Foundations of the Application of the Energy-Momentum Tensor in Relativity
The energy‑momentum tensor lies at the heart of relativistic physics. It gathers the scalar energy density, the vector momentum density, and the stress tensor into a single four‑dimensional object that transforms covariantly under Lorentz transformations. Understanding its definition, mathematical properties, and concrete realizations for different physical systems is a prerequisite for any work in relativistic field theory, astrophysics, or high‑energy physics.
In a flat Minkowski space with coordinates (x^{\mu}=(ct,\mathbf{x})) ((\mu=0,1,2,3)) the energy‑momentum tensor is denoted (T^{\mu\nu}). Its components have a clear mechanical meaning:
| Component | Interpretation |
|---|---|
| (T^{00}) | Energy density (\varepsilon) – total energy per unit volume |
| (T^{0i}=T^{i0}) | Momentum density (or energy flux) (S^{i}) |
| (T^{ij}) | Stress tensor (\sigma^{ij}) – pressure and shear stresses |
With the metric (\eta_{\mu\nu}= \mathrm{diag}(+1,-1,-1,-1)) the tensor satisfies two fundamental relations
[
T^{\mu\nu}=T^{\nu\mu},\qquad
\partial_{\mu}T^{\mu\nu}=0 .
]
The first expresses symmetry (which will be discussed below), while the second is the local conservation law for energy and momentum. Integrated over a spacelike hypersurface it yields the familiar global conservation of total energy and total momentum in the absence of external forces.
2. Lorentz‑Covariant Construction
2.1 Noether’s theorem
For a field (\phi(x)) described by a Lagrangian density (\mathcal{L}(\phi,\partial_{\mu}\phi)) that is invariant under spacetime translations, Noether’s theorem provides a canonical energy‑momentum tensor
[
T^{\mu\nu}{\text{can}}=
\frac{\partial\mathcal{L}}{\partial(\partial{\mu}\phi)},
\partial^{\nu}\phi-\eta^{\mu\nu}\mathcal{L}.
]
This object automatically fulfills (\partial_{\mu}T^{\mu\nu}_{\text{can}}=0) but is not guaranteed to be symmetric.
2.2 Belinfante–Rosenfeld symmetrization
When the underlying theory possesses intrinsic spin, the canonical tensor can be improved by adding a total divergence built from the spin current (S^{\lambda\mu\nu}):
[
\Theta^{\mu\nu}=T^{\mu\nu}{\text{can}}+\partial{\lambda}B^{\lambda\mu\nu},
\qquad
B^{\lambda\mu\nu}=-B^{\mu\lambda\nu}.
]
The corrected tensor (\Theta^{\mu\nu}) is symmetric, still conserved, and is the one that couples directly to the metric in general relativity.
3. Representative Forms for Common Systems
3.1 Point particle
A free particle of rest mass (m) follows a world‑line (x^{\mu}(\tau)) with four‑velocity (u^{\mu}=dx^{\mu}/d\tau). Its four‑momentum is (p^{\mu}=m u^{\mu}). The associated energy‑momentum tensor is a distribution concentrated on the trajectory:
[
T^{\mu\nu}(x)=\int p^{\mu}u^{\nu},
\delta^{4}!\bigl(x-x(\tau)\bigr),d\tau .
]
Integrating (T^{00}) over space reproduces the relativistic energy (\gamma mc^{2}); integrating (T^{0i}) gives the momentum (\gamma m\mathbf{v}).
3.2 Electromagnetic field
The Maxwell Lagrangian
[
\mathcal{L}= -\frac{1}{4\mu_{0}}F_{\alpha\beta}F^{\alpha\beta},
\qquad
F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu},
]
leads, after the Belinfante procedure, to the symmetric Maxwell stress‑energy tensor
[
T^{\mu\nu}= \frac{1}{\mu_{0}}!\Bigl(F^{\mu\lambda}F^{\nu}{}{\lambda}
-\frac{1}{4}\eta^{\mu\nu}F{\alpha\beta}F^{\alpha\beta}\Bigr).
]
Its components reproduce the familiar quantities:
[
\begin{aligned}
T^{00}&=\frac{1}{2}!\left(\varepsilon_{0}E^{2}+\frac{1}{\mu_{0}}B^{2}\right) &&\text{(energy density)}\[4pt]
T^{0i}&=\frac{1}{\mu_{0}}(\mathbf{E}\times\mathbf{B})^{i} &&\text{(Poynting vector)}\[4pt]
T^{ij}&=-\varepsilon_{0}E^{i}E^{j}
-\frac{1}{\mu_{0}}B^{i}B^{j}
+\frac{1}{2}\delta^{ij}!\left(\varepsilon_{0}E^{2}+\frac{1}{\mu_{0}}B^{2}\right)
&&\text{(Maxwell stress)} .
\end{aligned}
]
These expressions are indispensable for describing radiation pressure, photon momentum, and energy transport in plasma and astrophysical environments.
3.3 Perfect fluid
For an ideal relativistic fluid with proper energy density (\rho c^{2}) and isotropic pressure (p), the four‑velocity (u^{\mu}) (normalized as (u^{\mu}u_{\mu}=1)) yields
[
T^{\mu\nu}= (\rho c^{2}+p),u^{\mu}u^{\nu}-p,\eta^{\mu\nu}.
]
The term proportional to (u^{\mu}u^{\nu}) contains kinetic, internal, and rest‑mass contributions, while the (-p\eta^{\mu\nu}) part encodes the isotropic stress. This compact form underlies the relativistic hydrodynamics used to model neutron‑star interiors, gamma‑ray burst jets, and heavy‑ion collisions.
4. Conservation Laws and Symmetries
The equation (\partial_{\mu}T^{\mu\nu}=0) is a direct consequence of invariance under spacetime translations. If the action is also invariant under Lorentz rotations, an angular‑momentum current can be constructed:
[
M^{\lambda\mu\nu}=x^{\mu}T^{\lambda\nu}-x^{\nu}T^{\lambda\mu}+S^{\lambda\mu\nu},
\qquad
\partial_{\lambda}M^{\lambda\mu\nu}=0 .
]
Here (S^{\lambda\mu\nu}) is the spin density. When the energy‑momentum tensor is symmetric (as after the Belinfante correction) the spin term vanishes, and the total angular momentum reduces to the familiar orbital part (x^{\mu}T^{\lambda\nu}-x^{\nu}T^{\lambda\mu}).
5. Example: Photon Momentum in a Dielectric
Consider a monochromatic plane wave propagating through a homogeneous medium of refractive index (n). The fields satisfy (|\mathbf{E}|=c n |\mathbf{B}|). Using the Maxwell tensor, the energy and momentum densities become
[
\begin{aligned}
u &= \frac{n^{2}\varepsilon_{0}}{2}E^{2},\[4pt]
\mathbf{g} &= \frac{n^{2}\varepsilon_{0}}{c^{2}}\mathbf{E}\times\mathbf{B}
= \frac{n^{2}}{c^{2}},u,\mathbf{v},
\end{aligned}
]
with the phase velocity (\mathbf{v}=c/n). The momentum density is larger by a factor of (n) compared with the vacuum case, illustrating the classic Abraham–Minkowski controversy. The energy‑momentum tensor provides a unified framework that accommodates both viewpoints by distinguishing between the kinetic (Abraham) and canonical (Minkowski) momenta.
6. Key Take‑aways
- The energy‑momentum tensor is the relativistic carrier of energy, momentum, and stress; it is a symmetric, conserved rank‑2 tensor in flat spacetime.
- Noether’s theorem supplies a systematic route from a Lagrangian density to a canonical tensor; the Belinfante–Rosenfeld improvement yields the symmetric form required for coupling to gravity.
- Specific physical systems—point particles, electromagnetic fields, perfect fluids—have compact, widely used expressions for (T^{\mu\nu}) that make calculations of total energy, momentum flux, and pressure straightforward.
- Conservation of (T^{\mu\nu}) reflects spacetime translation invariance, while additional symmetries (rotations, boosts) generate corresponding angular‑momentum conservation laws.
- Mastery of the tensor’s structure is essential for advanced topics such as Einstein’s field equations, relativistic hydrodynamics, and quantum field theory in curved spacetime.
Armed with these foundations, the reader can move confidently from special‑relativistic applications to the full geometric language of general relativity, where the energy‑momentum tensor becomes the source term that shapes spacetime itself.