Hot answers tagged dimensions
18
Yes, absolutely. In fact, Gauss's law is generally considered to be the fundamental law, and Coulomb's law is simply a consequence of it (and of the Lorentz force law).
You can actually simulate a 2D world by using a line charge instead of a point charge, and taking a cross section perpendicular to the line. In this case, you find that the force (or ...
16
Dear asmaier, you shouldn't view $\vec L = \vec x \times \vec p$ as a primary "definition" of the quantity but rather as a nontrivial result of a calculation.
The angular momentum is defined as the quantity that is conserved because of the rotational symmetry - and this definition is completely general, whether the physical laws are quantum, relativistic, ...
14
Lubos Motl's answer is completely right, but I'll add my perspective anyway.
For many compound units, you shouldn't try to "visualize" the meaning of the unit, but you should think of it as reminding you about relationships between that quantity and others. Why are the units of Newton's constant $G$ ${\rm N\ m^2/kg^2}$? It's because $G$'s "purpose in life" ...
14
This is not paradoxical and it is not necessary for any physical phenomenon to a priori have to obey any particular law. Some phenomena do have to obey inverse-square laws (such as, particularly, the light intensity from a point source) but they are relatively limited (more on them below).
Even worse, gravity and electricity don't even follow this in ...
12
I think the best answer to your question is simply "because that's all we can see when we do experiments."
That is, no matter how hard anyone tries or how much energy they toss into the processes, electrons and quarks show no signs of any appendages, surfaces, hair-like structures, bumps, volume, whatever. When you model them mathematically as points, the ...
12
Specifically what that is referring to is the 'inverse-square law', nature of the gravitational force, i.e. the force of gravity is inversely proportional to the square of the distance:
$F_g \propto \frac{1}{d^2}$.
If you expand this concept to that of general power-law forces (e.g. when you're thinking about the virial theorem), you can write:
$F ...
11
A classical$^1$ theory of (relativistic) $p$-dimensional membranes exists for any non-negative integer $p$. Such classical membrane-like objects appears in the full non-perturbative formulation of string theory.
The problem arises if one tries to quantize a membrane (in a first quantized sense) using standard perturbative quantization methods, which have ...
10
There is no reason why you should be "imagining" a squared second. Most quantities in physics don't have any canonical "geometric" visualization and there is no reason why they should have. What matters is that you should be able to calculate with it.
For example, the gravitational acceleration on Earth is $9.81\,\,{\rm m/s}^2$. This simply means that the ...
9
The notion of negative dimension has appeared in various places of modern physics. For instance:
Grassmann-odd variables. Recall that the dimension ${\rm dim}(V)$ of a group representation $\rho: G \to GL(V)$ is given by the trace ${\rm dim}(V)={\rm Tr}(\rho(1))$ of the identity element. For a supergroup, one should use the supertrace, so Grassmann-odd ...
8
You are describing the idea of a relational space-time, so that one point particle would have 0 dimensions (because it cannot have position), two point particles would have only 1 dimension (because only their relation matters), three point particles would have 2 dimensions, and in general, N point particles would have N-1 dimensions, which describe their ...
7
Actually, let's give this a shot. This isn't evidence for extra dimensions (the non-obesrvation of extra dimensions/supersymmetry is one of the big reasons string theory is not accepted universally as true, after all), but this is an argument as to why small extra dimensions are unobservable.
Consider a particle in a box in quantum mechanics in n spatial ...
7
One example of such an approach is Ambjorn and Loll's Causal Dynamical Triangulations, which is very similar in many ways to the very old idea of Regge calculus, whereby spacetime is discretized. At small scales, non integer dimensions can emerge. For an introductory article , see
Jan Ambjørn, Jerzy Jurkiewicz and Renate Loll. The Self-Organizing ...
7
It is not so much as one single particle will be seen with different masses as it is that that one type of particle will be seen as having multiple different masses when it is detected multiple times.
For example if the extra dimension is like a rolled up microscopic cylinder, the particle can have an infinite number of discrete masses starting from the ...
6
I would say yes !
Actually some theories explaining quantum gravity use also this reasoning: gravity is a very weak interaction at a quantum level because it "leaks" into other dimensions, not observable at our scale, but that are present at this scale.
The mathematical tools are different, but if you just think about gauss's law you can imagine one ...
6
It's true that if you just have a set of points, with no additional mathematical structure, the notion of "dimension" is problematic as you say. But the spaces we deal with in physics usually have extra structures that make the notion well-defined. Often, the definition works by making precise a notion of different "directions" at a given point, and then ...
6
The formal statement would be that there is no one notion of time, and that one persons definition of time may be intermixed with another's definition of space. What is not correct is that time and space are wholly interchangeable, as Moshe says--there is a distinction between events separated in time and events separated in space. Causality theory is ...
6
Qmechanic's answer has the full details, but I figure it might be useful to break this down in a little more detail.
The proper way to generalize Maxwell's equations to higher-dimensional spaces is to use the field tensor $F^{\mu\nu}$. In our normal 3+1D space, it basically looks like this:
$$F = \begin{pmatrix}0 & E_1 & E_2 & E_3 \\ -E_1 & ...
6
Several authors (in particular Itzhak Bars) have written papers about two-time-physics that should help build intuition for the topic.
Infinitely many 'times' appear in integrable systems.
F-Theory, which is a 12 dimensional theory, has been described as having extra temporal dimensions, however see Wikipedia.
6
From the Wikipedia article for Reynolds number:
In fluid mechanics, the Reynolds number (Re) is a dimensionless number that gives a measure of the ratio of inertial forces to viscous forces and consequently quantifies the relative importance of these two types of forces for given flow conditions.
In addition to measuring the ratio of inertial to ...
6
There is nothing "wrong" with the Einstein field equations in $2+1$ as indicated by the comments, but they do have interesting, significantly restricted behavior in $2+1$ dimensions.
For example, the Wikipedia page referred to by Olof in the comments says that in $2+1$, every vacuum solution is locally either $\mathbb R^{2,1}$, $\mathrm{AdS_3}$, or ...
5
I will here only comment on the traditional superstring theory story, say, from the first superstring revolution in the 1980s, and leave it to others to include more recent developments.
Traditionally, the $10$-dimensional target space $(M^{10},g^{(10)})$ with a metric $g^{(10)}$ is viewed as a product $M^{10}=M^4 \times K^6$ with metric ...
5
The higher-dimensional version of Maxwell's equations is actually written explicitly in the very beginning of the linked answer. However, if you are only familiar with the traditional formulation of Maxwell equations, you will need to study two new subjects to appreciate this.
I) Special relativistic formulation. Even for standard $n=4$ spacetime ...
5
According to our best understanding of how Maxwell's equations would generalize to other dimensions, then yes, it is. The electromagnetic field is represented by an antisymmetric tensor,
$$F^{\mu\nu} = \begin{pmatrix}0 & -E_x & -E_y & -E_z \\ E_x & 0 & -B_z & B_y \\ E_y & B_z & 0 & -B_x \\ E_z & -B_y & B_x & ...
5
The composition law for quantum systems is always a tensor product. Your problem arises from a confusion over what the tensor product is applied to: you are trying to tensor product the spatial coordinates together, when it is in fact the basis vectors of the Hilbert space you should be tensoring together.
More formally, take two quantum systems A and B, ...
5
If you assume it is a lamp that is infinitely far away, the lamp size will also have to be infinitely large :)
What I think you want to do is let the actual light come from infinity, but place the lamp 1AU (=~150,000,000 km) away, with a radius of 695,500 km (actual values).
If this is not possible or impractical in your software for some reason, just ...
5
I'll try to answer it by considering radial deviations from a circular orbit. First we have to assume two things about our n-dimensional universe: Newton's second law still holds, that is,
for a particle's position vector in n-dimensions $\vec{x} = (x_1, x_2, \cdots x_n)$,
\begin{align}
m \ddot{\vec{x}} = \vec{F},
\end{align}
where $\vec{F}$ is some ...
5
In science fiction it's not uncommon to see phrases like "travel into the fourth dimension" as if the "fourth dimension" was something you could leave our universe and travel into. However this is not what the word dimension means in physics. The dimension of a spacetime is simply the number of co-ordinates needed to locate a point within it. For example to ...
4
Let me try to answer differently, in some sense more mathematically, and more directly to the point. Intuitively, the dimensionality is the number of independent numbers you need to identify a point. However, the redefinitions of the coordinates need to be continuous which prohibits your "labeling of the pixels"
By the way, one can easily define one-to-one ...
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