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Tensor describing the density and flux of energy in spacetime
Contravariant components of the stress-energy tensor.
The stress-energy tensor is defined as the tensorT of order two that gives the flux of the ?th component of the momentumvector across a surface with constant x?coordinate. In the theory of relativity, this momentum vector is taken as the four-momentum. In general relativity, the stress-energy tensor is symmetric,
Because the stress-energy tensor is of order two, its components can be displayed in 4 × 4 matrix form:
In the following, k and l range from 1 through 3.
a) The time-time component is the density of relativistic mass, i.e., the energy density divided by the speed of light squared. Its components have 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.
b) The flux of relativistic mass across the xk surface is equivalent to the density of the kth component of linear momentum,
c) 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
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 engineeringdiffers from the relativistic stress-energy tensor by a momentum-convective term.
Covariant and mixed forms
Most of this article works with the contravariant form, T of the stress-energy tensor. However, it is often necessary to work with the covariant form,
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.
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 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.
Note that the trace is defined to be . Note that
When we use our formula for the stress-energy tensor found above,
Using the raising and lowering properties of the metric, and the fact that ,
In general relativity, the partial derivatives used in special relativity are replaced by covariant derivatives. What this means is that the continuity equation no longer implies that the non-gravitational energy and momentum expressed by the tensor are absolutely conserved, i.e. the gravitational field can do work on matter and vice versa. In the classical limit of 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.
The Einstein field equations
In general relativity, the stress tensor is studied in the context of the Einstein field equations which are often written as
where is the mass-energy density (kilograms per cubic meter), is the hydrostatic pressure (pascals), is the fluid's four velocity, and is the reciprocal of the metric tensor. Therefore, the trace is given by
Noether's theorem implies that there is a conserved current associated with translations through space and time. This is called the canonical stress-energy tensor. Generally, this is not symmetric and if we have some gauge theory, it may not be gauge invariant because space-dependent gauge transformations do not commute with spatial translations.
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 pseudo-tensor.
Belinfante-Rosenfeld stress-energy tensor
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."
^Misner, Charles W.; Thorne, Kip S.; Wheeler, John A. (1973). Gravitation. San Francisco, CA: W.H. Freeman and Company. ISBN0-7167-0334-3.
^d'Inverno, R.A. (1992). Introducing Einstein's Relativity. New York, NY: Oxford University Press. ISBN978-0-19-859686-8.
^Landau, L.D.; Lifshitz, E.M. (2010). The Classical Theory of Fields (4th ed.). Butterworth-Heinemann. pp. 84-85. ISBN978-0-7506-2768-9.