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    LoyalLoyalJusticeJustice
    Made withinDC&Austin
    Statements
    321,452
    Perspectives
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    42
    Home/Original/inverse
    See Original
    Inverse View

    It is not the case that The Vierbein (tetrad or Lorentz-structure) formulation of general relativity is necessary to incorporate Dirac's spinor fields ψ(x)

    ?Set your confidence on the premises below to see your aggregate.

    Reasons For

    2 perspectives
    Reason for 1 of 2
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    • 1.Geroch (1968) proved spinor structures exist on a spacetime iff it admits a global tetrad field, making the vierbein a sufficient but not strictly necessary formalism.
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    • 2.Alternative fiber bundle formulations (e.g., spin structures on principal bundles) can incorporate Dirac spinors without privileging the tetrad as the fundamental object.
      ?

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    • 3.The necessity claim conflates a particular representational convenience with a metaphysically indispensable structural requirement of the theory.
      ?

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    Reason for 2 of 2
    ?
    • 1.Kibble and Sciama independently showed that torsion-based Poincaré gauge theories couple spinors to gravity without reducing the foundational role to the vierbein alone.
      ?

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    • 2.If the GL(4) incompatibility argument in P1 establishes necessity, it establishes only that some local Lorentz structure is needed, not specifically the vierbein formulation as distinct from equivalent spin-connection approaches.
      ?

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    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.The spinor representation of the orthogonal group O(1,3) cannot be extended to a representation of the general linear group GL(n) for n=4
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    • 2.Incorporating Dirac's spinor fields ψ(x) into general relativity requires a group representation compatible with spinors
      ?

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