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We consider a one-parameter family of composite fields -- bi-linear in thecomponents of the stress-energy tensor -- which generalise the$\mathrm{T}\bar{\mathrm{T}}$ operator to arbitrary space-time dimension $d\geq2$. We show that they induce a deformation of the classical action which isequivalent -- at the level of the dynamics -- to a field-dependent modificationof the background metric tensor according to a specific flow equation. Eventhough the starting point is the flat space, the deformed metric is generallycurved for any $d>2$, thus implying that the corresponding deformation can notbe interpreted as a coordinate transformation. The central part of the paper isdevoted to the development of a recursive algorithm to compute the coefficientsof the power series expansion of the solution to the metric flow equation. Weshow that, under some quite restrictive assumptions on the stress-energytensor, the power series yields an exact solution. Finally, we consider a classof theories in $d=4$ whose stress-energy tensor fulfils the assumptions abovementioned, namely the family of abelian gauge theories in $d=4$. For suchtheories, we obtain the exact expression of the deformed metric and thevierbein. In particular, the latter result implies that ModMax theory in aspecific curved space is dynamically equivalent to its Born-Infeld-likeextension in flat space. We also discuss a dimensional reduction of the lattertheories from $d=4$ to $d=2$ in which an interesting marginal deformation of$d=2$ field theories emerges.

Metric approach to a $\mathrm{T}\bar{\mathrm{T}}-$like deformation in arbitrary dimensions