Consider an object in an extremely eccentric elliptical orbit around a very high density central mass. Consider a spacecraft whipping around the central mass at periapse.
On one hand, objects in orbit are in free fall the whole time, so feel zero-g. An astronaut in the spacecraft with no windows would not be able tell where in orbit he or she was.
On the other hand, from far away, it looks like the spacecraft had been traveling in almost a straight line in one direction, then has suddenly changed directions to go the other way. Surely this had some noticeable effects on and inside the spacecraft. If the spacecraft were to change directions similarly by bouncing off a wall, passengers inside would get smashed against the walls.
Does the spacecraft undergo any stresses when making the turn at closest approach to the central mass? We're guessing not, like the astronaut inside: it's oblivious to the rapid change in direction. We ignore tidal forces.
If the central mass is a black hole, you can make painless U-turns even at near light speed, without losing any velocity at exit. This could be useful for sending a probe to go explore near a black hole and then come back, without having to carry fuel for the return trip (or power to communicate back home). It's the ultimate gravity assist. Acceleration to near lightspeed on the outbound trip could have been done with stationary lasers at the launch site, e.g., Breakthrough Starshot.
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