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But if we travel in a straight line through curved space-time, then our orbit around the sun should not be affected by velocity.
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I tend to think of it as likened to riding a bicycle around a steep ramp, you have to maintain the same velocity to stay in the same "orbit". Go to slow you fall down, to fast and you fly of the ramp like those punks on americas funnest home videos. Now I see it as matter warping space into a sort of conical ramp, that you can not see.