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In reference $[1]$ the author constructed a junction condition to an external Schwarzschild with cosmological constant and a traversable morris-thorne wormhole. The form of the energy-momentum tensor at the surface $S$ (and at the radius $r=a$ of junction) as given by:

$$T_{\hat{\mu}\hat{\nu}} = t_{\hat{\mu}\hat{\nu}}\delta(\hat{r}-\hat{a}) \tag{1}$$

where $\hat{r} = \sqrt{g_{rr}}r$ is the proper distant through the thin shell.

So, my doubt is:

Why the energy-momentum tensor have the form of $(1)$?

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$[1]$ LOBO.F.S.N, et al; Morris -Thorne wormholes with a cosmological constant, https://arxiv.org/abs/gr-qc/0302049.

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This form (1) is just an idealization for a matter-energy distribution which is very thin (in space), in such a way that the integrated quantities (energy, momentum flow, internal forces) are supposedly finite. This leads (through Einstein equations) to constraints on how "bad" the spacetime geometry can "vary" through boundaries (3d submanifolds) and still be considered physical.

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