Question on connections in general relativity and particle physics

Question

$1$ Introduction

It seems that when you learn General Relativity all the technology of bundles are irrelevant (at least in elementary discussions as $[1]$, $[2]$, $[3]$ and others). But, even the simplest field theory (electromagnetism) in a flat background like Minkowski spacetime, $\mathcal{L} = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$, do require a (curved) principal fiber bundle. Moreover, we can think in gluons $\mathcal{L} = -\frac{1}{4}G^{a}_{\mu\nu}G_{a}^{\mu\nu}$, with $a$ assuming values on the lie algebra, turning out the usage of fiber bundles even more necessary.But my question lies on the conceptual leap on the usage and definition of "connections in general relativity and particle physics".

$2$ Connections in Spacetimes

On in one hand, it seems that in general relativity we just need the knowledge of: the manifold $\mathcal{M}$ and the metric tensor $g$, $(\mathcal{M},g)$; the tangent spaces $T_{p}\mathcal{M}$, and the set of all tangent vectors $\mathfrak{X}(\mathcal{M})$ to settle down the formal concept of a (affine) connection:

$\nabla: \mathfrak{X}(\mathcal{M}) \times \mathfrak{X}(\mathcal{M}) \to \mathfrak{X}(\mathcal{M}),$

and then construct the "covariant derivative (of general relativity)":

Given a chart: $$\nabla_{X}Y = [X^{\mu}\nabla_{\mu}Y^{\nu}]\frac{\partial}{\partial x^{\nu}} = [X^{\mu}(\partial_{\mu}Y^{\nu}+\Gamma ^{\nu}\hspace{0.1mm}_{\mu \delta}Y^{\delta})]\frac{\partial}{\partial x^{\nu}}\tag{1}$$

$2.1$ Connections on Principal Fiber Bundles

On the other hand, it seems that in particle physics you need to:

1. Set your favorite principle fiber bundle: $\mathcal{P}= \big(\mathcal{E},\mathcal{M},\pi,G,G\big)$
2. Construct the tangent principal fibre bundle $\mathcal{TP}$
3. Divide the tangent bundle as $\mathcal{TP} = \mathcal{HP} \oplus \mathcal{VP} \implies T_{p}(\mathcal{P}) = H_{p}(\mathcal{P}) \oplus V_{p}(\mathcal{P})$
4. Use the push-forward $(\Phi_{g})_{*}$ to "push" the horizontal subspaces $H_{p}(\mathcal{P})$ along the fiber: $(\Phi_{g})_{*}[H_{p}(\mathcal{P})] = H_{(\Phi_{g})p}(\mathcal{P}) := H_{q}(\mathcal{P})$ creating a "connection of subspaces" in the bundle.

after all that,

5. Define (convince yourself/realize) the connection on $\mathcal{P}$ as a "horizontal distribution".

Equivalently:

1. Set your favorite principle fiber bundle: $\mathcal{P}= \big(\mathcal{E},\mathcal{M},\pi,G,G\big)$
2. Construct the tangent principal fibre bundle $\mathcal{TP}$
3. Divide the tangent bundle as $\mathcal{TP} = \mathcal{HP} \oplus \mathcal{VP} \implies T_{p}(\mathcal{P}) = H_{p}(\mathcal{P}) \oplus V_{p}(\mathcal{P})$
4. Construct the tangent spaces of $G$, $T_{q}G$
5. Use the fact that $T_{q}G \approx \mathfrak{g}$.
6. Use the $H_{p}(\mathcal{P})$ to project a $v \in T_{p}(\mathcal{P})$ in $V_{p}(\mathcal{P})$
7. Use the fact that $V_{p}(\mathcal{P}) = T_{p}G$
8. Realize, by $5)$, $6)$ and $7)$, the existence of a 1-form (or a $\mathfrak{g}$-valued differential form), $\omega_{p}: T_{p}\mathcal{P} \to \mathfrak{g}$, on $\mathcal{P}$.
9. Take a particular tangent space in the identity $e$ of $G$: $T_{e}G$
10. Use the fact $5)$, $T_{e}G \approx \mathfrak{g}$.
11. Take the vector space of "all left-invariant vector fields" $\mathfrak{X}_{L}(G)$
12. Take a $\xi \in \mathfrak{g}$
13. Using $11)$ and $12)$ to mount a vector field $X_{\xi} \in \mathfrak{X}_{L}(G)$, such that $X_{\xi}(e) = \xi$ (to be a unique flow).
14. Use the technology of flows in manifolds $[4]$, to describe a unique integral curve of $X_{\xi}$ passing at $t=0$ through $e \in G$: $g_{\xi}(t)$.
15. Set the famous exponential map to define the fundamental vector field: $\textbf{A} =: \xi_{\mathcal{P}}(p) = \frac{\mathrm{d}}{\mathrm{d}t}(p\cdot \mathrm{exp}[t\xi])|_{t=0} = p\xi$
16. Take a vector $v \in T_{p}(\mathcal{P})$, and then push (forward) it thorough the fiber $(\Phi_{g})_{*}|_{p}(v) = w$.
17. Then take the defined 1-form of $8)$ and form the adjoint action: $\omega_{\Phi_{g}(p)}[w] = \mathrm{Ad}_{g} [\omega_{p}(v)] := \Phi_{g}^{-1}\omega_{p}(v)\Phi_{g}$

after all that,

18. Define (convince yourself/realize) the (Ehresmann) connection on $\mathcal{P}$ as a $\mathfrak{g}$-valued 1-form $\omega_{p}: T_{p}\mathcal{P} \to \mathfrak{g}$ which satisfies, for each $p\in \mathcal{P}$,

$$\omega_{p}(\xi_{\mathcal{P}}(p)) = \xi \hspace{5mm} \forall \xi \in \mathfrak{g}$$

$$\mathrm{Ad}_{g} [\omega_{p}(v)] := \Phi_{g}^{-1}\omega_{p}(v)\Phi_{g} \hspace{5mm} \forall v\in T_{p}(\mathcal{P}), \forall g \in G $$

My Question

Well, my question is: why the texts books of general relativity don't define a connection in a principal fibre bundle? (or, why in general relativity we need "less math" to define a connection?).

$$----------------------------------------------------------$$

$[1]$ Carrol.S. Spacetime and Geometry

$[2]$ Weinberg. S. Gravitation and Cosmology

$[3]$ d'Inverno.R. Introducing Einstein's Relativity

$[4]$ Nakahara.M. Geometry, Topology and Physics

Answer 1

1. The claims in the question are very strange: Many people do particle physics without ever so much as mentioning the word "bundle". It is perfectly common in physics to circumvent the notion of a bundle for the gauge field by talking about the behaviour of the gauge field "at infinity". See e.g. this question for instances of that language. It is in practical terms absolutely not the case that the language of bundles would be more "required" for particle physics than for general relativity. In both GR and non-GR you can just define a covariant derivative/connection/parallel transport by writing $\nabla_\mu = \partial_\mu + \rho(A_\mu)$ for $A$ a Lie algebra-valued gauge field and $\rho$ the representation of the gauge field on the field being differentiated. In the case of GR you just have the Christoffel symbols thought of as a $\mathfrak{gl}(n)$-valued 1-form: $\Gamma_\mu\in\mathfrak{gl}(n)$ with matrix components $({\Gamma^\nu}_\sigma)_\mu$, acting in the natural way on tensors in $n$ dimensions.

2. However, there is a core of truth to the idea that one needs "more math" for the generic connection associated with a gauge theory than for the Levi-Civita connection in general relativity: In general relativity, the connection lives directly on the tangent bundle of the manifold, and the associated "principal bundle" is just the (orthogonal) frame bundle of the manifold. That is, even when using the language of bundles, there is no need to generalize to the abstract case of principal bundles and their associated bundles and representations and whatnot - all you need is already there "naturally" given to you by standard differential geometry: the (co)tangent bundles, the frame bundle and their tensor powers.

   In the language of generic gauge theories, what's special about GR as a theory of a connection on principal bundles is that the gauge theory here is not "abstract", but soldered onto the tangent bundle of the manifold. The d.o.f. that the connection acts upon are not "internal", wholly dissociated from anything more intuitively geometrical, they are instead the same d.o.f. of ordinary geometric vectors. Hence you do not need to develop a general theory of connections to formulate the Levi-Civita connection of GR - you only need to understand this special case of a completely soldered one.

Answer 2

This is because its takes a substantial amount of mathematics to get there as you have just outlined. They need to learn about topology, smooth structures, manifolds, connections, bundles, frame bundles, principal bundles and so on.

Instead, they take the traditional physics language of components which has its advantages although eschewed by mathematicians.

This allows them to get to the physics quickly. It's also sanctified as the method used by Einstein himself.

Undoubtedly this will change in the future but not anytime soon. It's a question of physical pedagogy changing to take into account all the mathematical technology that allows us to think about physics in a more invariant manner. This will take time.

Answer 3

I'm not sure I understand in what sense the technology of bundles is irrelevant in GR. Vector fields on a smooth manifold $M$ are sections of the tangent bundle $TM$, which is associated to the (generally curved) frame bundle $LM$; the connection $\Gamma$ and curvature $\mathrm{Riem}$ are (local representations of) the connection 1-form $\omega$ and its curvature $\Omega$ on $LM$.

In the typical low-level introduction to GR, the connection arises because we'd like to define the differentiation of vector fields in a way which is basis-independent. This requires the introduction of an auxiliary field $\Gamma$ with curvature $\mathrm{Riem}:= \mathrm d\Gamma + \Gamma\wedge \Gamma$. You seem to be asking why this low-level introduction does not require the language of principal bundles, but really we've done all the work to construct a principal bundle connection and simply haven't formalized it as efficiently as possible.

After all, you certainly don't need to talk about principal bundles to do electromagnetism (otherwise undergraduates would be dropping like flies). It's true that electromagnetism can be very elegantly understood via a Yang-Mills theory with structure group $\mathrm U(1)$, but that doesn't mean it has to be introduced in that language.

By the time one starts doing particle physics, one might more reasonably expect that they've encountered similar structures in different fields (e.g. when comparing electromagnetism vs. GR, there is a correspondence between $\mathcal A \leftrightarrow \Gamma$ and $\mathcal F \leftrightarrow \mathrm{Riem}$). Therefore, it is a reasonable time to formalize the idea of gauge invariance via principal bundles, understand past theories (GR and EM) through that lens, and then look forward to the more sophisticated and abstract gauge theories contained within the standard model (and extensions thereof).

Answer 4

As already mentioned in some other answers, it is mainly a matter of pedagogical choices. Deep down, one is indeed using all of the fiber bundle machinery to define the connection, but on the particular context of General Relativity one can take a few shortcuts, even when dealing with texts that are mathematically careful and avoid working in components. For some examples, Wald's General Relativity, for example, introduces the notion by introducing a "derivative operator" and then proceeds to use the metric to single out the Levi-Civita connection — he briefly discusses this pedagogical choice on a resource letter called Teaching General Relativity (see arXiv: gr-qc/0511073), p. 8. If I recall correctly, Hawking & Ellis' The Large Scale Structure of Space-time takes a similar approach.

In particular, let me highlight a bit of Wald's Teaching General Relativity, p. 8, which I believe provides a straightforward answer to your question:

In mathematical treatments, the notion of parallel transport is usually introduced in the more general context of a connection on a fiber bundle. The general notions of fiber bundles and connections have many important applications in mathematics and physics (in particular, to the description of gauge theories), but it would normally require far too extensive a mathematical excursion to include a general discussion of these topics in a general relativity course, even at the graduate level.

Although there is no unique notion of differentiation of tensors in a completely general context, when a metric is present a unique notion of differentiation is picked out by imposing the additional requirement that the derivative of the metric must be zero. In Euclidean geometry (or in special relativity), this notion of differentiation of tensors corresponds to the partial differentiation of the components of the tensors in Cartesian coordinates (or in global inertial coordinates). However, in non-flat geometries, this notion of differentiation—referred to as the covariant derivative—does not correspond to partial differentiation of the components of tensors in any coordinate system.

As an example of a reference that takes a different approach, I took a quick look and Baez & Muniain's Gauge Fields, Knots, and Gravity seems to take an approach a bit closer to the theory of bundles as far as I could tell (I do not know the reference in much detail, but I took a quick look at it since it discusses gauge theory before discussing gravity and it seems to follow such an approach).