Geometry without a potential is like a bed without a lover. The Riemann curvature tensor, with its divergence of two connections, is exclusively about geometry and all about the bed sheet. Newton’s scalar potential theory was the ﬁrst math to reach and direct the motion of the stars. It is only about the scalar potential. Unfortunately, it is too small, being inconsistent with special relativity. I will try to construct a uniﬁed ﬁeld theory for gravity and electromagnetism as a compromise between Newton and Einstein, the poten- tial and the metric, in a way that will get along with quantum mechanics. My guiding principle is provided by Goldielocks, who might ﬁnd a scalar theory too small, a rank 2 theory too large, so perhaps a rank 1 proposal will be just right.
I honestly love Newton’s potential theory. It is still in use today by rocket scientists who do not put an atomic clock onboard their ship. It gets half the answer right about light bending around the Sun. When a theory comes up short, we can either discard it or ﬁgure out the simplest way to lend a hand. A gravitational theory with a thousand poten- tials instead of one will be able to match every experimental test of gravity. Use Occam’s razor: a 4-potential should be more than adequate to match how gravity makes the mea- surement of time get a little smaller, while the measurement of 3-space gets a little larger.
I am torn between two lovers, Newton and Einstein, feeling like a fool. Thugs from Ulm will insist that gravity must be a metric theory. They have the experimental tests of the equivalence principle to prove it. They punch home the fact that the way to take the derivative of a connection that transforms like a tensor is through the Riemann curvature tensor. Drop the Ricci scalar into an action, vary it with respect to the metric, and out from the heavens ﬂies Einstein’s ﬁeld equations.
What’s wrong with that? A metric theory isn’t silly at all. One must be able to express gravity in terms of a metric. Based on my respect for Newton, I wonder if it is possible to ﬁnd a compromise between a larger 4-potential and a metric theory?
When we were young, we would write a covariant tensor as Aν. The differential ∂µ also transforms like a tensor. When we bring these two together, the 4-derivative of a 4-poten- tial, ∂µAν, the result does not transform like a tensor. The reason is that as we move around a manifold, the manifold - not the potential - might change. A means of accounting for a changing surface must be made. Here is the deﬁnition of a covariant derivative all students of gravity learn:
∇µAν = ∂µAν - ΓσµνAσ
Can you spot the symmetry and identify the group implied by this deﬁnition? Imagine we make a measure of one of these terms, say ∇0A0, and it happens to be 1.007. If one worked in ﬂat Euclidean spacetime, the connection would be zero everywhere, and every- thing would come from the change in the potential, ∂0A0. One could also decide to use a constant potential, so the dynamic metric’s connection would account for all the change seen, − Γσ00Aσ. One has the ability to continuously change the metric and thus the con- nection so long as there is a corresponding change in the potential which leaves the resulting covariant derivative invariant. This sounds like the group Diff(M) of all diffeo- morphisms of a 4D spacetime with the additional constraint that there are changes in the 4-potential such that the covariant derivative is unaltered.
Born background free, as free as general relativity, one must ﬁnd a differential equa- tion whose solution will dictate the terms of the dynamic metric. That is what the deriva- tive of the connection in the Riemann curvature tensor does: there are second derivatives of the metric whose solutions under simple circumstances can be found. I have chosen to study the simplest vacuum 4D wave equation:∇2Aµ = 0
It is vital to note that I did not write the D’Alembertian operator, which would have been a box without the 2. Instead this is a covariant derivative acting on a contravariant derivative acting on the 4-potential. The ﬁrst derivative will bring in a connection, and the second derivative will take the derivative of the connection, resulting in a second order differential equation of the metric, precisely what is needed to be background free. Can we ﬁnd interesting combinations of metrics and potentials that solve this differential equa- tion and is consistent with all tests of gravity to date?
Say we used a constant potential, where all the second derivatives were zero. Make the problem simple: a static, spherically symmetric, and non-rotating mass. For those skilled in the arts of differential geometry, it should be straightforward to show that the divergence of the connection of the exponential metric (below) is a non-trivial, entirely metric solution to the 4D wave equation. Compare the exponential metric in isotropic coordinates:
dτ2 = exp(-2 GM/c2R)dt2 − (exp(2 GM/c2R)(dx2 + dy2 + dz2)/c2
a nicely matched pair of exponentials, with the Schwarzschild solution in isotropic coordi- nates:
dτ2 = ((1-GM/2 c2R)/(1+GM/2 c2R))2dt2 − (1+GM/2 c2R))4(dx2 + dy2 + dz2)/c2
which is inelegant enough to rarely be seen in books on general relativity. Theorist prefer the Schwarschild coordinates while experimentalists must work with isotropic ones. Beauty may be in the eye of the beholder, but an exponential is the calling card of a deep insight into physics.
Either metric satisﬁes all tests of the equivalence principle because the solution is written as a metric. Either metric satisﬁes all tests of the weak ﬁeld because their Taylor series is the same to the terms tested. Either metric satisﬁes all strong ﬁeld tests because it is entirely about a metric, so there is no other ﬁeld to store energy or momentum. For an isolated system, the lowest mode of emission is the quadrapole moment. The metrics differ in second order effects by twenty percent in how much light is bent around the Sun, so it is a shame no one has been funded to get the data.
The 4D wave equation has been quantized, and written up in most books on quantum ﬁeld theory, in the section on relativistic quantization of the Maxwell equations. Two of the modes of emission are the transverse spin 1 ﬁelds of light. That is no surprise. The scalar and longitudinal modes are banished to a virtual state using a “supplementary con- dition” because the scalar mode would allow negative probabilities, a no-no. That is the way it is for a spin 1 ﬁeld theory where like charges repel. The ﬁeld strength reducible asymmetric tensor ∇µAν for this proposal can be split in two: an irreducible antisym- metric rank 2 tensor to do the work of electromagnetism with a spin 1 ﬁeld so like electric charges repel, and an irreducible symmetric rank 2 tensor to do the work of gravity with a spin 2 ﬁeld so like mass charges attract. Gravity couples to the 4-momentum, not the rank 2 stress-energy tensor. All forms of energy go into both sources, except one: the energy of a gravitational ﬁeld. To be consistent with electromagnetism, gravity ﬁelds do not gravitate. Should a gravity wave ever be detected and measured along six axes, the polarization of that wave will be transverse if general relativity is correct, but not if this uniﬁed ﬁeld proposal is accurate. Such data will be hard to get, but the difference would be unambiguous.
The speed of gravity is the speed of light, and so its ﬁeld strength tensor must be gauge invariant. The ﬁeld strength tensor ∇µAν is only gauge invariant if its trace hap- pens to be zero. That is where the massless graviton lives. When the trace is not zero, then the scalar ﬁeld formed from the trace of ∇µAν will break the U(1) symmetry of elec- tromagnetism. The Higgs particle is unnecessary. There is a quantum expression of the equivalence principle, a link between the spin 2 particle (∇µAν when tr(∇µAν) = 0) that mediates gravity and the scalar ﬁeld needed to establish inertia (tr(∇µAν) != 0).
There is an important beneﬁt to splitting the load for describing gravity between the connection and the changes in the potential. By using Riemann normal coordinates, an arbitrary point in spacetime can have a connection equal to zero. For that point, the energy will be zero. That has remained a technical problem for people trying to quantize general relativity. For this proposal, the energy contributed by the connection could be zero, but that contributed by the potential would be non-zero. Localized energy is a good thing.
Einstein had a great respect for Newton’s towering body of work. He might have appreciated this compromise between geometry and potentials which allows light to lay down with gravity in the same equation.