# Is there any synonym to the “flaring out condition” of travesable wormholes?

Given the knowledge of Einstein Field Equations, the one could ask if it is possible to connect asymptotically regions of a given spacetime, such that a time-like curve is possible to transit between this two regions.

This mechanism of a "connection between spacetime regions (a.k.a, a spacetime bridge)" is called travesable wormhole and is given by the metric:

$$ds^{2} = -e^{2\Phi(r)}dt^{2}+\frac{1}{1-\frac{b(r)}{r}}dr^{2}+r^{2}\{d\theta^{2}+sin^{2}d\phi^{2}\} \tag{1}$$

In order to visualize the geometry given by $$(1)$$ the one could create a embbeding diagram, and this diagram give the intuition of a particular region called throat. After some calculations with the embbeding diagram $$$$, the one realizes that far from the throat the spacetime becomes asymptotically flat in both directions. This requirement then implies that the embbeding surface "flares out" at the throat $$$$. This means then (at the throat) that:

$$\frac{\mathrm{d}^{2}r}{\mathrm{d}z^{2}} = \frac{b(r)-b'(r)r}{2b(r)^{2}} > 0 \tag{2}$$

Now, concerning the imposed energy-momentum tensor (the relativistic perfect fluid $$T_{\mu\nu} = Diag(\rho,p_{1},p_{2},p_{3})$$), the papers $$$$ and $$$$ gives the following Einstein Field Equations:

$$\rho = \frac{b'(r)c^{2}}{8\pi Gr^{2}} \\ \tau = \frac{c^{4}(b(r)/r - 2(r-b(r)))\Phi')}{8\pi G r^{2}} \\ p = (r/2)((\rho c^{2}-\tau )\Phi'-\tau ')-\tau \tag{3}$$

With equations $$(3)$$ it is possible to define a dimensionless function to study the tension in the throat $$$$:

$$\zeta_{0} = \frac{\tau_{0} -\rho_{0}c^{2}}{\mid \rho_{0}c^{2} \mid}>0 \tag{4}$$

Now comes my doubt. The equations $$(2)$$ and $$(4)$$ are called flaring out conditions and my doubt is solely about the meaning of "flaring out". I'm not a native speaker of English and the translate of "flaring out" to portuguese do not give me any inshigts. So, are there any synonym for "flaring out condition" in English, in this context?

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$$$$ MORRIS, THORNE. Wormholes in space-time and their use for interstellar travel: A tool for teaching general relativity

$$$$ BARUAH.A Travesable Lorentzian wormholes in higher dimensional theories of gravity