general relativity - Effect of gravity at near-lightspeeds

Let's say I'm in a space station, hurtling towards our galaxy nearly close to the speed of light. From my reference frame, I see the galaxy coming towards my ship at the same speed.

I pass the Sun, and am affected by its gravity.

From Earth's point of view, the gravity of the sun deflected my spaceship's trajectory. From my point of view in the spaceship, the trajectory of the entire galaxy changed very rapidly.

From my reference frame, how did my space station cause an entire galaxy to change course?

^{Originally had multiple questions, removed all but one}

Answer

This is sort of the same as Anna's answer, but I'd like to put a slightly different spin on it.

As Anna points out, there are two different co-ordinate systems involved: one for the observer sitting on Earth and one for an observer in the freely falling spaceship, and the situation looks very different for the two observers.

Each observer can (in principle) measure the stress energy tensor then solve the Einstein equation to give the curvature tensor. The key thing to note is that these tensors are co-ordinate independent i.e. both observers will calculate the same stress energy and curvature tensors.

However, although the tensors are co-ordinate independent the representations of them in the two co-ordinate systems will be different. We normally write the tensors as a 4 x 4 matrix, and the two different observers will calculate different values for the elements in the matrices because they're using different bases.

So it's not correct for the observer in the spaceship to think the galaxy is somehow being deflected by his gravity. Actually strictly speaking it's also incorrect for the observer on earth to think the spaceship is being deflected by the Sun's gravity. The gravity, i.e. the curvature, is not attached to any particular body. The solar system and the spaceship together (and in principle the rest of the universe) produce a curvature then both of them move in response to that curvature. The difference seen by the observers is purely down to them using different bases to represent the tensors.