updates made mainly in the introduction section and i would like to say more but my knowledge of the historical accounts on KK theory ran short besides, my plan was to make this draft as short as possible. the draft is still in the process of completion…
https://drive.google.com/file/d/1si4ifALrdrQUOaN9AUknjVBf_lMtV7Ao/view?usp=sharing
in the static setting, it's straightforward to establish correspondence to Schwarzschild solution yet, there is also the temptation to consider a setting where the fields evolve with both lower dimensional coordinates
in this (1+1)-dimensionally reduced form of GR, the form of the conformal field is handed down by an auxiliary field not from the equation of motion for the conformal field. this auxiliary field is put in by hand and the acceptable form must satisfy a constraint resulting from the equation of motion for the conformal field.
in this: the equation of motion for the conformal scalar field does not solve for a particular form of this scalar, instead this equation of motion yields an identity equation... otoh, this conformal scalar is a functional of an auxiliary scalar field which itself does not have a specific form that is the only acceptable, not just one form can satisfy the identity equation
https://foxtrotpaparomeo.wordpress.com/2023/11/27/27112023rough-draftstatic-solutions-to-einsteins-field-equations-in-11-dimensions/
updated it today, you may go directly to (22.11) for the de Sitter-like solution
https://foxtrotpaparomeo.wordpress.com/2023/12/10/10122023rough-draftstatic-solutions-to-einsteins-field-equations-in-11-space-time/
(additional resource material re G -- measuring G during the big bang nucleosynthesis and comparing it to its present value, results at 2sigma)
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When combining the measured abundances and the baryon density from CMB observations by Planck, we find G_BBN/G_0 = 0.99^+0.06_−0.05 at 2σ confidence level.
https://arxiv.org/abs/1910.10730
spotted typo in eqn((22.3) has been corrected and in this update, full spacetime solution (fields vary in both space and time) briefly explored
https://foxtrotpaparomeo.wordpress.com/2024/02/08/08022024rough-draftstatic-solutions-to-einsteins-field-equations-in-11-space-time/
with a revised title and the abstract made slightly shorter omitting some historical accounts
https://foxtrotpaparomeo.wordpress.com/2024/02/22/23022024rough-draftstatic-solutions-to-dimensionally-reduced-einsteins-field-equations-in-11-dimensions/
in later updated versions of this draft i shall include some concise details to illustrate through calculations that in (1+1) dimensions Einstein's tensor vanishes
https://foxtrotpaparomeo.wordpress.com/2024/07/03/04072024rough-draftstatic-solutions-to-einsteins-field-equations-in-11-space-time/
updated it today to include very short details that establish consistency of the equation of constraint (23.4) with the static metric solution (20.14) and such consistency is granted by the identity equation (22.5) in the static case
https://foxtrotpaparomeo.wordpress.com/2024/07/03/04072024rough-draftstatic-solutions-to-einsteins-field-equations-in-11-space-time/
additional basic resource material to my venture into 1 + 1 GR (you can refer to eqn (23) --- the vanishing of Einstein's tensor strictly in 1 + 1 dim)
https://digitalcommons.calpoly.edu/physsp/65/
What would be the properties of a universe with only one spatial dimension and one time dimension? General relativity in 1+1 dimensions is unique since the two curvature terms in the Einstein field equations cancel. This makes the Einstein field equations algebraic rather than differential equations. This special feature can make 1+1 dimensionality attractive as an instructional tool to simplify the mathematics that many beginners find opaque. We explore the implications and features of the Einstein field equations in 1+1 dimensions and find they provide a surprisingly rich and interesting model. We then study an alternate theory and its implications for a potentially stable wormhole solution.
digitalcommons.calpoly.edu