This makes no sense to me. Direction is a meaningless concept if motion is relative. We're not really going anywhere, as there's nowhere to go. Spacetime is spacetime. The only "place" we're going is through spacetime, and that's not a direction.
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Direction is a relative term. Therefore it exists in a relative space. (both mathematical entities)
spacetime is yet another mathematical entity. It is a non-Euclidean space where time is one of the axes. Since that axis is "well-ordered", it has a direction.
However, it is difficult to treat the concept of "direction" in space time, because the metric of space time is not Euclidean.
distance^2 = x^2 + y^2 + z^2 - (ct)^2
This allows for non-positive values for "distance squared" -- we then speak of "time-like" distances.
For direction (once you agree on a principal axis), you could use the same cosine function (involving the dot-product and the norm) with vectors, for example, but it is counter-intuitive (it still works, mathematically).
It is possible to mathematically project another vector onto this principal direction and measure the length of the projection and divide it by the length of the vector (another way to get the cosine). Of course, it is difficult to wrap your head around the value of the direction if you get a "time-like" distance.
Spacetime is NOT spacetime. It is a mathematical construct. It is a map of a 3-D universe on which one adds time as a 4th dimension. You do NOT go anywhere in space-time, the same way you do not go anywhere ON a map.** You can use a map to go somewhere, but your motion is in the reality, not IN the map.
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** OK, there is a subtle difference: you can unfold a map, lay it on the floor and, if it is big enough, you can travel ON the map. You cannot unfold spacetome and lay it on the floor.
spacetime is yet another mathematical entity. It is a non-Euclidean space where time is one of the axes. Since that axis is "well-ordered", it has a direction.
However, it is difficult to treat the concept of "direction" in space time, because the metric of space time is not Euclidean.
distance^2 = x^2 + y^2 + z^2 - (ct)^2
This allows for non-positive values for "distance squared" -- we then speak of "time-like" distances.
For direction (once you agree on a principal axis), you could use the same cosine function (involving the dot-product and the norm) with vectors, for example, but it is counter-intuitive (it still works, mathematically).
It is possible to mathematically project another vector onto this principal direction and measure the length of the projection and divide it by the length of the vector (another way to get the cosine). Of course, it is difficult to wrap your head around the value of the direction if you get a "time-like" distance.
Spacetime is NOT spacetime. It is a mathematical construct. It is a map of a 3-D universe on which one adds time as a 4th dimension. You do NOT go anywhere in space-time, the same way you do not go anywhere ON a map.** You can use a map to go somewhere, but your motion is in the reality, not IN the map.
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** OK, there is a subtle difference: you can unfold a map, lay it on the floor and, if it is big enough, you can travel ON the map. You cannot unfold spacetome and lay it on the floor.
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