## A Derivation of Einstein Gravity without the Axiom of Choice: Topology Hidden in GR

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Spaans, M.

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##### Abstract

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A derivation of the equations of motion of general relativity is presented
that does not invoke the Axiom of Choice, but requires the explicit
construction of a choice function q for continuous three-space regions. The
motivation for this (seemingly academic) endeavour is to take the background
independence intrinsic to Einstein gravity one step further, and to assure that
both the equations of motion and the way in which those equations of motion are
derived are as self-consistent as possible. That is, solutions to the equations
of motion of general relativity endow a three-space region with a physical and
distinguishing geometry in four-dimensional space-time. However, in order to
derive these equations of motion one should first be able to choose a
three-space region without having any prior knowledge of its physically
appropriate geometry. The expression of this choice process requires a
three-dimensional topological manifold Q, to which all considered three-space
regions belong, and that generates an equation of motion whose solutions are q.
These solutions relate the effects of curvature to the source term through the
topology of Q and constitute Einstein gravity. Q is given by 2T^3+3S^1xS^2, and
is embedded in four dimensions. This points toward a hidden topological content
for general relativity, best phrased as: Q and q provide a structure for how to
choose a three-space region irrespective of what geometric properties it has,
while at the same time Q and q determine that only GR can endow a three-space
with those geometric properties. In this sense, avoiding the Axiom of Choice
allows one to gain physical insight into GR. Possible links with holography are
pointed out.

Comment: final edits: more detail on derivation of equations of motion for q

Comment: final edits: more detail on derivation of equations of motion for q

##### Keywords

General Relativity and Quantum Cosmology, High Energy Physics - Theory, Mathematical Physics