Coney Island of the Non-Inertial Mind
"Geodesic" means straightest possible path. It's a relative idea depending on the space.
1. In Newton's 17th Century Theory of Gravity there is an objective gravity force field.
For example, the conservative gravity potential energy per unit test mass in Newton's geodesic global inertial frame of reference "the Lab frame" of Physics 101 for a sphere of mass M is
V(Newton) = - GM/r
The objectively real gravity force per unit test mass is
f(Newton) = - GradV(Newton) = - GM/r^2 pointing radially inward
A freely falling cannon ball is not on a Newtonian geodesic. It is generally on a parabolic path if it has initial speed perpendicular to the radial vector with the above f(Newton).
This interpretation changes completely in the switch to Einstein's General Relativity. Newton's above pure gravity force is completely eliminated because the path of the freely falling cannon ball is now a zero acceleration geodesic in the curved space-time. The Lab frame on surface Earth is not a Global Inertial Frame (GIF) it is a non-geodesic Local Non-Inertial Frame (LNIF) and the Lab Observer is objectively accelerating off geodesic from the quantum electrical forces pushing on him from the Earth's surface.
Locally frame-invariant objective accelerations vanish on geodesics.
Objective accelerations are contingent properties of contingent external quantun electrodynamical forces acting on the accelerating bodies on non-geodesic paths. Then and only then are 100% inertial g-forces detected as when a pilot "pulls g's" in a dogfight or when you step on a scale to weigh yourself. That is a non-intrinsic historical accident not a feature of the objective geometry of the curved spacetime.
Objective local frame invariant curved spacetime geometry is 100% geodesic deviation. It is only the zero objective acceleration geodesic structure that determines the objective geometry or geometrodynamic field. The accidental non-geodesics from external electrodynamic internal symmetry gauge fields have nothing to do with the objective geometry of the geometrodynamic field. The objectively real geometrodynamic field is the geodesic deviation tensor curvature field. That is the deep meaning of the equivalence principle misunderstood even by some professional physicists.
Now in the counter-intuitive example below from Vilenkin the situation is as follows.
The metric field guv looks different to different observers. For geodesic observers the metric field is actually globally flat Minkowski 3 + 1 space-time with zero curvature tensor everywhere except on an ideal zero thickness spherical 2D deSitter surface lightlike event horizon of coherent macroquantum vacuum topological vacuum defect that at time t = -infinity has infinite area. The sphere collapses and at time t = 0 it stops at a finite area /\^-1 = c^4/g^2 and then reverses expanding back to infinite area at t = + infinity. This is what the inertial GEODESIC observers INSIDE the collapsing-expanding vacuum defect sphere see. For them their spacetime patch is globally flat with zero 3+1 curvature tensor. The situation is very different for observers outside the sphere. Their spacetime patch is not globally flat.
Now there is a special class of uniformly non-accelerating LNIF "Rindler observers" with local frame invariant radial uniform acceleration g for which the really flat 3 + 1 Minkowski spacetime looks crazy with a complicated metric field. These observers see a plane wall instead of the sphere seen by the geodesic observers. If they compute the 3 + 1 curvature tensor they will get zero exactly like the geodesic observers. If they compute the radial slice they see a 1 + 1 Rindler submetric field. If they compute along planes parallel to the plane wall they get a 2 + 1 constant curvature /\ = c^4/g where to these non-inertial Rindler LNIF observers the geodesic test particles are in an effective Newtonian potential
V(Newton) = -gx + g^2x^2/2c^2
with illusional "gravity force" per unit test mass
f(Newton) = -dV/dx = +g (repulsive) - (g/c)^2x (attractive)
However, this is artificial because these Rindler LNIF observers must fire rockets (non gravity forces) to see this crazy artificial metric field. What they see is not fundamental but a wacky contingency in a Rube Goldberg contrived situation. The objective geometrodynamic field inside the deSitter spherical vacuum defect surface of Dirac delta function singularity is globally flat Minkowski spacetime. The so-called g-field above is 100% inertial not of fundamental geometrodynamic meaning.
Whenever the 3 + 1 curvature tensor vanishes in a region that region is Minkowski relative to geodesic local observers. There is no such thing as a Newtonian-like objective g-force in Einstein's theory ever. Any curvilinear metric representations that show a uniform g-force in particular must have zero objective curvature and such curvilinear representations are 100% illusionary artifacts of firing rockets in space in nutty ways like looking at an object through a Fun House Mirror in the Coney Island of a Demented Mind! ;-)
On Aug 23, 2006, at 10:07 PM, Jack Sarfatti wrote:
OK Paul you are right that Vilenkin does say the 3 + 1 curvature tensor R^uvwl = 0. But this 3 + 1 flat region is not the entire space-time of the exotic vacuum wall source tensor Tuv.
In fact he says that for inertial geodesic observers in this coordinate patch that does not cover the whole manifold
ds^2 = dt*^2 - dx*^2 - dy*^2 - dz*^2 Minkowski, but for them the wall is not flat, it is a sphere!
x*^2 + y*^2 + x*^2 = /\^-1 + t*^2
This vacuum bubble sphere for the geodesic observers is 2 + 1 DeSitter, it contracts from infinity to a minimum area /\^-1 then re-expands to infinity with constant acceleration g.
Also the metric in question only covers a fraction of the space-time of the source.
To review, the t-y,z slice is a 2 +1 DeSitter space. Therefore the 2 + 1 slice "parallel" to the plane is a subspace of constant positive curvature
/\ = g^2/c^4
embedded in the 3 + 1 flat space. So that's fine. No contradiction there. You can embed a curved subspace in a larger flat space. Also the 1 + 1 x-t slice is flat for a constantly accelerating Rindler observers. The metric field guv looks different for different sets of observers.
There are 3 separate metric fields here guv(3+1), gu'v'(2+1) & gu"v"(1 + 1) each with their own curvature tensors unto themselves.
We were talking apples and oranges.
The 3 + 1 curvature tensor of guv(3+1) can vanish and the curvature tensors of subspaces gu'v'(2+1) & gu"v"(1 + 1) not vanish! In particular the 2 + 1 slice parallel to the planar source is a DeSitter space of constant positive curvature /\. Its curvature tensor components are factors in the the larger 3 + 1 curvature tensor that vanishes. The algebra is complicated but the general idea is simple.