A Simpler Alternative to Einstein's General Relativity
Tantalizingly simpler, an exact analogy between Maxwell's Equations and an approximation to Einstein's Field Equations gives a theory that is accurate in many astronomical observations. But ultimately incorrect, as shown experimentally by the weariness of two cosmic ballerinas ….
I recently wrote two articles showing how special relativity applied to Coulomb's inverse square law can be used to derive all of Maxwell's equations and electrodynamics. It is a most beautiful introduction to the beginning theoretical physicist of the power of what Eugene Wigner famously called, "The Unreasonable Effectiveness of Mathematics in the Natural Sciences". That mathematics could offer such intimate insight into the soul of Nature Herself was a lifelong source of wonder and sense of beauty to the gentle and thoughtful Eugene Wigner, one of the twentieth century's great physicists.
It is a source of profound wonder and sublime beauty to me, too.
So, Coulomb's law is the Inverse Square Law for the electrostatic force between two charges as a function of their separation. When we apply special relativity to Coulomb's law, we find that the full glory of Maxwell's Electrodynamics results.
Hold On: Gravitation is Inverse Square Too. Can't We Do the Same Here?
An astute student will notice that there is another famous inverse square law in classical physics that is wholly analogous to Coulomb's Law. It is, of course, Newton's inverse square Law of Universal Gravitation. So, can we not derive a dynamic theory of gravity by applying special relativity to Newton's Law of Universal Gravitation?
We can, and indeed there is a vector theory of gravitation totally analogous to Maxwell's Equations. It is called Gravitoelectrodynamics, Gravitoelectromagnetism or GEM.
It is highly accurate for many dynamical effects in the gravitational phenomena. Including the accurate account for the famous apsidal precession of the orbit of Mercury around our Sun that Newtonian Gravitation failed to account for, but which Einstein's Gravitational Field Equations predicted perfectly. GEM can do the same.
In vector notation, analogous to Gibbs-Heaviside notation for Maxwell's electromagnetism, GEM is described as follows. The units are SI units, so that Eg is the gravitational force per unit mass, and Bg the gravitomagnetic field, whose units are Hertz and which physically is the famous Lense-Thirring frame dragging rate (more on this below).
The Gravitomagnetic and Gravitoelectric fields Bg and Eg are defined by their actions on gravitational mass m through the GEM analogy of the Lorentz Force Law. The gravitomagnetic force acts normal to both a particle's velocity and the gravitomagnetic field.
As in my second "Three Minute Electromagnetism" article, you can probably guess that an exterior / geometric calculus formulation of GEM would be more to my liking. We'd simply assemble the components of Bg and Eg into a two form — a Gravito-Faraday tensor — analogous to the electromagnetic Faraday Tensor, and write down:
The minus sign in the second equation arises because gravitational charge (rest mass) is positive and like masses attract.
So, what shakes? Why do we need Einstein's theory of gravitation at all, if GEM is so accurate? Why doesn't the Maxwellian approach of applying Special Relativity to the inverse square Newton's Gravitational Law work, whereas it works for electromagnetism.
The reason has ultimately to do mysterious factor of 4 in the Lorentz force equation that is absent in electromagnetism, and it is the result of the fundamental difference in symmetry between gravitation and electromagnetism. And no, it is not simply that GEM is "out by a factor of 4"; we simply use the right force equation on the left and get highly accurate results. This factor of 4 is what weak field general relativity tells us to do — it's the underlying root of the factor of 4, not the factor itself, that's the problem!
As we shall see below, the simplest gravitational radiator is, in fact, a quadrupole (a fourfold symmetrical pattern which is invariant under a 180º rotation, whereas a dipole requires 360º, like most general geometrical objects for invariance). This difference between radiation symmetry in the two theories is the key to experimental falsification of GEM, whereas General Relativity is found to be in keeping with experiment.
Another view of the reason for the need for a different theory of gravitation as compared to electromagnetism is that charge is truly Lorentz covariant, whereas, as we have already discussed in two articles on the outdated and obselete concept of mass in physics, mass is no longer useful in modern physics, since it is not Lorentz invariant, and only the Lorentz invariant concept of rest mass is regarded as useful.
What to Use as the Gravitational Source?
So we could use the Lorentz invariant rest mass as the gravitational source.
This, however, seems contrived, since total energy seems more likely to be a source, given that there no longer seems to be any fundamental distinction between mass and energy. "Rest Mass" is simply a name we give to a particle's total energy when it is in the same rest frame as us. And, for massless particles like the photon, there is no rest mass, but there's still an energy and momentum.
In contrast, charge is Lorentz invariant, and charge in motion is described by the current denstiy four-vector J. This four-vector contains the Lorentz invariant charge as its time component, and the current density vector as its spatial components. The "mass" analogy would be rest mass as the time component, and the rest mass current density as the spatial components. But where would the energy and momentum correctly come in?
For Maxwell's equations, electric charge is a Lorentz invariant scalar, and the source of the electromagnetic field is the four-current density.
For a relativistic gravitation theory, mass is not Lorentz invariant, the Four-Momentum is a Lorentz Invariant vector (one tensor) and so, analogously with the fact that the scalar charge becomes the four-current density source of the Maxwell's equations, the Lorentz invariant four momentum must become a two tensor as the correct, relativistic gravitational source. This is the ten independent component, symmetric 4×4 Stress Energy Tensor.
Let's Ask Nature Herself through Experiment
And, moreover, the ultimate test of correctness is experiment. Can we find an experiment that tells GEM apart from the prediction of the Einstein Field Equations?
There is, and it is the observation of the change with time of a Pas de Deux by two ancient cosmic ballet dancers. The famous Hulse — Taylor Binary Star. Before humanity's first direct momentous observation of gravitational waves on the 14th of September, 2015, when the gravitational wave telescopes LIGO in the US and VIRGO in Italy registered the first gravitational waves ever observed in the event GW150914, the slow decrease with time of the orbital period of the Hulse-Taylor Pulsar was our most direct confirmation of the existence of gravitational waves.
The pulsar in the Hulse Taylor binary is a highly magnetized neutron star, with a stupendous spin rate of seventeen times per second. This spinning magnetic field launches a 17Hz periodic radio signal towards Earth.
On closer examination, the 59 millisecond period itself was observed to speed up and slow down every so slightly every 7.75 hours. It was eventually realized that the pulsar belonged to a binary, and its period was being periodically modulated by Doppler shift, as it succesively hurtles towards and frowards the Earth in its orbiting motion. On the approach half of the orbit, the period is blue shifted, and red shifted during the recession half.
Since the Hulse-Taylor Binary was discovered in 1974, its mutual orbit period has steadily decayed, according to the plot below.
Between 1974 and 2004, the binary lagged a mere 40 seconds over 20 years behind the signal that would have been received if the period were truly constant at its 1974 value. The red dots in the image are the observation points.
General Relativity, and GEM both predict a slowdown in the mutual orbit. The blue curve in the figure below is the predictions of General Relativity for a mutual orbit between two neutron stars of almost equal masses of 1.4 Solar Masses.
GEM's prediction is a calculation generalized to a mutually orbiting binary system of the famous Larmor — Liénard Radiation Formula. And these two calculations differ in their predictions markedly — the Larmor — Liénard Fomula predicts a result that is way too big. For example, in the nonrelativistic example of the Earth's orbit of our Sun, General Relativity predicts that the Earth radiates gravitational waves at the tiny rate of roughly 200watts. Given that the Earth's total orbital kinetic energy is 1.14×10³⁶ joules, the inital 1/e decay time for the inspiral is 5.7×10³³ seconds, or 1.8×10²⁶ years, or about 1.3×10¹⁶ times the known age of the Universe!
In contrast, the Larmor — Liénard formula predicts a much heftier radiation of about 3 Gigawatts from our Earth. This sounds a great deal more, indeed the decay rate is 15 million times faster, but again, the 1/e decay time in this case would be 3.8×10²⁶ years, still a thousand million times the known age of the Universe. The vastly bigger slowdown in Earth's period of even 1 part in 3.8×10²⁶ per year (0.083 attoseconds or 83 zeptoseconds per year) predicted by GEM is still far too small to be detactable by present astronomical and timing methods.
But when we examine the predictions of the two theories with the Hulse Taylor binary, General Relativity yields the observed decay rate, whereas the equivalent of the Larmor — Liénard calculation — millions of times faster — is way off the scale. The latter is clearly, utterly wrong.
The fundamental difference here is that a Larmor — Liénard radiator, like a Maxwellian accelerated electron or a GEM accelerated mass, has a dipole shapen field. In contrast, a General Relativistic Gravitational Radiator radiates with a quadrupole pattern. This is related to the fact that the source in Maxwell's equations is a four-vector — the four current density, whereas the source in General Relativity is a two tensor — the Stress Energy Tensor. In a quantized version of gravity, it is, for the same reason, foreseen that the graviton is a spin-2 particle, whereas the electromagnetic mediator — the photon — is a spin-1 particle.
Incidentally, the factor of 4 in the Lorentz Force Law for a GEM mass above is also related to this difference in symmetry between the GEM and General Relativity systems. One can derive this factor of four directly from the weak field Einstein equations when they are used to derive the acceleration of test masses in the gravitational field.
Quadrupole radiation has much smaller scope to "escape" than dipole radiation. Quadrupoles tend to be much less efficient radiators than dipoles, and therefore the Larmor — Liénard radiation from a GEM Gravitational Radiator is much, much bigger that the gravitational radiation in General Relativity.
Accuracy is restored to GEM predictions if we disregard the dipole contribution to radiation and instead discard everything but the higher order quadrupole terms. But this is clunky, and the use of the linearized Einstein equations and working with the full stress energy tensor become mathematically simpler than sifting through terms in an interaction series derived from GEM. The Larmor formula and its symmetry assumptions are utterly useless.
Nature thus tells us which is the more correct theory, as long as we listen carefully to Her.
What is GEM Good For?
As such, the equations of GEM only approximate gravitation well when:
- Relative velocities between all gravitating masses are small compared to c, so that their rest masses dominate the total energy of all these masses and thus everything in the Stress energy tensor vanishes aside from its first row and first column (recall that it is symmetric).
- Thus one can effective use a vector "four-mass-current" as a source;
- Spacetime in between masses is substantially flat;
- Only quadrupole contributions to gravitational radiation are heeded;
GEM can correctly predict the Lense-Thirring Precession (see also here) at low spin rates (so that all masses have a relative velocity much smaller than c). Indeed, Eg in the GEM equations is the force per unit mass (the acceleration) of test masses, whereas the gravitomagnetic field Bg represents the Lense-Thirring frame dragging rate in Hertz. GEM's prediction of the Lense Thirring effect (which agrees with General Relativity) was rigorously tested by Gravity Probe B and found to be correct during the Gravity Probe B mission.
Indeed, the simplicity of GEM makes it popular amongst researchers in its régime of validity, particularly those specializing in frame dragging and Lense-Thirring phenomena. It is far easier to work with and visualize for simple general relativistic problems.
Geometry, Gravity and Electromagnetism
GEM, first proposed by Oliver Heaviside in 1893, is the best gravitational theory we can muster with the constraint of a flat, Minkowski space background spacetime. It is sometimes said that General Relativity is fundamentally different in being a purely geometrical theory of Gravitation, but this assertion is quite misleading.
Maxwellian Electromagnetism can equally well be given a geometric interpretation. For example, Maxwell's equations have a well known gauge symmetry. Since, in exterior calculus words, the Faraday — Gauss Magnetic laws can be stated that the Faraday tensor is a closed two — form, i.e. dF=0, the Poincaré Lemma means that there is a four potential A such that F=dA. The Ampère — Gauss Electric laws relate the Faraday's dual to the current, i.e. ★d★F = J. Therefore, we can add the Four Gradient, i.e. exterior derivative, d𝜃 of any twice differentiable scalar field 𝜃 to the four potential A, so that A↦A+d𝜃, yet the physical Maxwell equations are left unchanged. The universal Bianci Identity d²=0 of the exterior derivative enforces this invariance, so that F↦d(A+d𝜃)=dA+d²𝜃=dA=F. Maxwellian Electromagnetism is invariant modulo any twice differentiable, scalar gauge field.
Thus, for example, we have several Gravitational Theories from the 1910s and 1920s that are similar to Kaluza-Klein theory, whose underlying geometric manifold is five dimensional. The fifth dimension supports the scalar gauge field 𝜃 and it is "compact" or rolled up into a cylinder (see discussion of the Dirac-Maxwell equations below). The Electomagnetic field and the Lorentz Force Law describing its action on a "free falling" particle show up in the equations of geodesic (shortest path) motion for the particle. Maxwell's Electromagnetism can be understood as the curvature of the line bundle when 5D Kaluza-Klein space is thought of as a fiber bundle where the base space is the Einstein manifold and the fibers are the compact dimension at each point supporting the gauge field.
Kaluza and Klein regarded this compactness (their so-called "cylinder condition") as ugly and telling against their own theory, but later it was found to have a compelling physical interpretation. The Dirac equation for a single electron can be coupled to the Maxwell equations using suitable interpretations of the Dirac electron wavefunction for current and charge density. When this is done, the scalar gauge field 𝜃 has an interpretation as the local phase of the Dirac wavefunction. In quantum mechanics, phase only has physical significance when quantum entities interfere with one another. So the Dirac spinor wavefunction 𝜓(x) can be multiplied by any scalar pure phase function of the form 𝜓(x)↦exp(-i 𝜃(x)) 𝜓(x), where x is the spacetime position vector without changing the physics of the electron.
Lo and behold, the coupling with the Maxwell Equations is balanced by a Gauge Transformation of the Maxwell equations of the form A↦A+d𝜃 as above when we use the phase function exp( — i 𝜃(x)). The background electromagnetic field can thus be seen as actively "hiding" a global electron phase, and rendering it experimentally indetectible. The coupling with electrons is the reason for the Maxwellian Gauge invariance! The same is true for so called "minimal coupling" between the electron and either the relativistic Klein-Gordon equation or the nonrelativistic Schrödinger equation. The electromagnetic gauge field arises from the global phase common to the wavefunctions of a set of charges interacting with the field.
A minimally coupled Dirac electron in the Hydrogen Atom Potential reveals correct predictions for the Lamb Shift, which is the experimentally observed splitting into two distinct energy levels of electron orbitals which have the same (degenerate) energy levels when described by the bare Hydrogen Potential Dirac Equation. Its correction is not quite as accurate as full Quantum Electrodynamics. But the presence of the background electric field and its coupling to the Hydrogen Electron is revealed for this added complexity to the Dirac Electron and moreover makes a throroughly excellent approximation to the experimentally observed results.
Gravity is far from unique in having a geometrical interpretation, and indeed, it can probably be said that gravity is only special in being the physical phenomenon that alerted theoretical physicists to the power of geometrical interpretations of their theories.
References
I am mightily impressed that the following is a bachelor's thesis. This lady never fell for the trap i fell for, which was to go to the Larmor formula and its assumed incorrect symmetries. She develops the derivation of the GEM equations from the linearized Einstein equations, carefully always referring back to the latter for first-principles, fundamental guidance. I am sure she will have an illustrious career in physics. I have seen PhDs granted for considerably less insight and beauty….
Bahram Mushhoon seems to be one of the key researchers in ongoing interest in GEM:
Bahram Mushhoon, "Gravitoelectromagnetism: A Brief Review", Chapter 3 in the review almanac of research in the field by influential authors, "The Measurement of Gravitomagnetism: A Challenging Enterprise", compiled and edited by Lorenzo Iorio, Nova Publishers, 2007.
