Stress Energy TensorEdit
The stress-energy tensor, often called the energy-momentum tensor, is a central object in physics that packages how energy density, momentum density, and stresses are distributed and carried through spacetime. It is a rank-2 tensor field T^{μν} that, in a local inertial frame, organizes energy and momentum into a unified framework. In the setting of flat spacetime and Special relativity, its divergence vanishes for isolated systems (∂μ T^{μν} = 0); in curved spacetime, as described by General relativity, the covariant divergence ∇μ T^{μν} = 0 holds, and T^{μν} serves as the source term that shapes the geometry through the Einstein field equations G_{μν} = (8πG/c^4) T^{μν}.
This tensor collects all forms of energy and momentum: the energy density, the flux of energy and momentum, and the internal stresses such as pressure and shear. Different physical fields contribute characteristic forms of T^{μν}. In practice, one often encounters explicit expressions for specific matter contents, such as a perfect fluid, the electromagnetic field, or scalar fields, which illuminate how matter and fields couple to gravity and to each other.
Classical formulations
Perfect fluid
For a perfect fluid with energy density ρ, pressure p, and four-velocity u^μ, the stress-energy tensor takes the form T^{μν} = (ρ + p/c^2) u^μ u^ν + p g^{μν}. In the rest frame of the fluid, T^{00} corresponds to the energy density, while T^{ij} encodes the isotropic pressure. This simple model underpins much of cosmology and astrophysics, where large-scale matter behaves approximately like a fluid moving in a curved background.
Electromagnetic field
The electromagnetic field contributes its own stress-energy tensor, T^{μν}{EM} = F^{μλ} F^{ν}{}{λ} - (1/4) g^{μν} F^{λσ} F_{λσ}, where F^{μν} is the electromagnetic field strength tensor. This form expresses how electromagnetic energy and momentum flow through spacetime and how fields exert pressure and tension on matter.
Scalar field
For a real scalar field φ with potential V(φ), the tensor is T^{μν} = ∂^μ φ ∂^ν φ - g^{μν} [ (1/2) ∂_λ φ ∂^λ φ - V(φ) ]. Scalar fields are important in many contexts, from early-universe models to particle physics.
In a local inertial frame, the components T^{μν} have a clear physical interpretation: T^{00} is the energy density, T^{0i} are the components of the energy flux (or momentum density), and T^{ij} are the stress components (pressures and shear).
- The trace T^{μ}_{ μ} provides information about conformal properties. In conformally invariant theories, the classical trace vanishes, though quantum effects can introduce trace anomalies.
Conservation and coupling to gravity
The coupling of matter to gravity is encoded in the Einstein field equations, where T^{μν} acts as the source that tells spacetime how to curve. The mathematics behind diffeomorphism invariance ensures ∇_μ T^{μν} = 0, expressing local energy-momentum conservation in a curved spacetime. This statement is intertwined with the geometric side, G^{μν} = (8πG/c^4) T^{μν}, which relates curvature to energy and momentum.
In cosmology and high-energy physics, one often uses the semiclassical perspective in which the expectation value ⟨T^{μν}⟩ appears on the right-hand side of the field equations, linking quantum fields to the classical geometry of spacetime. The precise definition of ⟨T^{μν}⟩ can be subtle and depends on renormalization and regularization schemes, especially in curved backgrounds.
- Gravitational waves carry energy and momentum, and an effective description of their energy content is provided by the Isaacson stress-energy tensor in the appropriate high-frequency limit. This tensor encodes the averaged influence of rapidly varying gravitational fields on spacetime dynamics.
Applications and common forms
In the large-scale dynamics of the universe, the stress-energy tensor of a near-perfect fluid dominates, with the FRW (Friedmann–Lemaître–Robertson–Walker) metric providing a familiar setting. The equation of state w = p/ρ characterizes different eras (radiation-dominated with w ≈ 1/3, matter-dominated with w ≈ 0, and dark-energy–dominated with w ≈ −1).
In stellar interiors and compact objects, the precise T^{μν} reflects the matter content, radiation, and, in some models, superfluid components. The interplay between T^{μν} and the background geometry governs hydrostatic equilibrium and gravitational collapse.
In electromagnetism, the electromagnetic field tensor contributes through T^{μν}_{EM}, affecting how light and radiation pressure influence matter and curvature, particularly in high-energy astrophysical environments.
In quantum field theory, T^{μν} becomes an operator-valued object. Its expectation values and fluctuations have implications for vacuum energy,Casimir effects, and semiclassical gravity, where quantum fields source curvature.
Controversies and interpretive issues
A notable area of discussion concerns the localization of gravitational energy. Unlike other fields, gravity resists a unique, coordinate-independent local energy density. Various energy-momentum pseudo-tensors (for example, the Einstein and Landau–Lifshitz formulations) offer coordinate-dependent accounts of gravitational energy, but no single universally accepted local tensor exists. This has led to debates and alternative formulations based on quasi-local notions of energy, such as Brown–York or other quasi-local mass constructs, which attempt to capture the energy contained within a finite region of spacetime.
Another topic of interest is the behavior of T^{μν} in quantum and semiclassical settings. Defining and renormalizing ⟨T^{μν}⟩ in curved backgrounds introduces technical challenges, including trace anomalies and regularization dependence. In gravitational-wave physics, the use of an effective stress-energy tensor for waves relies on approximations, highlighting the tension between local field descriptions and global energy accounting.
The diversity of matter content and field theories also means that the precise form of T^{μν} is theory-dependent. For instance, beyond the Standard Model or in modified theories of gravity, the tensor may include additional fields or couplings that alter how energy and momentum contribute to spacetime dynamics, while still preserving the essential role of T^{μν} as a carrier of energy, momentum, and stress information.