It is a truism to note that historically, as the “laws of physics” were refined, many could be formulated elegantly in terms of the extrema of particular functionals. Today such “action principles” are routinely used as tools for seeking new laws of physics. In Chap. 20 we show how such extrema can be readily calculated using a variational calculus for local variations with compact support. The functionals will, in general, be defined in terms of a metric tensor \(\mathtt {g}\) with Euclidean or Lorentzian signature on an n-dimensional manifold and a connection \(\nabla \) that is not necessarily torsion-free or metric-compatible. An important role is played by the indexed Einstein \((n-1)\) -forms: \(\displaystyle \begin{aligned} \mathcal {G}_{c} \,=\, R_{ab} \wedge i_{X_{c}}\star (e^{a} \wedge e^{b}), \qquad (c\,=\,1,2,\ldots ,n) \end{aligned} \) defined in terms of the Hodge map \(\star \) , curvature 2-forms \(\{R_{ab}\}\) of \(\nabla \) and any \(\mathtt {g}\) -orthonormal basis \(\{X_{a}\}\) with dual cobasis \(\{e^{b}\}\) ( \(e^{a}(X_{b})=\delta ^{a}_{b}\) ) (defined in Sect. 5.5 ). With the aid of the covariant exterior derivative \(\mathfrak {D}\) (section \(17.4\) ) on indexed p-forms, a series of important relations between \(\{\mathcal {G}_{c}\}\) , the Ricci 1-forms \(\{P_{b}\}\) , the Ricci scalar \(\mathcal {R}\) , torsion forms \(\{T^{b}\}\) and \(\{\mathfrak {D} T^{b}\}\) is found using a contracted Bianchi identity. A variational calculus is then established in Sect. 20.2 for arbitrary p-forms that depend on the intrinsic geometry of a 4-dimensional manifold. This calculus is then illustrated in the context of gravitational theories with Newtonian limits in Sect. 20.3 and compared with the variational schemes pioneered by Cartan and Palatini. The chapter concludes with a detailed calculation in Sect. 20.4 of the extrema derived from a \(\text{U}(1)\) gauge invariant functional for a Einstein-Maxwell charged-scalar system where we comment on its relevance to the quantum field theory of charged particles and photons. In this chapter, the nomenclature associated with this terminology has been mainly introduced by physicists in their search for a “Standard Model” describing the relativistic fundamental particle interactions, largely gleaned from the data produced by particle collision in high-energy accelerators. The functionals accommodate classical general gravitational interactions so, strictly speaking, lie outside the Standard Model. They are formulated in the language of differential forms and a symmetric, non-degenerate metric tensor field \(\mathtt {g}\) with a specified signature on an n-dimensional manifold \(\mathcal {M}_{n}\) endowed with a linear connection \(\nabla \) . If such a \(\nabla \) exists for some n, a particular geometry on \(\mathcal {M}_{n}\) is then denoted by the triple \((\mathcal {M}_{n},\mathtt {g},\nabla )\) . If \(\nabla \) is a Levi-Civita connection ( \(\boldsymbol {\nabla }\mathtt {g} = 0\) , \(\mathtt {Tor}=0\) ) and \(\mathtt {g}\) has signature \((1,1,\ldots ,1)\) , the geometry is called Euclidean and \(\mathcal {M}_{n}\) is a Riemannian manifold. If \(\mathtt {g}\) has, with our signature conventions, the signature \((-1,1,1,\ldots ,1)\) and \(\nabla \) is a Levi-Civita connection, the geometry is Lorentzian and \(\mathcal {M}_{n}\) is called pseudo-Riemannian or sometimes Lorentzian pseudo-Riemannian. If \(\mathtt {g}\) has any other signature but \(\nabla \) is a Levi-Civita connection, the geometry is simply called pseudo-Riemannian with the specified signature. For a non-symmetric connection \(\nabla \) where any of the conditions \(\boldsymbol {\nabla }\mathtt {g}\neq 0\) , \(\mathtt {Tor}\neq 0\) arise then the above categories of geometry on \(\mathcal {M}_{n}\) are as above but with the prefix “non-” attached to the word “Riemannian”. For example, if the torsion tensor of \(\nabla \) is zero ( \(\mathtt {Tor}=0\) ) but \(\boldsymbol {\nabla }\mathtt {g} \neq 0\) , the geometry on \(\mathcal {M}_{n}\) is either non-Riemannian or non-pseudo-Riemannian with zero torsion and \(\mathcal {M}_{n}\) is either a non-Riemannian manifold or a non-pseudo-Riemannian manifold with zero torsion. Some authors call such a connection a Weyl connection. The word “spacetime” in this chapter refers to a 4-dimensional Lorentzian manifold with a metric-compatible connection \(\nabla \) that is not, however, necessarily torsion-free.

Variational Methods on n-Dimensional Manifolds

章节
发表日期：

2025年

查看全文

Robin W. Tucker,
Timothy J. Walton

摘要

It is a truism to note that historically, as the “laws of physics” were refined, many could be formulated elegantly in terms of the extrema of particular functionals. Today such “action principles” are routinely used as tools for seeking new laws of physics. In Chap. 20 we show how such extrema can be readily calculated using a variational calculus for local variations with compact support. The functionals will, in general, be defined in terms of a metric tensor \(\mathtt {g}\) with Euclidean or Lorentzian signature on an n-dimensional manifold and a connection \(\nabla \) that is not necessarily torsion-free or metric-compatible. An important role is played by the indexed Einstein \((n-1)\) -forms: \(\displaystyle \begin{aligned} \mathcal {G}_{c} \,=\, R_{ab} \wedge i_{X_{c}}\star (e^{a} \wedge e^{b}), \qquad (c\,=\,1,2,\ldots ,n) \end{aligned} \) defined in terms of the Hodge map \(\star \) , curvature 2-forms \(\{R_{ab}\}\) of \(\nabla \) and any \(\mathtt {g}\) -orthonormal basis \(\{X_{a}\}\) with dual cobasis \(\{e^{b}\}\) ( \(e^{a}(X_{b})=\delta ^{a}_{b}\) ) (defined in Sect. 5.5 ). With the aid of the covariant exterior derivative \(\mathfrak {D}\) (section \(17.4\) ) on indexed p-forms, a series of important relations between \(\{\mathcal {G}_{c}\}\) , the Ricci 1-forms \(\{P_{b}\}\) , the Ricci scalar \(\mathcal {R}\) , torsion forms \(\{T^{b}\}\) and \(\{\mathfrak {D} T^{b}\}\) is found using a contracted Bianchi identity. A variational calculus is then established in Sect. 20.2 for arbitrary p-forms that depend on the intrinsic geometry of a 4-dimensional manifold. This calculus is then illustrated in the context of gravitational theories with Newtonian limits in Sect. 20.3 and compared with the variational schemes pioneered by Cartan and Palatini. The chapter concludes with a detailed calculation in Sect. 20.4 of the extrema derived from a \(\text{U}(1)\) gauge invariant functional for a Einstein-Maxwell charged-scalar system where we comment on its relevance to the quantum field theory of charged particles and photons. In this chapter, the nomenclature associated with this terminology has been mainly introduced by physicists in their search for a “Standard Model” describing the relativistic fundamental particle interactions, largely gleaned from the data produced by particle collision in high-energy accelerators. The functionals accommodate classical general gravitational interactions so, strictly speaking, lie outside the Standard Model. They are formulated in the language of differential forms and a symmetric, non-degenerate metric tensor field \(\mathtt {g}\) with a specified signature on an n-dimensional manifold \(\mathcal {M}_{n}\) endowed with a linear connection \(\nabla \) . If such a \(\nabla \) exists for some n, a particular geometry on \(\mathcal {M}_{n}\) is then denoted by the triple \((\mathcal {M}_{n},\mathtt {g},\nabla )\) . If \(\nabla \) is a Levi-Civita connection ( \(\boldsymbol {\nabla }\mathtt {g} = 0\) , \(\mathtt {Tor}=0\) ) and \(\mathtt {g}\) has signature \((1,1,\ldots ,1)\) , the geometry is called Euclidean and \(\mathcal {M}_{n}\) is a Riemannian manifold. If \(\mathtt {g}\) has, with our signature conventions, the signature \((-1,1,1,\ldots ,1)\) and \(\nabla \) is a Levi-Civita connection, the geometry is Lorentzian and \(\mathcal {M}_{n}\) is called pseudo-Riemannian or sometimes Lorentzian pseudo-Riemannian. If \(\mathtt {g}\) has any other signature but \(\nabla \) is a Levi-Civita connection, the geometry is simply called pseudo-Riemannian with the specified signature. For a non-symmetric connection \(\nabla \) where any of the conditions \(\boldsymbol {\nabla }\mathtt {g}\neq 0\) , \(\mathtt {Tor}\neq 0\) arise then the above categories of geometry on \(\mathcal {M}_{n}\) are as above but with the prefix “non-” attached to the word “Riemannian”. For example, if the torsion tensor of \(\nabla \) is zero ( \(\mathtt {Tor}=0\) ) but \(\boldsymbol {\nabla }\mathtt {g} \neq 0\) , the geometry on \(\mathcal {M}_{n}\) is either non-Riemannian or non-pseudo-Riemannian with zero torsion and \(\mathcal {M}_{n}\) is either a non-Riemannian manifold or a non-pseudo-Riemannian manifold with zero torsion. Some authors call such a connection a Weyl connection. The word “spacetime” in this chapter refers to a 4-dimensional Lorentzian manifold with a metric-compatible connection \(\nabla \) that is not, however, necessarily torsion-free.