Multiple dimensions of space makes sense. We can understand multiple dimensions of space by lines ,planes,spaces etc. But what does a 2-D time,3-D time etc means. Visualizing time as a line makes sense ,where forward means future,backwards means past and present means the point we are in right now. But a 2-D or 3-D time would like a plane or something else. So what does it means to travel right ,left in time?
22139
We know that space-time dimensions are 3+1 macroscopically, but what if 2+2?  Obviously it is tough to imagine two time dimensions, but mathematically we can always imagine as either having two parameters $t_1$ and $t_2$ or else in Lorentz matrix $$\eta_{00} = \eta_{11} = -1$$ and, $$\eta_{22} = \eta_{33} = 1.$$  Is there any physical reason for not taking this, like the norms become negative or something else?
58587
Is there a single, more generic term that can be used to describe both a row and a column? In English, we can refer to a line as being horizontal or vertical, but unless we say ‘a line of something’, that implies a one-dimensional line—it doesn't communicate a row or a column of things. So what could we say instead…? A horizontal or vertical _______?
121416
I have read and heard in a number of places that extra dimension might be as big as $x$ mm. What I'm wondering is the following: How is length assigned to these extra dimensions?   I mean you can probably not get your ruler out and compare with the extent of an extra-dimension directly, can you? So if not how can you compare one dimension with the other? Does one have some sort of canonical metric? Could one also assign a length (in meters) to time in this way?
42763
Let's suppose I have a spacetime manifold $M$. Let $p$ be a point on my manifold. Now I move from $p$ to some other point $p'$. Presumably I should have moved some "distance" right? How can I speak of notions of space and time if I have no conception of distance?   But now consider light moving through spacetime. Suppose my light starts at $p = (0,0,0,0)$ and travels to $p' = (1,1,0,0)$. By the definition of the spacetime interval $ds^2 = dt^2 - dx^2 - dy^2 - dz^2$, this should mean $ds^2 = (1)^2 - (1)^2 - 0 - 0 = 0$. So $ds^2=0$.   Yet I have moved from point $p$ to $p'$. So I clearly have moved along some path along the curve, but the length of this path is zero. Shouldn't that mean $p$ and $p'$ are the same point?   Note: I think I may be suffering from an overly Euclidean mindset and my brain hasn't adapted yet enough to the non-Euclidean logic of semi-Riemannian manifolds.   My Question   Can someone resolve this contradiction?
41230
I've just started studying linear algebra, and I'm having trouble understanding something that apparentely seems obvious to everyone else. My question is, why does multiplying a vector by a positive scalar does not change its direction? I've seen some visual representations of this and it does seem reasonable to me that that should be the case. However, I wonder whether there is a way to actually show it mathematically? In the 2d coordinate plane, for instance, I guess way one of showing that vector $A$ has the same direction of $cA,c&gt;0$ would be to show that the line containing $A$ has the same slope as the one containing $cA$ (I'm not sure if this approach is correct though). However, how to proceed in the case of $n$ dimensions? I think I must be missing out something basic here, but I really can't seem to get my head around it. Any help will be highly appreciated. Thanks very much in advance.
55052
When trying to do physics on a more general pseudo-Riemannian manifold we want to require that there is a foliation of this manifold into three-dimensional subspaces. By this I mean we would like to have a way of splitting our spacetime $M$ into a part that we call space and one we call time. This is necessary, for example, in order to write the two-form $F$ that represents the electromagnetic field as $F = E \wedge dt + B$. I know that this is possible if we assume $M$ to be compact, but this implies that there are closed time-like geodesics (I have been told I would not know a proof). What are the necessary and sufficient conditions for such a foliation to exist? I would also appreciate it if you could inform me about books/resources that go into these sorts of questions. I have looked into books about GR, but they don't seem to talk about these sorts of things.
55509
I think the term "back and forth" gets thrown around a lot without much thought.  From Dictionary.com:         [fawrth, fohrth]      adverb    1.  onward or outward in place or space; forward: to come forth; go forth.   Wouldn't the term be more meaningful if "forth" was used first since you have to have a starting place before you can go "back"?  Examples:        I've been going back and forth from Houston to New York.   I've been going forth and back from Houston to New York.      Which is more correct?
116441
In a freshers lecture of 3-D geometry, our teacher said that 3-D objects can be viewed as projections of 4-D objects. How does this helps us visualize 4-D objects? I searched that we can at least see their 3-D cross-sections. A Tesseract hypercube would be a good example.  Can we conclude that a 3-D cube is a shadow of a 4-D tesseract?    But,  how can a shadow be 3-D? Was the screen used for casting shadow also 3-D; or else, what way is it different from basic physics of shadows we learnt?       edit: The responses are pretty good for 4th dimensional analysis, but can we generalize this projection idea for n dimensions, i.e. all n dimensional objects will have n-1 dimensional projections?    This makes me think about higher dimensions discussed in string theory.   What other areas of Mathematics will be helpful?
85305
In many pieces of literature, light is said to travel like a "wave". Does this mean the light literally propagates through space like a wave as in up and down and so on or does light move linearly through space? Also, if it move linearly, why do we say photons are a wave then?
83695
Space is just open 3D space, right? In Star Trek, ships are always coming from somewhere else: different solar systems, galaxies, etc. However, when they meet, they always seem to be on parallel planes (facing each other).  In some episodes I recall seeing ships that were adrift (and they appeared facing a different trajectory). How can there possibly be some standardized reference plane in space? Do ships just choose to align with each other to be polite? Or perhaps when entering a solar system, the pilot, or navigation system aligns orthogonally with the local orbital axis?
46322
I have been reading a lot about string theory and the necessity of extra dimensions (particularly as visualized in Calabi-Yau spaces), as "curling-ups" in our apparently 3-dimensional (or 4-dimensional, including time) world. I do not understand, though, how all these "extra dimensions" are actually adding dimensions to our world.  Even if these little multi-dimensional curls are everywhere and too tiny for us to see, how is it still adding extra dimensions? I mean, if our eyes were sharp enough, couldn't one still specify their location according to an $x,y,z$ axis? They will be a certain amount to the "left," a certain amount "high," and a certain amount front or backward (etc). It's still length, width and depth.   What am I missing here?
759
We are learning vectors in class and I have a question about parallel lines and coincident lines. According to wikipedia a parallel line is:      Two lines in a plane that do not intersect or touch at a point are called parallel lines.   But another reference says     Side by side and having the same distance continuously between them.   According to the above definition, you can have two parallel lines with a distance of zero but this contradicts the first definition. So which is it?  EDIT:  If it is the second definition then you can have two lines that are coincident and parallel right?
8314
To an an external observer it appears that time has stopped for photon. But this relation is reflexive, so for an observer travelling with the photon it appears the universe has stopped everywhere.  Is this right?  Space also gets distorted parallel to the direction of motion, but not perpendicular to it.   Does this mean that for an observer travelling with a photon sees spacetime as a flat plane?  Note 1: I'm using language vividly not literally when I say a photon experiences space and time. Not that I'm against idealist or panpyschist interpretations of matter or energy come to that.  Note 2: Since it's been pointed out that an inertial frame can't be boosted into the frame of a photon and moreover there is no limiting behaviour here - no matter what frame you are boosted into, light will always appear to travel at c. Nevertheless, when an external observer watches a train chasing a photon by gradually increasing its speed, he will note that the train speed is approaching the speed of the photon (but will never match it) and when he looks at the clock inside the train he will not too that it is gradually slowing down (but never actually stops). Using this picture, we can attempt the thought experiment - if not one that can be carried out - of what a world looks like to a photon.
23798
What is imaginary (or complex) time? I was reading about Hawking's wave function of the universe and this topic came up. If imaginary mass and similar imaginary quantities do not make sense in physics, why should imaginary (or complex) time make sense?    By imaginary I mean a multiple of $i$, and by complex I mean having a real and an imaginary part, i.e., $\alpha + i\beta$, where $\alpha, \beta \in {\mathbb R}$.
104403
How do you correctly abbreviate that something is in "three dimensions" in formal writing? As per the title, would you write either "3D", "3d", or "3-d"?  I want to write something like:     The figure contains a 2D widget (left) and a 3D gadget (right).   Additionally, if I use the full phrase in a sentence, I gather that I write     The three-dimensional gadget is composed of independent two-dimensional widgets.   i.e. using a hyphen and the word "three". Would it also be allowed to write "3-dimensional"? Especially in a context where I want to contrast it with something else "2-dimensional" in the same (or adjacent) sentence, as the numerals stick out more?
31067
I heard that there's several kind of geometries for instance projective geometry and non euclidean geometry besides the euclidean geometry. So the question is  what do you mean by a geometry, do you need truly many geometries and if yes what kind of results we can find in one geometry  and not in the others. Thanks a lot.
80619
I've looked up  and on Griffths' didatic Introduction to Electrodynamics, but, still, am struggling to understand why time has the opposite sign of the space coordinates. Any insights are welcome.