Now, if this energy-force 4-vector equation is to be covariant (so its transformed form is still a 4-vector) then the right hand sides must form a 4-vector too. antisymmetric, traceless field strength tensor 16.7: (16.152) In explicit component form, (16.153) The tensor with two covariant indices (formed by two contractions with ) is

The prescription, however, is not a unique one, since the conventionally renormalized stress-energy tensor contains an undetermined multiple of a term, which can be formally reinterpreted as arising from an addition to the gravitational action which is quadratic in the Weyl curvature tensor. Conditions for Traceless Symmetric Improved Stress-Energy Tensors Any new candidate for a stress-energy tensor must (1) conserve, (2) define the same energy-momentum, (3) be symmetric. These conditions set severe restrictions on stress-energy tensor’s further improvement. The simplest physical modification of Tµν is ET Cµν µν α Construction of the stress-energy tensor: second approach 219 Figure 77: A designated drop of liquid (think of a drop of ink dripped into a glass of water) shown at times t and t+dt.Every point in the evolved drop originated as a point in the initial drop. Not shown is the surrounding ﬂuid. A set of coupled equations, linear in the traceless tensors, for the shear rate and the rate of orientation as a function of the stress tensor and the degree of orientation, enables to derive expressions for the (complex) viscosity and the (complex) normal stress coefficients both in stationary and periodic shear, and, for the complex viscosity, also in parallel superposition of these two

## The following double sum generates all the terms of the stress tensor: The first line generates the energy density W, and part of the +0.5 delta(a, b)(E^2 + B^2) term of the Maxwell stress tensor. The rest of that tensor is generated by the second line. The third line creates the Poynting vector.

Conditions for Traceless Symmetric Improved Stress-Energy Tensors Any new candidate for a stress-energy tensor must (1) conserve, (2) define the same energy-momentum, (3) be symmetric. These conditions set severe restrictions on stress-energy tensor’s further improvement. The simplest physical modification of Tµν is ET Cµν µν α The Stress Tensor of the Electromagnetic Field The following double sum generates all the terms of the stress tensor: The first line generates the energy density W, and part of the +0.5 delta(a, b)(E^2 + B^2) term of the Maxwell stress tensor. The rest of that tensor is generated by the second line. The third line creates the Poynting vector.

### Conditions for Traceless Symmetric Improved Stress-Energy Tensors Any new candidate for a stress-energy tensor must (1) conserve, (2) define the same energy-momentum, (3) be symmetric. These conditions set severe restrictions on stress-energy tensor’s further improvement. The simplest physical modification of Tµν is ET Cµν µν α

If we’re working on a curved worldsheet, then the energy-momentum tensor is covariantly conserved, r↵T ↵ =0. The Stress-Energy Tensor is Traceless In conformal theories, T↵ has a very important property: its trace vanishes. To see this, let’s vary the action with respect to a … Covariant Formulation of Electrodynamics Now, if this energy-force 4-vector equation is to be covariant (so its transformed form is still a 4-vector) then the right hand sides must form a 4-vector too. antisymmetric, traceless field strength tensor 16.7: (16.152) In explicit component form, (16.153) The tensor with two covariant indices (formed by two contractions with ) is STRESS-ENERGY TENSOR FOR PERFECT FLUID: GENERAL We worked out the stress-energy tensor Tij for the case of a perfect ﬂuid in its rest frame, so now we want to generalize this to the case where the ﬂuid is viewed from some more general coordinate system, possibly in curved spacetime. The result is simply stated in Moore’s equation 20.16 and in pretty well every other source I looked at.