Flat sheet-like structure approaching a massive object

[EskWIRED]

I'm thinking about a flat sheet-like structure approaching a massive object (maybe a black hole?).

If it were to approach with the plane of the object tangent to the mass, it seems to me that eventually, with enough gravity, the sheet would end up looking much like those hokey "bowling ball on a sheet" illustrations of a gravity well. The tidal forces would spagettify the central portion of the object, while the outer areas would remain relatively uncurved, all proportional with the gravitational force affecting the various regions of the sheet-like object.

So here's what I am unsure about: If an astronaut aboard the sheet-like craft were to sight down the long surface of the craft near the middle, would the craft look uncurved to the astronaut? Would photons from the far side travel along the curved surface and strike him in the eye? All as if the craft were still in a region of comparatively flat spacetime?

If so, that is pretty cool. If not, why not?

 

[Naty1]

Most of spacetime distortion, say near earth, is that of time...stuff looks virtually straight in space...At extremes of gravity you'd see both space curvature and time variation with altitude...
Gravitational force varies inversely as the square of the distance. But you'd likely need some dynamics to observe the space curvature, like light passing a planet and curving...

 

[mfb]

The curvature of your sheet would show tidal forces. You can observe them, even if you do the experiment with Earth as a central object, where the bending of light is not relevant.

Imagine a sheet with a diameter of 300,000km, located 300,000km away from earth. The central part would hit Earth after a week, the outer parts would need a few more days. The astronaut would approach Earth with a velocity of ~11km/s, way too slow to see any relativistic effects.

 

[EskWIRED]

So if the astronaut were looking down an interior corridor of his sheet-like craft, from one end of the sheet to the other, would it appear to him as if the corridor were curving towards the source of gravity?

I take it from the replies that he wold lose sight of the far end of the corridor as the craft approaches the large mass. Is that correct?

 

[WannabeNewton]

Most of spacetime distortion, say near earth, is that of time...stuff looks virtually straight in space...

This is a meaningless statement. Where did you read this? Perhaps you're trying to say that the values of the temporal components of certain tensors (e.g. energy momentum tensor), relative to a suitable background global inertial frame, are of a much higher order than the values of the spatial components (with appropriate conversion factors) in the weak field limit.

 

[A.T.]

So if the astronaut were looking down an interior corridor of his sheet-like craft, from one end of the sheet to the other, would it appear to him as if the corridor were curving towards the source of gravity?

I take it from the replies that he wold lose sight of the far end of the corridor as the craft approaches the large mass. Is that correct?

What the common sheet with a dent represents is spatial geometry, not space-time geometry: 2D sheet = 2 spatial dimensions.

When you build a long straight rigid corridor in flat space-time, and somehow manage to place it statically in a strong non-uniform gravitational field without breaking it, it will follow that spatial geometry geodesically (localy straight), which results in a global "bend" of the corridor (unless placed radially). But a light ray is additionally affected by the gravitational time-dilation (temporal component of space-time distortion), so it will "bend" twice as much as the corridor "bends". So in this case you cannot see the other end of the corridor. The corridor will appear to bend outwards, away of the mass. The opposite of how it actually "bends" to follow a locally straight path in curved space. That's because compared to the light it "bends" less.

 

[mfb]

So if the astronaut were looking down an interior corridor of his sheet-like craft, from one end of the sheet to the other, would it appear to him as if the corridor were curving towards the source of gravity?

A rigid corridor? Not in a relevant way. A corridor made out of small particles? Sure.