# Magnetic monopoles and special relativity

I was thinking about magnetism as a product of special relativity and the result of this approach to the magnetic monopoles. So if magnetism is a product of electricity(like electricity from another point of view),then why do we need monopoles to exist?I know that many theories predict the existence magnetic monopoles but i am referring specifically to the concept of classical relativistic magnetism and magnetic monopoles,so do not give me answers that are mainly based on what other theories predict.

EDIT: why do we need electric AND magnetic monopoles to describe electromagnetism if the two are the same thing from another moving frame of reference?And if we do not NEED magnetic monopoles,why is there even a place for them to exist in relativistic electromagnetism?

EDIT: I know that the mathematics of the theory allow for magnetic monopoles, but the essence of the question is the following:
If I work from one frame of reference and change to any other frame of reference, there are no sources of magnetism that can be related to magnetic monopoles?

• I'm not sure we need them to exist, but they are problematic in that they can exist in theory but we don't find them in nature (conclusively as far as I'm aware); you can have free electric charges (say an electron), and you can even find in nature the analogous electric dipole (looks like a bar magnet conceptually, but swapping electric and magnetic fields / charges around), but no free magnetic charges? May 6, 2015 at 0:27
• – Paul
May 6, 2015 at 2:02
• @XerenNarcy I am asking something different..why is there even a place for magnetic monopoles to exist if magnetism can be described by special relativity as electricity May 6, 2015 at 8:49

The mathematical model for classical electromagnetism just doesn't forbid magnetic monopoles by construction.

Consider an arbitrary vector field $X$ in 3d. Such a vector field is totally characterized by its divergence and curl. Suppose the following is true:

$$\nabla \cdot X = \sigma, \quad \nabla \times X = Y$$

Then knowing $\sigma$ and $Y$ everywhere, one can reconstruct $X$ everywhere. The scalar field $\sigma$ and the vector field $Y$ are "sources" of $X$. That's important: it means that one only has so much freedom to add sources for a vector field.

Magnetic monopoles are one such freedom. In the context of EM, we know $\nabla \cdot B = 0$ from observations, but we still model $B$ as a vector field, and people still wonder what would happen if $B$ had nonzero divergence instead (corresponding to a magnetic monopole). This is something the math allows even though it might not be physical reality.

Such issues are very common in physics, though it's something of a matter of point of view. Why should we consider $B$ not being divergenceless, instead of, say, $B$ not being exactly a vector field? There's no real answer to that--it's just that physicists routinely must poke at the boundaries of a model to see if there's a prediction that might not hold, to see if there's a measurement or experimental result that could be confirmed more precisely.

Magnetic monopoles are just relatively easy to add to the theory.

• In relativity, B is not a vector field, and monopoles are extremely unnatural (they are the diagonal terms in an otherwise antisymmetric tensor). This whole "just add a monopole to Maxwell" philosophy only works accidentally in non-relativistic settings.
– user10851
May 6, 2015 at 5:13
• @ChrisWhite Not at all; a magnetic four-current is just a (four)-trivector source term, compared to the electric four-current that is a (four)-vector source term. The electric and magnetic four-currents can be transformed into each other through Hodge duality, just as the electric and magnetic fields can be (since bivectors, like the Faraday bivector, dualize to other bivectors). - Are you confusing the two sides of the equation? Adding magnetic monopoles in no way changes that the Faraday bivector...is a bivector (and thus can be represented with an antisymmetric matrix). May 6, 2015 at 5:20
• @Muphrid To comment on your answer,if i correctly understood you(because you have not mentioned relativity),magnetism can be said that is electricity using special relativity but if you add monopoles it just changes maxwell's equations and just give "extra magnetism"?i am a bit confused! May 6, 2015 at 8:54
• @LandosAdam That the electric and magnetic fields can be unified is true regardless of whether there are magnetic monopoles or not. If there can be magnetic monpoles, then the fields may take on different values, but they are still unified. - In my answer, I talk about adding a monopole source term in 3d; that doesn't change that the magnetic field is still a vector field in 3d. The same logic applies for special relativity. Adding a "magnetic four-current' doesn't change that the EM field is unified in 3+1 special relativity. May 6, 2015 at 15:05
• @Muphrid I understand that the mathematics allow for magnetic monopoles. But my question concerns special relativity. If I start working in any frame of reference and change to another frame of reference, there is always an absence of sources of magnetism that can be related to magnetic monopoles? This is the essence of my question. Aug 11, 2016 at 10:12

The idea that magnetism is a side-effect of electricity is deeply mistaken. The sooner you forget about that idea, the better.

Electricity and magnetism are the two aspects of a single phenomenon, electromagnetism. Read that sentence again: They are not two aspects of electricity, they are two aspects of electromagnetism. Electricity does not cause magnetism and magnetism does not cause electricity, but rather special relativity joins them together, just as it joins space and time into spacetime. (See also my related answer here).

For most electromagnetic phenomena, there is no frame of reference in which it is purely electric, or purely magnetic. It is always a mix of the two, although it's a different mix in different frames of reference. For example, there is no frame of reference in which a refrigerator magnet has no magnetic field. In some frames it will also have an electric field, but a magnetic field is there too.

So in conclusion, magnetism is a fundamental physical phenomenon in its own right, not merely a funny way of talking about certain aspects of electricity. For that reason, it's possible for magnetism to contain phenomena that cannot be directly extrapolated from everyday electricity plus SR.

PS: If there are no magnetic monopoles in a certain reference frame, then there are no magnetic monopoles in ANY reference frame. Conversely, if there ARE magnetic monopoles in one reference frame, then there are magnetic monopoles in every reference frame.

• So, you are saying that the "there are no magnetic monopoles" is true for every reference frame. So, not matter what reference frame I start working with(with its particular mix of electricity and magnetism), when I change the frame of reference, there are still no sources of magnetism by magnetic monopoles, right? Aug 11, 2016 at 10:11
• Indeed, if there are no magnetic monopoles in a certain reference frame, then there are no magnetic monopoles in ANY reference frame. Conversely, if there ARE magnetic monopoles in one reference frame, then there are magnetic monopoles in every reference frame. Aug 11, 2016 at 12:24

Neither classical electromagnetism nor special relativity requires monopoles. These theories merely allow them and can calculate what would happen if you had one. Some high energy physics theories do predict that magnetic monopoles actually have to exist somewhere and these theories will need modification if someone proves for sure that there are no magnetic monopoles in the universe.

Okay, so if classical electromagnetism doens't need magnetic monopoles, why is there a space for them in the equations? When most people write them, there isn't a space for monopoles $(\nabla\cdot B=0)$. When the space is left for them, it's for the fun of exploring what monopoles would do if there were any and also for completing the symmetry of Maxwell's equations.

Magnetic field appears in Relativity not in one step, but in several.

At first, one looks at two electrons in rest. They are subjects for Coulomb repulsion. The repulsion force is, say $F$ and it would cause acceleration of $a = F/m_e$

Now, one looks at the same two electrons from the moving frame.

According to relativity, they should repulse weaker, because time should flow slower inside the pair.

But according to electrodynamics, electrons are just moving. They have the same $q$ and should repulse at the same rate.

So, who it right? We know, that relativity IS right, then we should FIX electrodynamics and add magnetic field to it.

Moving electrons not only repulse electrically, but also attract magnetically.

Magnetic field -- is a patch to electrodynamics, which conserves relativity.

Since it is natural thing, it is not obliged to work only at the patching site -- it can work elsewhere. For example, it can belong to some particles -- which are monopoles.