding in that collection of symbols? Let electromagnetic theory be the guide. Coulomb's law is found in the covariant Lorentz force law if beta = A = 0 and the scalar potential for a point charge is q/r. For the Lorentz gravitational force, beta = P = P' = 0, so E E' ~= m m', gamma ~1, and the scalar potential for a point mass is m m'/r.

\(\((∂\/∂t, \(-▽\&⇀\))
            \) \((\(m\ m\^′\)\/r, 0\&⇀)\) 
          \((1, 0\&⇀)\) - \ 
        \((1, 0\&⇀)\) 
          \((∂\/∂t, ▽\&⇀)
            \) \((\(m\ m\^′\)\/r, 0\&⇀)\)
	 = 
      \(\((0, \(-▽\&⇀\) \(m\ m\^′\)\/r)
          \) = \((0, \(m\ m\^′\)\/r\^2)\)\)\)

Multiply by -G, and that is Newton's law.

Newton's law describes a 3-vector. How does one "repair" the law so that it gets along with the 4-force formulations of special relativity? This seemingly small job actually involves general relativity (see Misner, Thorne and Wheeler, chapter 7). No repairs are required for the Lorentz gravitational force law, since its form is every bit as covariant as the Lorentz force of electromagnetism. For a point source, the law becomes:

\(\((∂\/∂t, \(-▽\&⇀\))
            \) \((\(E\ E\^′\  - 
                P\&⇀.P\&⇀\^′\)\/r, 
            0\&⇀)\) \((γ, \(-γβ\))\)
			
         - \ \((γ, \(-γβ\))\) 
          \((∂\/∂t, ▽\&⇀)
            \) \((\(E\ E\^′\  - 
                P\&⇀.P\&⇀\^′\)\/r, 
            0\&⇀)\)
 = 
      \(γ 
          \((\(-β\)·▽\&⇀ 
                \((\(E\ E\^′\  - 
                      P\&⇀.P\&⇀\^′\)\/r)\), 
              \(-▽\&⇀\), 
              \((\(E\ E\^′\  - 
                    P\&⇀.P\&⇀\^′\)\/r)
                \), )\)
 = 
        γ 
          \((β·\((
                  \(E\ E\^′\  - 
                      P\&⇀.P\&⇀\^′\)\/r\^2)
                  \), \((\(E\ E\^′\  - 
                    P\&⇀.P\&⇀\^′\)\/r\^2)
                \), )\)\)\)

I do not know yet whether this form of the Lorentz gravitational force is consistent with the experimental tests designed for the Schwarzschild metric of general  relativity, but I do have high hopes :-)

Here's a subtle but fundamental distinction between electromagnetism and gravity.  For the Lorentz force of electromagnetism, the test charge q is independent of the 4-potential. Varying q does nothing to the 4- potential. For gravity, there is no way to separate the two momentum 4-vectors that together compose the gravitational source. Varying a test mass varies the 4-potential. This may play a similar function to the non-linearity seen in general relativity.

Now to the big difference: what is gravitational analogue of magnetism? Magnetism involves moving electric charges. The gravitational analogue must involve moving pairs of masses. Consider a spiral galaxy spinning around an axis. These equations predict a force generated by moving pairs of masses that cannot be accounted for by a scalar potential. The velocity profile of spiral galaxies is flat (V(r)  = k) as seen using neutral atomic hydrogen spectral lines. That is the data.

Newton's law predicts a Keplerian decline, V(r) = (k M(r)/r)^.5. Since this equation fails, people have proposed the dark matter hypothesis, where most of the matter in the universe is composed of "no-see-'ems", a type of bug that cannot be seen, but leaves a bite from its visit.

The B field is generated by DelxA. The gravitational analogue, let's call it the ME field since that sounds like B and implies the non-linearity, is Delx(E P' + P E' + PxP'). Do I know how to use the ME field to predict the velocity profile of a spiral galaxy? No. I will start playing with electromagnetic analogies, specifically electric charges on disks.  [note added later: I have heard that this sort of calculation has been made for general relativity, and the effect is not of the scale necessary to account for the effect seen.]

There is a small amount of data about gravity waves. Some of the data is based on the rate of decay of signals from binary pulsars due to energy emitted as gravity waves. The Maxwell gravitational wave equation also predicts such waves. I have yet to start doing calculations with this set of equations. One point I can make is that there are no ME field monopoles, just as there are no B field monopoles. However, the ME field is really more analogous to a dipole since it necessarily involves two 4-momenta. Therefore there are no ME field dipoles. Gravity waves may require a quadrupole source for this reason. General relativity makes a similar statement.

Quantum gravity

I have not broached the topic of quantum mechanics, yet the topic is critical for a modern gravitational theory. Here again, I must be brief since I am still formulating how to think about quantum mechanics with quaternions. My hope is to clone quantum electrodynamics, creating quantum gravitational dynamics, since the equations have identical forms, something like so:

((∂/∂)t, (Overscript[▽, ⇀]))
              ,[((∂/∂)t, 
                (Overscript[▽, ⇀])), 
              (E (E^′)  - 
                  (Overscript[P, ⇀]).((Overscript[P, ⇀])^′), 
                E (E^′)  - 
                  (Overscript[P, ⇀]).((Overscript[P, ⇀])^′))]} 
+[((∂/∂)t, (Overscript[▽, ⇀]))
              ,{((∂/∂)t, 
                -(Overscript[▽, ⇀])), 
              (E (E^′)  - 
                  (Overscript[P, ⇀]).((Overscript[P, ⇀])^′), 
                -E (E^′)  + 
                  (Overscript[P, ⇀]).((Overscript[P, ⇀])^′))}]
=(-(Overscript[▽, ⇀])·(Overscript[▽, ⇀]) X 
              ( E (E^′)  - 
                  (Overscript[P, ⇀]).((Overscript[P, ⇀])^′)) ,-(Overscript[▽, ⇀]) X (Overscript[▽, ⇀]) (E (E^′)  - 
                    (Overscript[P, ⇀]).((Overscript[P, ⇀])^′)))=(0,  (Overscript[0, ⇀]))
	
[((∂/∂)t, 
                  (Overscript[▽, ⇀])),[((∂/∂)t, 
                    (Overscript[▽, ⇀])), 
                  (E (E^′)  - 
                      (Overscript[P, ⇀]).((Overscript[P, ⇀])^′), 
                    E (E^′)  - 
                      (Overscript[P, ⇀]).((Overscript[P, ⇀])^′))]]
	+ {((∂/∂)t, 
                  (Overscript[▽, ⇀])),{((∂/∂)t, 
                    -(Overscript[▽, ⇀])), 
                  (E (E^′)  - 
                      (Overscript[P, ⇀]).((Overscript[P, ⇀])^′), 
                    -E (E^′)  - 
                      (Overscript[P, ⇀]).((Overscript[P, ⇀])^′))}}
=\(\((\ \(∂\^2 E\ E\^′\  - 
                    P\&⇀.P
                          \&⇀\^′\)\/∂t\^2 + 
                ▽\&⇀\^2\ 
                  \((E\ E\^′\  - 
                      P\&⇀.P\&⇀\^′\ )\), 

              				\(-
                  \(\(∂\^2 E\ E\^′\  - 
                        P\&⇀.P
                              \&⇀\^′\)\/∂t\^2
                          \)\) - \ 
                ▽\&⇀\^2\ 
                  \((E\ E\^′\  - 
                      P\&⇀.P\&⇀\^′\ )\))\)

            		 = 
            8  π\ \((mass, \(mass\ flow\)\&⇀)\)\)

That sounds like a reasonable, though unsupported, line of logic.

Implications

What has been accomplished? I have generated two equations for gravity--the Lorentz gravitational force and the Maxwell gravitational wave equation--with as many implications as their electromagnetic counterparts. The only solid connection to data has involved Newton's law of gravity. A link to the experimental tests of general relativity is merely a hope. A connection to quantum mechanics is more tenuous. The proof for this proposal, if it ever comes, will involve the ability to explain the velocity profiles of systems with masses in motion.