# Invariants of the stress energy tensor

• I
• Dale
In summary, the stress energy tensor has invariants that can be expressed as scalar fields, including the internal energy density and pressure in the local rest frames of fluid cells, as well as the trace of the tensor. These invariants are independent of coordinate systems and can be used to describe various physical situations, such as an ideal fluid or a free electromagnetic field. Additionally, the local conservation of energy is another invariant that can be constructed from the stress-energy tensor.
Dale
Mentor
Does anyone know of a set of invariants for the stress energy tensor? In particular, I would like to know if there is a small set of linearly independent invariants, each of which (or at least some of which) have a clear physical meaning.

It depends on which situation you look at.

Take an ideal fluid. It's characterized by an equation of state and a fluid four-velocity field ##u^{\mu}##. The energy-momentum tensor is defined through 2 invariants (or rather scalar fields): the internal-energy density and pressure in the local rest frames of the fluid cells, ##\epsilon## and ##P##. At one space-time point in the restframe the components read
$$T^{* \mu \nu}=\mathrm{diag}(\epsilon,P,P,P).$$
Since ##u^{* \mu}=(1,0,0,0)## and ##\eta^{*\mu \nu}=(1,-1,-1,-1)## you can write this in manifestly covariant form as
$$T^{* \mu \nu}=(\epsilon+P) u^{*\mu} u^{* \nu}-P \eta^{* \mu \nu}.$$
Since ##u^{\mu}## are four-vector components, and ##\eta^{* \mu \nu}=\eta^{\mu \nu}## are invariant tensor components (under Lorentz boosts), the equation holds in any frame,
$$T^{\mu \nu} =(\epsilon+P) u^{\mu} u^{\nu} - P \eta^{\mu \nu}.$$
The invariants (scalar fields) in this case are
$$u_{\mu} u_{\nu} T^{\mu \nu}=\epsilon$$
and
$$\eta_{\mu \nu} T^{\mu \nu}=\epsilon-3P.$$
Of course the latter scalar ("the trace") is one you can define for any energy-momentum tensor. For a free electromagnetic field or a fluid of massless particles it vanishes (in the classical-field theory approximation) because of the scale invariance of free electromagnetic fields and massless particles making up an ideal fluid.

Do you know if there are other invariants for a more general stress energy tensor or only for a perfect fluid?

Like any (1,1) tensor, the invariants should just be the eigenvalues or combinations thereof? This is reflected in #2 where the assumption of an ideal fluid makes three of the eigenvalues equal so that you only have two independent ones.

Dale and vanhees71
Does ##\nabla_\mu T^{\mu\nu}=0## count? It's the local conservation of energy, and is four invariants constructed from the stress-energy tensor.

Orodruin said:
Like any (1,1) tensor, the invariants should just be the eigenvalues or combinations thereof?
Is that true in all coordinate systems or just in locally inertial coordinates?

I rather thought the invariants would be ##T^u{}_u## and perhaps ##*T^u{}_u##, where ##*T^{cd} = \epsilon^{abcd}T_{ab}##. But that was mainly from a discussion of an anti-symmetric tensor, I'm not sure what difference symmetry might make.

pervect said:
I rather thought the invariants would be ##T^u{}_u## and perhaps ##*T^u{}_u##, where ##*T^{cd} = \epsilon^{abcd}T_{ab}##. But that was mainly from a discussion of an anti-symmetric tensor, I'm not sure what difference symmetry might make.
Well, to start ##T## is not a 2-form. Since it is symmetric, ##*T## would be zero trivially.

The trace of ##T## is indeed an invariant as it is the sum of eigenvalues.

Dale said:
Is that true in all coordinate systems or just in locally inertial coordinates?
Eigenvalues of a (1,1) tensor do not depend on the coordinates. The coordinate independent eigenvector equation is ##T(V)=\lambda V##.

Dale
Orodruin said:
The coordinate independent eigenvector equation is ##T(V)=\lambda V##.
So in coordinate notation that is ##g_{\nu\xi}T^{\mu\nu}V^{\xi}=\lambda V^{\mu}##

Dale said:
So in coordinate notation that is ##g_{\nu\xi}T^{\mu\nu}V^{\xi}=\lambda V^{\mu}##
Yes.

Dale

## 1. What is the stress energy tensor?

The stress energy tensor is a mathematical object used in the field of physics to describe the distribution of energy and momentum in a given system. It is a tensor quantity, meaning it has both magnitude and direction, and is commonly denoted by the symbol T.

## 2. What are the invariants of the stress energy tensor?

The invariants of the stress energy tensor are quantities that remain constant regardless of the reference frame or coordinate system used to describe the system. They are important because they provide a way to compare and analyze physical systems without being affected by the observer's perspective.

## 3. How are the invariants of the stress energy tensor calculated?

The invariants of the stress energy tensor are calculated using mathematical operations on the components of the tensor. These operations include contraction, multiplication, and differentiation. The resulting values are then compared to determine the invariants.

## 4. What is the significance of the invariants of the stress energy tensor?

The invariants of the stress energy tensor have significant implications in the study of relativity and the behavior of matter and energy in the universe. They allow for the understanding of conservation laws, the structure of spacetime, and the effects of gravity on matter.

## 5. How do the invariants of the stress energy tensor relate to Einstein's field equations?

The invariants of the stress energy tensor play a crucial role in Einstein's field equations, which describe the relationship between the curvature of spacetime and the distribution of matter and energy. The invariants help to determine the geometry of spacetime and how it is affected by the presence of matter and energy.

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