50
votes
Accepted
Why are magnetic field lines imaginary?
Most of us will have experimented with placing iron filings around a magnet to get this sort of thing:
This particular example is taken from Why iron filings sprinkled near a bar magnet aggregate ...
- 363k
34
votes
Accepted
Why is Spacetime described as flat even though we live in 3 dimensions of space?
"Flat space" means that on large scales, Euclidean geometry holds. All the angles in any triangle drawn in space add up to 180°; the total distance between points separated by $\Delta x$, $\...
- 11.2k
17
votes
How do you visualize a quantized electromagnetic field?
Warning: long answer ahead. To see pictures, scroll down.
A free field is just a cavity with infinitely large boundaries, right?
Right. This is how it's constructed: take a finite cavity with ...
- 29.6k
11
votes
Why are magnetic field lines imaginary?
This dates back to Faraday. He was a brilliant experimenter, and he discovered a whole lot about electromagnetic stuff. But he didn't do math, and his explanations for what he found were not in ...
- 3,086
11
votes
Why is Spacetime described as flat even though we live in 3 dimensions of space?
The meaning of "flat" here is not the colloquial meaning, and it certainly does not mean "in 2 dimensions with curvature".
Instead, the meaning of "flat" here is the ...
- 310
10
votes
Is there a way to visualize curvature of time?
If you are thinking of spacetime as a product of a three-dimensional space $S$ with one-dimensional time $T$, then time is necessarily flat, because all one-dimensional manifolds are flat. So no, ...
- 17k
8
votes
Accepted
Visualizing Tensors in the simplest possible way
The following answer is entirely my opinion: the best way to visualize tensors in Euclidean space is by considering them in their spherical basis. The spherical basis is standard in quantum mechanics, ...
- 39.5k
8
votes
Why is Spacetime described as flat even though we live in 3 dimensions of space?
Space being flat essentially means that the laws of Euclidean geometry hold. The interior angles of a triangle sum to 180 degrees. Parallel geodesics (straight lines) never intersect, etc.
Consider, ...
- 109k
7
votes
Is there a way to visualize curvature of time?
Here's one simple visualization. Imagine that the surface of the earth is spacetime. Time points north. There is one space dimension, which points east-west.
Imagine two objects with worldlines that ...
- 6,987
6
votes
Divergence of magnetic field
Perhaps it's more illuminating to write it in integral form.
$$\iint_S \mathbf{B} \cdot d\mathbf{A} = 0 $$
So you need to enclose the region in a gaussian surface. Once you do this, you can see ...
- 4,820
6
votes
Visualizing Tensors in the simplest possible way
This may be a controversial opinion. But I've actually found it unhelpful to try to directly visualize tensors (other than vectors, or special cases with a lot of symmetry).
I'd like to make an ...
- 55.3k
5
votes
How many eyes are needed to see a four-dimensional world?
One eye would be enough. (Try closing one eye and you can experience the result for a 3-D world).
But I guess you had in mind what is the value of $N$ such that $N$ eyes give geometric info not ...
- 61.8k
5
votes
Visualizing Tensors in the simplest possible way
The most common ways to think of a vector are a little arrow or an ordered tuple of coordinates such as
$$X = (x, y, z)$$
But this isn't what a vector really means.
To a mathematician, vectors are ...
- 45.7k
4
votes
Visualization of motion in black hole spacetimes
that is my little app ;)
Stopping at the horizon was an aesthetic decision, the effective potential allows horizon crossing no problem. It is easy to change this in the code but it looks a bit silly ...
- 41
4
votes
Can many-particle wave functions be imagined in 3 dimensions - instead of $3N$ dimensions?
This answer is simply to complement the one provided by J. Murray and discuss density functional theory, a much-used theory to study quantum systems with a 3-dimensional function: the electron density....
- 1,253
4
votes
Identity of dynamical D$p$-branes
D-branes, at first sight, look pretty useless - who cares about the hypersurface on which an open string can end? Classically this would probably be the case, but on quantisation of the oriented ...
- 8,570
4
votes
What is the "shape" of the electric potential of a moving charge? Can the field be derived as its gradient?
Congratulations! You have discovered an important fact about dynamic electromagnetic fields. For dynamic fields, the electric field is no longer simply given by the gradient of the potential $V$, ...
- 51.6k
4
votes
Why do magnetic fields exist as line?
They do not exist as lines. The shape of the field follows the curvature of those lines, which are used to visualize the field. Everywhere between two of those lines, there is another one inbetween ...
- 97.8k
4
votes
Accepted
Cosmological principle: can there be a center of the universe "outside" the universe?
This kind of confusion arises when you take the visualizations too seriously. By definition, the universe is all there is, so there can't be anything outside it.
Now, let's come to your balloon ...
- 1,354
3
votes
What do electromagnetic waves look like?
The fields are perpendicular, not the waves. Looking at a typical web search you get:
Let's say ${\bf \hat k} = {\bf \hat x}$ so the plane wave is:
$$ {\bf \vec E}(x, y, z, t) = E\sin{(kx-\omega t)} ...
- 39.5k
3
votes
Accepted
How does one imagine the curvature of spacetime in 3D?
The conventional intuitive understanding of curvature is actually not what we usually concern ourselves with in general relativity. You say that,
... The 2D flat drawings are very explanatory...
...
- 19.5k
3
votes
Accepted
What is the Fabric of Spacetime?
There isn't one. Probably the reason why this whole "fabric" thing was brought into existence was to attempt to make it possible for laypeople to visualize the fact that "spacetime geometry" is not ...
- 11.6k
3
votes
Accepted
How to interpret this image of $AdS_5\times S^5$?
This image from the paper by David Mateos schematically representing an embedding diagram for two-dimensional $(r,\phi)$ sector of the full 10 dimensional supergravity solution.
This is a visual aid ...
- 16.6k
3
votes
How can $n$-dimensional space be projected?
If you have a D-dimensional vector $\vec{d} \in \mathbb{R}^D$ you can project that vector into a (D-1)-dimensional hyperplane perpendicular to a unit-vector $\vec{n}$ in the following way:
The ...
- 3,129
3
votes
Accepted
What observable is visualized in lattice QCD visualizations?
What is plotted is iso-contours of the euclidean action density ${\rm Tr}[E^2+B^2]$ and topological charge density ${\rm Tr}[E\cdot B]$, where $E$ and $B$ can be defined using the plaquette $U_{\mu\nu}...
- 18.9k
3
votes
Better explanation of the common general relativity illustration (stretched sheet of fabric)
Are you familiar with the trouble with Map Projections? Every map of the Earth will necessarily distort its features. The surface of the earth is Intrinsically Curved. Among other things, in the ...
- 2,738
3
votes
How to plot Wigner functions for a quantum state
The so-called Wigner transform turns your density matrix into the corresponding Wigner function. An explicit formula is given at "What is the Wigner function of $|n\rangle\langle m|$?"
- 31
3
votes
Definition and visualization of a covector
The stacks of surfaces picture is just a helpful way of visualizing a covector which is compatible with the usual picture of vectors as arrows. Since a covector takes a vector and gives back a number, ...
- 426
3
votes
How to animate a ball rolling down an incline?
The acceleration of an object rolling down an incline is $$a=\frac{g\sin\theta}{1+\frac{I}{mr^2}}$$ where $I$ is moment of inertia about the centre of mass and $r$ is the radius. See slides 1-6 of ...
- 27.5k
3
votes
Accepted
Is there a way to separate 2D from 3D?
Using mathematics can mean different things. I will start explaining with a trivial example.
I give you a two dimensional image(or even series of them ) and ask you to reconstruct a 3-D object from ...
- 285
Only top scored, non community-wiki answers of a minimum length are eligible