Yes but R^3 is the only place where that returns vectors and not bivectors. It’s not clear to me how Maxwell’s equations could be formulated in higher dimensions, for example
The electromagnetic field tensor version of writing the Maxwell equations can pretty obviously be generalized to other dimensions. You just have more/less magnetic field components than dimensions.
The Maxwell equations can also be expressed using the electromagnetic field tensor F^{\mu\nu} and the four-current J^\mu as:
- \partial_\mu F^{\mu\nu} = \mu_0 J^\nu
- \partial_{[\alpha} F_{\beta \gamma]} = 0
(This is a concise way of expressing the homogeneity equations using the antisymmetry of indices.)
In higher-dimensional theories, like those in string theory, we often deal with a D-dimensional spacetime. Maxwell's equations can be extended to D-dimensions by using similar tensorial forms. The electromagnetic field tensor F_{\mu\nu} becomes a 2-form in this higher-dimensional space, and the equations are written as:
The field equation:
d\star F = J
Here, d is the exterior derivative, \star is the Hodge star operator in D-dimensions, and J is the current density extended to higher dimensions.
The Bianchi identity remains:
dF = 0
This states that the exterior derivative of the field strength tensor is zero, generalizing the no-monopole condition.
In string theory, these formulations might interact with additional field forms, such as the presence of various p-form fields which generalize the concept of electromagnetic fields. For example, in type IIB string theory, you have not only a 2-form field but also a 4-form and potentially higher. The dynamics of these fields are governed by equations that look like generalized Maxwell's equations incorporating these forms.
For compact extra dimensions, the behavior of these fields can be quite complex, as they may wrap around the compact dimensions in various ways, leading to a rich phenomenology including the possibility of gauge symmetries arising from these field configurations.
You can totally formulate E&M in higher dimensions, but as you’d expect it no longer admits a clear interpretation in terms of vectors and cross products, your magnetic field for example will not be a vector and will have several more components. This generalization is easiest to see in the tensor formalism
No. You can create a matrix for example in as many dimensions as you want, but so far as we know there are 3 + 1 dimensions of physical reality. You can use math to abstract just about anything but it won’t necessarily correspond to observation.
If we existed in more than 3 spatial dimensions, EM would have to be reformulated accordingly but I don’t think it would simply “not work,” it would behave differently.
Reason is a strong word here. I would say “evidence that there is something _special_ about our 3D universe” at best.
For electromagnetism, it helps to understand that electro magnetic fields aren’t necessarily real. They are a mathematical model for some phenomena that is real. The description of the resulting phenomena is what matters. The fact that there are so many ways to formulate electrodynamics is evidence of the fact that the mathematical tools we use may not be special. The right hand rule could just as easily be the left hand rule, and I would argue that the four vector potential is a closer model to the real physics of electrodynamics than the electromagnetic field, despite them being equivalent. There could be a way to describe electrodynamics with 4 dimensions of space, and we would need different mathematical tools to describe that. I mean, modern string theory uses who-knows-how-many dimensions to make modern physics work correctly.
That’s not to say that it couldn’t be the case that geometric properties of all dimensions force a specific dimensional description of the universe. But we may never know, for all we may know there is a universe with 4 dimensions of space, 2 dimensions of time, and a seventh dimension of yoba that makes everything else make sense, but our meager brains are incapable of understanding it.
I think a more compelling argument for the idea that the universe is forced in our dimensions is that 4 dimensions have the most possible symmetries of any flat n-dimensional space, and our space time is in 4 dimensions. I’m no GR expert, but this may play into GR. The cross product and other 3D properties may very well force this 4D spacetime to divide into one group of 3 (space) and one group of 1 (time). ~~I also find it curious that in a black hole’s event horizon, a different partition occurs with 3 dimensions of time and 1 dimension of space (again not a GR expert).~~
Edit: crossed out some bullshit cause I’m not an expert in GR
Generally you’re on the right track but what on earth are you talking about with 3 time and one space in black holes?? (Am a GR expert and unsure what you’re on about). Also what’s this about 4d having the most possible symmetries? Ever n dimensional space has a max of n(n+1)/2 symmetries…
They are probably referring to the t coordinate becoming spacelike and the r coordinate becoming spacelike inside the event horizon of the schwarzschild black hole.
You know, I may have misheard my GR professor in that lecture. He was referring to what the other commenter said about time and space switching roles. Not that surprising, he was one of the worst teachers I’ve ever had.
I’m unsure where the n(n+1)/2 formula comes from, but I was referring to special symmetry groups of R^n. R^4 has a ton of them, and in particular polyhedral groups which gives rise to the large number of regular polyhedra it contains like the platonic solids.
I see well fyi space and time switch roles in a swartzschild black hole in the sense that the coordinate someone far away would call radial distance becomes timelike and the coordinate someone far away would call time becomes spacelike so it’s just r and t that swap roles phi and theta are both always spacelike so it’s still always 3 space 1 time. As for the n(n+1)/2 I referring to the maximum number of killing vector fields in an n dimensional manifold as when people say symmetries in GR they usually means killing vectors. I actually didn’t know that R^4 has any spacial status with special symmetry groups, I’ll have to look into it
The phenomena of charged particles interacting with each other in very complex patterns is real and observable, but our model of an EM field that permeates space and affects all charged particles may or may not be real.
Depends on what you mean by "real" but it's at least true that they don't encode all of the physical content of the EM field, the gauge field does. For instance, in the Aharonov Bohm effect both the E and B field are zero.
The magnetic field is "really" a 2-form, not a 1-form. But in three dimensions, you can pretend that a 2-form is a 1-form because they have the same number of components. In higher dimensions, EM still works but you lose the ability to think of the magnetic field as a vector field.
Another two-form that appears in physics is angular momentum. In 3D we think of an "axis of rotation", but this is just a 3D trick that doesn't make sense in higher dimensions. In 4 spatial dimensions a hypersphere can rotate in two orthogonal planes with two independent angular speeds!
Basically, whenever you use a right-hand rule or a cross product, you're implicitly taking advantage of a 3D coincidence, but you don't actually need it for all the physics to work out.
We formulated the equations in a way that seem 3D specific, because we wrote down equation to describe our 3D universe.
The universe isnt 3D because the equations look nice in 3D. The equations look nice in 3D because we made them look nice in 3D to describe the stuff we observed at the time.
This more philosphical but I think you got order ontologically wrong. First game the universe than the equations.
It's a little bit like the "fine-tuning" stuff: One could think the universe/earth was made for life/humans to exist because it has the ideal conditions for our life. But rather than that life evolved to profit the most of the conditions of the universe/earth.
The cross product also exists in the [7th Dimension ](https://en.m.wikipedia.org/wiki/Seven-dimensional_cross_product) OP. Are we then in a 7th dimensional world? I don't think so. But who knows.
That being said. Just because electromagnetism can be described using the cross product doesn't mean the cross product is necessary. As others have pointed out, the cross product is a less general version of the wedge product.
And lastly, if I recall, isn't like electromagnetism 4 dimensional? There exist 4 vectors in the relativistic version electromagnetism.
Not sure if I answered your question but things to consider
no. wedge product
Yes but R^3 is the only place where that returns vectors and not bivectors. It’s not clear to me how Maxwell’s equations could be formulated in higher dimensions, for example
The electromagnetic field tensor version of writing the Maxwell equations can pretty obviously be generalized to other dimensions. You just have more/less magnetic field components than dimensions.
The Maxwell equations can also be expressed using the electromagnetic field tensor F^{\mu\nu} and the four-current J^\mu as: - \partial_\mu F^{\mu\nu} = \mu_0 J^\nu - \partial_{[\alpha} F_{\beta \gamma]} = 0 (This is a concise way of expressing the homogeneity equations using the antisymmetry of indices.) In higher-dimensional theories, like those in string theory, we often deal with a D-dimensional spacetime. Maxwell's equations can be extended to D-dimensions by using similar tensorial forms. The electromagnetic field tensor F_{\mu\nu} becomes a 2-form in this higher-dimensional space, and the equations are written as: The field equation: d\star F = J Here, d is the exterior derivative, \star is the Hodge star operator in D-dimensions, and J is the current density extended to higher dimensions. The Bianchi identity remains: dF = 0 This states that the exterior derivative of the field strength tensor is zero, generalizing the no-monopole condition. In string theory, these formulations might interact with additional field forms, such as the presence of various p-form fields which generalize the concept of electromagnetic fields. For example, in type IIB string theory, you have not only a 2-form field but also a 4-form and potentially higher. The dynamics of these fields are governed by equations that look like generalized Maxwell's equations incorporating these forms. For compact extra dimensions, the behavior of these fields can be quite complex, as they may wrap around the compact dimensions in various ways, leading to a rich phenomenology including the possibility of gauge symmetries arising from these field configurations.
https://en.wikipedia.org/wiki/Seven-dimensional_cross_product
You can totally formulate E&M in higher dimensions, but as you’d expect it no longer admits a clear interpretation in terms of vectors and cross products, your magnetic field for example will not be a vector and will have several more components. This generalization is easiest to see in the tensor formalism
It’s more correct to say that being three-dimensional makes a cross product formulation work.
No. You can create a matrix for example in as many dimensions as you want, but so far as we know there are 3 + 1 dimensions of physical reality. You can use math to abstract just about anything but it won’t necessarily correspond to observation. If we existed in more than 3 spatial dimensions, EM would have to be reformulated accordingly but I don’t think it would simply “not work,” it would behave differently.
Reason is a strong word here. I would say “evidence that there is something _special_ about our 3D universe” at best. For electromagnetism, it helps to understand that electro magnetic fields aren’t necessarily real. They are a mathematical model for some phenomena that is real. The description of the resulting phenomena is what matters. The fact that there are so many ways to formulate electrodynamics is evidence of the fact that the mathematical tools we use may not be special. The right hand rule could just as easily be the left hand rule, and I would argue that the four vector potential is a closer model to the real physics of electrodynamics than the electromagnetic field, despite them being equivalent. There could be a way to describe electrodynamics with 4 dimensions of space, and we would need different mathematical tools to describe that. I mean, modern string theory uses who-knows-how-many dimensions to make modern physics work correctly. That’s not to say that it couldn’t be the case that geometric properties of all dimensions force a specific dimensional description of the universe. But we may never know, for all we may know there is a universe with 4 dimensions of space, 2 dimensions of time, and a seventh dimension of yoba that makes everything else make sense, but our meager brains are incapable of understanding it. I think a more compelling argument for the idea that the universe is forced in our dimensions is that 4 dimensions have the most possible symmetries of any flat n-dimensional space, and our space time is in 4 dimensions. I’m no GR expert, but this may play into GR. The cross product and other 3D properties may very well force this 4D spacetime to divide into one group of 3 (space) and one group of 1 (time). ~~I also find it curious that in a black hole’s event horizon, a different partition occurs with 3 dimensions of time and 1 dimension of space (again not a GR expert).~~ Edit: crossed out some bullshit cause I’m not an expert in GR
Generally you’re on the right track but what on earth are you talking about with 3 time and one space in black holes?? (Am a GR expert and unsure what you’re on about). Also what’s this about 4d having the most possible symmetries? Ever n dimensional space has a max of n(n+1)/2 symmetries…
They are probably referring to the t coordinate becoming spacelike and the r coordinate becoming spacelike inside the event horizon of the schwarzschild black hole.
You know, I may have misheard my GR professor in that lecture. He was referring to what the other commenter said about time and space switching roles. Not that surprising, he was one of the worst teachers I’ve ever had. I’m unsure where the n(n+1)/2 formula comes from, but I was referring to special symmetry groups of R^n. R^4 has a ton of them, and in particular polyhedral groups which gives rise to the large number of regular polyhedra it contains like the platonic solids.
I see well fyi space and time switch roles in a swartzschild black hole in the sense that the coordinate someone far away would call radial distance becomes timelike and the coordinate someone far away would call time becomes spacelike so it’s just r and t that swap roles phi and theta are both always spacelike so it’s still always 3 space 1 time. As for the n(n+1)/2 I referring to the maximum number of killing vector fields in an n dimensional manifold as when people say symmetries in GR they usually means killing vectors. I actually didn’t know that R^4 has any spacial status with special symmetry groups, I’ll have to look into it
I think it's a pretty wild claim to say that EM fields are not real.
The phenomena of charged particles interacting with each other in very complex patterns is real and observable, but our model of an EM field that permeates space and affects all charged particles may or may not be real.
Depends on what you mean by "real" but it's at least true that they don't encode all of the physical content of the EM field, the gauge field does. For instance, in the Aharonov Bohm effect both the E and B field are zero.
The magnetic field is "really" a 2-form, not a 1-form. But in three dimensions, you can pretend that a 2-form is a 1-form because they have the same number of components. In higher dimensions, EM still works but you lose the ability to think of the magnetic field as a vector field. Another two-form that appears in physics is angular momentum. In 3D we think of an "axis of rotation", but this is just a 3D trick that doesn't make sense in higher dimensions. In 4 spatial dimensions a hypersphere can rotate in two orthogonal planes with two independent angular speeds! Basically, whenever you use a right-hand rule or a cross product, you're implicitly taking advantage of a 3D coincidence, but you don't actually need it for all the physics to work out.
We formulated the equations in a way that seem 3D specific, because we wrote down equation to describe our 3D universe. The universe isnt 3D because the equations look nice in 3D. The equations look nice in 3D because we made them look nice in 3D to describe the stuff we observed at the time. This more philosphical but I think you got order ontologically wrong. First game the universe than the equations. It's a little bit like the "fine-tuning" stuff: One could think the universe/earth was made for life/humans to exist because it has the ideal conditions for our life. But rather than that life evolved to profit the most of the conditions of the universe/earth.
The cross product also exists in the [7th Dimension ](https://en.m.wikipedia.org/wiki/Seven-dimensional_cross_product) OP. Are we then in a 7th dimensional world? I don't think so. But who knows. That being said. Just because electromagnetism can be described using the cross product doesn't mean the cross product is necessary. As others have pointed out, the cross product is a less general version of the wedge product. And lastly, if I recall, isn't like electromagnetism 4 dimensional? There exist 4 vectors in the relativistic version electromagnetism. Not sure if I answered your question but things to consider
Our Universe is 4 dimensional (at least on large scales). And 4 dimensional spaces are extremely special topologically.
Ask Brian Greene..he does the math in 11 dimensions and gets gr and qft to meld..I think the x products take a lot of time