Is there a novel Einstein–Gauss–Bonnet theory in four dimensions?
Abstract
Abstract No! We show that the field equations of Einstein–Gauss–Bonnet theory defined in generic $$D>4$$ <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> dimensions split into two parts one of which always remains higher dimensional, and hence the theory does not have a non-trivial limit to $$D=4$$ <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> . Therefore, the recently introduced four-dimensional, novel, Einstein–Gauss–Bonnet theory does not admit an intrinsically 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 $$D>4$$ <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> dimensional. As there is no canonical choice of 4 spacetime dimensions out of D dimensions for generic metrics, the theory is not well defined in four dimensions.