Curl or maxwell equations in higher dimensions

In summary, the conversation discusses the topic of Maxwell's equations in higher spatial dimensions and the possibility of representing the equations involving Curl in these dimensions. The discussion also touches on the book of Zwiebach on string theory, as well as the Helmholtz equation and the concept of redefining the 4-vector potential as a 5-vector. The conversation also mentions the behavior of the generalization of the crossproduct in higher dimensions and the potential emergence of magnetic monopoles. However, it is noted that further investigation may not result in the expected outcome.
  • #1
snowstorm69
13
0
Anyone know what topic, branch of math, book, or subject I should look up in order to find a formulation for Maxwell's equations in higher spatial dimensions? I don't mean having time as a 4rth dimension. I mean a 4rth (and more) spatial dimension. This would require the maxwell exquations involving Curl to be represented in higher dimensions, which would require that the curl itself be represented in higher dimensions. Does the curl (and do the 2 maxwell's equations involving curl) only apply to 3-D or is it extendable to higher dimensions? Where can I read about this?
Thanks!
 
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  • #3
Dear Vid,
Thank you so much! Can't wait to read this, and very cool that he was your high school physics teacher! Meanwhile I hope this doesn't discourage others from posting a response but... this response does look excellent! My heartfelt thanks for your help.
 
  • #4
The book of Zwiebach on string theory covers this topic quite early, somewhere in the first chapters. Maybe interesting for you.
 
  • #5
Thanks, haushofer,
Looks like a great book; I think I'm going to get it. Also... what about the Helmholtz equation that you get from the Maxwell's equations; any idea where to read about that in higher spatial dimensions? I may post that as a separate question but am very interested in an answer whether from this thread or another. Thanks guys.
 
  • #6
To do it 'right' you first might want to redefine the 4-vector potential as a 5-vector.

But it doen't partition as nicely as it does in spacetime. In the usual spacetime you get the electric and magnetic fields that both appear to be vectors (but don't really transform as vectors with a change in inertial frame).

In 5 dimensions you get something you might call an 'electric field' that 4 components--one that can be associated with each spatial dimension. But the other part that generalizes the magnetic field has 6 elements. (Each element is associated with two of your spatial dimensions rather than one-on-one.)

On top of all that, you get magnetic monopoles popping out of it, after taking another derivative.

This all has to do with the way the generalization of the crossproduct that utilizes the completely antisymmetric field tensor behaves in higher dimensions.

There's nothing wrong with investigating this, of course, it just might not turn out to be as you expect.
-deCraig
 

1. What is the significance of curl and Maxwell equations in higher dimensions?

The curl and Maxwell equations are fundamental equations in electromagnetism that describe the behavior of electric and magnetic fields in three-dimensional space. In higher dimensions, these equations still hold true and are crucial in understanding the behavior of electromagnetic fields in more complex environments.

2. How do curl and Maxwell equations change in higher dimensions?

The basic form of the curl and Maxwell equations remains the same in higher dimensions, but the number of variables and terms may increase. For example, in four dimensions, the curl equations have four variables instead of three, and there are additional terms in the Maxwell equations to account for the extra dimension.

3. Can the curl and Maxwell equations be used to describe electromagnetic fields in non-Euclidean spaces?

Yes, the curl and Maxwell equations can be extended to non-Euclidean spaces, such as curved spaces in general relativity. In these cases, the equations may have different forms, but they still describe the behavior of electromagnetic fields in these non-Euclidean spaces.

4. Are there any applications of curl and Maxwell equations in higher dimensions?

Yes, there are many applications of these equations in higher dimensions, particularly in fields such as astrophysics, where the behavior of electromagnetic fields in higher dimensional space is crucial in understanding phenomena like black holes and gravitational waves.

5. Are there any ongoing research efforts related to curl and Maxwell equations in higher dimensions?

Yes, there is ongoing research in this area, particularly in the field of string theory, which suggests that our universe may have more than three dimensions. Scientists are exploring how the curl and Maxwell equations may be modified or extended in these higher-dimensional theories to better understand the behavior of electromagnetic fields in these complex spaces.

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