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They are coordinates in the same field, but moving in any one direction doesn't necessitate movement in any other direction.
No, but it does mean an overall different location. The problem is you're thinking in 3d space. Well yeah, we can be motionless through 3d space. But why does motion only apply to 3d space? It applies to spacetime. If you move through time, you're moving through spacetime. Maybe not space, but that's only like saying you're not moving on a 2d graph if your x axis moves but not your y axis. Well, all we can say is we're not moving along the y axis. But we are moving through the graph.
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That's like saying, "You moved forward, so therefore you've moved up and to the side."
Not at all. It's like saying you moved forward, therefore you moved.
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Therefore, motion in one axis does not necessitate motion in another axis.
Ok, but my argument is that motion through one axis, whether that be x, y, z or t, is motion.
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"What do all observers agree on?"
Sure, but wasn't the underlying theory to this question based on the geometry of spacetime, and how we move through it?
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Specifically, and differentiating it from Special Relativity, GR considers accelerations.
Indeed, it argues gravity is acceleration caused by geometry... so when gravity accelerates an object, why is this different to another source of energy accelerating this object? What's the difference between graviational and intertial mass from a GR pov? Best I can tell, there isn't one... it's the same mechanism.
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Object follow orbits because those are logically straight lines in curved spacetime.
Yes, and this is provable because the same side of the moon constantly faces us (tidal locking), yet the moon doesn't rotate as it orbits in order to maintain this relationship... if it did, it would lose energy and crash into us. The moon is moving in a straight line through curved spacetime. I certainly do not dispute this.
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We cannot move freely in time the way we can in space, though. Proving that they are the same means we need to be able to do that.
Perhaps we can move freely in time if we have no mass. Can you swim against a current in a river that flows at light speed?
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Arguments for the arrow of time never really satisfy me. I don't have any good answers to these questions.
These questions are why I tend towards the idea that we're moving into expanded space. The future simply doesn't exist yet.
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Of course this is an excellent point, and maybe someday we will figure out how to move freely through time.
I'm not buying time travel, not backwards anyway. Of course we can travel to the future by moving fast. To move freely through time, perhaps it can be done in theory if you move at c, which we know is impossible in practise. But how do you ever prove such a thing? I guess in the same way we know it's impossible to move at c... we don't need to try to move that fast in order to figure out we can't.
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"If time travel is possible, where are all the tourists from the future?"
In the future future!