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1. Kaluza-Klein theory. This is similar to General Relativity, but instead of three space dimensions plus time, there are four space dimensions plus time. The fourth dimension is cyclic, and satisfies some symmetry conditions. The electromagnetic potential appears as the components of the metric in the fourth space dimension. It is usually rejected on the ...


9

When you write the five dimensional Kaluza-Klein metric tensor as $$ g_{mn} = \left( \begin{array}{cc} g_{\mu\nu} & g_{\mu 5} \\ g_{5\nu} & g_{55}\\ \end{array} \right) $$ where $g_{\mu\nu}$ corresponds to the ordinary four dimensional metric and $ g_{\mu 5}$ is the ordinary four dimensional vector potetial, $g_{55}$ appears as an ...


9

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 ...


9

If your definition of a lens space means $S^3/\Gamma$, a quotient of the three-sphere, then the $U(1)$, $SU(2)$ couplings may (classically) be derived from the parent theory on the full three-sphere whose isometry is $SO(4) \approx SU(2) \times SU(2)$ (at the level of Lie algebras). The subgroup $\Gamma$ acts only inside one of the $SU(2)$ factors, let's say ...


7

Annav gave the correct answer, but here's some help on visualisation. First thing: We cannot imagine more than 3 space dimensions. You can try, and get tantalizingly close, but it's extremely hard to wrap one's brain around it Due to this, I shall explain this in lower dimensions,and you can try to generalise it. Try. Alright. Let's imagine a thin hose. ...


7

In addition to what dmckee said, another hint at ("large") extra dimensions would be the detection of Kaluza-Klein particles at the LHC for example. Kaluza-Klein particles are in principle nothing but the known standard model particles which can propagate into the extra dimensions if these are large enough. It can be shown that the angular momentum in these ...


7

Yes, classically, we can unify gravity with electromagnetism. The theories that do so are the famous Kaluza-Klein theories. They are theories of pure gravity in $4+1$ dimensions rather than our usual $3+1$ dimensions. When such theories are viewed from a $3+1$ dimensional perspective, the effects of gravity in the fourth unseen dimension appear in the ...


6

Dilaton is the generic name of the Goldstone Boson (GB) associated with spontaneous breaking of scale invariance. Any model that break scale invariance spontaneously will give rise to a dilaton. N=4 SYM for example has a moduli space and any modulus away form the origin will break conformal invariance, and a massless dilaton would appear. The Radion is just ...


6

As the very formulation of your question makes clear, we know what the actual algebra of local symmetries is. It is the five-dimensional diffeomorphism invariance assuming the $M^4\times S^1$ topology of the five-dimensional spacetime. The term "Kač-Moody generalization of an algebra" is nothing else than an alternative name for this algebra, especially for ...


6

The definition of dimension used here is that of a dimension of a manifold - essentially, how many coordinates (=real numbers) we need to describe the manifold (thought of as spacetime). Manifolds may carry a notion of length, and one of volume. They may also be compact or non-compact, roughly1 corresponding to finite and infinite. E.g. a sphere of radius $...


5

On Unification I presume you're asking whether just classical gravity & classical EM can be unified. They sure can! Classical General Relativity and Classical Electromagnetism are unified in Kaluza-Klein-Theory, which proves that 5-dimensional general relativity is equivalent to 4-dimensional general relativity plus 4-dimensional maxwell ...


5

Make some assumptions about the physics associated with the dimensions in questions (say electric field strength goes by $r^{-(n-1)}$ over distances in which $n$ dimensions are significant). Make predictions on that basis Compare to experiment Many predictions can be made and tested in the realms of high energy particle physics, but so far all are null.


5

Let us quickly run through the standard KK compactification. We start with a $d+1$ dimensional theory $$ S = \frac{1}{16\pi G_{d+1}} \int d^{d+1}x \sqrt{G} R_{d+1} $$ More general actions on the $d+1$ dimensional space can be considered, but this will suffice for our purposes. The metric $G_{MN}$ can be decomposed as $$ ds^2 = G_{MN} dx^M dx^N = e^{2\Phi} \...


5

Are they identifying inertial mass with the ADM mass? There is an old style way of writing $g_{00}$ of a stationary metric, far from the matter source, as $-\left(1 - \frac{1}{2}\phi({\vec x})\right)$, where $\phi$ is the potential function for the metric, and then you can monopole expand this and identify the numerator of the $\frac{1}{r}$ term of the ...


4

The key is in the phrase following immediately after To us, the particle would not look like one particle, but a set of particles – all with different masses. which is The faster the particle moves along the extra dimension, the larger this apparent mass seems to be. (here is a link) I think that it means that the same particle in extra ...


4

The group manifold $U(1) \times SU(2)\times SU(3)$ is $1+3+8=12$-dimensional, not 7-dimensional. You probably meant the dimension of a manifold that may have this group as its isometry group. But one may show that no such low-dimensional manifold can be interpreted as the extra dimensions of string theory to produce a realistic model. The oldest Kaluza-...


4

Here is a nice illustration of the Calabi Yau manifold. One can visualize at each point of our 3 dimensional space as a tiny manifold like that which encloses the extra dimensions. Alternatively: if our third dimension were curled up we would be living in Flatland , without knowledge of the third dimension. One can rotate a two dimensional figure into the ...


4

The standard way to measure compactified dimensions is to test some inverse-square law (e.g. Newton's, electromagnetic, diffusion) at the scale and see if it breaks down and starts approaching some other (higher power) inverse-power law. In fact, the inverse-square law has only been verified down to a scale of 0.1mm -- here's a recent experimental paper ...


4

In differential geometry, a space of a given number of dimensions can be curved rather than Euclidean, so for example the surface of a sphere is understood to be a 2-dimensional space in spite of the fact that we can't help but visualize the sphere sitting in a higher-dimensional 3D Euclidean space. This 3D space that we imagine the 2D surface sitting in is ...


4

The Kaluza-Klein equations of motion (the geodesic equations) for a particle moving in the 5D spacetime contain the equations of motion of a particle in 4D spacetime under influence of electromagnetism if and only if one identifies $p^5 = mU^5 = \frac{1}{\sqrt{G}}cq$, i.e. relates the momentum in the fifth dimension $p^5$ to electric charge $q$. (And yes, ...


4

You are talking about the two ways to construct a consistent 10-dimensional heterotic string theory. The original paper "Heterotic string theory: I. The free heterotic string" by Gross, Harvey, Martinec and Rohm is rather accessible and describes the detailed construction. I'll address your specific question of how the compactification on a torus manages to ...


4

It is not really discredited, It is considered to be an important precursor to string theory. It was attractive in uniting electromagnetism with gravity, but could not accommodate the weak and strong interactions. The standard model of physics developed and the two could not be reconciled (that is the discredited part) until string theories with many ...


4

You need to be careful to define exactly what you mean by Kaluza-Klein theory. Originally it was the theory developed by Theodore Kaluza in 1919, and subsequently further developed by Oscar Klein, however the term has come to mean any theory involving compactified extra dimensions. In this latter sense it is most certainly not discredited because the idea is ...


4

General Relativity explains gravity, not the other three forces. Physicists have tried to extend General Relativity to higher dimensions to explain other forces, as in Kaluza-Klein theory, but this has not been successful. GR does “accommodate” other forces in the sense that we know how to write, for example, the Lorentz force law in curved spacetime, so ...


4

These extra dimensions are very small, on the order of the Planck length, they also wrap around. Eg, if w is a compactified dimension, if you travel in the w direction you quickly get back to where you started, like in those computer games (eg Asteroids) where if you go off the right side of the screen you reappear on the left side. The original theory that ...


3

The Cartan formalism is ideal for working in a great deal of generality, one does need even need to fix the dimension of spacetime. I will start with the Kaluza-Klein metric ansatz, namely, $$\mathrm{d}s^2 = g_{\mu\nu}\mathrm{d}x^\mu \mathrm{d}x^\nu - e^{2\sigma}\left[\mathrm{d}\psi + A_{\mu}\mathrm{d}x^\mu \right]$$ where $A$ is a potential $1$-form, $\...


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