Hurm,
The new popular term is orthogonal. In Euclidean geometry it does mean perpendicular. However, Euclidean geometry does not apply to the way you are using it in the context of space/time in the sense of Einstein's relativity. Orthogonal means something related in programming and information programming. It has always confused me and I remain that way.
Maybe you guys can help me out. Reading modern science papers this term appears often.
Sure! Two pieces of (potentially changing) information are orthogonal if they're not related in any way - if changing one doesn't alter the other. It's an analogy from geometry: Like two lines at absolute right angles, moving any distance along one line doesn't move you any distance along the other. The two dimensions are unrelated. But if one line is at a slope to the other, then moving any distance along that line *will* move you some distance along the other, so now there's a relation between them.
In software and maths, you normally want your simplest fundamental ideas to be orthogonal (have no relation to each other) because if they aren't, that means they're connected - and that means they're not actually the simplest possible ideas. At the very least, you now have a messy, complicated connection between them, and that means when you think about one idea you have to hold the entire cluster of ideas, not just that one idea. That's annoying so we try not to have to do it.
AS I understand it, in Einstein's Special and General Relativity the whole point is that the dimensions of space and time *aren't* orthogonal: that's what 'warping of spacetime' (in SR) and 'curvature of space-time' (in GR) means. So moving along one of the three space dimensions might also move you along one of the other two space dimensions, or the time direction, or both. And so on. And the amount of that curvature can change at any point in space and time. (And with weirder properties of distortion, for any observer, based on the relative velocity between events). So it's a whole very non-orthogonal thing (nonlinear, also). That's what makes it very hard to calculate with.
Einstein didn't originally think of time as a dimension, if I understand correctly: that was Minkowski, who took Einstein's SR equation and modelled it as geometry with time being 'imaginary' (eg, multiplied by i, the square root of negative 1 - this means that to calculate the distance in 'spacetime' you *subtract* the squared distance in time from the squared distance in space, before square-rooting). It's a mathematical trick and I'm not convinced it really does make time an actual dimension (I think it just really accounts for the idea that 'having more time to get somewhere is the same, in terms of speed, as having less time but the thing being closer'). Lots of things about relativity don't make intuitive sense to me, but I'm not a physicist. Einstein then took Minkowsi's geometry idea and used it to build General Relativity, where everything is geometry - but, crucially, he FAILED in the entire rest of his life's project, which was to extend the idea of 'spacetime curvature' to include the electromagnetic force as well as gravity. (His 'Unified Field Theory' project). And since we know EM forces exist - but we can't model them with Relativity but only by adding Quantum Field Theory over the top, using completely different ideas - then Relativity really is an unfinished house.
I think there are at least six levels of connectedness in maths/physics/computing (where 'thing' is usually a simply measurable quantity, a number or series of numbers)
* 'orthogonal': two things are completely unrelated, changing one doesn't change the other
* 'linear': two things are connected fairly simply: changing one changes the other in a small, predictable way, exactly related to how much the change is. Changes add up normally. It's a line at a slope, but still a line.
* 'nonlinear': two things are connected in a curve: changing one changes the other in a complicated way that also changes depending on how far you go along the line. A small change at one point may mean a bigger change at another point. And the exact shape of that curve could be really, really complicated. Things in Newtonian physics like acceleration and friction work like this: sorting out the maths for nonlinear quantities (curves) is what Calculus is about and is the breakthrough that Newton and Leibniz made.
* 'chaotic': two things are connected in a curve, and not only is it a really complicated curve but - although you can precisely model it step by step, in practice a small change makes such a big change that you lose track very quickly. The turbulent movement of air and water (eg, clouds, the weather) is a classic case of this in physics; everyone's seen the Mandelbrot Set fractal. Chaos maths is a really recent discovery, 1980s.
* 'random': two things are connected but there's no known formula that predicts the connection. Stuff just happens; though if it's still well-behaved you can measure just *how* unpredictable it is, which is what statistics and probability are all about (and they're very hard to learn). Quantum mechanics is all about probabilities, but also doing very weird things even by probability standards (eg complex and negative probabilities). Psi also seems to fall in to this basket, which is why people like Dean Radin think there's a connection; even if it's just a metaphor).
If Newton was correct and the universe is 'classical', then there really is nothing random, just chaotic. This may also be true if, eg, the Bohm interpretation of Quantum Mechanics is correct (the one with 'hidden variables'). But most QM people think that the QM probability wavefunction is fundamentally random, not chaotic. I'm not sure if there are any theological implications to either of these two positions.
(Also I'm not sure most QM people would agree that conscious minds are required to collapse the QM wavefunction; the general belief today seems to be that wavefunction collapse, or 'decoherence', is a real physical process that happens entirely on its own whether observed or not. Avoiding decoherence is a major problem in the design of functioning quantum computers, and it seems to be something related to heat. The colder you keep the computer the longer it will last before decohering. I don't think heat is especially psychic. But the Copenhagen Interpretation, like Special Relativity, does not sit well in my mind. I find the ideas so odd and slippery I just can't get a grasp.)
* 'infinite' (or 'singular' if all the infinity happens at one point in space/time): something is so big it literally can't be measured. The idea of infinity is not especially well-defined: Cantor argued that there are hierarchies of infinity (eg an infinitely big two-dimensional sheet of paper is literally more infinite than an infinitely long one-dimensional ball of string, such that you could not build that paper out of that string by folding the string an infinite number of times while keeping each fold infinitely long) but I'm not convinced there's a meaningful argument to be made either way on that idea; infinity, mathematically, is both equal to and not equal to itself. And that kind of breaks mathematics.
Although our physics equations are full of infinite quantities (eg the 'zero point energy' of quantum field theory, and black holes), I'm pretty sure nothing in our physical, measurable universe is actually infinite - but many religions hold that 'God/spirit/mind is infinite', and the idea of anything mental/spiritual ending doesn't make much intuitive sense, which suggests we do hold to that idea of infinity in our deepest self.
Regards, Nate