Stress-Energy Tensor for Fluids and Fields

1. What is the Stress-Energy Tensor?

The most direct way to understand $T^{\mu \nu }$ is through its physical interpretation:

The component $T^{\mu \nu }$ represents the flux of the $\mu$-component of the four-momentum vector $p^{\mu }$ across a surface of constant $x^{\nu }$.

This compact statement contains a great deal of physical information. To unpack it and build a solid intuition, we will explore it from a few complementary angles.

First, let's watch a video that provides an excellent visual and conceptual introduction. It starts by explaining why simple mass density is insufficient in a relativistic theory and then carefully builds up the meaning of each component of $T^{\mu \nu }$.

The video provides a great visual guide. Let's formalize the meaning of the components, using units where $c=1$.

• $T^{00}$: The Energy Density. This is the amount of energy per unit volume. It's the relativistic generalization of mass density, $\rho$.
• $T^{i0}$: The Momentum Density. $T^{i0}$ is the density of the $i$-component of momentum per unit volume.
• $T^{0i}$: The Energy Flux. This is the rate of energy flow per unit area in the $i$-direction. For the electromagnetic field, this is the Poynting vector.
• $T^{ij}$: The Momentum Flux (Stress Tensor). This is the flux of the $i$-component of momentum in the $j$-direction.
  • Diagonal components ($i=j$): $T^{11},T^{22},T^{33}$ represent pressure (or tension), a normal force per unit area.
  • Off-diagonal components ($i\ne j$): $T^{12}$, etc., represent shear stress, a tangential force per unit area.

An immediate and non-trivial consequence of relativity is that the stress-energy tensor is symmetric, $T^{\mu \nu }=T^{\nu \mu }$. The equality of the time-space and space-time components, $T^{0i}=T^{i0}$, means that energy flux is equivalent to momentum density. The symmetry of the spatial part, $T^{ij}=T^{ji}$, can be understood from a classical argument: if it were not symmetric, a small fluid element would experience a net torque, leading to infinite angular acceleration as its size goes to zero—a physical absurdity.

2. The Stress-Energy Tensor of a Perfect Fluid

Now that we have a general definition, let's construct the tensor for a specific, physically important case. A perfect fluid is an idealized fluid characterized completely by its energy density and isotropic pressure. It has no viscosity (no shear stresses) and no heat conduction. This is an excellent approximation for many cosmological and astrophysical sources, such as the matter in the early universe or the interior of a star.

Step 1: Pressureless Dust

The simplest fluid is "dust"—a collection of non-interacting particles all moving with the same velocity. In the rest frame of the dust, the particles are stationary. The only non-zero component is the energy density, which is just the rest mass density, ${\rho }_0$. So, in the rest frame, $T^{00}={\rho }_0$ and all other components are zero.

If we let $u^{\mu }$ be the four-velocity of the fluid (which is $(1,0,0,0)$ in the rest frame), we can write this in a covariant form that is valid in any frame:
$T_{\text{dust}}^{\mu \nu }={\rho }_0{u}^{\mu }{u}^{\nu }$
You can easily check that if you boost this tensor to a moving frame, you get the correct, non-zero components for momentum density and momentum flux.

Step 2: Adding Pressure

A perfect fluid is the next step up in complexity from dust. In its local rest frame, it is characterized by an energy density $\rho$ and an isotropic pressure $p$. "Isotropic" means the pressure is the same in all directions. Therefore, in the rest frame, the stress tensor is diagonal with the pressure on the diagonal: $T^{11}=T^{22}=T^{33}=p$.

The full tensor in the rest frame is:

Our task is to write this in a general, covariant form that works in any frame. The unique tensor that is built from the fluid's four-velocity $u^{\mu }$, the metric $g^{\mu \nu }$, $\rho$, and $p$, and reduces to the above matrix in the rest frame is:
$T^{\mu \nu }=(\rho +p)u^{\mu }{u}^{\nu }+p{g}^{\mu \nu }$

This is the stress-energy tensor for a perfect fluid. Note that in GR, pressure itself gravitates. It appears alongside energy density as a source of curvature.

To see a detailed derivation and discussion, the following lecture notes are an excellent resource. They build the fluid tensor from first principles by coarse-graining a collection of individual particles.

3. The Stress-Energy Tensor of the Electromagnetic Field

The other fundamental source we will consider is the electromagnetic field. Since you are familiar with the tensor formulation of electromagnetism, you know that the field is described by the antisymmetric Faraday tensor, $F_{\mu \nu }$. The stress-energy tensor must be constructed from $F_{\mu \nu }$ and the metric $g_{\mu \nu }$.

The correct expression, which can be derived from the electromagnetic Lagrangian, is:
$T^{\mu \nu }=\frac{1}{{\mu }_0}\left(F^{\mu \alpha }{F}_{\alpha }^{\nu }-\frac{1}{4}{g}^{\mu \nu }{F}_{\alpha \beta }{F}^{\alpha \beta }\right)$
(This is in SI units. In geometrized units, where ${ϵ}_0={\mu }_0=c=1$, the pre-factors simplify).

Let's unpack the components of this tensor. If you substitute the components of $F_{\mu \nu }$ in terms of the electric field $\vec{E}$ and magnetic field $\vec{B}$, you recover the familiar expressions from classical E&M:

• Energy Density: $T^{00}=\frac{1}{2}\left({ϵ}_0{E}^2+\frac{1}{{\mu }_0}{B}^2\right)$
• Energy Flux / Momentum Density: $T^{0i}=T^{i0}=$ the $i$-th component of the Poynting vector, $\vec{S}=\frac{1}{{\mu }_0}(\vec{E}\times \vec{B})$.
• Spatial Stresses: $T^{ij}$ are the components of the Maxwell Stress Tensor, ${\sigma }_{ij}={ϵ}_0(E_i{E}_j-\frac{1}{2}{\delta }_{ij}{E}^2)+\frac{1}{{\mu }_0}(B_i{B}_j-\frac{1}{2}{\delta }_{ij}{B}^2)$.

The following lecture notes provide a concise derivation and summary of these facts.

A fascinating property of the EM stress-energy tensor is that it is traceless: $T_{\mu }^{\mu }=g_{\mu \nu }{T}^{\mu \nu }=0$. This is a deep consequence of the fact that electromagnetism is a conformally invariant theory and photons are massless. For a gas of photons (radiation), this property implies a simple equation of state: $p=\rho /3$.

4. A Deeper Definition: The Variational Principle

For a student with your interest in the formal structure of physics, it's worth knowing that there is a more fundamental and elegant way to define the stress-energy tensor. In Lagrangian field theory, any matter action, $S_{\text{matter}}$, depends on the fields and the spacetime metric $g_{\mu \nu }$. The stress-energy tensor is defined as the functional derivative of the matter action with respect to the metric:
$T_{\mu \nu }\equiv -\frac{2}{\sqrt{-g}}\frac{\delta S_{\text{matter}}}{\delta g^{\mu \nu }}$
(Note: The sign and factor of 2 can vary with convention).

This definition is extremely powerful. It automatically guarantees that $T_{\mu \nu }$ is symmetric (since $g^{\mu \nu }$ is symmetric) and provides a direct, unambiguous procedure for calculating the stress-energy tensor for any field theory with a known action. It is the definition that reveals $T_{\mu \nu }$ as the canonical "source" that couples to the gravitational field (the metric). If you are interested, you can see this principle in action in Section 7 of the McGreevy notes (LINK, pages 92-95) you looked at earlier.

Conclusion

Today we have identified the source of gravity in General Relativity. It is not just mass or energy, but the full stress-energy tensor $T_{\mu \nu }$, a rank-2 tensor that encapsulates the complete energy-momentum-stress content of matter and fields.

Key Takeaways:

• The stress-energy tensor $T_{\mu \nu }$ is the relativistic source of gravitation.
• Its components describe the density of energy ($T^{00}$), the density of momentum ($T^{i0}$), the flux of energy ($T^{0i}$), and the flux of momentum, or stress ($T^{ij}$).
• For a perfect fluid, the tensor is given by $T^{\mu \nu }=(\rho +p)u^{\mu }{u}^{\nu }+p{g}^{\mu \nu }$. This shows that both energy density $\rho$ and pressure $p$ act as sources for gravity.
• For the electromagnetic field, the tensor is built from the Faraday tensor $F_{\mu \nu }$ and contains the familiar energy density, Poynting vector, and Maxwell stress tensor.
• Fundamentally, $T_{\mu \nu }$ can be defined as the variation of the matter action with respect to the spacetime metric.

We now have the two main conceptual pieces of our theory: we know that gravity is the curvature of spacetime (described by $g_{\mu \nu }$ and its derivatives), and we know what sources this curvature ($T_{\mu \nu }$). The next step is to connect them.

Preview of Next Lesson: The stress-energy tensor is not just a static object; it has its own dynamics. In flat space, the conservation of energy and momentum is given by ${\partial }_{\mu }{T}^{\mu \nu }=0$. In curved spacetime, this becomes the covariant conservation law, ${\nabla }_{\mu }{T}^{\mu \nu }=0$. In our next lesson, we will derive this law and see that it's not merely a statement of conservation, but that it actually contains the equations of motion for the matter itself. This will be the final ingredient we need before we can write down the full Einstein Field Equations.