Breakdown of analogy of gravity to electromagnetism
All local gauge theories come from localizing a rigid global symmetry of a physical dynamical action.
The compensating gauge potential field restoring the symmetry by adding additional dynamical degrees of freedom is a connection field for parallel transport. It is not a tensor relative to the gauge group.
The tensor "force" is the curvature of the connection.
Einstein's universal gravity comes from localizing the 4D translational symmetry of ALL physical actions. Translational symmetries are generated by energy-momentum. The equivalence principle breaks the usual Noether theorem connecting symmetry to conservation laws. All forms of energy-momentum including the virtual zero point vacuum quanta are gravity charges. This is why dark energy accelerates the expansion of space in our pocket universe of the megaverse of parallel worlds on the populated cosmic landscape of chaotic eternal inflation of sprouting baby pocket universes. It is also essentially how flying saucers work. The direct role of virtual zero point quanta in gravity is very different from its indirect role in internal symmetry gauge forces where it is indirect like in Casimir force and in the Lamb shift for example. The difference is the equivalence principle that adds a qualitative new feature to the physics making the gravity field non-renormalizable in top-down quantization. This is because the gravity field is bottom-up emergent from the renormalizable internal symmetry quantum fields forming the pre-inflationary false vacuum.
If you think "gravity field" think tensor curvature geodesic deviation on pairs of neutral test particle in same breath as tensor electromagnetic Lorentz force field on a single test particle. Do not think the 100% inertial g-force AKA "weight" because that is a connection field with gauge freedom exactly like the electromagnetic 4-potential that includes the Coulomb potential. Newton's potential is part of the metric field at the next level up from the connection Levi-Civita field. Therefore, the simplistic formal analogy of V(Newton) = -GM/r to V(Coulomb) = Q/r is very deceptive indeed - a Siren that has sunk Zielinski's ship!
Note raw alleged evidence for flying saucers on video.
On Aug 1, 2006, at 11:54 PM, Jack Sarfatti wrote:
It's also confusing because gravity really is a force in Newton's theory, though not in Einstein's. Einstein eliminates gravity as an intrinsic force like electromagnetic force replacing gravity force.
It's also confusing because gravity really is a force in Newton's theory, though not in Einstein's. Einstein eliminates gravity as an intrinsic force like electromagnetic force replacing g-force with curved space-time.
Note the "curvature" in internal fiber space is the gauge "force". Therefore, the correct analogy between gravity and electromagnetism
space-time curvature analogous to electromagnetic force
d^2(x^u - x'^u)/ds^2 = R^uvwl(dx^v/ds)(dx'^w/ds)(x^l - x'^l) gravity tidal acceleration for a pair of geodesic test particles
(This local frame-independent tensor equation can be calculated in either a LIF or a coincident LNIF)
is properly compared to the electromagnetic Lorentz force tensor equation
d^2x^u/ds^2 = (e/m)F^uvdx^v/ds for a single test particle with charge to mass ratio e/m.
In the language of Cartan's differential forms for space-time gravity for zero torsion limit
R = DW
R = curvature 2-form
W = spin-connection 1-form
D = d + W/\ = covariant exterior derivative
/\ = exterior multiplication
For general internal symmetry Yang-Mills fields (EM as trivial Abelian case)
D = d + A/
A = 4-potential connection 1-form
F = DA = Yang-Mills curvature 2-form ---> Yang-Mills 'force field" analog to Lorentz force eq.
DR = D^2W = 0 (Gravity) and DF = D^2A = 0 (Yang Mills) analog of Faraday's law & no magnetic monopoles
D*W = *J source eq. D*F = *J (Yang-Mills) source eq.
Here the analogy breaks down from the equivalence principle.
The Gravity source equation is a 4-form equation because W is a 1-form and *W is a 3-form. So *J is a 4-form.
J is 0-form local scalar.
With tetrad indices this is
D*Wab = *Jab
that maps to
Guv = kTuv
Local conservation of matter source stress-energy current density is from
D^2*Wab = D*Jab = 0
i.e. Tuv(Matter)^;v = 0
In contrast for Yang-Mills fields
D*F = *J is a 3-form eq. analog of Ampere's law and Gauss's law.
D*J = 0 local conservation of electrical current densities.