Heat flow components of Stress/Energy Tensor

Thread starter TerryW
Start date Oct 27, 2023
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Heat flow Physics

Oct 27, 2023

TerryW

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Homework Statement: Show that the Trace of the Heat Flux Stress/Energy Tensor is zero

Relevant Equations: T(heat) = q^i u^j + u^i q^j

I'm pretty sure I understand why q^i u^j + u^i q^i is the stress component (i \neq j) of the Heat Flux Stress/Energy Tensor but I can't think of, or find any explanation for why q^i u^i + u^i q^i = 0.

I found one reference in Lightman, Press, Price and Teukolsky - Problem Book in Relativity and Gravitation which just says
"Since q^α u_α = 0 (heat flux is spacelike in comoving frame)" but what does that really mean from a physics point of view?

Can anyone help?TerryW

PS - Preview doesn't seem to be working at the moment

Last edited: Oct 29, 2023

Related to Heat flow components of Stress/Energy Tensor

What is the Stress-Energy Tensor?

The Stress-Energy Tensor, often denoted as Tμν, is a mathematical object in physics that describes the density and flux of energy and momentum in spacetime. It is a key component in the field equations of General Relativity, representing how matter and energy influence the curvature of spacetime.

How is heat flow represented in the Stress-Energy Tensor?

Heat flow is represented in the off-diagonal components of the Stress-Energy Tensor. Specifically, components like T0i (where i represents the spatial coordinates x, y, z) describe the energy flux, which includes heat flow, in the i-th spatial direction. These components quantify how energy, including heat, is transferred through space over time.

What is the physical significance of the T00 component in the Stress-Energy Tensor?

The T00 component of the Stress-Energy Tensor represents the energy density of the system. It includes all forms of energy, such as mass-energy, kinetic energy, potential energy, and internal energy, including thermal energy. This component is crucial for understanding how energy is distributed in a given region of spacetime.

How do the components of the Stress-Energy Tensor relate to conservation laws?

The components of the Stress-Energy Tensor are closely tied to the conservation of energy and momentum. The covariant divergence of the Stress-Energy Tensor is zero, which mathematically expresses the local conservation of energy and momentum. This principle ensures that energy and momentum are conserved in all physical processes described by the tensor.

Can the Stress-Energy Tensor be used to describe non-thermal energy transfers?

Yes, the Stress-Energy Tensor describes all forms of energy and momentum transfer, not just thermal energy. It encompasses kinetic energy, potential energy, electromagnetic energy, and any other forms of energy and momentum present in the system. The tensor provides a comprehensive framework for analyzing how different types of energy and momentum interact and propagate through spacetime.