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  The generalized Einstein equations

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Let (M,g) be an EInstein manifold, Ricci flat Ric(g)=0 and X a vector field, I consider M.R and the metric gX:

gX=g+(X+dt)(X+dt)

The scalar curvature of gX is rX. The generalized Einstein equations are:

X=(drX)

Have we non trivial solutions of the generalized Einstein equations?

asked Nov 21, 2021 in Mathematics by Antoine Balan (-80 points) [ no revision ]

What is the purpose of this question, i.e, why do you ask? Is there any physics background to this question, a physics-based motivation to attempt such a generalisation? Is  X just any vector field? What is t? Probably it is related to "time", but is (dt) just an arbitrary time-like vector or is there a specific choice involved about which your question is silent? In this context also note that your metric gX and  curvature rX are only labelled by X, any choice of t does not show up. In what sense should X=(drX) be a generalisation of the Einstein equations?

What if in gX you choose X=(dt)?

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