In three dimensional space the three dimensions x,y,z…left/right, back/forth, up/down…are arranged at 90 deg. to each other.
Is Time…assumed to be a linear dimension…also at right angles the three spatial dimensions?
Is duration also direction?
This is something we used to talk about in grade school, someone would say, “oh, you could draw 4 lines that intersect at 90’, which would show 4 dinensions!” Then we would spend time trying, of course it’s impossible.
It’s unclear exactly what “orthogonal” means with respect to time vs space. Not just in the OP; nobody has a good handle on it.
But one working definition of “orthogonal” is that motion along one axis does not necessitate motion along another. IOW, going straight up/down does not alter your east/west or north/south position. Likewise going e.g. north does not alter your east/west or up/down position. etc.
In that approach, going forward in time doesn’t alter your 3D spatial position. So in a sense they’re orthogonal. But …
Of course what it really means to go backwards in time is rather difficult. So the timelike dimension is not simply another spatial dimension; it’s something inherently different and one-way. The 3 spatial dimensions might be orthogonal to the forward arrow of time, but not to the backwards arrow.
OTOH, movement in any 3D dimension necessarily takes time. So you can’t change position in space without movement in time. That breaks the rule of orthogonality: each dimension is independent.
Bottom line: it’s complicated and the traditional spatial geometric definition of “orthogonal” is only partly applicable to timelike dimensions(s).
As I understand it, time and space aren’t linear, what with distortions caused by gravity – so their being orthogonal is only approximate. I think. If that’s relevant. My guess is that duration should be considered a simple difference, but I suppose that 3 years from now should properly be considered as a different difference than 3 years ago, so yeah, properly expressed, a duration should be considered a direction.
I’ve got to go lie down now, I’ve got a sudden headache.
That’s wrong. Space is bent by gravity, but so are the angles. So it’s really 90 degrees in the context of the curved space.
I think it is relatively straightforward if we consider Minkowski space. Time is different to space by a sign, but you can put a “Minkowski dot product” on the space and say two vectors are orthogonal if the bilinear form evaluates to zero for them.
For (x_1,t_1) and (x_2,t_2), orthogonality amounts to something like t_1t_2=x_1x_2. So time and space are orthogonal in that sense.
Everything is always moving. If you were to move in time trying to remain in the same place after your movement in time, wouldn’t you need to purposely go in some x,y,z direction during your time travel? If you did not care about your destination position, just the time travel. Then everything else has moved, so in some way you did go in a direction too?
Barring the fact that there is no privileged reference frame, so there is no way to say for sure that standing still is actually standing still, an object could move in time, but not in space, by staying still in space. The passage of time does not require movement in space (in fact, rather the opposite; the passage of time is slower for moving objects).
This does not matter, because the time and space directions are orthogonal, and after a Lorentz transformation the new directions will remain orthogonal.
In four dimensions, it is always possible to choose four coordinates that are all orthogonal to each other, and in the usual choices of coordinates, they are. It is possible to choose coordinates that aren’t all orthogonal to each other, and sometimes this is is convenient for some particular problem.
I am trying to visualize this as any line going from a central point in a sphere or circle would intersect that circle or sphere at 90’? So there could be infinite points on a sphere where a line from the center intersects at 90’ to the sphere?
Yes, exactly. Every point on the surface of a sphere has that property. And by definition there are an infinite number of points on any sphere of any size.
But all of that is irrelevant to the idea of orthogonality of axes.
Think of the 3-dimensional version. You’ve got an X axis, a Y axis, and a Z axis. Make a model of those axes with Tinkertoys, with little Post-it notes on the sticks labeling them as X, Y, and Z.
Now turn that model any which way, and you’ll still have a valid coordinate system, that can still describe any point in space using three numbers, and the three coordinates are all still orthogonal to each other.
Usually, some coordinate systems will be more convenient than others, for any given problem, but there are still a lot of valid coordinate systems that could be used.