Generic Theory of Geometrodynamics from Noether’s Theorem for the $\mathrm {Diff}(M)$ Symmetry Group

Abstract

We work out the most general theory for the interaction of spacetime geometry and matter fields—commonly referred to as geometrodynamics—for spin-0 and spin-1 particles. Actually, we present a Hamilton–Lagrange–Noether formulation of the gauge theory of gravitation. It is based on the minimum set of postulates to be introduced, namely (i) the action principle and (ii) the form-invariance of the action under the (local) diffeomorphism group. The second postulate thus implements the Principle of General Relativity, also referred to as the Principle of General Covariance. According to Noether’s theorem, this physical symmetry gives rise to a conserved Noether current, from which the complete set of theories compatible with both postulates can be deduced. This finally results in a new generic Einstein-type equation, which can be interpreted as an energy-momentum balance equation emerging from the Lagrangian $\mathscr {L}_{R}$ for the source-free dynamics of gravitation and the energy-momentum tensor of the source system $\mathscr {L}_{0}$. Provided that the system has no other symmetries—such as SU(N)—the canonical energy-momentum tensor turns out to be the correct source term of gravitation. For the case of massive spin-1 particles, this entails an increased weighting of the kinetic energy over the mass as the source of gravity, compared to the metric energy momentum tensor, which constitutes the source of gravity in Einstein’s General Relativity. We furthermore confirm that a massive vector field necessarily acts as a source for torsion of spacetime. From the viewpoint of our generic Einstein-type equation, Einstein’s General Relativity constitutes the particular case for scalar and massless vector particle fields, and the Hilbert Lagrangian $\mathscr {L}_{R,\mathrm {H}}$ as the model for the source-free dynamics of gravitation.

Dedicated to the memory of Prof. Dr. Walter Greiner, our teacher, mentor, and friend.

In this contribution, we present the canonical transformation formalism in the realm of classical field theory, where spacetime is treated as a dynamical quantity, and apply it to formulate the gauge theory of gravitation. In this respect, it generalizes the Extended Hamilton–Lagrange–Jacobi formalism of relativistic point dynamics. Walter was very much interested in this formalism and therefore added several chapters on the matter to the second edition of his textbook “Classical Mechanics” [1]. To quote Walter from the Preface to the Second Edition: “It may come as a surprise that even for the time-honored subject such as Classical Mechanics in the formulation of Lagrange and Hamilton, new aspects may emerge.” We are sure, Walter would have loved the following elaboration.

Appendices

Identity for a Scalar-Valued Function S of an (n, m)-Tensor T and the Metric

Proposition 1

Let $S=S(g,T)\in \mathbb {R}$ be a scalar-valued function constructed from the metric tensor $g_{\mu \nu }$ and an (n, m)-tensor , where $(m-n)/2\in \mathbb {Z}$. Then the following identity holds:

(89)

Proof

The induction hypothesis is immediately verified for scalars constructed from second rank tensors and, if necessary, the metric, hence, , $S=T^{\alpha \beta }\,g_{\alpha \beta }$, and $S=T_{\alpha \beta }\,g^{\alpha \beta }$. Let Eq. (89) hold for an (n, m)-tensor . We first consider an $(n+1,m+1)$-tensor with the last indices contracted in order to again make up a scalar. Setting up S according to (89) with the tensor $\bar{T}$, one encounters the two additional terms

Equation (89) thus also holds for the scalar S formed from the $(n+1,m+1)$-tensor $\bar{T}$.

For the case that $\bar{T}$ represents an $(n+2,m)$-tensor , the scalar S must have one additional factor $g_{\alpha \beta }$. One thus encounters four additional terms:

For the case that $\bar{T}$ represents an $(n,m+2)$-tensor , the scalar S must have one additional factor $g^{\alpha \beta }$. Owing to

$$\begin{aligned} {\frac{\partial {S}}{\partial {g_{\mu \beta }}}}g_{\nu \beta }=-{\frac{\partial {S}}{\partial {g^{\alpha \nu }}}}g^{\alpha \mu },\qquad {\frac{\partial {S}}{\partial {g_{\alpha \mu }}}}g_{\alpha \nu }=-{\frac{\partial {S}}{\partial {g^{\nu \beta }}}}g^{\mu \beta }, \end{aligned}$$

one thus encounters the four additional terms:

The derivative of a Lagrangian $\mathscr {L}$ with respect to the metric $g_{\mu \nu }$ can thus always be replaced by the derivatives with respect to the appertaining tensors T that are made into a scalar by means of the metric. The identity thus provides the correlation of the metric and the canonical energy-momentum tensors of a given system.

Corollary 1

The contraction of Eq. (89) then yields a condition for the scalar S:

(90)

Proof

Contracting Eq. (89) directly yields Eq. (90).

Corollary 2

Let $\tilde{\mathscr {L}}=\tilde{\mathscr {L}}(g,T_k)\in \mathbb {R}$ be a scalar density (i.e. a relative scalar of weight one) valued function of the (symmetric) metric $g^{\mu \nu }$ and a sum of k tensors of respective rank $(n_k,m_k)$, where $(m_k-n_k)/2\in \mathbb {Z}$. Then the following identity holds:

(91)

Proof

Combine Eq. (89) with (58).

Equation (91) is obviously a representation of Euler’s theorem on homogeneous functions in the realm of tensor calculus.

Examples for Identities (89) Involving the Riemann Tensor

1.1 Riemann Tensor Squared

As a scalar, any Lagrangian $\mathscr {L}_{R}(R,g)$ built from the Riemann–Cartan tensor (41) and the metric satisfies the identity (89)

(92)

The factors “2” emerge from the symmetry of the metric and the skew-symmetry of the Riemann–Cartan tensor in its last index pair. The identity is easily verified for a Lagrangian linear and quadratic in the Riemann tensor:

The left-hand side of Eq. (92) evaluates to

which indeed agrees with the terms obtained from the right-hand side:

1.2 Ricci Scalar

The Ricci scalar R is defined as the following contraction of the Riemann tensor

(93)

With the scalar $\mathscr {L}_{R}=R$ and the tensor T the Riemann tensor, the general Eq. (89) takes on the particular form

$$\begin{aligned} {\frac{\partial {R}}{\partial {g^{\nu \beta }}}}g^{\mu \beta }+{\frac{\partial {R}}{\partial {g^{\beta \nu }}}}g^{\beta \mu }- {\frac{\partial {R}}{\partial {R_{\mu \alpha \xi \lambda }}}}R_{\nu \alpha \xi \lambda }- {\frac{\partial {R}}{\partial {R_{\eta \mu \xi \lambda }}}}R_{\eta \nu \xi \lambda }- {\frac{\partial {R}}{\partial {R_{\eta \alpha \mu \lambda }}}}R_{\eta \alpha \nu \lambda }- {\frac{\partial {R}}{\partial {R_{\eta \alpha \xi \mu }}}}R_{\eta \alpha \xi \nu }\equiv 0. \end{aligned}$$

(94)

Without making use of the symmetries of the Riemann tensor and the metric, this identity is actually fulfilled as

Similarly

The derivative terms of the Riemann tensor are

and

which obviously cancel the four terms emerging from the derivatives with respect to the metric.

Making now use of the skew-symmetries of the Riemann tensor in its first and second index pair and of the symmetry of the metric, Eq. (94) simplifies to

$$\begin{aligned} {\frac{\partial {R}}{\partial {g^{\nu \beta }}}}g^{\mu \beta }\equiv {\frac{\partial {R}}{\partial {R_{\mu \alpha \xi \lambda }}}}R_{\nu \alpha \xi \lambda }+ {\frac{\partial {R}}{\partial {R_{\eta \alpha \xi \mu }}}}R_{\eta \alpha \xi \nu }. \end{aligned}$$

(95)

For zero torsion, the Riemann tensor has the additional symmetry on exchange of both index pairs. Then

(96)

1.3 Ricci Tensor Squared

The scalar made of the (not necessarily symmetric) Ricci tensor $R_{\eta \alpha }$ is defined by the following contraction with the metric

(97)

With Eq. (97), the general Eq. (89) now takes on the particular form

$$\begin{aligned} {\frac{\partial {\mathscr {L}_{R}}}{\partial {g^{\nu \beta }}}}g^{\mu \beta }+{\frac{\partial {\mathscr {L}_{R}}}{\partial {g^{\beta \nu }}}}g^{\beta \mu }- {\frac{\partial {\mathscr {L}_{R}}}{\partial {R_{\mu \beta }}}}R_{\nu \beta }-{\frac{\partial {\mathscr {L}_{R}}}{\partial {R_{\beta \mu }}}}R_{\beta \nu }\equiv 0. \end{aligned}$$

(98)

Without making use of the symmetries of the Ricci tensor and the metric, this identity is actually fulfilled as

Similarly

The derivative terms of the Ricci tensor are

and

which obviously cancel the four terms emerging from the derivatives with respect to the metric.

For zero torsion, the Ricci tensor is symmetric. Then

(99)