X-Thread from Jonathan Gorard
"The boundary of a boundary is always empty." A huge amount of (classical) physics, including much of general relativity and electromagnetism, can be deduced directly from this simple mathematical fact. Yet, on the surface, it doesn't seem to have much to do with physics.
Some spaces (like spheres) don't have boundaries. But, when the boundary exists, it's always one dimension lower (codimension-1). A disc is a 2-dimensional space, but its boundary is a 1-dimensional circle. But what's the boundary of a circle? Well, it doesn't have one.
It turns out that this will always be true, for purely topological reasons: a space may or may not have a boundary, but its boundary never will. Yet physics is about differential equations, not topology, right? So how can this fact have any relevance to physics?
Well, the boundary operator, which maps a space to its boundary space, acts formally very much like a derivative (obeys the product rule, etc.). This is no coincidence: the boundary operator on submanifolds is dual to the exterior derivative operator on differential forms.
This allows us to translate topological statements about boundaries into analytic statements about exterior derivatives. So our "boundary of a boundary is empty" statement now becomes a statement about symmetries of the covariant derivative operator on certain tensors.
When applied to the Riemann tensor and then contracted, it yields the contracted Bianchi identities: the statement that the covariant divergence of the Einstein tensor vanishes. But, in GR, the Einstein tensor is equal to the stress-energy tensor (times a constant).
So the covariant divergence of stress-energy vanishes - in physical terms, this means that energy and momentum are always conserved in relativity! If instead we apply the Bianchi identities to the electromagnetic field tensor, we obtain the (homogeneous) Maxwell equations.
These encompass both Gauss' law for magnetism and the Maxwell-Faraday law of induction. In fact, under the gravitoelectromagnetic formalism, the entirety of general relativity can be represented in this way. Just start by choosing a unit timelike vector field...
This field allows us to decompose the Riemann/Weyl tensors into electrogravitic and magnetogravitic tensors (via the Bel decomposition), formally analogous to electric and magnetic field vectors. And, just as for Maxwell, the Bianchi identities give us the dynamical laws.
Within this formalism, the full system of Einstein equations emerge as rank-2 tensorial analogs of the rank-1 (vector) Maxwell equations. Yet, somehow, it all goes back to boundaries of boundaries, and their emptiness...
Comments from Leo C. Stein and discussion
The "much of GR and EM" mentioned here are just the Bianchi identities, which would be true regardless of the equations of motion. The real content of Maxwell's equations is not dF=0 (which is automatic from F=dA), but rather d*F=*J. Similarly for GR.
Though you only know that F=dA if you know that there are no magnetic monopoles, so there's some physical content there too.
Except if they're topological monopoles!!
But Faraday law in EM comes from the Bianchi identity.
I agree! And it is there for any other theory built from F=dA, not just Maxwell theory.
With all due respect, in Abelian gauge theories like E&M, the EoMs and topological identity e.g., the Bianchi Id. can be interchanged by a duality say, the electro-magnetic duality and hence these statements are not duality invariant statement. Hence, both are equally important!
The electric/magnetic duality is only in vacuum, where d*F=0.
I meant dualities ‘like’ the EM duality. To be specific the more general class of dualities are called Poincaré dualities which exchanges the electric and magnetic frames or takes F to its Hodge dual. In doing so, EoMs are exchanged with topological ids.
Leo C. Stein:
And again, this still only works in vacuum. One equation is dF=0, the other is d*F=*J. Sending F -> *F does not also send 0 -> *J.
And again: I am not saying it sends one equation to the other. I am saying it exchanges what we say EoMs with top. ids. In vacuum one gets the exact duality. But these dualities exist on general grounds even in 3d and 5d theories.
I am just not agreeing with the philosophy that: EoM have more content than top. ids. In Maxwell+elec. sources, rather it is the Bianchi Id. which reveals the true 1-form mag. symm. of the theory.
So, the contents of each of the eqn. are equally important.
This 1-form mag. symm. is the only global symm. of the theory and follows from mag. flux conservation. It has huge implications like the massless of the 4D photon is owing to the SSB of this mag. 1-form symm.
kash:
the point is that magnetic 1 form symmetry comes from assuming *F = da for a gauge field a different from A. the possibility of this being implied by d*F=0 is measured by the cohomology of spacetime so it’s not trivial always
The idea is not to exchange the eqns. The core idea is to exchange how we get them. What we get from the action now becomes a topological identity and vice-versa.
Hence, EoMs can’t be said to have the teal content and top. ids. don’t! Both are equal in their own right. Though it seems one comes from the action and the other is for free; but the way we get them are not ‘generically’ duality invariant statements.
these mathematical reformulations are worthless physically. They are good to unify notations and some formalism, but they actually obfuscate the physical meaning of the theories
Except it turns out they actually give you more physical insight, allowing you to generalize from Maxwell theory to non-Abelian Yang-Mills (necessary to formulate theories of the weak and strong forces). Improving notation leads to deeper understanding.
Maxwell originally wrote 3 equations instead of 1 vector equation. There is obviously a deeper understanding in going from that to a unified vector notation, as Heaviside (or Gibbs?) did.
If you look back at differential geometry calculations from the early 1900s, they are full of Christoffel symbols of the 1st and 2nd kind all over the place. Recognizing that we should prefer to express things in terms of a covariant connection not only simplifies but clarifies.