The stress–energy tensor, sometimes called the stress–energy–momentum tensor or the energy–momentum tensor, is a
Newtonian gravity
.
Definition
The stress–energy tensor involves the use of superscripted variables (not exponents; see
meters
.
The stress–energy tensor is defined as the
coordinate. In the theory of relativity, this momentum vector is taken as the four-momentum. In general relativity, the stress–energy tensor is symmetric,[1]
In some alternative theories like Einstein–Cartan theory, the stress–energy tensor may not be perfectly symmetric because of a nonzero spin tensor, which geometrically corresponds to a nonzero torsion tensor.
Components
Because the stress–energy tensor is of order 2, its components can be displayed in 4 × 4 matrix form:
where the indices μ and ν take on the values 0, 1, 2, 3.
In the following, k and ℓ range from 1 through 3:
The time–time component is the density of relativistic mass, i.e., the energy density divided by the speed of light squared, while being in the co-moving frame of reference.[2] It has a direct physical interpretation. In the case of a perfect fluid this component is
where is the
relativistic mass
per unit volume, and for an electromagnetic field in otherwise empty space this component is
where E and B are the electric and magnetic fields, respectively.[3]
The flux of relativistic mass across the xk surface is equivalent to the density of the kth component of linear momentum,
The components
represent flux of kth component of linear momentum across the xℓ surface. In particular,
(not summed) represents
normal stress in the kth co-ordinate direction (k = 1, 2, 3), which is called "pressure
" when it is the same in every direction, k. The remaining components
represent
stress tensor
).
In
solid state physics and fluid mechanics, the stress tensor is defined to be the spatial components of the stress–energy tensor in the proper frame of reference. In other words, the stress–energy tensor in engineering
differs from the relativistic stress–energy tensor by a momentum-convective term.
Covariant and mixed forms
Most of this article works with the contravariant form, Tμν of the stress–energy tensor. However, it is often necessary to work with the covariant form,
The divergence of the non-gravitational stress–energy is zero. In other words, non-gravitational energy and momentum are conserved,
When gravity is negligible and using a Cartesian coordinate system for spacetime, this may be expressed in terms of partial derivatives as
The integral form of the non-covariant formulation is
where N is any compact four-dimensional region of spacetime; is its boundary, a three-dimensional hypersurface; and is an element of the boundary regarded as the outward pointing normal.
In flat spacetime and using Cartesian coordinates, if one combines this with the symmetry of the stress–energy tensor, one can show that angular momentum is also conserved:
Christoffel symbol which is the gravitational force field
.
Consequently, if is any Killing vector field, then the conservation law associated with the symmetry generated by the Killing vector field may be expressed as
The integral form of this is
In special relativity
In special relativity, the stress–energy tensor contains information about the energy and momentum densities of a given system, in addition to the momentum and energy flux densities.[4]
Given a Lagrangian density that is a function of a set of fields and their derivatives, but explicitly not of any of the spacetime coordinates, we can construct the canonical stress-energy tensor by looking at the total derivative with respect to one of the generalized coordinates of the system. So, with our condition
By using the chain rule, we then have
Written in useful shorthand,
Then, we can use the Euler–Lagrange Equation:
And then use the fact that partial derivatives commute so that we now have
We can recognize the right hand side as a product rule. Writing it as the derivative of a product of functions tells us that
Now, in flat space, one can write . Doing this and moving it to the other side of the equation tells us that
And upon regrouping terms,
This is to say that the divergence of the tensor in the brackets is 0. Indeed, with this, we define the stress–energy tensor:
By construction it has the property that
Note that this divergenceless property of this tensor is equivalent to four
continuity equations
. That is, fields have at least four sets of quantities that obey the continuity equation. As an example, it can be seen that is the energy density of the system and that it is thus possible to obtain the Hamiltonian density from the stress–energy tensor.
Indeed, since this is the case, observing that , we then have
We can then conclude that the terms of represent the energy flux density of the system.
Trace
Note that the trace of the stress–energy tensor is defined to be , so
Newtonian gravity, this has a simple interpretation: kinetic energy is being exchanged with gravitational potential energy, which is not included in the tensor, and momentum is being transferred through the field to other bodies. In general relativity the Landau–Lifshitz pseudotensor is a unique way to define the gravitational field energy and momentum densities. Any such stress–energy pseudotensor
can be made to vanish locally by a coordinate transformation.
In curved spacetime, the spacelike integral now depends on the spacelike slice, in general. There is in fact no way to define a global energy–momentum vector in a general curved spacetime.
where is the mass–energy density (kilograms per cubic meter), is the hydrostatic pressure (pascals), is the fluid's four-velocity, and is the matrix inverse of the metric tensor. Therefore, the trace is given by
In general relativity, the translations are with respect to the coordinate system and as such, do not transform covariantly. See the section below on the gravitational stress–energy pseudotensor.
In the presence of spin or other intrinsic angular momentum, the canonical Noether stress–energy tensor fails to be symmetric. The Belinfante–Rosenfeld stress–energy tensor is constructed from the canonical stress–energy tensor and the spin current in such a way as to be symmetric and still conserved. In general relativity, this modified tensor agrees with the Hilbert stress–energy tensor.
By the equivalence principle gravitational stress–energy will always vanish locally at any chosen point in some chosen frame, therefore gravitational stress–energy cannot be expressed as a non-zero tensor; instead we have to use a pseudotensor.
In general relativity, there are many possible distinct definitions of the gravitational stress–energy–momentum pseudotensor. These include the Einstein pseudotensor and the
Landau–Lifshitz pseudotensor
. The Landau–Lifshitz pseudotensor can be reduced to zero at any event in spacetime by choosing an appropriate coordinate system.
^On pp. 141–142 of Misner, Thorne, and Wheeler, section 5.7 "Symmetry of the Stress–Energy Tensor" begins with "All the stress–energy tensors explored above were symmetric. That they could not have been otherwise one sees as follows."