Falling in the Milky Way

Let's imagine travel at very close to the speed of light. Let's say, for example, that your spaceship can reach a speed of 185,000 miles per second. You travel in the direction of the center of the Milky Way Galaxy, which is by some estimates about 25,000 light years away. So imagine that after six years of travel you start to feel a little claustrophobic. You decide to take a space walk. Unfortunately, you are a little absent minded and you forget your leash. Bye! You start to fall. My question is, which way are you falling? How can you describe direction in this scenario? Could you describe the direction as falling at such and such an angle in relation to the center of the galaxy?
Is it any better here on earth? That is, as we go about our lives, do we really know which way we're falling?

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