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I come from maths, not physics, but their construction g_{\mu \nu} = A_\mu A_\nu looks a little weird to me. The left side is a tensor, but isn't the electro

by curt15 5y ago

I come from maths, not physics, but their construction g_{\mu \nu} = A_\mu A_\nu looks a little weird to me. The left side is a tensor, but isn't the electromagnetic four-potential is a gauge field, not a tensor?

ajkjk 5y ago
A gauge field is tensor-valued; what's wrong with that? Although it does seems strange for other reasons. Primarily because it seems very unlikely that everyone else who developed this field wouldn't have considered the possibility and then discarded it.
- curt15 5y ago
  Although a gauge field has multiple components, it transforms differently from tensors under a change of coordinates: A -> O A O^{-1} - dO O^{-1} (O is the Jacobian matrix) The second term is absent for tensors (such as the left side of their equation 4). It also vanishes on a flat spacetime where one only considers linear (Lorentz) coordinate transformations, but based on my cursory reading they don't seem to be making that assumption.
  - codethief 5y ago
    I'm not sure I'm following. In General Relativity, the 4-electromagnetic potential A^μ is simply a vector field on spacetime, so what's wrong with taking the (symmetrization of) the tensor product of A_μ with itself to obtain a symmetric 2-tensor? (Whether or not that 2-tensor satisfies the requirements for a semi-Riemannian metric is another question.)
    - nilaykumar 5y ago
      Why is A_\mu a vector field on spacetime? In the standard treatment A is the (pullback to the base of the) connection 1-form of a connection on a principal U(1)-bundle on spacetime. Technically it's valued in the Lie algebra of U(1), but as that can be identified with i times the real numbers, we can ignore that here. Does the product A_\mu A_\nu happen to transform tensorially? Because as the parent pointed out, the transformation rule for A involves an extra term, so it's not obvious.
      - codethief 5y ago
        You're right – maybe I should have been more precise with my wording. Let me address curt15's (and your) question in two ways: ####### The physicist's argument ####### In classic electrodynamics it doesn't make much sense to talk about a U(1) gauge symmetry (whether global or local) – because there's no electron field exhibiting such a symmetry in the first place. Gauge symmetries really only become important when talking about quantum mechanics, Dirac's equation and, more generally, coupling particle fields to force-carrying boson fields in a QFT. There it turns out that imposing local gauge invariance would guarantee you a way to accomplish the coupling while preserving the conservation of the Noether current associated with the global gauge symmetry. But in classical electrodynamics, none of that is needed and the vector potential is simply a vector field on Minkowski space. In fact, the value of the vector field and the choice of gauge are not even important because there's no classic analogue of the Aharonov-Bohm effect. Now pretty much the same applies to Einstein-Maxwell theory: There is no electron field and no U(1) gauge symmetry. It's all classical. Indeed, in my experience relativists just view the vector potential as a vector field on spacetime and ignore the QFT-inspired classical gauge theory stuff (principal bundles, associated vector bundles, etc.) altogether. You might disagree with this approach but given that no one has yet managed to write down any 4-dimensional, coupled quantum field theory in a mathematically coherent fashion (let alone on a curved background), I wouldn't say there's too much pressure to incorporate results from QED into GR. Back to the paper: The authors basically start from the same perspective, i.e. the foundations of Special Relativity and classic electrodynamics where A was simply viewed as a vector field. They then propose an alternative to General Relativity and, in fact, electrodynamics(!) Like true physicists, they first worry about the functorial nature of their objects (what do they do & how do they relate to one another), not about the category the objects live in. If that means redefining[2] what A is in a mathematical sense, so be it. At the end of the day, experiments are what matters. ####### The mathematician's argument ####### Could it be that you and curt15 are thinking of the transformation behavior of A under local gauge transformations[0], not coordinate transformations? Because, unless I'm mistaken[1], the transformation of A under coordinate transformations is the same as for a vector fields, see definition 5.4.1 on p. 270 of [3]. The relevant portion reads: Let p: P -> M be a principal G-bundle on the manifold M and A a connection 1-form on P, let s: U -> P be a local gauge of the principal bundle on an open subset U of M. Then we define the *local connection 1-form* (or *local gauge field*) A_s on U by A_s := A ○ Ds = s* A If we have a manifold chart on U and {∂_μ} (μ=1,…,n) are the local basis vector fields on U, we set A_μ := A_s(∂_μ) Now since the basis vector fields ∂_μ transform in the usual way and A_s is linear, the components A_μ should transform just like those of a section of the cotangent bundle T*M. Am I missing something? --- [0]: See the transformation formula in https://en.wikipedia.org/wiki/Gauge_theory#An_example:_Scalar_O(n)_gauge_theory https://en.wikipedia.org/wiki/Gauge_theory#An_example:_Scala... or theorem 5.4.2 of [3]. [1]: It's been a while since I last did mathematical gauge theory, so please let me know in case I'm committing any errors here. [2]: Since g is defined in terms of A, A must necessarily be a globally defined vector field and cannot be a local gauge field. [3]: Mark J.D. Hamilton: "Mathematical Gauge Theory: With Applications to the Standard Model of Particle Physics" (Springer, 2017)
al2o3cr 5y ago
For one, it's going to produce a metric where all the diagonal elements are positive (or zero) - different from the -+++ or +--- signature of "normal" spacetime. The paper's calculation reminds me of Kaluza-Klein theory, which uses a similar construction as part of extending the metric from four dimensions to five: https://en.wikipedia.org/wiki/Kaluza–Klein_theory https://en.wikipedia.org/wiki/Kaluza–Klein_theory
- ozankabak 5y ago
  I was thinking about the signature issue as well. In flat space (i.e. Minkowski metric), this would imply a constant four-potential with an imaginary 0'th component, which I can not make sense of.

by curt15 5y ago

ajkjk 5y ago

A gauge field is tensor-valued; what's wrong with that? Although it does seems strange for other reasons. Primarily because it seems very unlikely that everyone else who developed this field wouldn't have considered the possibility and then discarded it.

curt15 5y ago
Although a gauge field has multiple components, it transforms differently from tensors under a change of coordinates: A -> O A O^{-1} - dO O^{-1} (O is the Jacobian matrix) The second term is absent for tensors (such as the left side of their equation 4). It also vanishes on a flat spacetime where one only considers linear (Lorentz) coordinate transformations, but based on my cursory reading they don't seem to be making that assumption.
- codethief 5y ago
  I'm not sure I'm following. In General Relativity, the 4-electromagnetic potential A^μ is simply a vector field on spacetime, so what's wrong with taking the (symmetrization of) the tensor product of A_μ with itself to obtain a symmetric 2-tensor? (Whether or not that 2-tensor satisfies the requirements for a semi-Riemannian metric is another question.)
  - nilaykumar 5y ago
    Why is A_\mu a vector field on spacetime? In the standard treatment A is the (pullback to the base of the) connection 1-form of a connection on a principal U(1)-bundle on spacetime. Technically it's valued in the Lie algebra of U(1), but as that can be identified with i times the real numbers, we can ignore that here. Does the product A_\mu A_\nu happen to transform tensorially? Because as the parent pointed out, the transformation rule for A involves an extra term, so it's not obvious.
    - codethief 5y ago
      You're right – maybe I should have been more precise with my wording. Let me address curt15's (and your) question in two ways: ####### The physicist's argument ####### In classic electrodynamics it doesn't make much sense to talk about a U(1) gauge symmetry (whether global or local) – because there's no electron field exhibiting such a symmetry in the first place. Gauge symmetries really only become important when talking about quantum mechanics, Dirac's equation and, more generally, coupling particle fields to force-carrying boson fields in a QFT. There it turns out that imposing local gauge invariance would guarantee you a way to accomplish the coupling while preserving the conservation of the Noether current associated with the global gauge symmetry. But in classical electrodynamics, none of that is needed and the vector potential is simply a vector field on Minkowski space. In fact, the value of the vector field and the choice of gauge are not even important because there's no classic analogue of the Aharonov-Bohm effect. Now pretty much the same applies to Einstein-Maxwell theory: There is no electron field and no U(1) gauge symmetry. It's all classical. Indeed, in my experience relativists just view the vector potential as a vector field on spacetime and ignore the QFT-inspired classical gauge theory stuff (principal bundles, associated vector bundles, etc.) altogether. You might disagree with this approach but given that no one has yet managed to write down any 4-dimensional, coupled quantum field theory in a mathematically coherent fashion (let alone on a curved background), I wouldn't say there's too much pressure to incorporate results from QED into GR. Back to the paper: The authors basically start from the same perspective, i.e. the foundations of Special Relativity and classic electrodynamics where A was simply viewed as a vector field. They then propose an alternative to General Relativity and, in fact, electrodynamics(!) Like true physicists, they first worry about the functorial nature of their objects (what do they do & how do they relate to one another), not about the category the objects live in. If that means redefining[2] what A is in a mathematical sense, so be it. At the end of the day, experiments are what matters. ####### The mathematician's argument ####### Could it be that you and curt15 are thinking of the transformation behavior of A under local gauge transformations[0], not coordinate transformations? Because, unless I'm mistaken[1], the transformation of A under coordinate transformations is the same as for a vector fields, see definition 5.4.1 on p. 270 of [3]. The relevant portion reads: Let p: P -> M be a principal G-bundle on the manifold M and A a connection 1-form on P, let s: U -> P be a local gauge of the principal bundle on an open subset U of M. Then we define the *local connection 1-form* (or *local gauge field*) A_s on U by A_s := A ○ Ds = s* A If we have a manifold chart on U and {∂_μ} (μ=1,…,n) are the local basis vector fields on U, we set A_μ := A_s(∂_μ) Now since the basis vector fields ∂_μ transform in the usual way and A_s is linear, the components A_μ should transform just like those of a section of the cotangent bundle T*M. Am I missing something? --- [0]: See the transformation formula in https://en.wikipedia.org/wiki/Gauge_theory#An_example:_Scalar_O(n)_gauge_theory https://en.wikipedia.org/wiki/Gauge_theory#An_example:_Scala... or theorem 5.4.2 of [3]. [1]: It's been a while since I last did mathematical gauge theory, so please let me know in case I'm committing any errors here. [2]: Since g is defined in terms of A, A must necessarily be a globally defined vector field and cannot be a local gauge field. [3]: Mark J.D. Hamilton: "Mathematical Gauge Theory: With Applications to the Standard Model of Particle Physics" (Springer, 2017)

al2o3cr 5y ago

For one, it's going to produce a metric where all the diagonal elements are positive (or zero) - different from the -+++ or +--- signature of "normal" spacetime. The paper's calculation reminds me of Kaluza-Klein theory, which uses a similar construction as part of extending the metric from four dimensions to five: https://en.wikipedia.org/wiki/Kaluza–Klein_theory https://en.wikipedia.org/wiki/Kaluza–Klein_theory

ozankabak 5y ago
I was thinking about the signature issue as well. In flat space (i.e. Minkowski metric), this would imply a constant four-potential with an imaginary 0'th component, which I can not make sense of.