Look, if you absolutely must understand the plumbing of the universe, we can talk about the stress-energy tensor. Just try to keep up. It's the universe's way of bookkeeping, a ledger of all the energy and momentum sloshing around in spacetime. And yes, it's what tells spacetime how to bend. Don't look so impressed; it's just physics.

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General relativity
!Contravariant components of the stress–energy tensor.
Contravariant components of the stress–energy tensor.
!Einstein field equations
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Wikiversity has learning resources about Gravitational stress-energy tensor

The stress–energy tensor, which also goes by the more cumbersome names stress–energy–momentum tensor or energy–momentum tensor, is a tensor field quantity in physics. Its job is to describe both the density and the flux of energy and momentum at every single point in spacetime. Think of it as a generalization of the simple stress tensor from your introductory Newtonian physics class, but built for a universe that's far more complicated. It’s a fundamental attribute of matter, radiation, and any non-gravitational force fields you might encounter.

Crucially, this density and flux of energy and momentum are the very sources of the gravitational field as described by the Einstein field equations of general relativity. In the old, comfortable world of Newtonian gravity, you only had to worry about mass density as the source of gravity. Relativity demands more. It recognizes that energy, pressure, and stress also gravitate, and it bundles all of that information into this one mathematical object.

Definition

Before we proceed, a note on notation. The stress–energy tensor uses superscripted variables. These are indices, not exponents, a detail you'll want to remember unless you enjoy being wrong. We're using Tensor index notation and the Einstein summation notation, so try to keep up. The four coordinates of an event in spacetime are labeled x⁰, x¹, x², x³. Conventionally, these correspond to t, x, y, z, where t is your time coordinate and the others are the three spatial dimensions you inhabit.

The stress–energy tensor is formally defined as the tensor T^αβ, an object of order two, that gives you the flux of the α-th component of the momentum vector passing through a surface of constant x^β. In the theory of relativity, this isn't just any momentum; it's the four-momentum. In the context of general relativity, the stress–energy tensor is symmetric. [a] This isn't an arbitrary choice; it's a deep consequence of the conservation of angular momentum.

${\displaystyle T^{\alpha \beta }=T^{\beta \alpha }.}$

There are, of course, exotic exceptions. In some alternative frameworks like Einstein–Cartan theory, the tensor might not be perfectly symmetric. This asymmetry is linked to a non-zero spin tensor, which geometrically corresponds to a non-zero torsion tensor, a twisting of spacetime itself. But for our purposes, assume symmetry. [ citation needed ]

Components

Being an order-2 tensor in four dimensions, the components of the stress–energy tensor can be laid out in a 4×4 matrix. It's not just a collection of numbers; each component has a direct, physical meaning. Don't just stare at the matrix; understand what it's telling you.

${\displaystyle T^{\mu \nu }={\begin{pmatrix}T^{00}&T^{01}&T^{02}&T^{03}\\T^{10}&T^{11}&T^{12}&T^{13}\\T^{20}&T^{21}&T^{22}&T^{23}\\T^{30}&T^{31}&T^{32}&T^{33}\end{pmatrix}}\,,}$

The indices μ and ν here run from 0 to 3. Let's dissect this piece by piece, with the spatial indices k and ℓ running from 1 to 3. [b]

• The T⁰⁰ component, sitting in the top-left corner, is the density of relativistic mass. This is simply the energy density divided by the speed of light squared. It tells you how much energy is packed into a given volume. ${\displaystyle T^{00}=\rho ~,}$ where ${\textstyle \rho }$ is the relativistic mass density.

• The components T⁰ᵏ represent the flux of relativistic mass across the xᵏ surface. This is equivalent to the k-th component of momentum density. Because the tensor is symmetric, T⁰ᵏ = Tᵏ⁰. So, T⁰¹ is the density of momentum in the x-direction, which is also the flow of energy in the x-direction.

• The components ${\displaystyle T^{k\ell }}$ make up the 3x3 spatial block of the tensor. These represent the flux of the k-th component of momentum across the x^ℓ surface. This is the relativistic version of the classical Cauchy stress tensor.

  • The diagonal components, ${\displaystyle T^{kk}}$ (with no summation implied), represent normal stress in the k-th coordinate direction. You can think of this as pressure. T¹¹ is the pressure in the x-direction.
  • The off-diagonal components, ${\displaystyle T^{k\ell }}$ where ${\displaystyle k\neq \ell }$, represent shear stress. This is the force per unit area acting parallel to a surface.

In fields like solid state physics and fluid mechanics, the term "stress tensor" usually refers only to these spatial components, viewed in the proper frame of reference. The relativistic stress-energy tensor used in engineering contexts is thus subtly different from the one in general relativity, often differing by a momentum-convective term.

Covariant and mixed forms

While this entire discussion has focused on the contravariant form, T^μν, it's often necessary for calculations to use the covariant form, T_μν, or the mixed form, T^μ_ν. You get these by lowering indices with the metric tensor g_αμ. It's a bit of mathematical bookkeeping.

The covariant form is: ${\displaystyle T_{\mu \nu }=T^{\alpha \beta }g_{\alpha \mu }g_{\beta \nu },}$

And the mixed form: ${\displaystyle T^{\mu }{}_{\nu }=T^{\mu \alpha }g_{\alpha \nu }.}$

For the record, this article uses the spacelike sign convention (− + + +) for the metric signature. Get it wrong, and all your signs will be flipped.

Conservation law

In special relativity

The stress–energy tensor is not just some arbitrary construct; it's the conserved Noether current that arises from the fundamental symmetry of spacetime under translations. In simpler terms, because the laws of physics don't change if you shift your experiment in space or time, there must be a corresponding conserved quantity. That quantity is the four-momentum, and its density and flux are captured by this tensor.

The conservation law is expressed by saying that the four-divergence of the non-gravitational stress–energy tensor is zero. This means non-gravitational energy and momentum are locally conserved.

${\displaystyle 0=T^{\mu \nu }{}_{;\nu }\ \equiv \ \nabla _{\nu }T^{\mu \nu }.}$

In integral form, this statement becomes: ${\displaystyle 0=\int _{\partial N}T^{\mu \nu }\mathrm {d} ^{3}s_{\nu }}$ Here, N is any compact four-dimensional region of spacetime, ${\textstyle \partial N}$ is its three-dimensional boundary, and ${\textstyle \mathrm {d} ^{3}s_{\nu }}$ is an element of that boundary with an outward-pointing normal.

In the simple case of flat spacetime and using linear coordinates, this conservation law, combined with the symmetry of the tensor, also implies that angular momentum is conserved: ${\displaystyle 0=(x^{\alpha }T^{\mu \nu }-x^{\mu }T^{\alpha \nu })_{,\nu }\,.}$

In general relativity

When you introduce gravity, things get more complicated. The universe is no longer flat. The divergence of the stress–energy tensor still vanishes, but the divergence itself is now defined using the covariant derivative, which accounts for the curvature of spacetime.

${\displaystyle 0=\operatorname {div} T=T^{\mu \nu }{}_{;\nu }=\nabla _{\nu }T^{\mu \nu }=T^{\mu \nu }{}_{,\nu }+\Gamma ^{\mu }{}_{\sigma \nu }T^{\sigma \nu }+\Gamma ^{\nu }{}_{\sigma \nu }T^{\mu \sigma }}$

The new terms, ${\textstyle \Gamma ^{\mu }{}_{\sigma \nu }}$, are the Christoffel symbols. They represent the gravitational force field. [ citation needed ] This equation doesn't mean energy and momentum are vanishing; it means they are being exchanged with the gravitational field itself. The tensor only accounts for matter and other fields, not gravity.

If spacetime possesses a symmetry, represented by a Killing vector field ${\textstyle \xi ^{\mu }}$, then this leads to a genuine conservation law, which can be expressed as: ${\displaystyle 0=\nabla _{\nu }\left(\xi ^{\mu }T^{\nu }{}_{\mu }\right)={\frac {1}{\sqrt {-g}}}\partial _{\nu }\left({\sqrt {-g}}\ \xi ^{\mu }T_{\mu }^{\nu }\right)}$

The integral form of this conservation law is: ${\displaystyle 0=\int _{\partial N}\xi ^{\mu }T^{\nu }{}_{\mu }{\sqrt {-g}}\ \mathrm {d} ^{3}s_{\nu }\,.}$

In special relativity

In the framework of special relativity, the stress–energy tensor is a powerhouse of information, encapsulating the energy density, momentum density, and their respective fluxes for any given system.

If you have a Lagrangian density ${\textstyle {\mathcal {L}}}$ that depends on a set of fields ${\textstyle \phi _{\alpha }}$ and their derivatives, but not explicitly on the spacetime coordinates (a condition expressed as ${\displaystyle {\frac {\partial {\mathcal {L}}}{\partial x^{\nu }}}=0}$), you can derive the canonical stress-energy tensor.

By applying the chain rule to the total derivative of the Lagrangian with respect to a coordinate x^ν, we get: ${\displaystyle {\frac {d{\mathcal {L}}}{dx^{\nu }}}=d_{\nu }{\mathcal {L}}={\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\phi _{\alpha })}}{\frac {\partial (\partial _{\mu }\phi _{\alpha })}{\partial x^{\nu }}}+{\frac {\partial {\mathcal {L}}}{\partial \phi _{\alpha }}}{\frac {\partial \phi _{\alpha }}{\partial x^{\nu }}}}$

Or, in a more compact notation: ${\displaystyle d_{\nu }{\mathcal {L}}={\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\phi _{\alpha })}}\partial _{\nu }\partial _{\mu }\phi _{\alpha }+{\frac {\partial {\mathcal {L}}}{\partial \phi _{\alpha }}}\partial _{\nu }\phi _{\alpha }}$

Now, we invoke the Euler–Lagrange Equation: ${\displaystyle \partial _{\mu }\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\phi _{\alpha })}}\right)={\frac {\partial {\mathcal {L}}}{\partial \phi _{\alpha }}}}$

Substituting this into our expression and using the fact that partial derivatives commute, we find: ${\displaystyle d_{\nu }{\mathcal {L}}={\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\phi _{\alpha })}}\partial _{\mu }\partial _{\nu }\phi _{\alpha }+\partial _{\mu }\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\phi _{\alpha })}}\right)\partial _{\nu }\phi _{\alpha }}$

The right-hand side is just the result of the product rule for derivatives. This allows us to write: ${\displaystyle d_{\nu }{\mathcal {L}}=\partial _{\mu }\left[{\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\phi _{\alpha })}}\partial _{\nu }\phi _{\alpha }\right]}$

In flat spacetime, we can also write ${\textstyle d_{\nu }{\mathcal {L}}=\partial _{\mu }[\delta _{\nu }^{\mu }{\mathcal {L}}]}$. Combining these gives: ${\displaystyle \partial _{\mu }\left[{\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\phi _{\alpha })}}\partial _{\nu }\phi _{\alpha }\right]-\partial _{\mu }\left(\delta _{\nu }^{\mu }{\mathcal {L}}\right)=0}$

Regrouping the terms inside the derivative: ${\displaystyle \partial _{\mu }\left[{\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\phi _{\alpha })}}\partial _{\nu }\phi _{\alpha }-\delta _{\nu }^{\mu }{\mathcal {L}}\right]=0}$

This equation tells us that the divergence of the quantity in the brackets is zero. We define this quantity as the canonical stress–energy tensor: ${\displaystyle T^{\mu }{}_{\nu }\equiv {\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\phi _{\alpha })}}\partial _{\nu }\phi _{\alpha }-\delta _{\nu }^{\mu }{\mathcal {L}}}$

By its very construction, it satisfies the conservation law: ${\displaystyle \partial _{\mu }T^{\mu }{}_{\nu }=0}$

This single tensor equation is equivalent to four continuity equations, one for each value of ν. For instance, the ν=0 component shows that ${\textstyle T^{0}{}_{0}}$ is the energy density of the system, allowing one to derive the Hamiltonian density. The conservation law for ν=0 is: ${\displaystyle {\frac {\partial {\mathcal {H}}}{\partial t}}+\nabla \cdot \left({\frac {\partial {\mathcal {L}}}{\partial \nabla \phi _{\alpha }}}{\dot {\phi }}_{\alpha }\right)=0}$

From this, we can identify the term ${\textstyle {\frac {\partial {\mathcal {L}}}{\partial \nabla \phi _{\alpha }}}{\dot {\phi }}_{\alpha }}$ as the energy flux density of the system.

Trace

The trace of the stress–energy tensor is the contraction ${\displaystyle T^{\mu }{}_{\mu }}$, which is calculated as: ${\displaystyle T^{\mu }{}_{\mu }={\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\phi _{\alpha })}}\partial _{\mu }\phi _{\alpha }-\delta _{\mu }^{\mu }{\mathcal {L}}.}$

Since the trace of the four-dimensional Kronecker delta is ${\displaystyle \delta _{\mu }^{\mu }=4}$, this simplifies to: ${\displaystyle T^{\mu }{}_{\mu }={\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\phi _{\alpha })}}\partial _{\mu }\phi _{\alpha }-4{\mathcal {L}}.}$

In general relativity

In general relativity, the symmetric stress–energy tensor takes center stage. It acts as the source of spacetime curvature. It is the current density associated with the gauge transformations of gravity, which are simply general curvilinear coordinate transformations. (Again, if torsion is present, the tensor may not be symmetric, corresponding to a non-zero spin tensor as in Einstein–Cartan gravity theory.)

The transition from special to general relativity requires replacing all partial derivatives with covariant derivatives. This has a profound consequence: the continuity equation no longer implies absolute conservation of non-gravitational energy and momentum. The gravitational field can perform work on matter, and vice versa. In the weak-field limit of Newtonian gravity, this is easily understood as an exchange between kinetic energy and gravitational potential energy. In full general relativity, defining gravitational energy is notoriously difficult. The Landau–Lifshitz pseudotensor is one attempt to define gravitational energy and momentum densities, but any such stress–energy pseudotensor can be made to vanish locally with a clever choice of coordinates.

In curved spacetime, a spatial integral depends on the chosen spatial slice. There is no unambiguous way to define a global energy–momentum vector in a general curved spacetime.

Einstein field equations

The role of the stress–energy tensor is immortalized in the Einstein field equations, which are often written as:

${\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }=\kappa T_{\mu \nu },}$

where:

• ${\textstyle G_{\mu \nu }=R_{\mu \nu }-{\tfrac {1}{2}}R\,g_{\mu \nu }}$ is the Einstein tensor, which describes the geometry of spacetime.
• ${\textstyle R_{\mu \nu }}$ is the Ricci tensor.
• ${\textstyle R=g^{\alpha \beta }R_{\alpha \beta }}$ is the scalar curvature.
• ${\textstyle g_{\mu \nu }\,}$ is the metric tensor.
• Λ is the cosmological constant, generally negligible on scales smaller than a galaxy.
• ${\textstyle \kappa =8\pi G/c^{4}}$ is the Einstein gravitational constant.

These equations articulate a fundamental principle: matter tells spacetime how to curve, and spacetime tells matter how to move. The T^μν on the right side is the "matter" part of that statement.

Stress–energy in special situations

Isolated particle

In special relativity, the stress–energy for a single, non-interacting particle with rest mass m following a trajectory ${\textstyle \mathbf {x} _{\text{p}}(t)}$ is given by:

${\displaystyle T^{\alpha \beta }(\mathbf {x} ,t)={\frac {m\,v^{\alpha }(t)v^{\beta }(t)}{\sqrt {1-(v/c)^{2}}}}\;\,\delta \left(\mathbf {x} -\mathbf {x} _{\text{p}}(t)\right)={\frac {E}{c^{2}}}\;v^{\alpha }(t)v^{\beta }(t)\;\,\delta (\mathbf {x} -\mathbf {x} _{\text{p}}(t))}$

Here, ${\textstyle v^{\alpha }}$ is the velocity vector ${\displaystyle v^{\alpha }=\left(1,{\frac {d\mathbf {x} _{\text{p}}}{dt}}(t)\right)\,,}$ (not to be confused with the four-velocity), ${\textstyle \delta }$ is the Dirac delta function which ensures the tensor is zero everywhere except at the particle's location, and ${\textstyle E={\sqrt {p^{2}c^{2}+m^{2}c^{4}}}}$ is the particle's total energy.

In a more classical language, this tensor represents the packaging of relativistic mass, momentum, and the dyadic product of momentum and velocity: ${\displaystyle \left({\frac {E}{c^{2}}},\,\mathbf {p} ,\,\mathbf {p} \,\mathbf {v} \right)\,.}$

Stress–energy of a fluid in equilibrium

For an idealized perfect fluid, the stress–energy tensor has a particularly clean and useful form:

${\displaystyle T^{\alpha \beta }\,=\left(\rho +{p \over c^{2}}\right)u^{\alpha }u^{\beta }+pg^{\alpha \beta }}$

Here, ${\textstyle \rho }$ is the mass density and ${\textstyle p}$ is the isotropic pressure as measured in the fluid's rest frame, ${\textstyle u^{\alpha }}$ is the fluid's four-velocity, and ${\textstyle g^{\alpha \beta }}$ is the inverse of the metric tensor. The trace of this tensor is ${\displaystyle T^{\alpha }{}_{\,\alpha }=g_{\alpha \beta }T^{\beta \alpha }=3p-\rho c^{2}\,.}$

The four-velocity is normalized such that ${\displaystyle u^{\alpha }u^{\beta }g_{\alpha \beta }=-c^{2}\,.}$

In the fluid's own proper frame of reference, the four-velocity is simply ${\displaystyle u^{\alpha }=(1,0,0,0)\,,}$ and the inverse metric tensor is ${\displaystyle g^{\alpha \beta }\,=\left({\begin{smallmatrix}-{\frac {1}{c^{2}}}&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{smallmatrix}}\right)}$. In this frame, the stress–energy tensor becomes a simple diagonal matrix:

${\displaystyle T^{\alpha \beta }=\left({\begin{matrix}\rho &0&0&0\\0&p&0&0\\0&0&p&0\\0&0&0&p\end{matrix}}\right).}$

This form is intuitive: the top-left component is the energy density, and the three diagonal spatial components represent the equal pressure exerted in all directions.

Electromagnetic stress–energy tensor

The Hilbert stress–energy tensor for an electromagnetic field in a vacuum (source-free) is:

${\displaystyle T^{\mu \nu }={\frac {1}{\mu _{0}}}\left(F^{\mu \alpha }g_{\alpha \beta }F^{\nu \beta }-{\frac {1}{4}}g^{\mu \nu }F_{\delta \gamma }F^{\delta \gamma }\right)}$

where ${\textstyle F_{\mu \nu }}$ is the electromagnetic field tensor. This tensor describes the energy, momentum, and stress stored in electric and magnetic fields.

Scalar field

For a complex scalar field ${\textstyle \phi }$ that obeys the Klein–Gordon equation, the stress–energy tensor is:

${\displaystyle T^{\mu \nu }={\frac {\hbar ^{2}}{m}}\left(g^{\mu \alpha }g^{\nu \beta }+g^{\mu \beta }g^{\nu \alpha }-g^{\mu \nu }g^{\alpha \beta }\right)\partial _{\alpha }{\bar {\phi }}\partial _{\beta }\phi -g^{\mu \nu }mc^{2}{\bar {\phi }}\phi ,}$

When the metric is flat (Minkowski), its components are:

${\displaystyle {\begin{aligned}T^{00}&={\frac {\hbar ^{2}}{mc^{4}}}\left(\partial _{0}{\bar {\phi }}\partial _{0}\phi +c^{2}\partial _{k}{\bar {\phi }}\partial _{k}\phi \right)+m{\bar {\phi }}\phi ,\\T^{0i}=T^{i0}&=-{\frac {\hbar ^{2}}{mc^{2}}}\left(\partial _{0}{\bar {\phi }}\partial _{i}\phi +\partial _{i}{\bar {\phi }}\partial _{0}\phi \right),\ \mathrm {and} \\T^{ij}&={\frac {\hbar ^{2}}{m}}\left(\partial _{i}{\bar {\phi }}\partial _{j}\phi +\partial _{j}{\bar {\phi }}\partial _{i}\phi \right)-\delta _{ij}\left({\frac {\hbar ^{2}}{m}}\eta ^{\alpha \beta }\partial _{\alpha }{\bar {\phi }}\partial _{\beta }\phi +mc^{2}{\bar {\phi }}\phi \right).\end{aligned}}}$

Variant definitions of stress–energy

Because physicists enjoy complicating things, there are several different, inequivalent definitions of the non-gravitational stress–energy tensor.

Hilbert stress–energy tensor

This is arguably the most elegant definition. It is defined as the functional derivative of the non-gravitational part of the action, ${\textstyle S_{\mathrm {matter} }}$, with respect to the inverse metric tensor:

${\displaystyle T_{\mu \nu }={\frac {-2}{\sqrt {-g}}}{\frac {\delta S_{\mathrm {matter} }}{\delta g^{\mu \nu }}}={\frac {-2}{\sqrt {-g}}}{\frac {\partial \left({\sqrt {-g}}{\mathcal {L}}_{\mathrm {matter} }\right)}{\partial g^{\mu \nu }}}=-2{\frac {\partial {\mathcal {L}}_{\mathrm {matter} }}{\partial g^{\mu \nu }}}+g_{\mu \nu }{\mathcal {L}}_{\mathrm {matter} },}$

where ${\textstyle {\mathcal {L}}_{\mathrm {matter} }}$ is the non-gravitational part of the Lagrangian density. This definition automatically produces a symmetric and gauge-invariant tensor. For more, see the Einstein–Hilbert action.

Canonical stress–energy tensor

As derived earlier, Noether's theorem gives us a conserved current associated with spacetime translations. This is the canonical stress–energy tensor. It's the most "natural" definition from a symmetry perspective, but it isn't always symmetric. Furthermore, in a gauge theory, it may not be gauge invariant because gauge transformations don't generally commute with spatial translations.

Belinfante–Rosenfeld stress–energy tensor

When dealing with fields that have intrinsic spin or angular momentum, the canonical Noether tensor fails to be symmetric. The Belinfante–Rosenfeld stress–energy tensor is a modification. It's constructed from the canonical tensor and the spin current in a specific way that produces a new tensor which is both symmetric and conserved. In general relativity, this modified tensor agrees with the Hilbert stress–energy tensor.

Gravitational stress–energy

This is a subtle but critical point. According to the equivalence principle, the effects of gravity can be locally eliminated by choosing a freely-falling frame of reference. In such a frame, the gravitational stress–energy vanishes at that point. This means that gravitational energy cannot be described by a true tensor, which must be independent of the coordinate system. Instead, one must use a pseudotensor.

In general relativity, several such gravitational stress–energy–momentum pseudotensors have been proposed, including the Einstein pseudotensor and the Landau–Lifshitz pseudotensor. These objects are not true tensors; their values depend on the chosen coordinate system, and they can be made to vanish at any given event in spacetime. They are a necessary but awkward tool for discussing the energy of the gravitational field itself.

See also

• Electromagnetic stress–energy tensor
• Energy condition
• Energy density of electric and magnetic fields
• Maxwell stress tensor
• Poynting vector
• Ricci calculus
• Segre classification

Notes

• ^ "All the stress–energy tensors explored above were symmetric. That they could not have been otherwise one sees as follows." — Misner, Thorne, and Wheeler
• ^ For a convention in which the coordinates of a displacement vector x^μ are [ ct, x, y, z ], T⁰⁰ will be energy density, and T⁰ᵏ will be areal density of the rate of momentum transfer.