On Apr 3, 2014, at 2:09 PM, Paul Zelinsky wrote:

"Also, what do you think the Minkowski fiber bundle represents in modern formulations of GR? It represents a local mapping of the curved base space inner products determine by g_uv onto the Minkowski fibers defined by the globally flat Minkowski metric n_uv."

To which I replied:

To which I replied:

frame field for LNIF is eu(LNIF) with curvilinear metric guv(LNIF).

One can always find LNIFs where in Taylor series about origin

g^u^v(LNIF) ~ n^u^v(Minkowski) + {Levi-Civita Connection}^u^vw&x^w + {Riemann Curvature Tensor}^u^vwl&x^w&x^l + ….

ds^2 = guv(LNIF)e^u(LNIF)e^v(LNIF)

frame field for LIF (Cartesian coordinates a must as Einstein stipulates in his papers) eI(LIF) tangent bundle fiber metric Taylor expansion is

n^I^J(LIF) = n^I^J(Minkowsk) + {Riemann Curvature Tensor}^I^JKL&x^K&x^L + ….

ds^2 = nIJ(LIF)e^I(LIF)e^J(LIF)

Small font indices u,v ... are in the LNIF base space

Caps I,J are in the LIF fiber

The tetrad transformation is

e^u(LNIF) = e^uIe^I(LIF) etc.

EEP means that

{Levi-Civita Connection}^I^JK = 0

though in general

{Riemann Curvature Tensor}^I^JK =/= 0

{Levi-Civita Connection}^u^vw =/= 0

Riemann Curvature Tensor}^I^JKL = 0

Riemann Curvature Tensor}^I^JKL =/= 0

Note that e^I(LIF) is a set of 4-vectors with components e^Iu

e^u(LNIF) is a set of 4-vectors with components e^uI

e^uIe^Iv = kronecker delta uv etc. ORTHOGONAL GROUP O(1,3)

ds^2 = nIJ(LIF)e^Ie^J = guv(LNIF)e^ue^v

LIF Alice and LNIF Bob are COINCIDENT

LIF Alice has zero proper acceleration

LNIF Bob has non-zero proper acceleration

ds is invariant space interval between 2 neighboring events measured simultaneously by both Alice and Bob.

Since we impose COINCIDENCE no problem with simultaneity.

Also clock postulate that proper acceleration of clocks in LNIF can be synchronized to clocks in LIF if they are coincident.

Category: MyBlog

Published on Monday, 31 March 2014 17:35