Multi-dimensional grand unification?

[Enola Straight]

Einstein used Reimann geometry (a kind of non-Euclidean geometry which involves things like parallel lines don't exist an' stuff ) to describe how mass can distort a four dimensional field ( length, width, depth, time), thus describing gravity.

This is the General Theory of Relativity

http://library.thinkquest.org/27930/med ... tensor.gif

Later, the Kaluza-Klein theory arose by simply adding a fifth dimension to 4-d Relativity, which then yielded both Gravity AND Electromagnetism, the force which manifests itself as electricity, magnetizm, light, and chemical bonding.

http://feynman.physics.lsa.umich.edu/se ... ture11.gif

Eventually, you could add two more dimensions to describe the Weak Nuclear Force...responsible for certain kinds of nuclear decay...and then four more dimensions to describe the Strong Nuclear Force, which binds together quarks into protons and neutrons.

http://members.aol.com/yggdras/paraphysics/image2.gif
http://www.ldolphin.org/matrix2.gif

Now there is this other little thing called the Standard Model:

http://boomeria.org/physicslectures/sec ... dmodel.jpg

which arranges various elementary particles by the properties of mass, charge, and spin.

A dimension is basically a linear measurement.

If space and time are dimensions, are mass, charge, and spin dimensions, too?

If so, can the 11-d supergravity/supersymmetry/superstring/M-theory have added to it three more dimensions of the Standard Model?

[TheLemur]

If so, can the 11-d supergravity/supersymmetry/superstring/M-theory have added to it three more dimensions of the Standard Model?

Er, no, it doesn't work that way. For one thing, dimensions do not add neatly like that; note that Maxwell's theory is 3D and Einstein's is 4D, yet Kaluza-Klein is 5D, not 7D. For another thing, 11D string theory already includes Einstein's four dimensions. For a third thing, I believe the Standard Model is 4D, not 3D.

[Kuroneko]

Mass is not a dimension, but a particular kind of curvature of spacetime. General relativity cannot represent spin natively, although its cousin, Eistein-Cartan theory, can. There is actually a relativistic form of Maxwell's equations, so GTR can handle classical electromagnetism natively; the difficulty is with mimicking the quantum version. Kaluza-Klein theory described electromagnetism by adding a dimension; this dimension is compact, with the motion of a charged particle looping in it to overall make something life a helix, with the handedness of this helix corresponding to the sign of the charge. Regarding the standard model, we have four spacetime dimensions with strong and electroweak interactions adding seven compact dimensions.

I'm not certain whether I understand your question, but I hope that answers it. [edit: corrected number of dimensions]

[TheLemur]

Regarding the standard model, we have four spacetime dimensions with strong interactions (color being described by three quarks with SU(3) symmetry) adding three and electroweak interactions three more.

The Harvard physics people say the Standard Model is 5D; see here.

[Kuroneko]

The Harvard physics people say the Standard Model is 5D; see here.

Have you even read the abstract? The standard model has four dimensions--three space and one time. The "Harvard physics people" were extending the model by having more than that. The standard model's gauge group SU(3)×SU(2)×U(1) actually has twelve dimensions. It takes seven extra compact dimensions (not six, as I erroneously said above) to have a Kaluza-Klein theory of that type, which is what I thought the original poster was asking.

[TheLemur]

The standard model has four dimensions--three space and one time.

I thought you just said that the Standard Model had seven additional dimensions to handle strong and electroweak interactions. I'm not a physicist; I simply remembered hearing that the SM had four dimensions and Googled it.

[Kuroneko]

I thought you just said that the Standard Model had seven additional dimensions to handle strong and electroweak interactions.

Not quite. Perhaps this wasn't clear, but since the discussion was in the context of Kaluza-Klein theories, that type of theory needs at least seven extra dimensions to represent the interactions in the standard model. To have gravity and electromagnetism together can be done in plain GTR already--it's just another type of field on a curved spacetime background. The significance of Kaluza-Klein for electromagnetism is that it treats EM not as a field but as another type of spacetime curvature just like gravity by introducing an extra dimension.

[Surlethe]

The significance of Kaluza-Klein for electromagnetism is that it treats EM not as a field but as another type of spacetime curvature just like gravity by introducing an extra dimension.

As a wetter-behind-the-ears pup a while ago learning about the conceptual basis for GR, I wondered why Einstein didn't treat each force-field as curved four-dimensional spacetime. After all, to my mind, the "elevator" thought experiment works equally well in a gravitational field or wearing metal boots in an elevator with a magnet on the bottom. Obviously, there are good reasons why Kaluza-Klein has to add a fifth dimension to curve spacetime for EM; could you perhaps please explain them?

[Thinkmarble]

Accelerated electrons radiate

[Kuroneko]

After all, to my mind, the "elevator" thought experiment works equally well in a gravitational field or wearing metal boots in an elevator with a magnet on the bottom. Obviously, there are good reasons why Kaluza-Klein has to add a fifth dimension to curve spacetime for EM; could you perhaps please explain them?

The conclusion is based on the fact that locally, there is no observable distinction between a uniform gravitational field and acceleration. This is not the case for, say, a uniform electric field, in which a charge would emit radiation.

In five dimensions, the metric tensor g_μν has 5² = 25 components. The initial 4×4 (g_{0-3,0-3}) block corresponds to a standard GTR metric, the g_{0-3,4} to a Maxwell field (those terms are the electromagnetic four-current) and the g_44 term another scalar field (actually, isn't not simply pasted together, but those terms can be extracted via simple algebra); the remaining g_{4,0-3} terms are exactly the same as g_{0-3,4} because GTR assumes g to be symmetric. The neat part about this is that the Einstein field equation in five dimensions automatically gives rise to the Maxwell equations for the g_{0-3,4} four-current term if the scalar field term is constant. So in essence, (classical) electromagnetism is a compactification of five-dimensional gravity.

Accelerated electrons radiate

Well, it depends. Electrons undergoing uniform proper acceleration do not radiate in the comoving frame. But uniform fields are what we should be looking at for this gedankenexperiment.