I'm confused. My understanding is that by metric we understand the spacetime metric of SM or MM, owing to the relatively massive shell, in which the small test body is immersed. And that mechanical distortion of said body does not constitute some added metric, or appreciably alter the embedding metric owing to the shell. Besides, it was stipulated that mechanical distortions are either eliminated or compensated for. And one can simply apply a limiting process, ie let specific stiffness -> 'near infinite', a common enough idealization surely.