Canonical Energy Momentum TensorEdit

The canonical energy-momentum tensor is a central concept in relativistic field theory. It encodes how energy, momentum, and their fluxes are distributed in space and time for a given physical system described by a Lagrangian. It arises naturally from Noether’s theorem as the conserved current associated with spacetime translations. In flat spacetime this object provides a clear bookkeeping device for energy and momentum densities, and it plays a foundational role in both classical field theory and quantum field theory. However, the canonical form is not the end of the story: it is not guaranteed to be symmetric or gauge-invariant, which matters for certain applications—especially when gravity is involved.

In discussions of fundamental physics, the canonical energy-momentum tensor is often the starting point for understanding how fields carry energy and momentum. Its conservation is tied to the translation symmetry of the underlying theory, and its integrated components yield the total energy and total momentum of a system. For gravitation and curved spacetime, the tensor that couples directly to the metric is symmetric and comes from a different construction, which has led to a rich set of refinements and alternatives. Across disciplines, the canonical tensor remains a robust reference object for local densities and fluxes, while the symmetric, gauge-invariant versions address additional requirements in gravitation and certain gauge theories.

Definition

The canonical energy-momentum tensor T_c^{μν} is derived from the Lagrangian density L(φ, ∂μφ) of the fields φ as the Noether current associated with spacetime translations. In general form, it can be written as T_c^{μν} = ∑_r (∂L/∂(∂μ φ_r)) ∂^ν φ_r − η^{μν} L, where φ_r runs over the fields in the theory and η^{μν} is the Minkowski metric in flat spacetime.
For a scalar field φ with L = ½ ∂_μφ ∂^μφ − ½ m^2 φ^2, this leads to T_c^{μν} = ∂^μ φ ∂^ν φ − η^{μν} L.
In more complicated theories with spinor or vector fields, the explicit form generalizes in a way that still embodies the Noether construction, though the resulting T_c^{μν} need not be symmetric or gauge-invariant in general.

For reference, see Noether's theorem and Energy-momentum tensor for the broader context of currents and conservation laws, and consult Lagrangian density for the starting point of these expressions.

Derivation

The starting point is a spacetime translation x^μ → x^μ + ε^μ. Fields transform accordingly, and the action S = ∫ d^4x L transforms in a way that yields a conserved current.
Noether’s theorem then identifies a conserved current j^μ with ∂_μ j^μ = 0 when the equations of motion hold (on-shell). The spatial components of this current provide the fluxes of energy and momentum, while the time component encodes the densities.
The resulting canonical energy-momentum tensor T_c^{μν} is conserved in flat spacetime: ∂_μ T_c^{μν} = 0 on-shell. This conservation underpins the global charges, such as total energy ∫ d^3x T_c^{00} and total momentum ∫ d^3x T_c^{0i}.

For a deeper mathematical view, see Noether's theorem and Conservation law.

Symmetry and issues

A notable property (or lack thereof) is symmetry. T_c^{μν} is not guaranteed to be symmetric in μ and ν, and for many physical questions—most notably angular momentum in a relativistic setting—a symmetric tensor is desirable.
Gauge theories exacerbate the issue: the canonical tensor for gauge fields can be non-gauge-invariant, and its separation into orbital and spin parts can be ambiguous once gauge freedom is involved.

For alternatives and improvements, see Belinfante–Rosenfeld stress-energy tensor and Hilbert energy-momentum tensor.

Belinfante-Rosenfeld improvement and the Hilbert tensor

The Belinfante–Rosenfeld procedure adds a total divergence of a suitably defined spin current to T_c^{μν} to produce a symmetric, conserved tensor T_B^{μν}: T_B^{μν} = T_c^{μν} + ∂_λ X^{λ μ ν}, where X is constructed from the spin density of the fields.
The symmetrized tensor T_B^{μν} has the same conserved charges as T_c^{μν} when integrated over space, so total energy and momentum remain unchanged, even though the local density has been reorganized.
In many practical contexts, the resulting symmetric tensor coincides with or is closely related to the Hilbert (or metric) energy-momentum tensor, T_H^{μν}, which is defined by variation of the action with respect to the spacetime metric: T_H^{μν} = −(2/√−g) δS/δg_{μν}.
The symmetric tensor is what typically serves as the source in Einstein’s equations of General relativity; this is one reason the symmetric form is preferred in gravitational contexts.

For more detail, see Belinfante–Rosenfeld stress-energy tensor and Hilbert energy-momentum tensor.

In curved spacetime and gravity

In general relativity, the metric itself determines spacetime geometry, and the gravitational field equations couple to a symmetric, covariantly conserved energy-momentum tensor. The standard choice is the Hilbert tensor T_H^{μν}, which reduces to the symmetric Belinfante form in many field theories.
The canonical form T_c^{μν} by itself is typically not adequate as a gravitational source because it is not guaranteed to be symmetric or locally well-behaved under general coordinate transformations.
The broader issue of energy localization in gravity is subtle. While energy-momentum can be defined locally in special relativity, the equivalence principle blurs a local gravitational energy density in a coordinate-independent way. This has led to the development of quasi-local and pseudotensor constructs (for example, the Landau–Lifshitz pseudotensor) to discuss energy in gravitational fields, though these are often coordinate-dependent and conceptually distinct from a true tensor.

See General relativity and Landau–Lifshitz pseudotensor for related discussions.

Physical interpretation and applications

In flat-space quantum field theory, the canonical tensor provides a natural starting point for deriving conserved currents and deformation symmetries. It underpins how fields carry energy and momentum and how those quantities transform.
In electromagnetism, the energy-momentum content of the field can be described by a well-known tensor that, in its symmetric form, matches the standard energy density and Poynting flux. The distinction between the canonical and symmetric formulations becomes important when one seeks a tensor that cleanly couples to gravity or that cleanly separates orbital and spin contributions in a gauge-invariant way.
In many practical calculations, physicists move between the canonical form and its improved, symmetric versions depending on the problem at hand, trusting that the essential conserved charges remain consistent and that the symmetric form is appropriate for coupling to gravity or for defining angular momentum in a relativistic setting.

For broader context on how tensors encode physical content, see Ward identities and Angular momentum.

Controversies and debates

Local energy density in gravity: A long-standing issue is whether gravity admits a locally meaningful energy density. Because gravity is geometry in general relativity, attempts to assign a local tensorial energy density to the gravitational field run into conceptual difficulties, which is why quasi-local and pseudotensor approaches exist alongside the standard Hilbert tensor. This area remains nuanced, with different approaches offering complementary viewpoints on what it means to talk about energy in curved spacetime.
Symmetry vs. canonical form: The canonical tensor is rooted in Noether’s theorem and translation symmetry, but many applications require a symmetric tensor to couple consistently to gravity and to support a clean decomposition of angular momentum. The Belinfante improvement provides a widely accepted path from the canonical form to a symmetric one, but the process highlights that local densities can be reshaped by adding total derivatives without changing global charges.
Gauge theories and gauge invariance: In gauge theories, care is needed to separate physical content from gauge artifacts. While the canonical tensor encodes the Noether currents, the physically meaningful energy and momentum densities are tied to gauge-invariant and, in gravitational contexts, symmetric formulations. This tension motivates using the Hilbert or Belinfante forms in many settings.
Woke criticisms and the physics core: In the context of physics discourse, the debate about how energy and momentum are defined tends to center on mathematical consistency, experimental relevance, and compatibility with established theories, rather than sociopolitical interpretations. The canonical tensor remains a foundational construct for understanding symmetry-driven conservation, while its improvements and alternatives reflect the demands of coupling to gravity and ensuring gauge-invariant, symmetric descriptions.

For foundational links, see Noether's theorem, Conservation law, and General relativity.