Is there a novel Einstein–Gauss–Bonnet theory in four dimensions?

Abstract

<jats:title>Abstract</jats:title><jats:p>No! We show that the field equations of Einstein–Gauss–Bonnet theory defined in generic <jats:inline-formula><jats:alternatives><jats:tex-math>$$D&gt;4$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>D</mml:mi><mml:mo>&gt;</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:math></jats:alternatives></jats:inline-formula> dimensions split into two parts one of which always remains higher dimensional, and hence the theory does not have a non-trivial limit to <jats:inline-formula><jats:alternatives><jats:tex-math>$$D=4$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>D</mml:mi><mml:mo>=</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:math></jats:alternatives></jats:inline-formula>. Therefore, the recently introduced four-dimensional, novel, Einstein–Gauss–Bonnet theory does not admit an <jats:italic>intrinsically</jats:italic> four-dimensional definition, in terms of metric only, as such it does not exist in four dimensions. The solutions (the spacetime, the metric) always remain <jats:inline-formula><jats:alternatives><jats:tex-math>$$D&gt;4$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>D</mml:mi><mml:mo>&gt;</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:math></jats:alternatives></jats:inline-formula> dimensional. As there is no canonical choice of 4 spacetime dimensions out of <jats:italic>D</jats:italic> dimensions for generic metrics, the theory is not well defined in four dimensions.</jats:p>