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59

Moving through space at a uniform pace does not require energy, or force (Newton's 1. law), but accelerating through space does (Newton's 2. law). Similarly, moving through time at a uniform pace does not require a force, but if you're accelerating, your time will change wrt. a non-accelerating observer, so in a way you might say that you accelerate through ...


48

From a mathematical point of view you will never make something two dimensional by squeezing it because it will always have a thickness greater than zero. The limit would be something like graphene that is a single atom thick. This is pretty thin, but it still has a non-zero thickness so it's still 3D. However in the quantum world it is possible to produce ...


34

Coming from a math perspective, I would define a dimension as "any property which is orthogonal to all other properties." "Orthogonal" here means you cannot get to one property by applying scalar operations on another. For example, the x-axis dimension can never become a y-axis value, and similarly for time vs. spatial dimensions. For that matter, ...


29

By your own definition, "one atom thick" is not two dimensional. In that case, you would have to squish something so hard that the atoms stop existing. In which case it is not two dimensional any more, either. As John pointed out, graphene is often considered a model for a 2D material - it is SO much thinner in the thickness dimension that we have to ...


29

I think that you are referring to space-filling curves and how they can map a line segment to more than one dimension. For example the Hilbert space filling curve can be used to map the interval $[0,1]$ to $[0,1]\times[0,1]$. I am afraid while a continuous bijection is possible one-way, it is not possible to have a homeomorphism between two different ...


27

In this context, I usually explain it (non-mathematically) by saying that the number of dimensions is the number of values you need to specify where an event occurs. For most people this involves space and time (but for particle physicists it might involve more values ;). Anyway, certainly even people before Einstein would need to specify the time as well ...


24

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, ...


24

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 ...


17

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 ...


17

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 ...


15

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 ...


14

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 & ...


14

The dimensionality of a system in practice means the number of dimensions in which objects confined to that system are free to move. For graphene we are generally talking about the motion of electrons within it (though I guess we could be talking about phonons). Anyhow, the thickness of the sheet is around one atom, which means that in the direction normal ...


14

Varun's answer basically tells it all, but perhaps it's useful to also explain why we want a homeomorphism rather than just a bijection. The whole idea of modelling "real physical space" by an Euclidean $\mathbb{R}^3$ is so we can make predictions about physical processes, like how an electromagnetic wave spreads. Typically, we use differential equations ...


14

Maxwell's equation can be given in the form $$\text dF = 0$$ $$\text d\star F + J = 0$$ where $F$ is a 2-form and $J$ an $n-1$-form (a current density) which in principle can be generalised to any manifold (for physical reasons one might want to consider pseudo-Riemannian manifolds with signature $(+,-,\cdots,-)$). In the four dimensional theory one usually ...


13

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 ...


13

A classical1 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 ...


13

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 ...


13

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.


12

Dimensional analysis can help to "guesstimate" the form of many important results but it can, for instance, not produce general solutions to equations of motion. It's an invaluable tool to understand the structure of physical theory, including quantum mechanics and relativity, and to check results for consistency, but it can rarely replace complex ...


11

The terms elliptic and ultrahyperbolic are technical terms used in the classification of partial differential equations, and going into their precise meaning wouldn't be very illuminating. Basically for these types of PDEs a set of initial data doesn't uniquely determine the evolution of the system and any particular solution, like the planetary orbits in ...


11

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 ...


11

Related answer: http://math.stackexchange.com/a/532746/24293 Looking at the comments, you seem to be asking why there are chiral 'pairs' and not chiral multiplets. Looking at the tag, it looks like you want an analysis of higher dimensions as well. TL;DR Short answer: In any number of dimensions, chiral objects come in pairs. This is because numbers ...


11

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 & ...


10

I) 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. II) 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 ...


10

This is perhaps best summarized by a review that was left on his book's amazon site: I bought the book, because I am a graduate student in string theory and was curious about "new" ways of thinking in ten dimensions. I knew the author of the book was actually a musician (some research with google was required for that), but so is Brian May of Queen, and ...


9

This idea is explored at great length in the novella "Flatland", which is a classic of pedagogical mathematics. It was published in the late 19th century, and the themes reappear in H.G. Welles "The Time Machine", and in Einstein's relativity. The four-dimensional pyramid will, if it crossed our 3-dimensional universe, as a cube growing from a point, then ...


9

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 ...



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